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Energy is not an invariant, meaning that it will be different in different inertial frames. However, it will be conserved in all frames.

This is not something new in relativity as it is also the case in classical mechanics. The new thing in relativity is the assumption that the speed of light* is invariant, ie, the same regardless of the observer. This turns out to be incompatible with the assumption of absolute and universal time in classical mechanics.

* Really, the assumption is the existence of an invariant finite speed. Light just happened to be the thing we already knew that travelled at this speed.

- #3

A.T.

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Why should one value for total kinetic energy be more special than any other value?If an experiment can be devised to detect this difference in the energy of the whole system, can't it say if the space ship is moving or the rest of the universe is moving?

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Dale

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First, even if the universe were some rigid object there is no reason to assume it has low KE. As long as energy and momentum are conserved, there is nothing about a universe with large KE and net momentum that violates any known law.If the rest of the universe is moving in the opposite direction, won't it require huge energy?

Second, the universe is not a rigid object. If you say that it is moving at 100 m/s here then there is another location where it is stationary by parallel transport. Why shouldn’t that location be the stationary one instead of this?

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There is no such thing as "moving at 100 m/s" without qualification. You have to specify what the ship is moving at 100 m/s relative to. Note that this is just as true in Newtonian mechanics as in SR.If a space ship is moving at 100 m/s

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In Galilei-Newton spacetime you have the kinetic of a particle ##E=\vec{p}^2/(2m)##, and going from one frame of reference to another moving with speed ##\vec{v}## you have ##\vec{p}'=\vec{p}-m \vec{v}## and thus

$$E'=\frac{\vec{p}^{\prime 2}}{2m}=\frac{(\vec{p}-m \vec{v})^2}{2m},$$

while in SR ##(E/c,\vec{p})## are Minkowski-four-vector components, and ##E^2/c^2-\vec{p}^2=m^2 c^2## give the relation between ##E## and ##\vec{p}##. Note that in SR and Galilei-Newtonian physics the mass is the same quantity (which is why we use only this invariant mass ##m## in modern formulations of SR) but the relativistic energy also contains the rest energy ##E_0=m c^2##, which is not included in the kinetic energy of Galilei-Newtonian physics, but that's just a convention, because it's only an additive constant.

All that counts are energy differences, and these are interesting because of energy-conservation law (i.e., the energy-work theorem) valid for closed systems in both SR and Galilei-Newton physics, because both space-time models imply time-translation invariance and the corresponding conserved "Noether charge" of this symmetry is by definition energy.

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Moderator's note: Some posts which did not add any real value to the discussion have been deleted.

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haushofer

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There is also another way to look upon it: the speed of light is invariant in both classical mechanics and relativity. But in classical mechanics the speed of light is infinite, while in relativity it's finite.Energy is not an invariant, meaning that it will be different in different inertial frames. However, it will be conserved in all frames.

This is not something new in relativity as it is also the case in classical mechanics. The new thing in relativity is the assumption that the speed of light* is invariant, ie, the same regardless of the observer. This turns out to be incompatible with the assumption of absolute and universal time in classical mechanics.

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This only works if you drop "speed of light" for "invariant speed". The speed of light was known to be finite long before (1676) special relativity came along (1905). Actually, even before Newton's classical mechanics came along (1687).There is also another way to look upon it: the speed of light is invariant in both classical mechanics and relativity. But in classical mechanics the speed of light is infinite, while in relativity it's finite.

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In classical mechanics, light takes about eight minutes to travel from the Sun to the Earth. That was definitely understood pre-SR.There is also another way to look upon it: the speed of light is invariant in both classical mechanics and relativity. But in classical mechanics the speed of light is infinite, while in relativity it's finite.

- #11

Ibix

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I agree with Orodruin's point that haushofer would definitely be correct if he said "invariant speed" instead of "speed of light". But whether "speed of light" is wrong seems to me to depend on whether you interpret things like Rømer's measurement as part of classical mechanics or as the first (unrecognised) crack in the edifice.

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On the contrary, Rømer's measurement was in no way a crack in the theory if you see it in the context of how the theory was laid out at the time. Instead, it was reasonable to assume that light was carried by a medium (if a wave) or that its speed depended on the emitter (if particulate). Only when Maxwell’s theory of electromagnetism was laid out and searches for relative motion vs the aether failed did the cracks start to appear. There was no a priori contradiction between infinite speed being invariant and light speed being finite. That distinction comes only a posteriori after identifying the invariant speed with the speed of light, which did not happen until 1905.

I agree with Orodruin's point that haushofer would definitely be correct if he said "invariant speed" instead of "speed of light". But whether "speed of light" is wrong seems to me to depend on whether you interpret things like Rømer's measurement as part of classical mechanics or as the first (unrecognised) crack in the edifice.

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haushofer

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Sure, but this is because "the speed of light", c, plays a double rôle: as the speed of wave propagations of the electromagnetic field, and as determining the causal structure of spacetime. I was talking about the latter ;)This only works if you drop "speed of light" for "invariant speed". The speed of light was known to be finite long before (1676) special relativity came along (1905). Actually, even before Newton's classical mechanics came along (1687).