Two trains moving in opposite directions around the earth

[MnagurnyIII]

You have two trains sitting on the equator (one on each side of the equator so they don't collide -.-). Train one is moving East with the rotation of the earth, train two in moving west against it. both trains accelerate to a speed close to the speed of light, using the same amount of energy. So train one is moving near the speed of light plus the energy of the spin of the earth, train two is moving near the speed of light minus the energy of the spin of the earth.

my question is would an outside observer on Earth be able to measure a difference in the total speed of the two trains, and would an observer who is not on Earth be able to measure a difference?

My understand of this would state that an observe on Earth would not be able to notice a difference and the two trains would cross at the same points, one for each side of the earth. But somebody who was not on Earth would observe train one as moving faster than train two and would observe them as crossing at different points which has me... thinking I need to understand this concept better.

 

[PeterDonis]

both trains accelerate to a speed close to the speed of light, using the same amount of energy.

Using the same amount of energy from what starting point? Energy is frame-dependent.

So train one is moving near the speed of light plus the energy of the spin of the earth, train two is moving near the speed of light minus the energy of the spin of the earth.

But if they both used the same amount of energy to accelerate, they should have the same amount of energy when the acceleration is done, as long as you measure the energies relative to the same frame. If they have different energies, it must be because you have switched frames of reference between this statement and the previously quoted one.

Also, the difference in energy, in a frame where the energies are different, is not "the energy of the spin of the Earth"--that energy is huge, much, much larger than the energy of a train even if it is moving close to the speed of light. What I think you mean here is the kinetic energy that a train at rest relative to the rotating Earth would have, as measured by an observer who was not rotating with the Earth.

would an outside observer on Earth be able to measure a difference in the total speed of the two trains, and would an observer who is not on Earth be able to measure a difference?

It depends on the answer to the question I asked above. If the answer is that both trains start from rest relative to the (rotating) Earth, and both use the same amount of energy to accelerate relative to the (rotating) Earth, then when the acceleration is done, they will both be moving at the same speed (but in different directions) relative to the (rotating) Earth, so an observer at rest relative to the (rotating) Earth will not measure any difference in their speed, but an observer who is not rotating with the Earth *will* measure a difference in speed (and will measure a difference in energy between the trains as well).

 

[MnagurnyIII]

OK you hit what I was asking. The very last part is what I am having issues with. To a person on the earth, the trains are moving at the same speed and have the same energy because they are moving with the earth. To an outside observer who is not on Earth - they are moving at different speeds with different energy. Does that mean the outside observer would observe the trains passing each other at different points then a person standing on the Earth would?

edit: nevermind. the answer is no. He would observe them as crossing at the same point. but that point would be at a different location relative to the rest of the universe because the Earth rotated that point to a different spot - but it is still the same spot on the earth.

 

[PeterDonis]

OK you hit what I was asking. The very last part is what I am having issues with. To a person on the earth, the trains are moving at the same speed and have the same energy because they are moving with the earth. To an outside observer who is not on Earth - they are moving at different speeds with different energy. Does that mean the outside observer would observe the trains passing each other at different points then a person standing on the Earth would?

If you mean different points in space relative to the two observers, yes, they will. But that's because "space" itself is relative; different observers in different states of motion will assign, in general, different spatial coordinates to the same event.

For example, suppose the trains both start out at 0 degrees longitude and move along Earth's equator. They will cross at 180 degrees longitude on the equator, which, to an observer rotating with the Earth, is a fixed point in space. But to an observer not rotating with the Earth, 180 degrees longitude on the equator moves, so it can't possibly be "the same point in space" as it is for the observer on Earth.