Energy conservation and General Relativity

Question

Generally, we know that energy is conserved and there is Hamiltonian mechanics which describes particle motion by energy conservation and conversion. Quantum mechanics is completely based on the conservation of energy. And everything works out very well.But, in general relativity, there seems to be no conservation of energy. Why is this? What does it mean by no energy conservation? Where does the energy come from?

Answer 1

Energy in General Relativity

Energy conservation arises due to invariance under translations in time, and in general this will not hold. In general relativity, we do have the analogue,

$$\nabla_\mu T^{\mu\nu} = 0$$

however this does not imply energy is conserved, because one cannot bring the expression into an integral form, as one can normally when applying Noether's theorem to field theories, where we could define a conserved current $\partial_\mu j^\mu = 0$ and a conserved charge,

$$Q = \int d^3x \, j^0.$$

Locality

It is possible to define a Landau-Lifshitz pseudo-tensor $\tau_{\mu\nu}$ which ascribes stress-energy to the gravitational field, such that,

$$\partial_\mu (T^{\mu\nu} + \tau^{\mu\nu}) = 0,$$

from which one can define momentum $P^\mu$ and angular momentum $J^{\mu\nu}$. However, the modified stress-energy and $\tau_{\mu\nu}$ itself has no geometric, coordinate free significance. It may vanish in one coordinate system and not in another.

The trouble boils down to the fact that gravitational energy cannot be localised. For electromagnetism, one can speak of a region of space-time with some energy density due to the electromagnetic field, which is responsible for inducing curvature and changing the wordlines passing through it.

Due to the equivalence principle, locally we can always define a coordinate system wherein the gravitational field vanishes, which means it does not make sense to speak of a local gravitational energy density.

Alternate Definitions

Nevertheless, there are further analogues of energy and other quantities of the Hamiltonian formalism in general relativity which are sometimes useful, though there are issues with these as aforementioned, such as either coordinate dependence, or other ambiguities. One expression is the quasilocal energy, defined as,

$$E = -\int_B d^2 x \, \frac{\delta S_{\mathrm{cl}}}{\delta N} = \frac{1}{\kappa}\int_B d^2x\, \sqrt{\sigma}k \big\rvert_{\mathrm{cl}}$$

where $B$ is the boundary of a spatial hypersurface $\Sigma$, with $\sigma$ and $k$ the metric and extrinsic curvature respectively. Contributions from flat space must be subtracted off typically, and there is an ambiguity in this due to two possible signs of the normal.

If we parametrize a system by a coordinate $\lambda$ for a path in the state space, then the action of a system is, $$S = \int_{\lambda'}^{\lambda''}d\lambda \, \left( p\dot x - \dot t H(x,p,t)\right).$$

For a classical history,

$$H_{\mathrm{cl}}\big\rvert_{\lambda''} = -\frac{\delta S_{\mathrm{cl}}}{\delta t''}$$

which is to say the energy at the boundary $\lambda''$ is minus the change in the classical action due to an increase in the final boundary time, $t(\lambda'') = t''.$ The expression for the quasilocal energy in general relativity is precisely the closest analogue to this Hamilton-Jacobi equation.

For more details and an illuminating discussion of energy conservation in general relativity, see Quasilocal Energy in General Relativity by D. Brown and J.W. York in the text, Mathematical Aspects of Classical Field Theory 132 by the AMS.

Answer 2

No, energy is not always conserved in general relativity. There is no way to define the energy of an isolated system as a function of the state of the system such that total energy and momentum is conserved when ever 2 systems combine or undergo a hyperbolic orbit and the definition simplifies to the definition in special relativity for low mass and density. An electromagnetic field can't affect a gravitational field directly but can accelerate a particle which in turn affect the gravitational field and in the absense of matter, an electromagnetic field can't affect a gravitational field. According to the Wikipedia article No hair theorem, the observable state of a black hole is completely described by its mass, charge, and angular momentum. An electric field can't accelerate a charged black hole because there's no matter outside its event horizon for it to accelerate and in turn affect the fabric of space outside the event horizon so energy is not always conserved in general relativity.