I have once heard that the definition of energy is "the ability to do work". However, that is a counterfactual definition, because a physical system can have that ability without actually doing work. So, has someone proposed a non-counterfactual, non-modal definition of energy? I want a rigorous definition.
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$\begingroup$ Check its relation to Noether's theorem, for example. $\endgroup$– Tobias FünkeCommented Jul 11, 2023 at 15:14
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2$\begingroup$ Does this answer your question? What's the real fundamental definition of energy? $\endgroup$– Tobias FünkeCommented Jul 11, 2023 at 15:14
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$\begingroup$ Energy can be defined generally as the conserved Noether charge corresponding to time translations physics.stackexchange.com/q/59719/133418. Defining energy in general relativity is subtle and difficult, however. $\endgroup$– AvantgardeCommented Jul 11, 2023 at 15:16
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1 Answer
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The modern definition is that energy is the conserved quantity associated to time translation.
Loosely speaking, whenever the mathematical description of a physical system remains the same after some transformation is done to the system, the system is said to be symmetric under this transformation. The prime example of such a symmetry is rotating a perfect ball arounds it's center; it looks exactly the same after the rotation as it did before. There is a mathematical theorem (Noether's theorem) which says that for every symmetry of your model, there will be a correspondent quantity which is conserved along the time evolution of the system.
If your model is symmetric under time translations, that is, under taking the time variable $t$ and adding a constant to it, $t \to t + \Delta t$, the correspondent conserved quantity is what we call energy. Informally, time translation symmetry basically means that you can expect the same experimental results if you do a measurement today, tomorrow, or one year from now.
There are physical systems which do not exhibit time translation invariance, like the expanding Universe. In that case, the metric of the Universe, an object which measures the distances between points, is changing continously in time, so doing $t \to t + \Delta t$ actually produces a measurable change (in the interval $\Delta t$ the Universe has expanded a bit more). In this case there is no standard notion of energy, and we do expect some difference in doing experiments today or tomorrow.
