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Where does energy conservation come from? In modern understanding, energy is the Noether charge of time translation symmetry, as found by Noether's first theorem. But in general relativity, the metric is dynamical, and so in general we simply don't have any time translation symmetry. Static spacetimes do, and there is also a form of energy conservation for spacetimes that regain time translation symmetric far away from the gravitating system (e.g. ADM energy of asymptoptically flat spacetimes). But those are the exceptions, not the rule.

Addendum: It's notable that there is yet another sense in which the total energy of the a spatially finite universe is exactly zero. Intuitively, one can try to measure the content inside some closed surface, and then expand that surface to try to enclose everything in the universe. However, for a closed universe, that surface will contract to a point, thuse enclosing nothing (picture a circle around the north pole of the Earth, and expand it to try to enclose all of Earth's surface--it just contracts to a point at the south pole).

Where does energy conservation come from? In modern understanding, energy is the Noether charge of time translation symmetry, as found by Noether's first theorem. But in general relativity, the metric is dynamical, and so in general we simply don't have any time translation symmetry. Static spacetimes do, and there is also a form of energy conservation for spacetimes that regain time translation symmetric far away from the gravitating system (e.g. ADM energy of asymptotically flat spacetimes). But those are the exceptions, not the rule.

Addendum: It's notable that there is yet another sense in which the total energy of the a spatially finite universe is exactly zero. Intuitively, one can try to measure the content inside some closed surface, and then expand that surface to try to enclose everything in the universe. However, for a closed universe, that surface will contract to a point, thus enclosing nothing (picture a circle around the north pole of the Earth, and expand it to try to enclose all of Earth's surface--it just contracts to a point at the south pole).

Where does energy conservation come from? In modern understanding, energy is the Noether charge of time translation symmetry, as found by Noether's first theorem. But in general relativity, the metric is dynamical, and so in general we simply don't have any time translation symmetry. Static spacetimes do, and there is also a form of energy conservation for spacetimes that regain time translation symmetric far away from the gravitating system--e.g., ADM energy of asymptoptically flat spacetimes). But those are the exceptions, not the rule.

One thing you can do in the context of comsology is look at the Friedmann equations as some sort of analogue of energy conservation, by making a balance between the terms describing cosmic expansion and the energy density, pressure, and cosmological constant. The Friedmann equations come from the components of the Einstein field equation connecting the Einstein curvature tensor and the stress-energy tensor: $G_{\mu\nu} = 8\pi T_{\mu\nu}$. According to this interpretation, Einstein curvature always exactly balances the stress-energy of the matter in spacetime. But this is just a restatement of a dynamical law, so it's not really a "conservation" law.

Where does energy conservation come from? In modern understanding, energy is the Noether charge of time translation symmetry, as found by Noether's first theorem. But in general relativity, the metric is dynamical, and so in general we simply don't have any time translation symmetry. Static spacetimes do, and there is also a form of energy conservation for spacetimes that regain time translation symmetric far away from the gravitating system (e.g. ADM energy of asymptoptically flat spacetimes). But those are the exceptions, not the rule.

One thing you can do in the context of cosmology is look at the Friedmann equations as some sort of analogue of energy conservation, by making a balance between the terms describing cosmic expansion and the energy density, pressure, and cosmological constant. The Friedmann equations come from the components of the Einstein field equation connecting the Einstein curvature tensor and the stress-energy tensor: $G_{\mu\nu} = 8\pi T_{\mu\nu}$. According to this interpretation, Einstein curvature always exactly balances the stress-energy of the matter in spacetime. But this is just a restatement of a dynamical law, so it's not really a "conservation" law.

Addendum: It's notable that there is yet another sense in which the total energy of the a spatially finite universe is exactly zero. Intuitively, one can try to measure the content inside some closed surface, and then expand that surface to try to enclose everything in the universe. However, for a closed universe, that surface will contract to a point, thuse enclosing nothing (picture a circle around the north pole of the Earth, and expand it to try to enclose all of Earth's surface--it just contracts to a point at the south pole).

More formally, one can find a sequence of asymptotically flat universes (for which, again, the energy actually is defined) that approximate a spatially finite universe. In the limit in which the approximating universes "pinch off" and separate from the asymptotically flat region (thus becoming actually finite), the ADM energy goes to $0$.