Dirac's statement about conservation

[exmarine]

In Dirac's book on GRT, he says the following, p. 45: "In curved space the conservation of energy and momentum is only approximate. The error is to be ascribed to the gravitational field working on the matter and having itself some energy and momentum."

Yet when I work my way from the Schwarzschild metric to the geodesic equations, one of them produces the conservation of angular momentum. There is no approximation that I am aware of.

What am I missing? Was Dirac wrong? Can anyone explain? Thanks.

 

[HomogenousCow]

I think he was referring to arbitrary metrics. A general metric without angular symmetry will not conserve angular momentum along the geodesic.

 

[WannabeNewton]

In Dirac's book on GRT, he says the following, p. 45: "In curved space the conservation of energy and momentum is only approximate. The error is to be ascribed to the gravitational field working on the matter and having itself some energy and momentum."

Say instead we were talking about the EM field. Then $T_{\mu\nu} = T^{\text{charges}}_{\mu\nu} + T^{\text{EM}}_{\mu\nu}$ hence $\nabla^{\mu}T_{\mu\nu} = 0$ already takes into account the energy-momentum of the EM field when talking about the local conservation of energy-momentum of the charge distribution interacting with this EM field. It is immediate from this why there comes a difficulty when wanting to take into account the energy-momentum of the gravitational field interacting with a given matter distribution in speaking of the local conservation of energy-momentum because we cannot write down the local energy-density of the gravitational field; $\nabla^{\mu}T_{\mu\nu} = 0$ does not account for this! In fact in GR we can only talk about the energy-momentum of the gravitational field using global quantities (such as the Komar energy/angular momentum, ADM energy-momentum etc.) or pseudo-tensorial quantities (such as the LL pseudo-tensor).

So Dirac's statement is much deeper and much more general than that of an axially symmetric space-time possessing an axial Killing field conserving angular momentum.

 

[bcrowell]

There are two different things here:

(1) Conservation of angular momentum for a test particle moving in a fixed background metric.

(2) Conservation of angular momentum for the field theory in general, including the angular momentum contained in the gravitational fields itself.

Dirac is talking about #2.

GR doesn't have general conservation laws for vectors and higher-order tensors. As an alternative to Dirac's explanation, this can be seen from the fact that parallel transport is path dependent in GR. Therefore you can't define any unique way to add up the conserved quantity.

 

[sardar]

Dirac's statement about conservation in curved space is not incorrect, but it may require some clarification. In general relativity, energy and momentum are still conserved, but the way they are conserved is different from that in flat space. In flat space, energy and momentum are conserved due to the symmetry of space and time, known as Noether's theorem. However, in curved space, this symmetry is broken by the presence of a gravitational field. As a result, the conservation of energy and momentum is no longer exact, but only approximate.

This approximation can be seen in the geodesic equations, which describe the motion of particles in a gravitational field. While one of the equations does produce the conservation of angular momentum, this is only true for a specific type of motion known as circular orbits. In more general cases, the conservation of energy and momentum will not hold exactly.

Furthermore, as Dirac pointed out, the gravitational field itself has energy and momentum, which must be taken into account when considering the conservation laws. This is known as the energy-momentum pseudotensor and it represents the energy and momentum of the gravitational field.

So, in short, Dirac's statement about the conservation of energy and momentum in curved space is not incorrect, but it may require some clarification and understanding of the differences between conservation in flat and curved space.