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The thought experiment:

Two spaceships are passing each other some distance from a star. Both ships are at relativistic speed, one toward and one away from the star. Should the total energy observed by each ship be equal?

I observe that the light reaching each ship is not shifted to the same frequency, from which I suspect a difference in energy.

If the energy involved is small but nonzero, does it go anywhere specifically?

Speculation: Gravity and lumens both use the inverse square, so perhaps this involves gravitational potential? Is this just a case breaking energy conservation?

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    $\begingroup$ "Energy" is a scalar coordinate quantity. The true coordinate-invariant conserved quantity is the stress-energy-momentum tensor, which obeys $\nabla_\mu T^{\mu\nu}=0$. $\endgroup$ Commented 2 days ago
  • $\begingroup$ Thank you controlgroup. If anyone is reading from the future, I apologize that the title is inaccurate. Red/Blueshifting changes the frequency of light and given the same amplitude should have less/more energy respectively. $\endgroup$ Commented 2 days ago
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    $\begingroup$ No need to go relativistic. Alice is at a train station and Bob is travelling in a train, northbound. Both observe the ongoing train, southbound. $\endgroup$ Commented yesterday

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Energy conservation only means that within each reference frame, energy does not change in time. It does not mean that energy is frame-invariant. In fact, energy depends on the reference frame. If you are walking past me, you have kinetic energy in my reference frame, but in your reference frame you are at rest and have no kinetic energy.

Since energy is frame-dependent, there is no reason why the two observers in your scenario should agree on the energy of the starlight.

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That's kind of like saying I am running with respect to the world. So, my kinetic energy is, say, x. Now, from my frame, the world is moving at me, with its own kinetic energy, say y. Now, because of energy conservation, x=y.

Now, this is obviously not true.

Energy conservation is frame-dependent. Energy is conserved in a frame with respect to time. Different observers can assign different energy values to the same system.

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