### Conservation of Energy

This is the best technical introduction I know to the problems of energy conservation in General Relativity. In my comments on Keneffick’s book I describe some subtleties in energy conservation even in Newtonian physics. I try here to demystify some of the problems.

When we speak of conservation of mass in Newtonian physics, we reason about an integral of a scalar field, the density, over a portion of space. Two observers in different Galilean frames will agree on the numerical magnitude of this integral. When we speak of conservation of momentum, two observers will not agree on the vector value of this magnitude, for the integral may be the vector zero in one frame and not in the other. This does not bother the Newtonian physicist for momentum is conserved in either frame and the equations to convert the vector components in one frame to another frame are well known.

Energy density seems at first blush like a scalar field and we would expect the situation to be like mass conservation. My story of the wheel in the Kennefick note shows that Newtonian energy conservation is more complex however. Also the two balls story shows that it is problematic to localize energy. Two physicists will not agree on the numerical values of these energy densities. They will, however, know how to convert their numbers so as to confirm each other’s results.

With special relativity energy conservation and momentum conservation are replaced by a single conservation for a tensor field called the ‘Stress-Energy tensor’. This tensor is typically introduced only in general relativity, but it is implicit in its full form in special relativity. In special relativity this tensor field is conserved in four dimensions; the flux of this tensor across a 3D surface surrounding a 4D portion of space time sums to 0. Summing this flux amounts to adding vectors that are displaced from each other in space and time. This is not a problem in Minkowski’s flat geometry of special relativity. It is not generally possible in the curved space-time of general relativity.

There is a close analog to this conundrum in Newtonian physics. Consider a perfectly round planet that is not spinning. There is an ocean on this planet that would be of constant depth if it were not moving, but it is indeed moving with a local velocity parallel to the local horizon. Locally we can reason about the flow of this ocean by appealing to local conservation of momentum and energy. Globally (planet wide) we cannot. We cannot add momentum vectors in the tangent plane at one point on the planet, to vectors in the plane at another point. This is the same problem that plagues the concept of what it even means to conserve energy in general relativity.

The analogy breaks down at this point. In the case of our fictitious planet we can resurrect conservation of energy and momentum by considering the whole planet including its ocean. There is no similar tactic in the case of GR where there is no notion of a greater space in which to redo our calculations. Efforts to posit such a space lead to inconsistent theories, or theories that lead to false predictions. With our planet the ocean may trade momentum with the planet’s solid core. In general relativity the motions of stress-energy may trade something with the underlying space. I know no name nor formalism for this something but I fear that our analogy may mislead—it is not merely stress-energy in some higher dimensional flat embedding space where the vectors can be added.