Gravity and Strings

172 Conserved charges in general relativity
Apart from the problem of the gravitational energy–momentum tensor, the definition
of conserved quantities in GR has many interesting points. Several approaches have been
proposed and here we are going to study two: the construction of an energy–momentum
pseudotensor for the gravitational field and the Noether approach. In both approaches there
is a great deal of arbitrariness and in Section 6 we will study and compare several different
results given in the literature in the weak field limit, finding complete agreement and a deep
relation to the massless spin-2 relativistic field theory studied in Chapter 3.
6.1 The traditional approach
As we have stressed several times, the metric (or Rosenfeld) energy–momentum tensor of
any general-covariant Lagrangian always satisfies (on-shell) the equation
∇µTmatter µν = 0, (6.1)
as a direct consequence of general covariance. This equation is crucial for the consistency of
the theory. Furthermore, it is the covariantization of the Minkowskian energy–momentumconservation
equation
∂µTmatter µν = 0, (6.2)
which is discussed at length in Chapter 2, and from which we can derive local conservation
laws of the mass, momentum, and angular momentum and, in general, of those charges
related to the invariance of a theory under certain coordinate transformations.
In curved spacetime, however, Eq. (6.1) is not equivalent to a continuity equation for the
tensor density √ |g| Tmatter µν that holds in Minkowski spacetime. Actually, we can rewrite
Eq. (6.1) in the form
|g| Tmatter µν
=−Ɣρσ ν Tmatter ρσ , (6.3)
∂µ
and, in general, the r.h.s. of this equation does not vanish. From this equation we cannot
derive any local conservation law.
In a sense this was to be expected: only the total (matter plus gravity) energy and momentum
should be conserved 3 and, therefore, we can only hope to be able to find local
conservation laws for the total energy–momentum tensor. Now, how is the gravity energy–
momentum tensor defined in GR? This is an old problem of GR. 4 It is clear that we cannot
use the same definition (Rosenfeld’s) as for the matter energy–momentum tensor because
that leads to a total energy–momentum tensor that vanishes identically on-shell. On the
other hand, if we found a covariantly divergenceless gravitational energy–momentum tensor,
the total energy–momentum tensor would have the same problem as the matter one.
In fact, it can be argued, on the basis of the PEGI, that it is impossible to define a fully
general-covariant gravitational energy–momentum tensor: according to the PEGI we can
remove all the physical effects of a gravitational field locally, at any given point, by using an
appropriate (free-falling) reference frame. This means that we could make the gravitational
3 Actually, the coupling of gravity to the total, conserved, energy–momentum tensor was the main principle
leading in Chapter 3 to GR.
4 Some early references on the energy–momentum tensor of the gravitational field are [90, 355–627, 660,
839].