Question regarding explanation of impossiblity of faster then light travel speed

[sharpstones]

I've just gotten around to reading "The Elegant Universe" and it has an explanation of why it is impossible to travel faster then light through the use of Einstein's equation E=mc^2 (pg 52). Greene says that the faster an object moves the more energy it has, and because of Einstein's equation the more energy something has the more mass it must have.

My confusion here comes from having taken a basic modern physics course a couple months back where the total energy of a particle was determined by adding its Kinetic energy and the energy associated with its mass: 1/2mv^2 + mc^2. From the problems that my proffesor gave me it seemed that there was a separation between the two. I understand that the higher an object's velocity the higher its Kinetic energy, but is this really interchangeble with its mass? What am I missing here?

 

[mathman]

One of the fundamental principles of special relativity is the equivalence of mass and energy. When things are traveling at high speed, acceleration leads mostly to mass increase, with small increases in speed. The total energy equation (mc²+mv²/2) only applies at low energy.
The exact expression is Mc², where M is the total mass, given by m/sqrt(1-(v/c)²), where m is the rest mass. If you expand the denominator in a power series in (v/c), the first two terms of the total energy expression are the low energy terms you are familiar with.

 

[DW]

Originally posted by mathman
One of the fundamental principles of special relativity is the equivalence of mass and energy. When things are traveling at high speed, acceleration leads mostly to mass increase, with small increases in speed. The total energy equation (mc²+mv²/2) only applies at low energy.
The exact expression is Mc², where M is the total mass, given by m/sqrt(1-(v/c)²), where m is the rest mass. If you expand the denominator in a power series in (v/c), the first two terms of the total energy expression are the low energy terms you are familiar with.

As I recently showed the equation should be
$$E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$
The mass is m and the "relativistic mass" M should not even enter into the paradigm as you are actually referring to the energy E which does not need to be renamed.
The power series then yields
$$E = mc^2 + \frac{1}{2}mv^2 + ...$$
The first term is the rest energy $$E_{0}$$ which is what energy the mass m is equivalent to and the second is the Newtonian expression for kinetic energy. The higher order terms become negligably small in the Newtonian limit. The mass m never increases with speed, only the kinetic energy terms do.

 

[mathman]

The use of M(=E/c²) seems perfectly reasonable to me. In modern day particle accelerators, the magnetic fields have to be adjusted for M in order to keep the particles on track.

 

[StarThrower]

Consider the time dilation formula:

$$\Delta t = \frac{\Delta t'}{\sqrt{1-v^2/c^2}}$$

As you can see, when v=c we have division by zero error. This can only be avoided, if when v=c delta t' is equal to zero. That would mean that in one of the systems, all clocks stopped ticking, even though the object is still moving in someone else's frame.

Thus, if a spaceship were to be accelerated to the speed of light, time would pass slower and slower in the ship, until it didn't pass at all in the ship, and thus all relative motion inside the ship would cease. That would mean that the temperature of the ship reached absolute zero, which would violate thermodynamical law.

The above argument can be used to show why it is that the theory of relativity predicts that no object can travel at the speed of light.

 

[Jimmy]

Originally posted by StarThrower
Consider the time dilation formula:

$$\Delta t = \frac{\Delta t'}{\sqrt{1-v^2/c^2}}$$

Thus, if a spaceship were to be accelerated to the speed of light, time would pass slower and slower in the ship, until it didn't pass at all in the ship, and thus all relative motion inside the ship would cease. That would mean that the temperature of the ship reached absolute zero, which would violate thermodynamical law.

The above argument can be used to show why it is that the theory of relativity predicts that no object can travel at the speed of light.

It thought time as measured on the ship would remain the same and would be measured to slow down in a different frame of reference.

Or is that what you meant?

 

[DW]

Originally posted by mathman
The use of M(=E/c²) seems perfectly reasonable to me. In modern day particle accelerators, the magnetic fields have to be adjusted for M in order to keep the particles on track.

No, modern accelerators are not completely circular.

 

[Arcon]

Originally posted by mathman
The use of M(=E/c²) seems perfectly reasonable to me. In modern day particle accelerators, the magnetic fields have to be adjusted for M in order to keep the particles on track.

That is exactly correct. You're, of course, speaking of relativistic mass and the cyclotron frequency. The derivations are found here

http://www.geocities.com/physics_world/sr/inertial_mass.htm
http://www.geocities.com/physics_world/sr/cyclotron.htm

 

[Njorl]

The prohibition is for travel at the speed of light. That yields an undefined value for gamma.

But there is conjecture about the use of complex velocities so as to "go around" the value c to superluminal velocities. These would yield an imaginary, but defined, value for gamma. There is not, as yet, a good explanation for what the physical analog of an imaginary component to velocity might be though. That sort of throws a wet blanket on the whole concept.

"Complex speeds and special relativity", Asaro, C., AM J PHYS 64 (4): 421-429 APR 1996

Njorl

 

[russ_watters]

Originally posted by Njorl
But there is conjecture about the use of complex velocities so as to "go around" the value c to superluminal velocities. These would yield an imaginary, but defined, value for gamma. There is not, as yet, a good explanation for what the physical analog of an imaginary component to velocity might be though. That sort of throws a wet blanket on the whole concept.

A velocity component not in the x,y, or z directions...time travel?

 

[Njorl]

Possibly, haven't checked it. If you do a gedanken experiment of something flying away from you a greater than the speed of light, it actually appears to start far away, and fly toward you at subluminal velocity. I believe it also acts as if it has had it's charge flipped too. I think it acts indistinguishably from its antiparticle with a velocity of -c²/v.

That's a real, superluminal velocity though. What an imaginary or complex v does... I'll think about.

Njorl

 

[that_guy]

For the observers inside the ship, time does not slow down. Observers in the 'at rest' frame will see time as passing slower in the ship.

Originally posted by StarThrower
Consider the time dilation formula:

$$\Delta t = \frac{\Delta t'}{\sqrt{1-v^2/c^2}}$$

As you can see, when v=c we have division by zero error. This can only be avoided, if when v=c delta t' is equal to zero. That would mean that in one of the systems, all clocks stopped ticking, even though the object is still moving in someone else's frame.

Thus, if a spaceship were to be accelerated to the speed of light, time would pass slower and slower in the ship, until it didn't pass at all in the ship, and thus all relative motion inside the ship would cease. That would mean that the temperature of the ship reached absolute zero, which would violate thermodynamical law.

The above argument can be used to show why it is that the theory of relativity predicts that no object can travel at the speed of light.

 

[DW]

Originally posted by Arcon
That is exactly correct. You're, of course, speaking of relativistic mass and the cyclotron frequency. The derivations are found here

http://www.geocities.com/physics_world/sr/inertial_mass.htm
http://www.geocities.com/physics_world/sr/cyclotron.htm

He didn't say cyclotrons. For straight sections of track and linear injectors and for any parts producing tangential acceleration at all the force expression is two orders of $$\gamma$$ bigger than one would get by mistakenly arbitrarily replacing the mass in expressions with relativistic mass. Replacing mass with relativistic mass is simply not the correct physics.