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For the sake of simplicity, let's imagine that the entire universe is empty except for a single lump of (classical) matter with mass $m$. In its center of momentum frame, it is clear that the total energy is simply $E_\text{CoM}=mc^2$. However, in a frame moving relative to it with speed $v$, we have $E_\text{moving}=\sqrt{m^2c^4+p^2c^2}> mc^2$ for all nonzero $v$.

Are we to infer that the energy content of a system can be different relative to the observer? Does this not violate the first law of thermodynamics?

ACuriousMind
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Caio Belussi
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  • I'm going to edit your question to leave only the essence of it. I hope you don't mind this; if you do, you can 'rollback' my edit. – Danu Dec 24 '14 at 13:17
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    Are we to infer that the energy content of a system can be different relative to the observer? Of course. Does this not violate the first law of thermodynamics? No. What makes you think that it does? – David Hammen Dec 24 '14 at 13:24

3 Answers3

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Yes, the observed energy content differs between different observers:

Energy is manifestly no Lorentz invariant quantity, as it is the zeroth component of the momentum four vector, and hence differs between different inertial frames. Thus, rather trivially, different frames will observe different energy contents for the same system.

This does not mean that conservation of energy is violated, since conservation of energy simply states that the time derivative of energy is zero - and it is, within every frame. Only when you change frames, the energy you observe changes, and again, in every momentarily comoving frame in that process, energy is indeed conserved.

ACuriousMind
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  • as far as we observing different energy contents for the same system should we also observe different spacetime curvature effects? – mirt Aug 14 '19 at 17:20
  • @mirt I don't know what you mean by "different spacetime curvature effects", but maybe you're thinking along the lines of https://physics.stackexchange.com/q/3436/50583. – ACuriousMind Aug 14 '19 at 17:23
  • by "different spacetime curvature effects" i mean for example "geometrical quantities characterizing spacetime curvature, such as the Ricci scalar" or anything else you can suggest. (they are supposed to be different because you stated that we are observing different energy) – mirt Aug 14 '19 at 17:25
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    @mirt No, you won't observe different Ricci scalars, precisely because it's a scalar. That's what being a scalar means - being the same in all frames. Of course you will get different values for the components of the curvature tensors in different frames, exactly like you get different values for the components of the 4-momentum vector here. – ACuriousMind Aug 14 '19 at 17:28
  • Well i want to admit i know nothing about Ricci scalars. But will there be gravitational lensing effect around me (because you know "observed energy content differs...") – mirt Aug 14 '19 at 18:10
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This is really just a footnote to ACuriousMind's answer, since I can't improve on his discussion of energy.

In relativity, both special and general, there is an invariant quantity analogous to energy called the stress-energy tensor. This does not depend on the observer, that is all observers in all frames will measure the same stress-energy tensor (though its representation will vary for different observers).

John Rennie
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Newtonian Mechanics: A particle of mass $m$ is moving with constant speed $v$ in a given inertial frame A. The energy of the particle is:

$$E_A=E^{kin}_A=\frac{1}{2}m v^2 $$

Energy of the same particle in a frame B moving with speed $v$ relative to the first frame:

$$E_B=E^{kin}_B=0 $$

Rexcirus
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