Relative Nature of Speed: Einstein and Orbits

[matternat968]

TL;DR Summary

I have no education beyond high school, but i have always had this question about Einstein's theory that nothing can travel faster than the speed of light in a vacuum.

At the risk of sounding stupid, this question has always perplexed me. Einstein theorized that mass can not travel faster than the speed of light. I don't really understand it, I assume it has something to do with mass just being energy. Anyway imagine two objects in orbit of something with enough mass (or gravitational pull) to keep those objects in orbit traveling more than 50% the speed of light. Now imagine they are in orbits opposite of each other.

At one point when the pass near each other would they not in relation to each other be traveling faster than the speed of light?

Now does Einsteins theory only relate to mass moving faster than the speed of light in orbit of another body?

 

[Orodruin]

While an observer at rest in the frame where both objects travel at > 0.5c will see the separation between them change by more than c, this does not mean that the objects will see each other having speeds greater than c in their respective reference frames. In relativity, velocities do not add as you are used to, instead you need to use relativistic velocity addition.

 

[matternat968]

While an observer at rest in the frame where both objects travel at > 0.5c will see the separation between them change by more than c, this does not mean that the objects will see each other having speeds greater than c in their respective reference frames. In relativity, velocities do not add as you are used to, instead you need to use relativistic velocity addition.

Thanks for the quick reply. I'm doing my best to understand a subject that I have zero formal education in. Ill do more research and see if I can come to understand this different kind of velocity addition. But what might help is one more explanation. When Einstein says matter can not move faster than light, what would that matter be moving faster than light in relation to? Can we not only measure speed from a fixed point since vacuum is nothing?

 

[strangerep]

[...] Einstein says matter can not move faster than light, [...]

That's a misleading statement. A more accurate statement is that the relative speed between 2 inertial observers is limited by the speed of light. (An "inertial" observer is an observer who feels no acceleration.)

Can we not only measure speed from a fixed point since vacuum is nothing?

There is no such thing as a "fixed point" in the sense you mean. Everything is relative.

 

[PAllen]

That's a misleading statement. A more accurate statement is that the relative speed between 2 inertial observers is limited by the speed of light. (An "inertial" observer is an observer who feels no acceleration.)

I think you can say a lot more than this. In SR, relative speed of two bodies is less than c irrespective of their inertial vs noninertial trajectory, and irrespective of the simultaneity convention used. To clarify the latter, pick any points on their world lines connected by some arbitrary spacelike geodesic. Parallel transport the 4 velocity from one event to the other over any path whatsoever. Then take inner product to get gamma of relative speed. The result is the same for all paths, and independent of which vector you transport to the other. Further, no matter what spacelike connecting geodesic you use (which chooses simultaneity and events on the world lines), you always get speed less than c. Thus relative speed between two bodies is always less than c, no exceptions or caveats, in SR.

In GR, relative speed of two nearby bodies is always less than c, and is simply undefined for distant bodies due to path dependence of parallel transport.

[oops - sorry for I level answer in B level thread. But the conclusion is B level, even if the justification is not]

 

[matternat968]

I think you can say a lot more than this. In SR, relative speed of two bodies is less than c irrespective of their inertial vs noninertial trajectory, and irrespective of the simultaneity convention used. To clarify the latter, pick any points on their world lines connected by some arbitrary spacelike geodesic. Parallel transport the 4 velocity from one event to the other over any path whatsoever. Then take inner product to get gamma of relative speed. The result is the same for all paths, and independent of which vector you transport to the other. Further, no matter what spacelike connecting geodesic you use (which chooses simultaneity and events on the world lines), you always get speed less than c. Thus relative speed between two bodies is always less than c, no exceptions or caveats, in SR.

In GR, relative speed of two nearby bodies is always less than c, and is simply undefined for distant bodies due to path dependence of parallel transport.

[oops - sorry for I level answer in B level thread. But the conclusion is B level, even if the justification is not]

I know I'm at a huge disadvantage here because I don't understand the math that explains all this. But what I gather is because of the Lorentz transformation or "time dilation" two objects in relation to each other can never be moving faster than c from their perspective, and this has all been practically observed with atomic clocks on space stations correct?

 

[Ibix]

I know I'm at a huge disadvantage here because I don't understand the math that explains all this. But what I gather is because of the Lorentz transformation or "time dilation" two objects in relation to each other can never be moving faster than c from their perspective, and this has all been practically observed with atomic clocks on space stations correct?

You are mixing up quite a few things here, although I think you're on the right track.

One of Einstein's postulates was that the speed of light is always the same in inertial frames (based on previous experiments in electromagnetism and some inspired insight). This instantly precludes anything that travels slower than light from traveling faster than light - at some point it would have to be traveling at the same speed as light, which would mean the light was stationary for it, which is not the same speed as normal. Relativity is basically Einstein figuring out the implications of that postulate and showing that it is logically consistent and matches experiment.

One of the implications of relativity is that velocities do not add - if I say you are going at 60mph in one direction and someone else is going at 60mph in the opposite direction that does not mean you see him going at 120mph (although you'll never notice the difference at that speed - which is why we never realized before Einstein). But when you get to near lightspeed the errors become very obvious.

The above is all Special Relativity. Einstein (and others) expanded it into General Relativity, which includes the effects of gravitation. There, things get much more complicated - it's quite often impossible to define velocity in any meaningful way. When it is possible to define it, however, the same velocity addition rules apply and nothing exceeds the speed of light.

You mentioned time dilation. That isn't really relevant here, although it does come into the explanation of how I understand your measurements when I see you and a mate moving in opposite directions near light speed.

One of the simplest demonstrations of the speed of light limit is Bertozzi's experiment. You can see it on YouTube:
He accelerates electrons, but no matter how hard he tries they never reach light speed. Lots of other related experiments have been done, including putting clocks on airliners and showing that they tick at a different rate compared to one left at home (by Hafele and Keating, originally), so we're pretty confident in the theory, but I don't think they're quite so directly applicable to your question.

 

[PeroK]

I know I'm at a huge disadvantage here because I don't understand the math that explains all this. But what I gather is because of the Lorentz transformation or "time dilation" two objects in relation to each other can never be moving faster than c from their perspective, and this has all been practically observed with atomic clocks on space stations correct?

The maths underpinning SR is not particularly advanced and the basic ideas of SR are not highly mathematical.

Note that you can't through "pure thought" alone "prove" SR. If you could, the ancient Greeks might have come up with it. SR, and all of modern physics, is guided by experiment and observed natural phenomena.

You need, therefore, an experimental starting point for relativity. That starting point is the invariance of the speed of light in all inertial reference frames. Although, you could also take the inability of high-energy accelerators to accelerate particles up to the speed of light, no matter how much energy they give them.

Using these experimental facts you can then construct a consistent theory of time and space that has:

The invariance of the speed of light in all inertial reference frames
The speed of light as an (unattainable) upper limit for relative motion between particles
The relativistic velocity addition law
Time dilation
The Lorentz Transformation
And so on.

The alternative "classical" or "Newtonian" physics is equally logical and plausible. So, you cannot rule this out by thought or logic alone. You have to go out and conduct an experiment to determine whether relativity applies in our universe or not.

It turns out that it does apply! And the equally logical foundations of classical physics do not apply.