# Expansion again: where does the energy come from?

Is it sensible to ask for the overall increase in potential energy when looking at the baryonic mass in the universe moving away from each other, that is, moving against the direction of the force towards a higher energy state in each others however weak gravitation field? I can't imagine that nature has the same take on this matter like a numerical simulation (a distance which it deems "large enough to make effects negligible so let's forget about it completely"), i.e. the energy can't be zero... where does it come from?

Conceptually there are several things going on here.

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.

In other words, in general relativity we don't have any scalar notion of "energy" that's applicable globally. Vacuously, it is neither conserved nor violated.

But what about locally? In a local inertial frame, energy is exactly conserved, but the gravitational forces exactly vanish.

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.

The Einstein field equation itself can be found from the Einstein-Hilbert action, and trying to apply Noether's second theorem simply shows that the covariant derivative of the stress-energy tensor vanishes: $\nabla_\nu T^{\mu\nu} = 0$, which is analogous to $\nabla\cdot\mathbf{B} = 0$ of electromagnetism: "there are no local sources or sinks of [stress-energy/magnetic field] anywhere." This is actually trivial, because the covariant derivative of Einstein curvature always vanishes (a theorem of geometry lacking any physics), so Noether's second theorem didn't tell us much more than we would have known otherwise.

Because the derivative is covariant rather than partial, many people don't consider this to be a true conservation law either. Certainly it gives no information as to "how much" energy there in spacetime--that's still undefined.

So we have the following issues:

• There is no global energy conservation is general relativity, except for very special spacetimes, and the FRW family used for Big Bang models don't qualify.
• In a local inertial frame, energy is exactly conserved, but there are no gravitational forces. (Local inertial frames only exist as first-order approximations anyway.)
• One can interpret the Einstein field equation as Einstein curvature exactly balancing the stress-energy of matter, which is also motivated by interpreting the Friedmann equations of cosmology as a balance between cosmic expansion and local energy, pressure, and cosmological constant. However, this is actually a dynamical law.
• The vanishing of the covariant derivative of stress-energy can be construed as an analogue of local energy conservation, although doing so is conceptually misleading.

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).

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 $0$.

• This is really a great answer and I keep rereading it from time to time, groking a bit more each turn :D – Vroomfondel Aug 11 '15 at 15:45

The totally unsatisfying answer is: Dark energy. It's formally quantified by the cosmological constant.

There are lots of hypotheses about the nature of this energy. Quantum theoretical explanations are considered as likely candidates; the Casimir effect is at least an experimentally accessible way to show the existence of vacuum energy.