Paper about special theory of relativity

[NNSSA]

Hi there,

I'm doing a paper about the special theory of relativity. The paper consists of two parts:
- explaining the theory
- explaining what kind of consequences the special theory will encounter, if particles move faster than the speed of light. Where does the theory fail?

I've already finished the first part, but...

Homework Statement

I just don't know where to start with the second part. I've already asked for help in another science forum, but nobody replied. I just want to know what will happen with the special theory of relativity. I hope there are some equations/graphs that I can use for this.
Are there useful books/articles about this subject?

Homework Equations

?

The Attempt at a Solution

I just don't know where to start. I've already searched for books, but couldn't find one.


Thank you in advance.

 

[oli4]

Hi NNSSA, welcome to pf
If you already finished the first part, you must have relevant equations, you must have seen this everywhere: [tex]\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}[/tex]
So what would happen for v>c ?
There are probably several ways to 'address the problem', but at least you can clearly see that if the same equations hold above c, this will have to involve imaginary numbers.
If you didn't already, google for "tachyon", that should give you something to get started on your paper

Cheers...

 

[andrien]

C is the ultimate speed.it can not be crossed.

 

[Fredrik]

SR doesn't say that FTL motion is impossible. One thing that it does say is that the work required to accelerate a mass m from speed 0 to speed v<c goes to infinity as v goes to c. So no matter how much energy you use to accelerate a massive particle, its speed will always be less than c. This is something you may want to include in your paper. I think I've done that calculation in a couple of places in this forum. This is the one I could find. Note that γ→∞ as v→1. (I'm using units such that c=1).

In SR, space and time are represented mathematically by something called (Minkowski) spacetime. The motion of a (classical) particle is represented by a curve in spacetime. Since there are curves in spacetime that correspond to speeds faster than c, it's impossible to argue that Minkowski spacetime forbids FTL motion. (FTL=faster than light). However, SR is more than just a way to represent space and time mathematically. It's also a framework in which we can define theories of matter. (My definition of "matter" includes both particles and fields, like the electromagnetic field). It's certainly possible to define a theory that includes point particles or waves in a field that move FTL. But such theories are usually not taken very seriously. None of the established theories of matter in Minkowski spacetime (like classical electrodynamics or quantum electrodynamics) includes particles or waves that move FTL.

So why don't we take such theories seriously? Mainly because experiments agree so well with the theories that don't include particles or waves at FTL speeds. But there are other reasons as well. A big one is that we can derive a contradiction from the assumption that these statements are all true:

a) It's possible to build a device that that sends out a message containing 1 bit of information, at a speed v>c relative to the device, when a button on the device is pushed.
b) It's possible to build a device that can receive such 1-bit messages and correctly interpret them as "1" or "0" in a time that doesn't grow at least linearly with the time or distance the message has traveled.
c) People can freely choose to build these devices, where to put them, what message to send (either "1" or "0"), and when to push the buttons.

So any theory of matter in Minkowski spacetime that includes FTL motion must somehow make (at least) one of these statements false. See this post for more details. I chose v=∞ to keep things as simple as possible, but a similar argument can be made for any speed >c.

 

[NNSSA]

Thank you all for responding.

I know that the speed of light can't be exceeded. But it's a ''what-if'' situation.

Oli4, I knew that we will get a imaginary number. What does it actually mean? There are many equations in the theory that use the Lorentz factor.

Fredrik, thank you for your reply. I saw the calculations you made, but I can't follow it completely. I understand the calculations, but what can we conclude from it?

Some time ago I read that weird things happen with simultaneity if we exceed the speed of light.

I've also found an article about faster-than-light-particles, but it's quite difficult...
(see: http://www.relativitycalculator.com/images/superluminal_velocities/possibility_faster_than_light.pdf)

Thanks again.

 

[Fredrik]

I know that the speed of light can't be exceeded. But it's a ''what-if'' situation.

You have to be careful when you ask "what if" questions. Keep in mind that the only thing that can answer a question is a theory. A question of the form "What does theory X say will happen if I do Y?" doesn't make sense when theory X says that you can't do Y.

A question like "What would happen if I do Y?" only makes sense when it's clear from the context what theory we're supposed to use to answer the question. If it's asked during a conversation about SR, it would be interpreted as "What does SR says would happen if I do Y?", so it only makes sense if SR says that Y can be done.

I suppose the question could also be interpreted as "Are there any theories that allow me to do Y, and what do they say will happen if I do Y?" Questions of this type are problematic too. For example, consider "What would I see if I travel faster than light?", interpreted as "Are there any theories that say what would happen if I travel faster than 299792458 m/s?". The answer is yes, there's one, but it's far less accurate than SR, so we probably shouldn't care what it says. (The theory is Newtonian mechanics, and it says that there's nothing special about that speed).

Oli4, I knew that we will get a imaginary number. What does it actually mean? There are many equations in the theory that use the Lorentz factor.

You won't ever get an imaginary number, since that v is by definition the relative velocity of two inertial coordinate systems, and that's always <c, by definition of "inertial coordinate system".

Some time ago I read that weird things happen with simultaneity if we exceed the speed of light.

This is one of the what-ifs that don't make sense. Since SR says that we can't do that, it doesn't make sense to ask what SR says happens if we do that.

If we input speed ≥c into formulas that were derived from the assumption that the speed is <c, we can get absurd results, but that is of course to be expected. It only makes sense to think about inputting speeds ≥c into formulas that can be derived without the assumption that the speed is <c.

One such formula is the one for velocity addition. If u is the velocity of B relative to A, v is the velocity of C relative to B, and w is the velocity of C relative to A, then we have
$$w=\frac{u+v}{1+\frac{uv}{c^2}}.$$ The derivation of this formula relies on the assumption that u<c, but no assumption is made about v. (Note that none of the velocities is "relative to C", so C can be a particle that moves FTL). So it holds for v>c. If you input large values of v, you get weird results, but those results are the actual predictions of SR.

I've also found an article about faster-than-light-particles, but it's quite difficult...
(see: http://www.relativitycalculator.com/images/superluminal_velocities/possibility_faster_than_light.pdf)

Interesting article. I may have to check it out sometime. It's very possible that he (or someone else) has thought of things that I haven't.

I saw the calculations you made, but I can't follow it completely. I understand the calculations, but what can we conclude from it?

The conclusions are what I said in my post above. In particular,

no matter how much energy you use to accelerate a massive particle, its speed will always be less than c.
...
So why don't we take such theories seriously? Mainly because experiments agree so well with the theories that don't include particles or waves at FTL speeds.
...
any theory of matter in Minkowski spacetime that includes FTL motion must somehow make (at least) one of these statements false.

 

[NNSSA]

Thank you for your reply Fredrik, I really appreciate it.

Let me tell you how I got into this part of my paper. Last year there was that OPERA experiment with neutrinos going faster than the speed of light (it was incorrect though). I was shocked how the media criticized Einstein's special theory of relativity.

So I thought what would happen to the special theory if these neutrinos would go faster than the speed of light. Would it really break down?

Unfortunately, I see that there's not that much to tell about it...

 

[berkeman]

We do not waste people's time with "what if this impossible thing were possible" discussions here at the PF. This thread is closed.