9.3: Special Cases

For the two special cases I will just give the solution. It requires a substantial amount of algebra to study these two cases.

Case I: Two equal roots

If the indicial equation has two equal roots, (gamma_1=gamma_2), we have one solution of the form [y_1(t) = t^{gamma_1} sum_{n=0}^infty c_n t^n.] The other solution takes the form [y_2(t) = y_1(t)ln t +t^{gamma_1+1} sum_{n=0}^infty d_n t^n.] Notice that this last solution is always singular at (t=0), whatever the value of (gamma_1)!

Case II: Two roots differing by an integer

If the indicial equation that differ by an integer, (gamma_1-gamma_2=n>0), we have one solution of the form [y_1(t) = t^{gamma_1} sum_{n=0}^infty c_n t^n.] The other solution takes the form [y_2(t) = ay_1(t)ln t +t^{gamma_2} sum_{n=0}^infty d_n t^n.] The constant (a) is determined by substitution, and in a few relevant cases is even (0), so that the solutions can be of the generalised series form.

Example (PageIndex{1}):

Find two independent solutions of [t^2y''+ty'+ty=0] near (t=0).


The indicial equation is (gamma^2=0), so we get one solution of the series form

[y_1(t) = sum_n c_n t^n.]

We find

[egin{align*} t^2y''_1 &= sum_n n(n-1) c_n t^n onumber ty'_1 &= sum_n n c_n t^n onumber ty_1 &= sum_nc_n t^{n+1} =sum_{n'}c_{n'-1} t^{n'}end{align*}]

We add terms of equal power in (x),

[egin{array}{rclclclclcl} t^2y''_1 &= 0&+& 0 t&+&2c_2t^2&+&6c_3t^3&+&ldots ty'_1 &= 0&+& c_1t&+&2c_2t^2&+&3c_3t^3&+&ldots ty_1 &= 0&+& c_0t&+&c_1t^2&+&c_2t^3&+&ldots hline t^2y''+ty'+ty&= 0&+&(c_1+c_0)t&+&(4c_2+c_1)t^2&+&(9c_3+c_2)t^2&+&ldots end{array}]

Both of these ways give [t^2y''+ty'+ty=sum_{n=1}^infty (c_n n^2+c_{n-1})t^n,]

and lead to the recurrence relation

[c_n = -frac{1}{n^2} c_{n-1}] which has the solution [c_n = (-1)^n frac{1}{n!^2}] and thus [y_1(t) = sum_{n=0}^infty (-1)^n frac{1}{n!^2} x^n] Let us look at the second solution

[y_2(t) =ln(t) y_1(t)+underbrace{tsum_{n=0}^infty d_n t^n}_{y_3(t)}]

Here I replace the power series with a symbol, (y_3) for convenience. We find

[egin{align*} y_2' &= ln(t) y_1' + frac{y_1(t)}{t}+y_3' onumber y_2'' &= ln(t) y_1'' + frac{2y'_1(t)}{t}-frac{y_1(t)}{t^2}+ +y_3''end{align*}]

Taking all this together, we have,

[egin{align*} t^2y_2''+ty_2'+ty_2 &= ln(t)left(t^2 {y_1}''+t {y_1}'+t{y_1} ight) -y_1+2ty'_1+y_1 + t^2 {y_3}''+t {y_3}'+y_3 onumber &= 2t{y_1}'+t^2{y_3}''+t{y_3}'+ty_3=0.end{align*}]

If we now substitute the series expansions for (y_1) and (y_3) we get [2c_n+d_n(n+1)^2+d_{n-1}=0,] which can be manipulated to the form

Here there is some missing material

Example (PageIndex{2}):

Find two independent solutions of [t^2{y'}'+t^2{y}'-ty=0] near (t=0).


The indicial equation is (alpha(alpha-1)=0), so that we have two roots differing by an integer. The solution for (alpha=1) is (y_1=t), as can be checked by substitution. The other solution should be found in the form

[y_2(t) = atln t + sum_{k=0}d_k t^k]

We find

[egin{align*} y_2' & = & a+aln t + sum_{k=0}kd_k t^{k-1} onumber y_2'' & = & a/t + sum_{k=0}k(k-1)d_k t^{k-2} onumber end{align*}]

We thus find [egin{align*} t^2y''_2+t^2y'_2-ty_2= a(t+t^2)+ sum_{k=q}^infty left[d_k k(k-1)+d_{k-1}(k-2) ight] t^kend{align*}]

We find

[d_0 = a,;;;2 d_2+a=0,;;;d_k = (k-2)/(k(k-1))d_{k-1};;(k>2)]

On fixing (d_0=1) we find [y_2(t) = 1 + t ln t + sum_{k=2}^infty frac{1}{(k-1)!k!}(-1)^{k+1} t^k]

9.3: Special Cases


[OK, now you don't have to make your own all-brass .410 hulls. Midway has 2 1/2" hulls already made! But I will not delete the information below because you may have to make your own in the future.]

See below for more information from a reader of this page. Annealing.

Long term reloading of .410's means using all-brass cases. That requires more knowledge and skill, but the brass cases will last virtually forever. Extremely sturdy brass cases can be made from .303 British cases, .444 Marlin brass, or for full length 3" cases, 9.3 x 74R brass can be used. First the empty cases should be annealed, then fireformed with 8.0 grains of Herco and a case full of Cream of Wheat (load single shot, straight up, then fire). The Cream of Wheat will blow the case walls straight, including the shoulder and neck, resulting in a straight sided all brass case.

For reference, the base diameter of factory .410's is 0.469", and the rim diameter is 0.524". .444 Marlin cases are straight walled and can be used as .410 brass without the need for fireforming, but have the penalty of a 2.162" case length. The base diameter of the .444 Marlin is 0.469", whereas the .303 British has a base diameter of 0.458". The rim diameter is smaller with the .444 Marlin - 0.514" instead of the 0530" rim diameter of the .303 British - so it is theoretically possible that a really loose extractor might not catch the rim. The 9.3 x 74R European brass has a base diameter of 0.465", a rim diameter of 0.524, and an overall length of 3.47". The rim thickness (headspace) of the 9.3 x 74R, however, can be a tight fit in minimum .410 bore shotgun chambers, and may need to be thinned before use. Obviously we're only talking a few hundreds or thousands of an inch differences here, but it is a complication not taken with abandon.

Let me digress a little. By now you are wondering why the cases mentioned above all have essentially the same base diameter. The reason is the cost of the draw dies when the cases were designed and originally manufactured. In 1869, when the .44 S & W American was designed, draw dies for making brass were very expensive, so they were used for other cartridges as the need arose. In 1870, Smith & Wesson developed the .44 Russian - so manufacturers used the same draw dies. When development work was underway in the early 1890's on the .303 British and .30-40 Krag, again those same draw dies were used to form the longer rifle cases. Then in 1907 came the .44 Special, etc. In the finishing process, of course, rims of different diameter and thickness could turned, so the .30-40 Krag rim is 0.540" in diameter, as opposed to the 0.530" rim diameter on the .303 British. Shotshell cases were originally drawn brass, not paper or plastic. The base diameter of .410 shotshells is 0.469" and the rim diameter 0.524". not surprising that brass cases will work, as they were made with essentially the same draw dies!

For single shot shotguns, an all-brass case made by fireforming .303 British brass seldom requires resizing, needing only a new primer before reloading. The resulting case is 2.25" in overall length. As shotshell length is measured as fired, a 2 " .410 case actually measures 2 1/4" when loaded. the same as the all brass shell made from a .303 British case! Wonder of wonders. Like it was meant to be. Actually, it was, as you will see. Occasional sizing may be needed, and can be performed using a .44 Special size die and a .303 British shell holder. Heck, I knew a geezer who sized only the top 5/8" or so of the brass with an old .30-40 Krag sizer die! He had an oversize chamber, so it made sense to use the seater die instead of the actual size die, as the brass was worked less and lasted longer. With strong, thick walled rifle cases designed for 38,000 PSI or more, only the top third of the case expands at the 12,000 to 14,000 PSI working pressures of .410 shotgun loads.

Tubular magazine shotguns like the Mossberg HS410 and Winchester 9410 need a slight crimp in the case mouth and mild sizing for reliable feeding, which can easily be done with a .44 Special/Magnum crimp die (or .444 Marlin) with the bullet seater insert removed.

Shotgun primers are actually pistol primers minus the anvil, the rest of the shotgun primer assembly actually being a Berdan primer-type holder. In the 1950's and 60's, it was common practice for shotgun reloaders to simply replace the primer in the assembly, and Cascade (CCI) made primers without anvils just for that purpose - I still have some - but it was tedious work. When loading brass cases for use in the .410 bore shotgun, pistol primers should be used. It all has to do with the pressure generated by the loads and the designed impact of the firing pin: pistol primers are designed to ignite with only 6 to 9 pounds of impact, and resist pressures only half that of rifle primers.

A powder charge of 8.0 grains of Herco under to 5/8 ounce of shot will work well. Card wads are needed over the powder and the shot column, and felt wads are needed over the powder wad. These wads can be cut from thick felt or solid cardstock using a home made punch. A 300 H & H case can be sized in a .44 Special seater die deep enough to create about 3/8" of straight wall on the side of the case. Cut the case at the top of the straight section, then again just above the solid web at the base of the case. Sharpen the outside of the case mouth with a deburring tool, leaving the inside edge straight. Card stock is then placed on a relatively soft but flat surface, like a piece of truck inner tube on a kitchen cutting board, to protect the cutting edge of the home made wad cutter. Use a rubber hammer on the top of the card cutter, and disks of the correct diameter will be cut perfectly.

The card wads should be left "dry," but felt undershot wads should be lubricated to reduce leading in the barrel. There are nice commercial products available for this, or you can soak the felt wads in a 50-50 mix of melted beeswax and Vaseline, then let them cool and dry on paper.

Wad pressure is a controversial subject. Of more importance is seating resistance, where the top wad is sealed firmly enough to enhance pressure buildup before releasing. The old way, when all brass cases were the norm, was to put 8 to 10 drops of waterglass (also called "egg keep") on the top wad. In an emergency, use what you have, such as Elmer s glue, and clean the barrel more often.

Obviously, the height of the load within an all brass case depends upon the thickness of the felt wads used, but the actual height doesn t matter much, really, as long as the top wad is securely sealed. This is a total departure from loading recommendations for plastic shotshell cases, but all brass cases don t need internal support for stability of the top folding or roll crimp, as there isn t one. As long as you weigh (or measure) the powder charge and shot charge, the thickness of the felt wads is not that critical. Of course you can experiment and add of subtract felt wads to reach the top of the case if you want too, just as it is possible to cut plastic strips (similar in thickness to gallon milk jugs or bleach bottles) to surround the shot column inside the case to reduce lead shot scrubbing against the barrel.

Making shot is not hard: making it round and uniform is very tricky. Drop melted lead through a sieve into water and you have made shot. Teardrop shapes, odd sizes, but it is shot. Dropped from a considerable height through a chimney (or stove pipe) into water makes the shot more uniform in diameter, but in emergency situations may not be worth the trouble. Patterns with home made shot are usually twice as large as with nice, round commercial shot, but it works!

Simulated "buckshot" can even be made using .225" to .309" light weight cast bullets or round balls by using split shot sinker molds. The resulting "buckshot" does not have a great pattern, but does have greater penetration than smaller shot. When loading, the "buckshot" column should be buffered with Cream of Wheat filler or coarsely ground wheat to reduce leading and provide feed for more birds. For a really devastating load, use split shot crimped over strong monofilament, so the buckshot is essentially tied together with a two inch or so separation.

Of course I recommend having reloading tools and bullet molds for all of your rifle and handgun cartridges. You may not be able to get jacketed bullets, but if you have the primers, powder and a bullet mold, your rifle, pistol, revolver and shotgun can still work in hard times.

My booklet, Survival Reloading, has reloading tables for virtually every cartridge you might ever encounter, for both cast and jacketed bullets. Order here.

Brass shells come in two basic flavors:

1) Extruded -- which is the "old" way of making them. These are (relatively) cheap but, reloading them is a bit of an art (as I am sure you know!). Apparently, one needs to know the proper wad sizings (or make their own as described on your page) because the case wall is thinner than today's plastic (or yesterday's paper) shells and standard wad sizes will not fit. 12 gauge, for example, typically requires an 11 gauge "under shot" wad and a 10 gauge over shot wad to work properly.

2) Lathe-turned -- these shells are a bit "spendy" [20 three-inch (my interest) .410 shells are about $60] but have the advantage of matching the inner diameters of the plastic & paper shells. This makes it possible for one to use all of the "modern" components -- except for needing an over-shot card or, one can load black powder & traditional wadding or, one can mix & match as needed. See for sample pricing better (read "cheaper") sources may be available.

If you visit the Rocky Mountain Cartridge site, check out the page -- the best part of that page, I think, is where they mention Duco Cement as the over shot wad glue.

Unless one is "into" making their own wads as per your procedure, seems to be a great place to find components. Even if one wants to make his own for self-sufficiency purposes, a sample order or two might save tyros like me a lot of time getting the materials and sizes "right."

Finally, I tumbled to most of this stuff by reading the posts at -- for what it is worth.

If you do decide to edit your pages to include the "factory-available" brass shells, please DON'T removed the information on fire-forming, et cetera. As I said, it looks like it would be interesting to try some of your techniques & I'd hate to see the information get "lost" because there's a commercial product available.




ANSWER: I anneal all my brass in hot lead. Wheelweight alloy (89 lead, 1 tin, 10 antimony) melts at 619 F. The reason for using lead for annealing is to keep the temperature low enough for proper annealing AND have uniform annealing, and that is simply not possible using the torch method. With the propane torch, you stand the cases upright in a pan of water, heat the shoulder and neck, and when it glows the case is tipped over into the water. The case is heated on one side more than the other, and in falling over into the water, one side is quenched before the other side.

I use primed cases, using fired primers, as that forms an airlock that keeps lead from entering the case. Then I dip the case mouth (and about a half inch below the shoulder) down into the molten lead for about a count of two, pull it up out of the lead, tap on the side of the case to remove any bits of lead (if the lead is really sticking, the case isn't annealed!), then drop it mouth down (straight) into a 3 pound coffee can mostly full of ice water. I have another can with ice cubes, and every 10 rounds or so I add a few ice cubes to keep the water cool. I don't use gloves, as if the case head I'm holding got hot enough to require gloves, I would be annealing the case head and primer pocket too -- bad news.

Usually I don't tell people about this method because they may not be mentally organized. Water and molten lead do not mix, and I worry about the liability angle I don t assume any liability because people can t follow directions properly. Being left handed, I have the cases on the right side, the lead in the middle, and the ice water on the left. The cases go only one direction -- to the left -- and I use only one hand. Because it only takes a few seconds per case, I can anneal hundreds of cases in an hour with this method.

Over three decades ago I experimented with various methods of annealing brass, including the torch method. The reason was that I was that I was making brass as a commercial reloader specializing in obsolete and wildcat cartridges. I needed the cases to last, and fireforming had to be easy and reliable. I made a lot of 7 mm Weatherby out of anything belted, from 375 H & H on down to 300 Win Mag, and that entailed actually reducing the body diameter/taper to that of the 300 H & H case, trimming to 2.555", inside neck reaming, then fireforming with 15 grains of Herco and a case of Cream of Wheat. Same thing with 7 mm Ackley Improved and the various Gibbs cartridges made from 30-06 brass (with 10 or 12 grains of Herco depending upon case volume) -- fireforming was a must.

In my trails, annealing in lead gave the best results. But I know of one dude who dropped an ice cube into the lead pot and got himself (and everything close) covered with a thin film of hot lead, so I hesitate to tell just anyone about this method.

QUESTION: Dave: Being a bit lazy, I was looking for an easier shell to form that the 303. Taking a look at the 405 Win, I think most of the work is already done.

ANSWER: I looked at the .405 Win, and Hornady does make it again, but I don't think it would work without some extra work. The rim diameter of the .405 Win is 0.543", whereas the rim diameter of .410's is 0.524". As you know, the rim is countersunk into the rear face of the chamber, and .303 Br with a rim diameter of 0.530" is about maximum to fit the rim recess in my three .410's. ergo, the rims of the .405 Win would have to be filed down. True, the .405 Win has a case length of 2.58", but .410, 2 1/2" shells are only 2.25 OAL, only being .2 1/2" when the crimp is opened. So the extra length doesn't gain much. Nice try! Good idea, but .444 Marlin cases are probably better if you don't want to fireform.

QUESTION: Dave: My point being this Brass would load 410 x 2.5 virgin right out of the box. Now my question is, if I decided to size it and load it with 45 Win bullets, as the pistol is riffled, what charge would you recommend. For that matter I could load it like a shot shell, that just has a heavier slug, wads and all?

ANSWER: Won't work right out of the box because of the rim diameter. But if loaded with 0.454" bullets of 250 grain weight, a load of 12.5 grains of Herco would be about maximum. 410's are NOT strong actions.

QUESTION: Dave: I found your Herco burned at

59% Bullseye. If I wanted to load it as a metallic cartridge, would I go with a slower burn rate or a faster one?

ANSWER: On a burning rate chart where Bullseye is 1 and H-4831 is 100, Herco is 15. It burns in about 9 inches, is a very bulky powder, so it is about ideal when loading a huge capacity case with a small amount of powder. 2400 is too slow for your 9" barrel (burning rate about 22), and Unique is too dense, burns too hot (temperature), and burns too fast (10 on the burning rate chart). Bullseye is way to fast burning for anything but plinker loads. So we're back to Herco as being the right burning rate, bulky to help fill the case, and has a very predictable pressure curve, so I would stick with Herco and simply load a .45 Colt load but in the longer case. A little Dacron fluff (steal from a pillow) will hold the powder against the primer as well as cushion the base of the soft lead, plain base bullet. Right. NOT a jacketed bullet, but a cast bullet, as the pressure limitation imposed by the action itself mitigates for cast bullets over jacketed bullets with full loads.

QUESTION: Dave: Of course I could just be committing suicide too.. hehe.

ANSWER: Nah. Just take it easy and use loads designed for OLD .45 Colt revolvers, and you should be OK. There are loads on the Internet for .454 Magnums based on the .45 Colt solid head case, but the revolvers that fire them weigh more than a .410 shotgun and have twice as much steel around the chamber area as a .410 shotgun. Those magnum loads generate up to 54,000 PSI, whereas the .410 shotgun is designed for about 14,000 PSI chamber pressures. There is no way you can duplicate a .454 Magnum load in a "normal" .410 shotgun without going into orbit.

QUESTION: From Richard. Thank you so much for your very informative web Site. I found it a pleasure to read I have a New England Firearms .410/.45 Colt. It is a rifled 3 slugger. I have successfully made some nice .45 Colt Magnum rounds. But I am WANTING MORE ..HEHEH I am having trouble finding 9.3x74R cases so I can use a full 3 or at least 2.75 of the chamber I know from your report that a .444 Marlin and the .303 Brit will work, but those are still a bit short Any other ideas. Or maybe suppliers of the 9.3 s Also can the 9.3 s cases withstand the pressures that I may be putting on it My round of choice is a 240 JHP if that helps, but may be willing to try something in the 300 gr range if I can find some good hard cast lead

ANSWER: You have a top break, single shot action designed in the 1890's as a maritime 10 gauge line throwing gun. With modern steel and heat treating techniques, it will take pressures of 50,000 PSI, however, the pressures should be reasonably limited to about 42 to 45,000 PSI to keep the action from getting loose. The 9.3 x 74R case was designed to approximate 375 H&H flanged ballistics in a rimmed case for drillings, so that brass can take MORE pressure than your rifle can handle! But that isn't the problem.

You want, I think, to duplicate .444 Marlin ballistics in your .410/45 Colt. The best powders for the .454" expansion ratio of your bore are in the medium burning rates, NOT the bulky slower burning rate powders, so you most likely would be using 4198, 4320 or 4895 (heaviest bullets), and there is more than enough case capacity in a .444 Marlin case to achieve the ballistics you want. The 2.162" length of the case is not a problem for holding enough powder, and the powders mentioned, even being double based, have sufficient deterrent coating to not cause any appreciable chamber erosion, so using the shorter case would not harm your rifle's chamber, and the cases do not need much alteration of the rim to fit the rim recess in your rifle. The 9.3 x 74R rim is both a larger diameter AND thicker (headspace), so it needs more alteration. I'd go with the .444 or .303 cases, in my opinion.

Of more significance is the primer pressure limitation. No matter what the load INSIDE the case, a .45 Colt case is designed with a shallower primer pocket for large pistol primers. Those primers only take 6 to 9 lbs of pressure to ignite because the cup is thinner, BUT that thinner cup limits your pressures to well under 50,000 PSI or the firing pin indentation will blow through. The .303 or .444 Marlin cases are designed to accept the deeper large rifle primers, which take 12 to 15 lbs of striking force to ignite, and can handle pressures to at least 60,000 PSI.

Given that your rifle will produce enough energy to reliably ignite rifle primers, and you are going to be using rifle powders, I'd be inclined to use the readily available .303 or .444 Marlin cases and STANDARD (not magnum!) large rifle primers.

You also have very shallow rifling, and that mitigates AGAINST cast bullets, even hard cast bullets. You also need to consider bullet expansion at the velocities you will be achieving, and that means using 240 to 300 grain bullets with thick jackets designed for upwards of 2,000 FPS, NOT the thin jacketed bullets designed to expand at 1350 to 1500 FPS in the .45 Colt or even the .454 Casull. I really hate to write this, as I'm a cast bullet fan from way back. I use them exclusively in my 1881 .303 Martini (at 2415 FPS!) and even use them in my 7mm Wby Mag. But I've got .004" rifling in those barrels, and you do not have that in the New England SS, so cast bullets would not generally give very good accuracy. Sorry.

QUESTION : How do I bell the case mouth, or remove a dent, without having a special belling die? And what is the least expensive way to decap cases?

ANSWER : If you have to round out the mouth of bent cases, or perhaps even to slightly bell the case mouth for removing a previous crimp that is not completely blown out on firing, use a .30-06 case. It has a shoulder diameter of 0.441 to 0.447", and a shoulder angle of 17 degrees. Fill the case with melted lead for weight and to add inner strength so the shoulder has enough strength to last for years. Place the case base down on a solid surface, put the belling tool in the case mouth, and rap gently with a light hammer.

NOTE: Obviously I believe that everyone interested in survival should have reloading and bullet casting equipment and supplies, plus reloading data for every cartridge normally encountered. Keeping your own rifles and handguns working - as well as those of your neighbors - would be an obvious advantage in most any survival scenario. Full power loads are not needed for this purpose. Cast bullets at moderate velocities would be just fine, and far better than nothing at all! See Survival Reloading.

Keep your rifles and handguns shooting with properly reloaded ammunition using portable reloading equipment. Includes tool selection, adaptable shell holders, and complete reloading data for virtually every caliber, including cast bullet loads -- all with only three different smokeless powders for ease of storage and versatility. Twenty four pages.

[Note: The focus of this booklet is very specific - Survival Reloading. I make no attempt to list max loads for every caliber with every powder - you can get that from any reloading manual. I do list loads for virtually every cartridge manufactured in the past 120 years, including many long obsolete, for both jacketed and cast bullets with only 3 powders: 4985, Herco and Red Dot.]

A little nostalgia for long time reloaders. Nosler only produced bullets in Ashland, OR for a few months in the late 1940's before moving to Bend, OR. Winchester made the high pressure 9mm's for submachine guns only for the military 50 years ago. The Barnes bullet box on the right was from one of his first lots of bullets, well over 40 years ago.

"Those who hammer their guns into plows will plow for those who do not."
Thomas Jefferson


1. An armed man is a citizen. An unarmed man is a subject.
2. A gun in the hand is better than a cop on the phone.
3. Gun control is not about guns it's about control.
4. If guns are outlawed, can we use swords?
5. If guns cause crime, then pencils cause misspelled words.
6. Free men do not ask permission to bear arms.
7. If you don't know your rights, you don't have any.
8. Those who trade liberty for security have neither.
9. The United States Constitution (c)1791. All Rights Reserved.
10. What part of "shall not be infringed" do you not understand?
11. The Second Amendment is in place in case the politicians ignore the others.
12. 64,999,987 firearms owners killed no one yesterday.
13. Guns only have two enemies rust and politicians.
14. Know guns, know peace, know safety. No guns, no peace, no safety.
15. You don't shoot to kill you shoot to stay alive.
16. 911: Government sponsored Dial-a-Prayer.
17. Assault is a behavior, not a device.
18. Criminals love gun control it makes their jobs safer.
19. If guns cause crime, then matches cause arson.
20. Only a government that is afraid of its citizens tries to control them.
21. You have only the rights you are willing to fight for.
22. Enforce the gun control laws we ALREADY have don't make more.
23. When you remove the people's right to bear arms, you create slaves.
24. The American Revolution would never have happened with gun control.


"Calling an illegal alien an "undocumented immigrant" is like calling a drug
dealer an "unlicensed pharmacist."


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(i) General

All Motion Papers, Oppositions, and Replies must be served on all parties and filed with the clerk in accordance with the procedure set forth in this Paragraph (b). Compliance with this Paragraph shall constitute compliance with the “reasonable time” provisions of the first sentence of Mass. R. Civ. P. 5(d)(1).

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Papers must be served on specifically named non-parties in compliance with this Rule if (a) the Motion seeks to add the non-party as a party to the case (b) the Motion seeks an order or other relief against the non-party (c) the issues affect the personal information or other interests of the non-party. The non-party need not be served, however, if excused by a court order issued in advance for cause or if a statute or rule expressly authorizes ex parte relief.

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The parties must cooperate in filing with the court a “Rule 9A Package.” The Rule 9A Package consists of the original Motion Papers, the Opposition, and the Reply, any other papers for which leave of court is granted under Paragraph (a)(6), and any appendices or other papers permitted or required by this Rule, statute, or order of the court.

(ii) Time for filing or withdrawal of the motion

Within 10 days of service of the Opposition, the moving party must either (1) file the Rule 9A Package with the court or (2) notify all parties that the motion has been withdrawn and will not be filed. If the moving party does not receive an Opposition within 3 business days after expiration of the time permitted for service of an Opposition, then the moving party must file with the clerk the Motion Papers together with an affidavit reciting compliance with this Rule and receipt of no Opposition in a timely fashion, unless the moving party withdraws the motion and has so notified all parties.

(iii) Notice of filing

The moving party must give prompt notice of the filing of a Rule 9A Package by serving all parties with a copy of a notice of filing in a separate document that lists the title of each document included in the Rule 9A Package, and by filing the notice with the Rule 9A Package.

(3) Time periods in general

The time periods prescribed below apply unless a different time period is set by statute or order of the court. Where papers are served by mail, these time periods are extended by 3 days in accordance with Mass. R. Civ. P. 6(d).

(4) Motions except motions for summary judgment

(i) Time for service of opposition

All Oppositions must be served no later than 10 days after service of the Motion Papers.

(ii) Effect of cross-motion/motion to strike

The provisions of Paragraph (b)(4)(i) apply to cross-motions (including motions to strike) served with the Opposition to a motion. When a cross-motion is brought, the time for filing the Rule 9A Package for the original motion is extended to be coterminous with the date for filing the cross-motion. The Rule 9A Packages for the original motion and the cross-motion must be filed together by the original moving party.

(5) Motions for summary judgment

(i) Statement of facts

A motion for summary judgment must be accompanied by a statement of the material facts as to which the moving party contends there is no genuine issue to be tried, set forth in consecutively numbered paragraphs, with page or paragraph references to supporting pleadings, depositions, answers to interrogatories, responses to requests for admission, affidavits, or other evidentiary documents (“Statement of Facts”). Only such facts as are material to deciding the motion shall be included in the Statement of Facts.

The Statement of Facts as served shall not exceed 20 pages in length and shall not include:

a. Background facts not material to decision of the motion. Such facts may be included in a party’s memorandum of law even though they are not in the statement.

b. Quotations from any contract, trust, agreement, or other transactional document, or any characterizations of the document (except if admissible through percipient witnesses). The Statement of Facts may only establish the existence and authenticity of the document and the date it became effective.

c. Quotations from any statute, regulation or rule.

Quotations from material described in paragraphs b and c may be included, without argument or commentary, in an addendum to the party’s memorandum of law.

This Statement of Facts must be a separately captioned document. Failure to include the Statement of Facts constitutes grounds for denial of the motion. The Court may disregard a Statement of Facts in whole or part if it is unnecessarily long or otherwise materially out of compliance with this rule.

(ii) Service of motion papers

The moving party must serve a copy of its Motion Papers, and the Moving Party’s Statement of Facts, on every other party. The Moving Party’s Statement of Facts must also be sent contemporaneously in electronic form by email to all parties in Rich Text Format (RTF) or such other format as to which the parties agree. The email transmission of the Moving Party’s Statement of Facts is excused if (1) the moving or any opposing party is self-represented, (2) the attorney for the moving party certifies in an affidavit that he or she does not have access to email, or (3) the attorney for the moving party certifies in an affidavit that an opposing party’s attorney has no email address or has not disclosed his or her email address.

(iii) Opposition

Within 21 days after service of the Motion Papers, any party opposing the motion must serve on the moving party the original and one copy of the Opposition, and must serve on all other parties one copy of the Opposition.

(A) Response to moving party’s statement of facts

The Opposition may include a response to the Moving Party’s Statement of Facts. The opposing party must reprint the Moving Party’s Statement of Facts and set forth a response directly below the appropriate numbered paragraph, including, if the response relies on opposing evidence, page or paragraph references to supporting pleadings, depositions, answers to interrogatories, responses to requests for admission, affidavits, or other evidentiary documents. The response to the numbered paragraphs shall be limited to stating whether a given fact is disputed and, if so, cite to the specific evidence, if any, in the Joint Appendix that demonstrates the dispute. It shall not:

a. Deny a fact unless the party has a good faith basis for contesting it.

b. Include a statement that a fact is not supported by the materials cited by the moving party, unless the responding party has a good faith basis for contesting it.

c. Include commentary on whether the fact asserted is relevant or material to any issue raised in the case, although a responding party may indicate, where appropriate, that the fact is admitted only for the purposes of the summary judgment motion.

d. Assert any additional facts. Additional facts may be included in the response only in the manner provided in section (b)(5)(iii)(B) below.

e. Make legal arguments or advocacy-oriented characterizations concerning the sufficiency, relevance or materiality of the moving party’s factual proffers.

Where the obligation to send the Moving Party’s Statement of Facts in electronic form has been excused, the response thereto may be in a separate document. For purposes of summary judgment, each fact set forth in the moving party’s statement of facts is deemed to have been admitted unless properly controverted in the manner forth in this Paragraph (b)(5)(iii)(A).

(B) Statement of additional facts

Opposing parties who argue that additional facts warrant denying summary judgment shall include those facts in the opposition memorandum, each to be supported with page or paragraph references to supporting pleadings, depositions, answers to interrogatories, responses to requests for admission, affidavits, or other evidentiary documents. They may not submit a separate statement of additional facts, except in support of a cross-motion for summary judgment.

(C) Service of response to statement of facts

The opposing party’s response to the Moving Party’s Statement of Facts must be served contemporaneously by email as described in (b)(5)(ii) above, unless such service is excused.

(D) Exhibits for joint appendix

Where the opposing party relies upon evidence not included in the exhibits served with the Motion Papers, the opposing party must serve the moving party with such evidence in the form of new exhibits for inclusion in the Joint Appendix, in accordance with Paragraph (b)(5)(v) below.

(E) Citation of evidence

The opposing party must cite to the Joint Appendix in accordance with Paragraph (b)(5)(v) below.

(iv) Filing of Rule 9A package

(A) Joint appendix and statement of facts

The Rule 9A Package must also include the Joint Appendix and a Consolidated Statement of Facts, which must include the opposing party’s responses to the Moving Party’s Statement of Facts. Similarly, in cases with multiple parties, all parties moving or opposing summary judgment shall coordinate their statements and responses so that there shall be a single statement and response covering all motions. Unless the obligation to send the Moving Party’s Statement of Facts or the response thereto in electronic form has been excused, only the Consolidated Statement of Facts (and not any intermediate versions thereof) may be filed so that the court has only a single document.

(B) Service of statement of facts and joint appendix

Upon filing the Rule 9A Package, the moving party must serve on the opposing parties the Notice of Filing described below and the following, in paper and electronic form, unless electronic form is excused: (1) the Consolidated Statement of Facts filed with the clerk (2) the Joint Appendix, unless the parties otherwise agree

(C) Effect of cross-motion/motion to strike .

The provisions of Paragraph (b)(5)(i)-(iv) apply to cross-motions for summary judgment and any other cross-motion (including a motion to strike) served with the Opposition to a motion for summary judgment. A separate Consolidated Statement of Facts must be served with any cross-motion for summary judgment. All parties moving for or opposing summary judgment shall coordinate their statements and responses so that there shall be a single consolidated document containing the respective statements of material facts and responses thereto. When a cross-motion (including motion to strike) is brought, the time for filing the Rule 9A Package for the original motion is extended to be coterminous with the date for filing the cross-motion. The Rule 9A Packages for the original motion and the cross-motion must be filed together by the original moving party.

(v) Joint appendix

(A) Contents, format, citation, and service

All exhibits referred to in the memoranda supporting or opposing a motion or cross-motion for summary judgment, or in the Consolidated Statement of Facts, must be filed as a single joint appendix, which must include an index of the exhibits (“Joint Appendix”). The initial moving party, with the cooperation of each opposing party, is responsible for assembling the Joint Appendix and index. Unless all the pages of the Joint Appendix are consecutively numbered by page, each exhibit must be separated by an off-set tab divider. The exhibits served by the moving party with its Motion Papers must include either the consecutive numbering or offset tabs. Where an opposing party relies upon any evidence included in the moving party’s exhibits, the opposing party must cite to that evidence using the form of designation of the moving party. If the opposing party designates new exhibits in accordance with Paragraph (b)(5)(iii)(D), it must serve those new exhibits, together with an index of the new exhibits, on the moving party with the Opposition, and it must serve the index on the moving party in electronic form (unless electronic service is excused). Those new exhibits must begin with the next consecutive designation following the last designation by the initial moving party (whether consecutive page numbering or off-set tab dividers). The opposing party must serve the original and one copy of those new exhibits with its Opposition.

(B) Certification

The initial moving party must certify that the Joint Appendix includes all exhibits served with the Opposition, except for any exhibit(s) designated by the opposing party but not provided to the moving party. The burden is on the opposing party to move to file any designated exhibit not timely submitted. All memoranda of law filed in support of or in opposition to a motion for summary judgment shall reference the exhibit numbers as well as a paragraph in the statement of material facts.

(vi) Decision on certain motions without a hearing

The following types of summary judgment motions may, in the court’s discretion, be denied on the papers without a hearing notwithstanding Rule 9A(c)(3) (but shall not be granted without a hearing unless the hearing is waived):

(1) Multiple summary judgment motions by a single party, or subsequent summary judgment motions by parties sharing similar interests and making the same arguments as those the court has already resolved.

(2) Motions for partial summary judgment that will save little or no trial time, will not simplify the trial and will not promote resolution of the case.

(3) Motions for summary judgment where a genuine dispute of material fact is obvious on the face of the papers.

(vii) Sanctions for noncompliance

The court need not consider any motion or opposition that fails to comply with the requirements of this Rule, may return non-compliant submissions to counsel with instructions for re-filing, and may impose other sanctions for flagrant violations of the Rule.


Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies". [p 1] The incompatibility of Newtonian mechanics with Maxwell's equations of electromagnetism and, experimentally, the Michelson-Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized luminiferous aether did not exist. This led to Einstein's development of special relativity, which corrects mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities ). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible. [3] [4] Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.

Special relativity has a wide range of consequences that have been experimentally verified. [5] They include the relativity of simultaneity, length contraction, time dilation, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession. [1] [2] It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = m c 2 > , where c is the speed of light in a vacuum. [6] [7] It also explains how the phenomena of electricity and magnetism are related. [1] [2]

A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into a single continuum known as "spacetime". Events that occur at the same time for one observer can occur at different times for another.

Until Einstein developed general relativity, introducing a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity" "special" really means "special case". [p 2] [p 3] [p 4] [note 1] Some of the work of Albert Einstein in special relativity is built on the earlier work by Hendrik Lorentz and Henri Poincaré. The theory became essentially complete in 1907. [4]

The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, the curvature of spacetime, described by the energy–momentum tensor and causing gravity, is negligible. [8] [note 2] In order to correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate accelerations as well as accelerating frames of reference. [9] [10]

Just as Galilean relativity is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, that is, at a sufficiently small scale (e.g., when tidal forces are negligible) and in conditions of free fall. General relativity, however, incorporates non-Euclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as Minkowski space. As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.

Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light, [11] a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics. [12]

Albert Einstein: Autobiographical Notes [p 5]

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in a vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as: [p 1]

  • The Principle of Relativity – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other. [p 1]
  • The Principle of Invariant Light Speed – ". light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface). [p 1] That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.

The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment. [13] [14] In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.

The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history. [p 6]

Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations. [15] However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively to K. [16]

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.

Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles. [p 7]

Reference frames and relative motion Edit

Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space that is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).

An event is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in spacetime. Since the speed of light is constant in relativity irrespective of the reference frame, pulses of light can be used to unambiguously measure distances and refer back to the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.

For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.

In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called transformation equations.

Standard configuration Edit

To gain insight into how the spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration. [17] : 107 With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-1, two Galilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′.

  • The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
  • Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.
  • The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.

Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.

Lack of an absolute reference frame Edit

The principle of relativity, which states that physical laws have the same form in each inertial reference frame, dates back to Galileo, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance they called "aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an absolute reference frame against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether was supposed to be sufficiently elastic to support electromagnetic waves, while those waves could interact with matter, yet offering no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelson–Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist. [18] Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

Relativity without the second postulate Edit

From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity) it can be shown that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum. [p 8] [19]

Alternative approaches to special relativity Edit

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:

The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events . The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws . [p 5]

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime. [p 9] [p 10]

Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations. [20] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler [21] and by Callahan. [22] This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.

Lorentz transformation and its inverse Edit

Define an event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in a reference frame moving at a velocity v with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:

is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′, relative to S, is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.

Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:

Enforcing this inverse Lorentz transformation to coincide with the Lorentz transformation from the primed to the unprimed system, shows the unprimed frame as moving with the velocity v′ = −v, as measured in the primed frame.

There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction parallel to the motion (which are warped by the γ factor) and perpendicular see the article Lorentz transformation for details.

A quantity invariant under Lorentz transformations is known as a Lorentz scalar.

Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x1, t1) and (x1, t1) , another event has coordinates (x2, t2) and (x2, t2) , and the differences are defined as

If we take differentials instead of taking differences, we get

Graphical representation of the Lorentz transformation Edit

Spacetime diagrams (Minkowski diagrams) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario. [19]

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1. [19] [23] : 155–199

While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane, but the frames are actually equivalent.

The consequences of special relativity can be derived from the Lorentz transformation equations. [24] These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.

Invariant interval Edit

In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as Δ s 2 > :

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.

The form of Δ s 2 , ,> being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances. [note 7] The invariance of this interval is a property of the general Lorentz transform (also called the Poincaré transformation), making it an isometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, Lorentz boosts, in the x-direction) with all other translations, reflections, and rotations between any Cartesian inertial frame. [28] : 33–34

In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration: [19]

In considering the physical significance of Δ s 2 > , there are three cases to note: [19] [29] : 25–39

  • Δs 2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that | Δ x / Δ t | < c , and given the Lorentz transformation Δ x ′ = γ ( Δ x − v Δ t ) , it is evident that there exists a v less than c for which Δ x ′ = 0 (in particular, v = Δ x / Δ t ). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, Δ s / c , is called the proper time.
  • Δs 2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that | Δ x / Δ t | > c , and given the Lorentz transformation Δ t ′ = γ ( Δ t − v Δ x / c 2 ) , ),> there exists a v less than c for which Δ t ′ = 0 (in particular, v = c 2 Δ t / Δ x Delta t/Delta x> ). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, − Δ s 2 , >>,> is called the proper distance, or proper length. For values of v greater than and less than c 2 Δ t / Δ x , Delta t/Delta x,> the sign of Δ t ′ changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. The temporal order of timelike-separated events, however, is absolute, since the only way that v could be greater than c 2 Δ t / Δ x Delta t/Delta x> would be if v > c .
  • Δs 2 = 0: In this case, the two events are said to be lightlike separated. This implies that | Δ x / Δ t | = c , and this relationship is frame independent due to the invariance of s 2 . .> From this, we observe that the speed of light is c in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.

Relativity of simultaneity Edit

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).

From Equation 3 (the forward Lorentz transformation in terms of coordinate differences)

It is clear that the two events that are simultaneous in frame S (satisfying Δt = 0 ), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0 ). Only if these events are additionally co-local in frame S (satisfying Δx = 0 ), will they be simultaneous in another frame S′.

The Sagnac effect can be considered a manifestation of the relativity of simultaneity. [30] Since relativity of simultaneity is a first order effect in v , [19] instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity. [p 14]

Time dilation Edit

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels).

Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0 . To find the relation between the times between these ticks as measured in both systems, Equation 3 can be used to find:

This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory. [31]

Length contraction Edit

The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).

Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0 , which can be combined with Equation 4 to find the relation between the lengths Δx and Δx′:

This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S).

Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system.

Lorentz transformation of velocities Edit

Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector u . .> What is its velocity u ′ > in frame S′ ?

Substituting expressions for d x ′ and d t ′ from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed u to u ′ :

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing v with − v .

The forward and inverse transformations for this case are:

We note the following points:

  • If an object (e.g., a photon) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±c), then it would also be moving at the speed of light in any other frame, moving at | v | < c .
  • The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.
  • If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g., | u / c | ≪ 1 ), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
  • Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.

There is nothing special about the x direction in the standard configuration. The above formalism applies to any direction and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.

Thomas rotation Edit

The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.

Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion. [29] : 169–174

Thomas rotation provides the resolution to the well-known "meter stick and hole paradox". [p 15] [29] : 98–99

Causality and prohibition of motion faster than light Edit

In Fig. 4-3, the time interval between the events A (the "cause") and B (the "effect") is 'time-like' that is, there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).

The interval AC in the diagram is 'space-like' that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. However, there are no frames accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result.

For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams). [32] [p 16] A variety of causal paradoxes could then be constructed.

Consider the spacetime diagrams in Fig. 4-4. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ct′), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.

  1. Fig. 4-4a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the − x ′ axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.
  2. Fig. 4-4b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the + x axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality. [33]

Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in vacuum.

This is not to say that all faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light. [35] For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon). [36] [37]

Dragging effects Edit

In 1850, Hippolyte Fizeau and Léon Foucault independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory. [38] The speed of light was measured in still water. What would be the speed of light in flowing water?

In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light.

According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed through the medium plus the speed of the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If u ′ = c / n is the speed of light in still water, and v is the speed of the water, and u ± > is the water-bourne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then

Fizeau's results, although consistent with Fresnel's earlier hypothesis of partial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since n depends on wavelength, the aether must be capable of sustaining different motions at the same time. [note 8] A variety of theoretical explanations were proposed to explain Fresnel's dragging coefficient that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies. [39]

From the point of view of special relativity, Fizeau's result is nothing but an approximation to Equation 10, the relativistic formula for composition of velocities. [28]

Relativistic aberration of light Edit

Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the aberration of light. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light. [40] (2) If the source is in motion, the displacement would be the consequence of light-time correction. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver. [41]

The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light, [42] and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope. [43] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag, [44] but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag. [45]

Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include [28] : 57–60

Relativistic Doppler effect Edit

Relativistic longitudinal Doppler effect Edit

The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here. [46] [47]

For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source. [48] [19]

Transverse Doppler effect Edit

The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.

Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver.

Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments. [19] In Fig. 5-3a, the receiver observes light from the source as being blueshifted by a factor of γ . In Fig. 5-3b, the light is redshifted by the same factor.

Measurement versus visual appearance Edit

Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.

Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. The visual appearance of an object, however, is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation. [p 19] [p 20] [49] [50] A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted. [51]

Fig. 5-4 illustrates a cube viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect. [note 9]

Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects, quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel. [52] [53] [54] In Fig. 5-5, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5-4 has been stretched out. [55]

Section Consequences derived from the Lorentz transformation dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.

Equivalence of mass and energy Edit

As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.

The energy content of an object at rest with mass m equals mc 2 . Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.

In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc 2 .

Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0) : it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c 2 , 0, 0) . The momentum is equal to the energy multiplied by the velocity divided by c 2 . As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c 2 .

The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations. [p 1] The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905. [p 21] Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong. [56] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions. [57]

Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy. [p 22] [note 10]

How far can one travel from the Earth? Edit

Since one can not travel faster than light, one might conclude that a human can never travel farther from Earth than 40 light years if the traveler is active between the ages of 20 and 60. One would easily think that a traveler would never be able to reach more than the very few solar systems which exist within the limit of 20–40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1g, it will, after a little less than a year, be travelling at almost the speed of light as seen from Earth. This is described by:

where v(t) is the velocity at a time t, a is the acceleration of 1g and t is the time as measured by people on Earth. [p 23] Therefore, after one year of accelerating at 9.81 m/s 2 , the spaceship will be travelling at v = 0.77c relative to Earth. Time dilation will increase the travellers life span as seen from the reference frame of the Earth to 2.7 years, but his lifespan measured by a clock travelling with him will not change. During his journey, people on Earth will experience more time than he does. A 5-year round trip for him will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for him (5 years accelerating, 5 decelerating, twice each) will land him back on Earth having travelled for 335 Earth years and a distance of 331 light years. [58] A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy. [58] This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its half-life (when at rest). [59]

Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.

The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.

Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, that is, in the language of tensor calculus. [60]

Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. How general relativity and quantum mechanics can be unified is one of the unsolved problems in physics quantum gravity and a "theory of everything", which require a unification including general relativity too, are active and ongoing areas in theoretical research.

The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time. [61]

In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour, [p 24] that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only describe the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron), [p 24] [p 25] and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics.

On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary in which particles can be created and destroyed throughout space and time.

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c 2 in the region of interest. [62] In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10 −20 ) [63] and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.

Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).

Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.

Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905, [64] and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory. [14]

  • The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
  • The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
  • The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
  • The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction does not lead to birefringence for a co-moving observer, in accordance with the relativity principle.

Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:

    – testing the limiting speed of particles – testing relativistic Doppler effect and time dilation – relativistic effects on a fast-moving particle's half-life – time dilation in accordance with Lorentz transformations – testing isotropy of space and mass – various modern tests
  • Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
  • Experiments to test the aether drag hypothesis – no "aether flow obstruction".

Geometry of spacetime Edit

Comparison between flat Euclidean space and Minkowski space Edit

Special relativity uses a 'flat' 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.

In 3D space, the differential of distance (line element) ds is defined by

where dx = (dx1, dx2, dx3) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X 0 derived from time, such that the distance differential fulfills

where dX = (dX0, dX1, dX2, dX3) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1). [66] Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.

The actual form of ds above depends on the metric and on the choices for the X 0 coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X0 = ict (this is called a Wick rotation). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X 0 = ct , rather than a "disguised" Euclidean metric using ict as the time coordinate.

Some authors use X 0 = t , with factors of c elsewhere to compensate for instance, spatial coordinates are divided by c or factors of c ±2 are included in the metric tensor. [67] These numerous conventions can be superseded by using natural units where c = 1 . Then space and time have equivalent units, and no factors of c appear anywhere.

3D spacetime Edit

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space

we see that the null geodesics lie along a dual-cone (see Fig. 10-2) defined by the equation

which is the equation of a circle of radius c dt.

4D spacetime Edit

If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:

As illustrated in Fig. 10-3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = c dt.

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance d = x 1 2 + x 2 2 + x 3 2 ^<2>+x_<2>^<2>+x_<3>^<2>>>> away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the Fig. 10-2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

The cone in the −t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought experiments in special relativity.

Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see Center of mass (relativistic).

Physics in spacetime Edit

Transformations of physical quantities between reference frames Edit

Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.

The Lorentz transformation in standard configuration above, that is, for a boost in the x-direction, can be recast into matrix form as follows:

In Newtonian mechanics, quantities that have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "four vectors", in Minkowski spacetime. The components of vectors are written using tensor index notation, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.

The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z) , in a contravariant position four vector with components:

where we define X 0 = ct so that the time coordinate has the same dimension of distance as the other spatial dimensions so that space and time are treated equally. [68] [69] [70] Now the transformation of the contravariant components of the position 4-vector can be compactly written as:

where the Lorentz factor is:

The four-acceleration is the proper time derivative of 4-velocity:

The transformation rules for three-dimensional velocities and accelerations are very awkward even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.

The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:

which is the transpose of:

only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.

More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:

The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.

More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law [71]

An example of a four-dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.

The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.

Metric Edit

The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates) which can be arranged in a 4 × 4 matrix:

The Poincaré group is the most general group of transformations which preserves the Minkowski metric:

and this is the physical symmetry underlying special relativity.

The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is:

Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself:

One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:

similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.

Relativistic kinematics and invariance Edit

The coordinate differentials transform also contravariantly:

so the squared length of the differential of the position four-vector dX μ constructed using

is an invariant. Notice that when the line element dX 2 is negative that √ −dX 2 is the differential of proper time, while when dX 2 is positive, √ dX 2 is differential of the proper distance.

The 4-velocity U μ has an invariant form:

which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces:

So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

Relativistic dynamics and invariance Edit

The invariant magnitude of the momentum 4-vector generates the energy–momentum relation:

We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.

The rest energy is related to the mass according to the celebrated equation discussed above:

The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.

To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.

If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:

In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, dp/dt while the four force is defined by the rate of change of momentum with respect to proper time, that is, dp/dτ.

In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.

How to Improve Your NCOER Score

Read DA Pamphlet 623-3 and AR 623-3. Understanding how the process works is the first step in improving your score. Don't waste time on areas that you can't control and concentrate on those that you can. Target areas that provide the most points first.

Talk with your supervisor about your NCOER. Ask him or her how you're doing and what you can do to improve. This may be the single most effective step you can take and can't be over-emphasized. Listen carefully to what he or she says. Often, it's difficult for supervisors to directly criticize their troops and they may offer advice in a tone that sounds more like a suggestion than an order. Try to read between the lines and ask questions if their meaning isn't clear. Then act on what your supervisor said. Follow up periodically and don't hesitate to ask for further advice. Supervisors love that.

Fill out an NCOER on yourself and see what areas are rated and which areas might be improved. Do this as soon as possible because it takes time to change behavior and even longer for others to become aware of it.

Keep track of your performance. Write down your accomplishments as they occur so that you can remember them when it comes time to provide material for your NCOER. You can be the best troop in the world but if you can't remember what you've accomplished, it's the same as if you didn't do anything. The best way to do this is to develop a habit that works for you -like every Friday after lunch, write down what you accomplished during the week. Or make a habit of recording your accomplishments when you have some other writing requirement, like a weekly report or weekly checks. Whatever method you choose, just make sure you do it regularly. This will pay big dividends when it's time to provide material for your NCOER or even a quarterly award.

Different Time Limits for Different Types of Claims

In some states, the type of personal injury claim may also affect the time limit. For example, certain defamation cases and claims involving minors (persons under age 18) may be granted longer time limits, while medical malpractice statutes of limitations may grant shorter time limits.

Typically, the statute of limitations in a lawsuit for injuries to a minor does not begin to run until he or she reaches the age of 18. For example, suppose Pat is injured in a car accident on his 17th birthday. In a state that has a two-year statute of limitations for personal injury lawsuits, Pat will have three years to file suit for injuries suffered in that accident.

7.0 Rules for Continuing Employment and Other Special Rules

You must complete a new Form I-9 when a hire takes place, unless you are rehiring an employee within three years of the date of the employee’s previous Form I-9. However, in certain situations, a hire is not considered to have taken place despite an interruption in employment. In case of an interruption in employment, you should determine whether the employee is continuing in his or her employment and has a reasonable expectation of employment at all times.

These situations constitute continuing employment:

  • Approved paid or unpaid leave on account of the employee’s illness or pregnancy maternity or paternity leave vacation study, union business a family member’s illness or disability or other temporary leave that you have approved.
  • Promotions, demotions, or pay raises.
  • Temporary layoff for lack of work.
  • Strikes or labor disputes.
  • Reinstatement after disciplinary suspension for wrongful termination found unjustified by any court, arbitrator, or administrative body, or otherwise resolved through reinstatement or settlement.
  • Transfer from one distinct unit of an employer to another distinct unit of the same employer you may transfer the employee’s Form I-9 to the receiving unit.
  • Seasonal employment.
  • Continuing employment with a related, successor, or reorganized employer, as long as the employer obtains and maintains records and Forms I-9, where applicable, from the previous employer. A related, successor, or reorganized employer includes:
  • The same employer at another location
  • An employer who continues to employ any employee of another employer’s workforce, where both employers belong to the same multi-employer association and the employee continues to work in the same bargaining unit under the same collective bargaining agreement. For these purposes, any agent designated to complete and maintain Form I-9 must enter the employee’s date of hire and/or termination each time the employee is hired and/or terminated by an employer of the multi-employer association.
  • Employers who have acquired or merged with another company have two options:
  • Option A: Treat all acquired employees as new hires and complete a new Form I-9 for every individual. Enter the effective date of acquisition or merger as the employee’s first day of employment in Section 2 of the new Form I-9. If you choose Option A, avoid engaging in discrimination by completing a new Form I-9 for all of your acquired employees, without regard to actual or perceived citizenship status or national origin.
  • Option B: Treat all acquired individuals as employees who are continuing in their uninterrupted employment status and retain the previous owner’s Form I-9 for each acquired employee. You will be liable for any errors or omissions on the previously completed Form I-9. You and/or the employee should make any corrections to the acquired employee’s existing Form I-9. For more information, see Section 8: Correcting Form I-9.
  • Employees hired on or before Nov. 6, 1986, who are continuing in their employment and have a reasonable expectation of employment at all times are exempt from completing Form I-9 and cannot be verified in E-Verify. For help with making this determination, see 8 CFR 274a.2(b)(1)(viii) and 8 CFR 274a.7. If you determine that an employee hired on or before Nov. 6, 1986, is not continuing in their employment or does not have a reasonable expectation of employment at all times, the employee may be required to complete a Form I-9.

    Federal contractors with the FAR E-Verify clause are subject to special rules regarding the verification of existing employees. For more information, see the E-Verify Supplemental Guide for Federal Contractors at

    To determine whether an employee continuing their employment had a reasonable expectation of employment at all times, you should consider several factors, including (but not limited to) whether:

    • The individual was employed on a regular and substantial basis. You can determine “regular and substantial basis” by comparing other workers who are similarly employed by the employer.
    • The individual complied with the employer’s established and published policy regarding his or her absence.
    • The employer’s past history of recalling absent employees for employment indicates the likelihood that the individual in question will resume employment with the employer within a reasonable time.
    • Another worker has not permanently taken the individual’s former position.
    • The individual has not sought or obtained benefits during their absence from employment that are inconsistent with an expectation of resuming employment within a reasonable time.
    • Your financial condition allows the individual to resume employment within a reasonable time.
    • The oral and/or written communication between you, your supervisory employees, and the individual indicates the individual will likely resume employment within a reasonable time.

    Inspect the previously completed Form I-9 (and, if necessary, update the form or reverify the employee) and store the form as if there was no interruption in employment.

    If you determine that your employee was terminated and is now rehired, and the rehire occurs within three years from the date the original Form I-9 was completed, you have an option to complete a new form or rely on the original one.


    MC-38509 Create New Customer Account patch

    This hotfix resolves an issue with Magento Commerce and Open Source 2.4.1 and 2.3.6 where the "Create an Account" button on the Create New Account page remains disabled if a shopper has entered invalid data. This prevents shoppers from re-attempting to create an account after making an error.

    Affected Magento versions: Magento Commerce and Open Source v2.4.1 / v2.3.6 (on prem and Cloud).

    See Applying patches for specific instructions on downloading and applying Magento patches.

    BUNDLE-2670 Braintree Virtual Terminal patch

    BUNDLE-2683 Braintree Settlement Report patch



    Patch for specific country payment method issue for Magento 2.3.5-p1

    This patch resolves an issue in Magento 2.3.5 and 2.3.5-p1 where the storefront checkout workflow did not display any payment method that has been enabled for specific countries with the exception of the Klarna and Amazon Pay payment methods.

    Affected Magento versions: Magento Commerce and Open Source v2.3.5 / v2.3.5-p1 (on prem and cloud).

    See Applying patches for specific instructions on downloading and applying Magento patches.

    Patch for Amazon Pay issue with payment method selection on checkout for Magento 2.3.5-p1

    This patch resolves the issue with inability to change a payment method on checkout "Review & Payments" step from the payments widget, while checking out with Amazon Pay.

    Affected Magento versions: Magento Commerce and Open Source v2.3.5 / v2.3.5-p1 (on prem and cloud).

    See Applying patches for specific instructions on downloading and applying Magento patches.

    Resend account confirmation email link issue patch for Magento 2.3.5

    This patch resolves an inability to re-send an account confirmation email link from storefront account login page. (This known issue was first identified in Magento 2.3.5.)

    Affected Magento versions: Magento Commerce and Open Source v2.3.5 / v2.3.5-p1 / v2.3.5-p2 (on prem and cloud).

    See Applying patches for specific instructions on downloading and applying Magento patches.

    Remove failed login attempts from the database patch for Magento 2.3.0 - 2.3.2p1

    This patch addresses a lingering issue created by the fix for CVE-2019-8118 (PRODSECBUG-2452) included in Magento 2.3.3 and 2.2.10.

    While the fix for that bug stopped the logging of failed login attempts, information collected prior to updating to these current versions may still exist, and previous, unpatched versions of Magento may still have this issue. This patch clears the login attempts that were previously collected. See Remove failed login attempts from the database for information on how to download and install this patch.

    PayPal Express Checkout issue with region patch for Magento 2.3.4

    This patch resolves the issue which affects orders placed with PayPal Express Checkout where the order’s shipping address specifies a country region that has been manually entered into the text field rather than selected from the drop-down menu on the Shipping page.

    Affected Magento versions: Magento Commerce and Open Source v2.3.4 (on prem and cloud).

    See Applying patches for specific instructions on downloading and applying Magento patches.

    Catalog pagination issue on Elasticsearch 6.x patch for Magento 2.3.3

    This patch resolves issues that users of Magento 2.3.3 experience in deployments where Elasticsearch 6.x is used as the catalog search engine. Users who attempt to navigate past the first page of search results are unsuccessful, and Magento displays an error message. After this patch is installed, users will be able to page through all search results.

    Affected Magento versions: Magento Commerce and Open Source v2.3.3 (on Prem and Cloud).

    EmailMessageInterface backward compatibility issue patch for Magento 2.3.3

    This patch addresses backward compatibility issues that extension developers may have experienced after the introduction of MagentoFrameworkMailEmailMessageInterface , which was released in Magento 2.3.3. In the scope of this patch, the new EmailMessageInterface inherits from the old MessageInterface , and core modules are changed back to rely on MessageInterface . Merchants should apply this patch as soon as possible, especially if their deployments include extensions or customizations that use the mail interface.

    Affected Magento versions: Magento Commerce and Open Source v2.3.3.

    See the Magento forum DevBlog post for much more information.

    Fixed method chaining contract for Product Collection patch for Magento 2.3.3

    This patch addresses changes that were introduced in Magento 2.3.3 that resulted in problems with extensions and customizations of the product collection feature that rely on method chaining contracts. The addAttributeToFilter method (in file app/code/Magento/Catalog/Model/ResourceModel/Product/Collection.php ) was refactored without a return statement, which broke the method-chaining that is used extensively in customizations of this feature. This patch refactors the method to add the missing return statement and ensure that method chaining works.


    An unauthenticated cross-site scripting vulnerability combined with an authenticated Phar deserialization vulnerability has left older versions of Magento Commerce and Magento Open Source open to serious exploit. An attacker can use these vulnerabilities to inject JavaScript into the Magento Admin, and subsequently launch malicious code in a store user’s browser. We strongly recommend that all users of the affected versions of Magento download and apply the appropriate patch as soon as possible.

    • Magento Open Source v2.3.1, 2.3.0, 2.2.8, and earlier 2.2.x releases
    • Magento Commerce v2.3.1, 2.3.0, 2.2.8, and earlier 2.2.x releases
    • Magento Commerce Cloud v2.3.1, 2.3.0, 2.2.8, and earlier 2.2.x releases

    Scope parameter for Async/Bulk API patch

    This patch resolves an issue with the Async and Bulk APIs, which in certain versions of Magento do not provide the information needed to update or create data for specific stores. Without this patch, the Async/Bulk REST APIs will support the default store view scope only.

    Affected Magento versions are: Magento Open Source v2.3.2, 2.3.1


    An issue has been discovered in Magento Open Source and Magento Commerce that can be used to disclose the URL location of a Magento Admin panel. While there is currently no reason to believe this issue would lead to a compromise directly, knowing the URL location could make it easier to automate attacks. To help prevent against potential attacks, Magento has released patches for this issue. For complete details, install instructions, and recommendations, see:

    Admin Dashboard Image-Charts API Composer Patch

    This patch provides a replacement for the deprecated Google Image Charts service that Magento uses for all 2.x instances and replaces it with the Image-Charts free service. Users of Magento 2.x deployments will not be able to view static charts in Magento 2.x instances unless they download and apply this patch. See Switch from deprecated Google Image Charts to Image-Charts for Magento for more information.


    This patch provides protection against the SQL injection vulnerability described under PRODSECBUG-2198 here. To quickly protect your store from this vulnerability only, install this patch. However, to apply protection against this vulnerability and others, you must apply the 2.3.1, 2.2.8, or 2.1.17 patch code. We strongly suggest that you install these full patches as soon as you can Direct Post Signature Key patch

    This patch updates Authorize.Net Direct Post integration to continue processing payments beyond March 14th 2019 (see MD5 Hash End of Life & Signature Key Replacement). There are additional steps that you need to execute after installing this patch to ensure continued use of Authorize.Net &ndash read more Update Authorize.Net Direct Post from MD5 to SHA-512.









    Physical Therapy Management [ edit | edit source ]

    Physiotherapy is the mainstay in the treatment of the Pes Anserine syndrome. To temper the pain caused by the bursitis, the most important thing of all is rest. Avoid stairs, climbing, or other irritating activities to quiet down the bursa and the related pain [1] . Nonsteroidal anti-inflammatory drugs (NSAID) can be taken to alleviate the pain. Restrict movement and alternately apply ice during the inflammatory phase. An ice massage of 15 minutes every 4-7 hours will reduce the inflammation. An elastic bandage can be wrapped around the knee to reduce any swelling or to prevent swelling from occurring [1] . Be careful not to increase friction. Teach the patient muscle-conditioning exercises [19] . These may include leg stretching exercises such as hamstring stretch, standing calf stretch, standing quadriceps stretch, hip adductor stretch, heel slide, quadriceps isometrics, hamstrings isometrics [1] . Progression of these exercises may involve closed-kinetic chain exercises such as single-knee dips, squats and leg presses. Resisted leg-pulls using elastic tubing are also included.

    The closed-kinetic chain exercises are also a recommended method to prevent the development of collateral knee instability, which occurs to be a risk factor of Pes Anserine bursitis. [20]

    Ultrasound has been documented as effective in the reduction of the inflammatory process in pes anserine bursitis. [9] Some patients receive an injection which consists of a solution of anesthetic and steroid. Afterwards, a physiotherapist will give a hamstring stretching program and a concurrent closed-chain quadriceps strengthening program that has to repeated several times a day. This will result in less pain at about 6-8 weeks.

    Kinesiotaping is more effective than naproxen or physical therapy in reduction of pain and swelling. [21]


    Defendants Goettsche, Balaci, and Weeks have been charged with conspiracy to engage in wire fraud in connection with their roles in BitClub Network. From April 2014 through December 2019, BitClub Network was a fraudulent scheme that solicited money from investors in exchange for shares of purported cryptocurrency mining pools and rewarded investors for recruiting new investors into the scheme. Goettsche, Balaci, Weeks, and others conspired together to solicit investment in BitClub Network through fraudulent means, including by providing false and misleading figures that BitClub investors were told were “bitcoin mining earnings” purportedly generated by BitClub Network’s bitcoin mining pool. Goettsche, Balaci, Weeks, and others obtained the equivalent of at least $722 million from BitClub Network investors.

    Defendants Goettsche, Balaci, Weeks, and Abel also conspired to sell BitClub Network shares—which were securities—notwithstanding that BitClub Network did not register the shares with the U.S. Securities and Exchange Commission. Weeks and Abel created videos and traveled around the United States and the world to promote BitClub Network and recruit others to invest.

    If you believe you may have been a victim of this fraud scheme, please complete and email this Questionnaire to [email protected]

    Status conference scheduled for July 16, 2020 at 2:00 pm

    12/3/2020 Detention Appeal Hearing

    9/3/2020 Guilty Plea Hearing

    Sentencing Hearing (ADOURNED)

    8/19/2021 11:00 am Sentencing Hearing

    11/5/2020 Guilty Plea Hearing

    7/8/2020 Initial Appearance

    7/9/2020 Guilty Plea Hearing

    9/14/2021 10:00 am Sentencing Hearing

    Indictment: On December 5, 2019, a federal grand jury returned a two count Indictment.

    District of New Jersey Victim-Witness Program Information:

    All case updates will be posted on this website.

    The public is reminded that an indictment contains only charges and is not evidence of guilt. The defendants are presumed innocent and are entitled to a fair trial at which the government has the burden of proving guilt beyond a reasonable doubt.

    Watch the video: Special Cases (December 2021).