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I would like to clarify that this idea that mass increases at relativistic speeds is an explanatory tool only (usually to explain why you cannot exceed the speed of light). It comes from the fact that in some equations, the factor gamma = 1/sqrt(1-v^2/c^2) multiplies the mass. However, I think it is a flawed explanatory tool and I never use it in explanations of special relativity. Not only is it completely unnecessary but I think it creates more confusion than understanding.

The idea of mass increasing at relativistic velocities leads to the unavoidable conclusion that mass is relative just like velocity, which I find absurd. Reading some of the other post I see that it also leads to the conclusion that mass is different in different directions, which is even more absurd. This idea of mass increasing has forced us, for the sake of clarity, to rename the sensible concept of mass as "rest mass" which is not relative and does not change. In short the absurdites and confusion promoted by this idea of increasing mass warrant giving the whole idea a decent burial.

The total energy of a mass m at a relative velocity v and thus lorentz contraction factor gamma is given by E = gamma m c^2.

Well what about kinetic energy? We can extract the classical kinetic energy from a binomial expansion of gamma.

gamma = 1 + .5(v/c)^2 + .375(v/c)^4 + ...

when you put this into the above equation you get

E = m c^2 + .5 m v^2 + .375 m v^4/c^2 + ...

The first term is the famous mass energy, and the second term is the classical kinetic energy.

To handle the relativistic correction, we typically write

E = m c^2 + (gamma-1) m c^2

and we say that the first term is the mass energy (or rest energy) and the second term here is the relativistic kinetic energy, KE = (gamma-1) m c^2

In this case we are obviously not thinking that the mass has increased by a factor of gamma at all, because the energy associated with mass has not changed. To say that the mass has increased by a factor of gamma would mean that all of the energy is a part of the mass and there is no energy of motion, and I don't think this helps in understanding special relativity at all.

I do not even like the idea as an explanation of why you cannot exceed the speed of light because it is too stuck in the thinking of motion as absolute. What I mean is that it puts too much emphasis on one particular relative velocity as if that were special. It is true that increasing the relative velocity with respect to something requires and increasing amount of energy for the same increase in that relative velocity, but I think this misses the point.

I guess the only way to make what I am saying clear is to look at an example. Suppose you accelerate a big ship to 1/2 the speed of light relative to the earth. If you have a medium ship inside the big ship then you can accelerate that medium ship to 1/2 the speed of light relative to the big ship. Then if you have a small ship inside the medium ship you can accelerate the small ship 1/2 the speed of light relative to the medium ship.

The energy requirements of all these acceleration depend on the rest masses of these ships (lets call them mbig, mmed, and msmall) in exactly the same way, using the KE shown above KE = (gamma-1) m c^2.

In each of the three cases gamma = 1/sqrt(1-.25) = 1.1547

First acceleration: energy required was KE = .1547 mbig c^2

Second acceleration: KE = .1547 mmed c^2

Third acceleration: KE = .1547 msmall c^2

If you want to talk about the resulting velocity with respect to the earth then you need the velocity addition formula v3 = (v1+v2)/(1+ v1 v2/c^2).

So the velocity of the medium ship with respect to the earth is (c/2+c/2)/(1+.25) = 0.8 c, and the velocity of the small ship with respect to the earth is (.8 c + .5 c)/(1+ .8x.5) = .92857 c. If you look carefully at the velocity addition formula you will see that if both v1 and v2 are less than c then v3 will be less than c, but if either v1 or v2 is equall to c then v3 will also be c.

The point is there is no increase of mass in this explanation nor should there be. The idea of mass increase promotes a misconception that something changes as you accelerate making an increase of speed more difficult. Absolutely nothing changes. The only limit is on relative velocity at which you see objects receding behind you. It does not even limit how fast you can travel to a destination.

The speed of light is unreachable because it is like an infinite speed in the sense that if you chase after a light beam your accelertion never reduces the relative velocity between you and the light beam you are chasing, the light continues to speed away from you at 3x10^8 m/s. You cannot catch the light no matter how fast you go, just as if the light were traveling infinitely fast. In fact, we know from the relativity of simultaneity that any travel faster than light would be equivalent to arriving at your destination before you left, leading to the same paradoxes as in time travel. Also if you think of the infinite speed as the limiting case where you get to your destination in no time at all, the speed of light is exactly such a limiting case.

P.S. Check out my relativistic flight simulator at www.relspace.astahost.com[/URL]

The idea of mass increasing at relativistic velocities leads to the unavoidable conclusion that mass is relative just like velocity, which I find absurd. Reading some of the other post I see that it also leads to the conclusion that mass is different in different directions, which is even more absurd. This idea of mass increasing has forced us, for the sake of clarity, to rename the sensible concept of mass as "rest mass" which is not relative and does not change. In short the absurdites and confusion promoted by this idea of increasing mass warrant giving the whole idea a decent burial.

The total energy of a mass m at a relative velocity v and thus lorentz contraction factor gamma is given by E = gamma m c^2.

Well what about kinetic energy? We can extract the classical kinetic energy from a binomial expansion of gamma.

gamma = 1 + .5(v/c)^2 + .375(v/c)^4 + ...

when you put this into the above equation you get

E = m c^2 + .5 m v^2 + .375 m v^4/c^2 + ...

The first term is the famous mass energy, and the second term is the classical kinetic energy.

To handle the relativistic correction, we typically write

E = m c^2 + (gamma-1) m c^2

and we say that the first term is the mass energy (or rest energy) and the second term here is the relativistic kinetic energy, KE = (gamma-1) m c^2

In this case we are obviously not thinking that the mass has increased by a factor of gamma at all, because the energy associated with mass has not changed. To say that the mass has increased by a factor of gamma would mean that all of the energy is a part of the mass and there is no energy of motion, and I don't think this helps in understanding special relativity at all.

I do not even like the idea as an explanation of why you cannot exceed the speed of light because it is too stuck in the thinking of motion as absolute. What I mean is that it puts too much emphasis on one particular relative velocity as if that were special. It is true that increasing the relative velocity with respect to something requires and increasing amount of energy for the same increase in that relative velocity, but I think this misses the point.

I guess the only way to make what I am saying clear is to look at an example. Suppose you accelerate a big ship to 1/2 the speed of light relative to the earth. If you have a medium ship inside the big ship then you can accelerate that medium ship to 1/2 the speed of light relative to the big ship. Then if you have a small ship inside the medium ship you can accelerate the small ship 1/2 the speed of light relative to the medium ship.

The energy requirements of all these acceleration depend on the rest masses of these ships (lets call them mbig, mmed, and msmall) in exactly the same way, using the KE shown above KE = (gamma-1) m c^2.

In each of the three cases gamma = 1/sqrt(1-.25) = 1.1547

First acceleration: energy required was KE = .1547 mbig c^2

Second acceleration: KE = .1547 mmed c^2

Third acceleration: KE = .1547 msmall c^2

If you want to talk about the resulting velocity with respect to the earth then you need the velocity addition formula v3 = (v1+v2)/(1+ v1 v2/c^2).

So the velocity of the medium ship with respect to the earth is (c/2+c/2)/(1+.25) = 0.8 c, and the velocity of the small ship with respect to the earth is (.8 c + .5 c)/(1+ .8x.5) = .92857 c. If you look carefully at the velocity addition formula you will see that if both v1 and v2 are less than c then v3 will be less than c, but if either v1 or v2 is equall to c then v3 will also be c.

The point is there is no increase of mass in this explanation nor should there be. The idea of mass increase promotes a misconception that something changes as you accelerate making an increase of speed more difficult. Absolutely nothing changes. The only limit is on relative velocity at which you see objects receding behind you. It does not even limit how fast you can travel to a destination.

The speed of light is unreachable because it is like an infinite speed in the sense that if you chase after a light beam your accelertion never reduces the relative velocity between you and the light beam you are chasing, the light continues to speed away from you at 3x10^8 m/s. You cannot catch the light no matter how fast you go, just as if the light were traveling infinitely fast. In fact, we know from the relativity of simultaneity that any travel faster than light would be equivalent to arriving at your destination before you left, leading to the same paradoxes as in time travel. Also if you think of the infinite speed as the limiting case where you get to your destination in no time at all, the speed of light is exactly such a limiting case.

P.S. Check out my relativistic flight simulator at www.relspace.astahost.com[/URL]

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