# Does SR really set a speed limit ?

## Main Question or Discussion Point

I recently studied the data which claims to be evidence of the need for an infinite amount of energy to accelerate a body to the speed of light. This consisted of accelerating particles at different energy levels and then graphing the resultant speeds attained by the accelerated particles. The resulting graph clearly showed the claimed result. However, this experiment was premised on the equipment which is doing the accelerating remaining at rest in the original reference frame. In some systems the means of acceleration remain in the frame of reference of the accelerated object. A rocket is a good example of such a system, and I cannot see why relativity precludes a rocket from reaching c with respect to its initial reference frame.

For example, say an object of initial mass m1 is able to convert this mass to energy and then use this energy so that it accelerates until it reaches the speed of light relative to its initial framework. The kinetic energy needed to do this is infinite when viewed from the initial frame of reference, but when viewed from the frame of the mass itself there would appear to be little difference in energy needed for each increment of speed. As the energy for acceleration is coming from the mass itself and is not delivered from the original reference frame, then it would appear only to need a maximum of 0.5*m1*c^2 to reach the speed of light. The energy stored in the object initially is m1*c^2 so it only takes half the available energy to reach the required speed.

Another twist here is that one would only need to reach 0.5c in each mass as two such experiments could be set up, with each mass moving towards each other, giving a closing speed of c.

Given that it is widely accepted that attaining such a speed is not possible, can someone please tell me what is wrong with this argument.

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PeterDonis
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There are several things wrong that I can see:

(1) The accelerating object is always at rest relative to itself, so the fact that any given small increment of speed takes a small increment of energy relative to the rocket is irrelevant; it doesn't tell us anything about the increment of energy relative to the initial (fixed) frame.

(2) As the rocket accelerates, its momentary rest frame *changes*, so your argument that a given small increment of speed takes only a small increment of energy is being made with respect to a constantly changing reference point. Again, that says nothing about whether small increments of speed continue to require only small increments of energy with respect to a *fixed* reference point, which is the question at issue.

(3) The "closing speed" of two masses moving towards each other, as viewed from a third (fixed) frame, is *not* the same as the relative velocity of either mass with respect to the other. Velocities don't add linearly in relativity. In the case you give, with each mass moving at 0.5c relative to the "lab" frame, one to the left and one to the right, the relative velocity of either one with respect to the other would be 0.8c by the relativistic velocity addition law.

There are several things wrong that I can see:

(1) The accelerating object is always at rest relative to itself, so the fact that any given small increment of speed takes a small increment of energy relative to the rocket is irrelevant; it doesn't tell us anything about the increment of energy relative to the initial (fixed) frame.

(2) As the rocket accelerates, its momentary rest frame *changes*, so your argument that a given small increment of speed takes only a small increment of energy is being made with respect to a constantly changing reference point. Again, that says nothing about whether small increments of speed continue to require only small increments of energy with respect to a *fixed* reference point, which is the question at issue.

(3) The "closing speed" of two masses moving towards each other, as viewed from a third (fixed) frame, is *not* the same as the relative velocity of either mass with respect to the other. Velocities don't add linearly in relativity. In the case you give, with each mass moving at 0.5c relative to the "lab" frame, one to the left and one to the right, the relative velocity of either one with respect to the other would be 0.8c by the relativistic velocity addition law.
Although viewed from the initial frame the energy consumption of the ship would be observed to diminish to virtually zero.Ejected mass, photons, or whatever. But am I correct in assuming that internally the energy consumption would neither increase nor decrease in maintaining constant proper acceleration?

PeterDonis
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Although viewed from the initial frame the energy consumption of the ship would be observed to diminish to virtually zero.
No, it wouldn't. See below.

But am I correct in assuming that internally the energy consumption would neither increase nor decrease in maintaining constant proper acceleration?
No, because the rest mass of the rocket is decreasing as it burns fuel and ejects exhaust. As its rest mass decreases, the amount of energy required to produce a small increment of velocity in the rocket's momentary rest frame also decreases. However, it won't decrease to zero unless the rocket is carrying no payload! Assuming that some portion of the rocket is not engine or fuel or reaction mass, some finite minimum "energy burn" will always be required to accelerate that small portion.

Also, when viewing things from the initial frame, you have to remember that "energy consumption" is frame-dependent. When the rocket has attained relativistic velocities, even a small "energy burn" in the rocket's rest frame corresponds to a huge "energy burn" in the initial frame, because of the large relative velocity. Basically, even the small amount of fuel being "burned" and the small amount of exhaust being ejected has a huge kinetic energy in the initial frame, so "energy consumption" in the initial frame is not small.

russ_watters
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The issue here isn't the energy, its the speed measurement. If the astronout uses an onboard accelerometer and clock and applies a Newtonian speed calculation, he may indeed calculate a speed greater than C. But then when he goes and measures it externally wrt his starting point, he'll find it isn't.

Dale
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this experiment was premised on the equipment which is doing the accelerating remaining at rest in the original reference frame. In some systems the means of acceleration remain in the frame of reference of the accelerated object. A rocket is a good example of such a system, and I cannot see why relativity precludes a rocket from reaching c with respect to its initial reference frame.
A rocket works by accelerating the fuel relative to the rocket, so the accelerator and rocket data are closely related. If it takes infinite energy to accelerate mass to c relative to a fixed apparatus then you cannot have superluminal exhaust. If you cannot have superluminal exhaust then you cannot get a superluminal rocket.

I never thought of it from this perspective, it's pretty cool.

Would any measurements taken by the traveler indicate that c is unachievable? (excluding accelerating, idk about that stuff, I mean the traveler's "independent" measurements of time & distance. of course those measurements are never truly "independent", at least with comparative motion)

opps nvm,

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DrGreg
Gold Member
Would any measurements taken by the traveler indicate that c is unachievable? (excluding accelerating, idk about that stuff, I mean the traveler's "independent" measurements of time & distance. of course those measurements are never truly "independent", at least with comparative motion)
If it were possible to reach the speed of light, the accelerating traveller would expect to measure the speed of light relative to himself as decreasing and eventually reaching zero. But it doesn't. It remains a constant 299 792 458 m/s no matter what happens. The traveller never gets any nearer his target.

If it were possible to reach the speed of light, the accelerating traveller would expect to measure the speed of light relative to himself as decreasing and eventually reaching zero. But it doesn't. It remains a constant 299 792 458 m/s no matter what happens. The traveller never gets any nearer his target.

Sorry I shouldn't have used the term c, should have said a speed close to c.

The SR postulate "all physics the same" tells me the traveler would not have observations that 299 792 458 m/s is unachievable, based on their own measure of length / time. Pretty much the same as c is the same for everyone, regardless of inertial motion.