Mathematical explanation of maximum speed limit c'

[sanko]

mathematical explanation of maximum speed limit "c'

i have tried many texts and searched but can't find particular mathematical proof or explanation of how sp.theory led to conclusion that maximum velocity limit is "c"

 

[tiny-tim]

hi sanko!

it comes from the velocity-addition formula …

adding velocities < c can never get a result ≥ c

see http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity"

 

[f todd baker]

The energy of a particle of rest mass m₀ and with speed v is E=m₀c²/sqrt(1-(v/c)²). If v=c, E is infinite. It takes an inifinite amount of energy to get to c, and, of course, we do not have that much to give! The previous answer shows you can't get to or beyond c by velocity addition either.

 

[bcrowell]

FAQ: Why can't anything go faster than the speed of light?

In flat spacetime, velocities greater than c lead to violations of causality: observer 1 says that event A caused event B, but observer 2, in a different state of motion, says that B caused A. Since violation of causality can produce paradoxes, we suspect that cause and effect can't be propagated at velocities greater than c in flat spacetime. Special relativity is one of the most precisely and extensively verified theories in physics, and in particular no violation of this speed limit for cause and effect has ever been detected -- not by radiation, material particles, or any other method of transmitting information, such as quantum entanglement. Particle accelerators routinely accelerate protons to energies of 1 TeV, where their velocity is 0.9999996c, and the results are exactly as predicted by general relativity: as the velocity approaches c, a given force produces less and less acceleration, so that the protons never exceed c.

The corresponding speed limit in curved spacetime is far from being established. The argument from causality is not watertight. General relativity has spacetimes, such as the Godel solution, that are valid solutions of the field equations, and that violate causality. Hawking's chronology protection conjecture says that this kind of causality violation can't arise from realistic conditions in our universe -- but that's all it is, a conjecture. Nobody has proved it. In fact, there is a major current research program that consists of nothing more than trying to *define* rigorously what the chronology protection conjecture means.

There are certain things we *can* say about faster-than-light (FTL) motion, based on the fundamental structure of general relativity. It would definitely be equivalent to time travel, so any science fiction that has routine FTL without routine time travel is just plain wrong. It would probably require the existence of exotic matter, which probably doesn't exist. If it were possible to produce FTL artificially, it would certainly require the manipulation of godlike amounts of matter and energy -- so great that it is unlikely that beings able to carry it out would have anything like ordinary human concerns.

There are many ways that velocities greater than c can appear in relativity without violating any of the above considerations. For example, one can point a laser at the moon and sweep it across, so that the spot moves at a speed greater than c, but that doesn't mean that cause and effect are being propagated at greater than c. Other examples of this kind include a pair of cosmic-sized scissors cutting through a gigantic piece of paper at greater than c; phase velocities greater than c; and distant, observable galaxies receding from us at greater than c, which can be interpreted as an effect in which space itself is expanding in the space in between.

 

[pervect]

I'd like to add one thing to bcrowell's excellent summary on the status of FTL travel in GR. That is that it's possible that some mechanism, for example infinite (or at least very large) vacuum fluctuations destroys physical systems (such as wormholes, for example) that would otherwise become time machines. This is the famous "chronology protection conjecture" attributed to Hawking.

This provides a science fiction writer who doesn't want to deal with time machines but wants to have the laws of physics at least somewhat based on General Relativity an "out". Or, for someone who isn't interested in fiction, the ability to consider interesting and novel space-time geometries without worrying about the resulting "time paradoxes".

See for instance Visser's http://arxiv.org/abs/gr-qc/0204022 for a more detailed discussion (though it may be dated now). Visser's conclusion is that semi-classical analysis is probably not sufficient to answer the question, so it will be open until we have a full theory of quantum gravity.

 

[John232]

The energy of a particle of rest mass m₀ and with speed v is E=m₀c²/sqrt(1-(v/c)²). If v=c, E is infinite. It takes an inifinite amount of energy to get to c, and, of course, we do not have that much to give! The previous answer shows you can't get to or beyond c by velocity addition either.

I saw a problem with this explanation in that if the object was traveling at a constant speed then he wouldn't detect any increase in mass or any forces acting on him that would allow him to distinguish himself as haveing a true speed. If the the principal of equivalence of being at rest and traveling at a constant speed was to hold true then an object traveling at a constant speed wouldn't be able to detect his own mass increase. If he couldn't detect his own mass increase traveling at a constant speed then he wouldn't require an extra amount of energy in order to accelerate from that constant speed.

Maybe if somehow someone using an afterburner accelerated close to the speed of light then the mass increase of the exhaust would give extra energy viewed for the observer at rest in order for him to agree on the mass and energy used in order to give the viewed amount of acceleration. Sending a greater amount of mass aft of the ship would give an extra push to the increased mass of the ship.

This "law" has already been said to been broken all over television recently in the claims that the universe can expand at speeds greater than the speed of light. They never mention if and how these relative speeds have an effect on anything. I think it just comes from an attempt to describe the rapid periods of inflantion that should have existed from looking at the cosmic background radiation. The universe should have expanded faster than light and if the universe is currently larger than the visable universe then galaxies outside the visable universe would be currently traveling faster than light. Yet, there is no mention of any relativistic effects acting on these galaxies at the edge of the visable universe. I would have thought it would be some interesting stuff for television or any book. It is always completely left out.

 

[sanko]

Tiny-tim,f todd,bcrowell Thank you very much for replys.This is more than i expect. @Most the of books & other places i find maximum speed explanation as casuality violation and 'mass tends to infinity' and i was looking for mathematical explanation e.g. Asymptotic function or c=limit->infinity of function
sry,i should have mention this in question. Tiny tim's explanation clears my confusion also gives mathematical basis why maximum limit is c

 

[sanko]

i am having conceptual problems again.
In relation to my "maximum velocity c"question.
According to RT measure of length,time,mass depends upon velocity of observer and if observer is static v->0 and observer will find gallion measures

law relativistic additions of velocity states if for frame moving with velocity 'v' speed of third frame is 'u' Then actual speed of third frame with respect to static frame is given by stated formula,
suppose we call static frame observer"a" and moving frame "b". If we increase speed of b by u ,increase in speed will change observations of b but should not change observations of a as he is still static.Meaning he will observe b moving with velocity v+u .So it can be increased above c.

 

[tiny-tim]

hi sanko!

i am having conceptual problems again.
In relation to my "maximum velocity c"question.
According to RT measure of length,time,mass depends upon velocity of observer and if observer is static v->0 and observer will find gallion measures

i assume you mean "Galilean"?

… moving frame "b". If we increase speed of b by u ,increase in speed will change observations of b but should not change observations of a as he is still static.Meaning he will observe b moving with velocity v+u .So it can be increased above c.

you can't change the speed of a frame

a frame is a frame, it has only one speed, if it had a different speed it would be a different frame!

you have called "b" a frame, so you can't increase its speed

if you are talking about a third frame "d" whose speed relative to "b" is u, do you mean as observed by "a" or as observed by "b"?

if the former, then yes, d will have speed v+u … but that's begging the question! … if u+v ≥ c, then it can't be done!