Judging by the comments on the previous article I have not yet succeeded in convincing anyone that energy is conserved. Luboš Motl has posted a response contradicting my viewpoint and agreeing with an older blog post by Sean Carroll. Fortunately I have an advantage, I’ve done the maths and the outcome is clear and unambiguous. Since my detractors are people who understand equations I should have no trouble convincing them if I take a more technical approach. So, no more analogies, let’s start with Einstein’s equation.
is the Einstein tensor given by
is the Ricci curvature tensor, is the metric tensor, is the cosmological constant, is a gravitational coupling constant and is the energy-stress tensor for matter.
A time translation is generated by a time-like vector field with a small parameter
All the tensor quantities have corresponding transformation rules and the field equations are covariant under the transformation. Using Noether’s theorem a formula for the corresponding conserved energy current can be derived. The details of this calculation are quite lengthy and can be found in arXiv:gr-qc/9701028 (without the cosmological constant which is easy to add in.) The result obtained is
Where is the energy current from the matter contribution given by
is the dark energy component given by
and is the gravitational contribution given by
is the Komar superpotential given by
Given the Einstein field equations we can eliminate the matter term, the dark energy term and the first part of the gravitational term to leave just the Komar superpotential
Then it is easy to see that
Since the current is divergenceless it defines a conserved energy. Some people claim that this result is trivial. Clearly it is not because it requires the gravitational field equations to prove conservation of energy. You should not make the mistake of defining energy as just the Komar superpotential, even though it is equal to that when the dynamics are taken into account. Energy must be defined as a sum over contributions from each field including the gravitational field, and dark energy.
The energy contribution from matter is a sum of contributions from each type of matter field. It includes all the non-gravitational forms of energy including heat, electrical energy, rest mass equivalent of energy, radiation etc. This form of the law of energy conservation in general relativity tells us that the energy from all these contributions plus the gravitational contribution and the dark energy contribution is conserved. In other words energy can be transformed from one form to another but is never created or destroyed. There is nothing approximate, trivial or ambiguous about this result. It is energy conservation in the same old form that we have always known it but with the contribution from gravity included.