Conservation of Relativistic mass and thus energy is easily proven by considering an inelastic collision of two bodies while invoking the conservation of momentum. As such the momentum law appears more primitive. Is this somehow an artefact of momentum - space and energy- time as conjugate variables and the implicit necessity to assume causality in constructing such a scenario?
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Are you looking for Noether's theorem for special relativity: http://physics.stackexchange.com/q/12559/? – CuriousOne Sep 05 '14 at 04:41
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3I don't see how a relativistic momentum-conservation law could be considered more fundamental than a relativistic energy-conservation law. Energy and momentum are different parts of the same four-vector in relativity, so surely they have equal logical precedence. – Sep 05 '14 at 05:41
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Emmy Noether discovered a fundamental connection between symmetries and conservation laws, embodied in her famous theorem. In simple terms, Noether's theorem is that
For every symmetry in a physical system, there must be a conserved quantity.
The proof requires neither Lorentz invariance nor causality.
By applying Noether's theorem, we find that energy conservation is connected to the symmetry of time translations, $t \to t + \delta t$, and that momentum conservation is connected to the symmetry of space translations, $x \to x + \delta x$.
Within a relativistic theory, we can write space and time translations collectively as translations of a position four-vector, $x_\mu \to x_\mu + \delta x_\mu$. By applying Noether's theorem, we find that energy-momentum, $p_\mu = (E,\vec p$), is conserved.
The conservation of energy and momentum appear on an equal-footing from Noether's theorem; neither one is more "primitive" than the other.
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