Exploring Acceleration and Spatial Distances Near the Speed of Light

[jerromyjon]

I'm wondering, starting with an extreme example, what would happen if two masses accelerate away from each other towards the speed of light. Imagine we could do this very quickly like large hadron separator and the distance is short (in terms of light travel per time) and we compare this to an accelerating frame wouldn't we see a difference in acceleration distance?

 

[PeterDonis]

I don't understand what you're asking. What is "acceleration distance"? What does "compare this to an accelerating frame" mean?

 

[jerromyjon]

I have a logical concept in my head that we suppose two objects of equal mass are accelerated in opposite direction simultaneously with the same amount of force. Wouldn't velocity of the launch site in the direction of either mass cause a quicker acceleration in that direction?

 

[mfb]

Wouldn't velocity of the launch site in the direction of either mass

Velocity of the launch site relative to what?

cause a quicker acceleration in that direction?

No, as seen by the launch site the setup is completely symmetric.

 

[jerromyjon]

Relative to light speed in either direction, which is the same regardless of motion for EMR but not for matter? I visualize that only in absolute rest could two objects accelerate to as close to c as given propulsion force allows in a calculable distance which progressively contracts. If the launcher is moving .5c and you launch one ahead and one behind you ahead is progressively gaining lead and behind you gladly screeches to a halt and reverses direction?

 

[PeterDonis]

I have a logical concept in my head that we suppose two objects of equal mass are accelerated in opposite direction simultaneously with the same amount of force.

Ok so far.

Wouldn't velocity of the launch site in the direction of either mass cause a quicker acceleration in that direction?

mfb asked "velocity relative to what?", and you answered:

Relative to light speed in either direction,

This doesn't make sense; there is no such thing as "velocity relative to light speed". Velocities of objects are relative to other objects. As you have specified the scenario, in the rest frame of the launch site, the two objects accelerate in opposite directions with the same acceleration; therefore their velocities at any instant, in the rest frame of the launch site, will have equal magnitudes but opposite directions. The velocity of the launch site itself relative to something else (like, say, a distant planet or star) is irrelevant to any of that. (I am assuming, btw, that the launch site itself is moving inertially, in free fall.)

If you look at this scenario in a frame in which the launch site is moving, it will no longer look symmetric, true; but that's because different frames give different descriptions of any sequence of events.

 

[jerromyjon]

Doesn't it take more energy to accelerate up to infinity at c so accelerating ahead of launcher moving .9c would take a lot of energy but decelerating from that velocity would also take the same energy?

 

[PeterDonis]

Doesn't it take more energy to accelerate up to infinity at c so accelerating ahead of launcher moving .9c would take a lot of energy but decelerating from that velocity would also take the same energy?

Once again, moving relative to what?

Perhaps it would help if you took a step back and explained why you are asking these questions. What are you trying to figure out?

 

[mfb]

You cannot "accelerate up to infinity", whatever that might mean.

There is no absolute motion. In the system of the launcher, both objects accelerate exactly in the same way.
In a system that is moving relative to the launcher, things can look different. The object launched "ahead" will gain less velocity (if relativistic effects become relevant), while the object "behind" might reverse its direction (that depends on the numbers).

 

[jerromyjon]

You cannot "accelerate up to infinity"

What I was trying to say was the energy or force required to accelerate mass tends to infinity as you approach c.
I think what I might be missing is that we can't measure how close to c we are?

For two objects to accelerate to 300,000 km/s relative to each other would take greater than infinite energy for each. Therefore there is a maximum speed in which the similar energy in similar accelerating masses could accelerate relative to each other, because at some point the energy to accelerate faster is greater than the energy of the propulsion.

 

[PeterDonis]

What I was trying to say was the energy or force required to accelerate mass tends to infinity as you approach c.

Relative to some fixed inertial frame, yes.

I think what I might be missing is that we can't measure how close to c we are?

Not in any absolute sense, no. There's no such thing.

For two objects to accelerate to 300,000 km/s relative to each other would take greater than infinite energy for each.

I would just say "infinite", or more precisely, I would say that it's impossible to do this with any finite amount of energy. Your phrasing this time is OK because you say "relative to each other" instead of "relative to c".

Therefore there is a maximum speed in which the similar energy in similar accelerating masses could accelerate relative to each other

Let me rephrase what I think you're saying so it actually makes sense. Suppose we have two objects of identical rest mass which start out at rest relative to each other. And suppose that we somehow manage to convert all of the rest mass of each object into energy by means of rocket engines attached to each object that accelerate them in opposite directions. Then, at the end of this process, the two objects will be moving relative to each other at some finite velocity that is less than c. The larger the original rest masses of the objects, the closer their final relative velocity will be to c. (More precisely, what matters is the ratio of original rest mass to final rest mass--the rest mass of the "payload". See below.)

The above is true, but I'm still not sure why you're asking about it.

because at some point the energy to accelerate faster is greater than the energy of the propulsion.

This, however, is not true; it doesn't even make sense. The point at which the two objects stop accelerating is the point at which there is no more rest mass in either one to convert into energy. (We are idealizing the objects, including their rocket engines, as having negligible rest mass for the "payload", which means anything, like the structure of the rockets or the people aboard the rockets, that can't get converted into energy by the rocket engines.) This has nothing to do with "energy to accelerate" (which doesn't make sense to me either) compared to "energy of the propulsion" (which is frame-dependent anyway).

 

[jerromyjon]

This, however, is not true; it doesn't even make sense. The point at which the two objects stop accelerating is the point at which there is no more rest mass in either one to convert into energy. (We are idealizing the objects, including their rocket engines, as having negligible rest mass for the "payload", which means anything like the structure of the rockets or the people aboard the rockets that can't get converted into energy by the rocket engines.) This has nothing to do with "energy to accelerate" (which doesn't make sense to me either) compared to "energy of the propulsion" (which is frame-dependent anyway).

This was what I was trying to avoid with the original example. Suppose we have opposing linear accelerators which accelerate particles up to the maximum velocity their energy can accomplish very quickly, and so in the length of the apparatus both particles reach peak velocity. Would we be able to measure the distance or the time for each particle to achieve this velocity?

 

[mfb]

Would we be able to measure the distance or the time for each particle to achieve this velocity?

Sure. You can just track where the particle is as function of time.

 

[jerromyjon]

So wouldn't the velocity of the apparatus where the two particles are accelerated away from have an effect on the time to reach max velocity in opposite directions?

 

[mfb]

"The velocity of the apparatus" does not exist.

The velocity of the apparatus, as seen by the apparatus, is zero => no.
The velocity of the apparatus, as seen by others, can be non-zero => yes the time and distance can vary.

 

[jerromyjon]

How about two identical launchers moving wrt each other in parallel to the line of acceleration, would all four particles reach max velocity simultaneously?

 

[mfb]

Simultaneously in which frame?
No, independent of the frame.

 

[jerromyjon]

Simultaneously in which frame?

Is one frame capable of containing the entire experiment? Ideally we'd want flat space but since that isn't possible could we have a "close enough to flat" and at least perpendicular to gravitational potential? Of course each mass will have a unique path affected differently by surroundings, but would the error be too significant?

No, independent of the frame.

I'm just trying to pin down a physical concept that proves an absolute rest frame, something which to my knowledge doesn't exist for light but which I believe could exist for matter. I can't even think of any practical use for this knowledge, I just want to prove or disprove the concept to advance my understanding. If there is no scientific evidence disproving this claim then I would like to devise a test of the hypothesis by the simplest means available. What I don't have is a good understanding of is the accuracy required to obtain definitive results or whether any data from other experiments might provide evidence supporting my hypothesis.

 

[PeterDonis]

I'm just trying to pin down a physical concept that proves an absolute rest frame, something which to my knowledge doesn't exist for light but which I believe could exist for matter.

It doesn't exist for matter either. Repeated experiments have shown this; the best-known is the Michelson-Morley experiment, which has been repeated with much greater accuracy several times.

 

[jerromyjon]

These experiments only measure the relative velocity of light, how does it apply to accelerating masses?

 

[russ_watters]

These experiments only measure the relative velocity of light, how does it apply to accelerating masses?

Not accelerating, but moving: the purpose of the experiment was to measure the absolute motion of the apparatus.

 

[PeterDonis]

These experiments only measure the relative velocity of light

No, they measure whether the velocity of light is different in different directions, and they do it at multiple times of the year so that the Earth's own state of motion is different. As russ_watters said, the purpose of all this is to measure the absolute motion (if any) of the experimental apparatus; if any such absolute motion existed, the velocity of light would have been different in different directions when the experiment was run. It wasn't.

how does it apply to accelerating masses?

The apparatus was accelerated; it was sitting at rest on the surface of the Earth, so it was not in free fall, and its proper acceleration was nonzero.

Alternatively, if you insist on looking at coordinate acceleration instead of proper acceleration, the apparatus was accelerated with respect to the center of the Earth, because the Earth is rotating. It was also accelerated with respect to the Sun, because it was moving along with the Earth in a circular orbit about the Sun. And it was accelerated with respect to the center of the galaxy, because the solar system is orbiting the center of the galaxy. And so on...

 

[jerromyjon]

What about testing particle accelerations in different directions, in all those cosmic relative motions?

 

[jerromyjon]

Ah, nevermind, the results are always the same.

 

[russ_watters]

Yeah, particle accelerators are fast enough that such effects would show up if they existed.

 

[James_Harford]

Does the following thought experiment illustrate your concern?

First, perform this experiment in a remote region of space far from detectable gravitational fields.

Second, *temporarily* forget about special and general relativity. This is a *relatively* simple Newtonian experiment.

Consider two equal masses, say cannon balls. Place between them a large coil spring. Press the balls towards one another compressing the spring tightly and tie with a cord so that the spring remains compressed and motionless. The spring now has a finite potential energy of compression, a quantity equal to the chemical energy lost by your muscles in compressing it.

Now cut the cord. What happens?

The spring expands to full size, and accelerates both balls in opposite directions. When the spring reaches its maximum size, it stops expanding, the acceleration ends, both balls simultaneously separate from the spring to fly in opposite directions at constant speed, and the spring is left quivering, but at rest.

• The duration from the moment the cord was cut to the moment the spring reaches its maximum size is your "acceleration time", and is the *same* for both masses.
• The distance traversed by the balls during their acceleration is your "acceleration distance" and is the same for both balls for an observer at rest with respect to the spring. Since your question is about forces, we need not discuss it further.
• The average force applied by the spring to the ball is the ball's change in momentum divided by the acceleration time. Since the spring applies equal and opposite forces, the changes in momentum of the two balls are likewise equal and opposite.

Now for an observer moving relative to the spring and parallel to the path of the two balls, the observed changes in momentum are the same, the acceleration time is the same, and therefore the magnitude of the average force on each ball (the ratio of these quantities) is unchanged. In other words, the force applied during acceleration to each ball is completely independent of the speed of the spring in the observer's frame of reference.

Now let's restore special relativity. Because (and only because) all motions and forces in this problem are parallel to the direction of the spring, we can continue for now to use the definition of average force as change in momentum divided by acceleration time into the relativistic case. And as with the Newtonian case, the average force of the spring on each ball is unchanged. So, once again, the force applied during acceleration to each ball is always equal and opposite and *completely independent* of the speed of the spring in the observer's frame of reference. This is the point upon which I believe you have been in doubt.

 

[jerromyjon]

What I can't understand is how you could be moving .5c relative to whatever and launch projectiles or rockets or whatever and see that the one accelerating ahead of you isn't getting as far as the one "slowing down through space" behind you. It takes increasingly more energy to continue to accelerate faster towards c and less energy to decelerate? Perhaps at the slow speeds here on Earth it doesn't make much difference but as far as I can imagine if one particle accelerator moving .5c is approaching another one we'll call "way slower than the one coming at us" and as they are side by side they both fire a proton one is already at .5c and the other is starting from somewhere near 0.

 

[jerromyjon]

Yeah, particle accelerators are fast enough that such effects would show up if they existed.

But would the relative motion of the Earth be significant enough to see any difference if it is like .00001c? Wouldn't that also be in the "least significant" stage of acceleration? Up to .1c would be significantly faster than .8-.9c, right?

 

[PeterDonis]

as far as I can imagine if one particle accelerator moving .5c is approaching another one we'll call "way slower than the one coming at us" and as they are side by side they both fire a proton one is already at .5c and the other is starting from somewhere near 0.

One is at .5c and the other is somewhere near 0 relative to you. That is not what determines how "difficult" it is for each accelerator to launch particles relative to itself. And the only way an accelerator, or anything else, can launch anything is relative to itself.

 

[jerromyjon]

So two launchers approach each other at relativistic velocity and as they pass one launches forwards and the other backwards parallel. They launch same acceleration?

 

[PeterDonis]

So two launchers approach each other at relativistic velocity and as they pass one launches forwards and the other backwards parallel. They launch same acceleration?

Same acceleration relative to what?

If you mean the same proper acceleration, which in this scenario is equivalent to the coordinate acceleration in the rest frame of each launcher, if that's the way the launchers are set up, and you have stipulated that they are, then yes. The proper acceleration is determined by the actual thrust felt by the objects being launched, and you have stipulated that that's the same for both launchers. Since each object starts out at rest relative to the launcher that launches it, each object's coordinate acceleration in the rest frame of its launcher will be the same as its proper acceleration.

If you mean coordinate acceleration in a particular frame, then of course the coordinate acceleration of the object launched by launcher #2, in launcher #1's rest frame, will be different than the coordinate acceleration of the object launched by launcher #1, in launcher #1's rest frame. The coordinate acceleration of the two objects will also be different in launcher #2's rest frame. If you work it out, it turns out that the same will be true in any frame.

None of this has anything to do with whether there is such a thing as absolute motion or not, so I'm still not sure why you're asking about it.

 

[russ_watters]

But would the relative motion of the Earth be significant enough to see any difference if it is like .00001c? Wouldn't that also be in the "least significant" stage of acceleration? Up to .1c would be significantly faster than .8-.9c, right?

I don't know what you mean by "least significant stage of acceleration", but particle accelerators get their masses closer than .00001c from c, so it would be noticeable.

 

[TrickyDicky]

Maybe the OP's confusion would diminish by clarifying that the existence of examples of absolute motion like proper acceleration or rotation doesn't imply the existence of an absolute rest.

 

[Jonathan Scott]

I'm just trying to pin down a physical concept that proves an absolute rest frame, something which to my knowledge doesn't exist for light but which I believe could exist for matter.

There is no such frame in Special Relativity. As I mentioned recently on another thread, when you look at things from a different angle, you see them wider, deeper or whatever, but nothing is different physically. Special relativity extends that to looking at them from another velocity, by extending the mathematics of rotations in space to the Lorentz transformations in space and time, which include "boost" transformations as well as rotations. Other quantities related to time and space are transformed in systematically corresponding ways, including mass and energy.

 

[jerromyjon]

If something is at rest, yeah its always at rest relative to itself, ok. So relative to this "something supposedly at rest" the fastest something can move away in any direction is .99999999c (or c if massless). Now you imagine something approaches then passes at very high velocity, we'll say .5c, and relative to this something in the direction of its travel something else could still appear to accelerate from 0 to .99999999c and this would appear the exactly the same as something accelerating from -.5c, past 0, and to .99999999c in the opposite direction.