Gravity and Strings

3.3 General relativity 101
local and we can say that observable effects of the gravitational field can be eliminated
locally in a small enough neighborhood of a given point. A longer discussion with examples
can be found in [242].
There is an ongoing debate on the validity and interpretation of the PEGI into which we
will not enter. Some interesting criticisms can be found in [659].
So far we have seen that the PGR forces us to use general spacetime metrics gµν and
that these encode gravitational and inertial forces on the same footing, implying the PEGI
in its medium-strong form. Any theory making use of a metric in this way would do the
same. Now we want to find an equation of motion for the metric field which determines the
dynamics of the gravitational field.
The PGR tells us that the equation of motion of the metric field must be a general
tensor equation, A αβ = T αβ
matter. Wehave tofind a suitable two-index, symmetric, tensor
A αβ = A (αβ) that is a function only of the metric and its first and second derivatives,
A αβ = A αβ (gµν,∂ρgµν,∂σ ∂ρgµν). Now comes a very important point: in special relativity
the matter energy–momentum tensor is always conserved: ∂µT µν
matter = 0. Now we require
that the covariant generalization (as required by the PGR) of this equation
∇αT αβ
matter = 0 (3.269)
also holds. The connection is the Levi-Cività connection. It has to be stressed that this
equation is no longer a conservation equation, as we will explain in detail in Chapter 6.
However, it is the covariant generalization of the special-relativistic continuity equation
and reduces to it in locally inertial frames and it seems a plausible requirement. Thus, we
have to ask that A αβ be covariantly divergence-free.
The problem of finding the most general tensor A αβ satisfying these conditions was
solved by Lovelock in [662] and the solution is
A α β =
p=[ d+1
2 ]
p=1
cpg αγ1···γ2p
βδ1···δ2p Rγ1γ2
δ1δ2 δ2p−1δ2p
···Rγ2p−1γ2p + c0g α β, (3.270)
where the cs are arbitrary constants and the Riemann tensor is the one associated with the
Levi-Cività connection. If we also want to recover the Fierz–Pauli equation in the linear
limit gµν = ηµν + χhµν, A αβ has to be linear in second derivatives of the metric. In that
case, the only possibility is, as originally proven in [215, 924, 952],
A αβ = aG αβ + bg αβ , (3.271)
where G αβ is the Einstein tensor. This is also the only possibility in d = 4evenifwedonot
impose the requirement of linearity in second derivatives of the metric. The vanishing of
its covariant divergence is due to the contracted Bianchi identity ∇µG µν = 0 when the connection
is the Levi-Cività connection as we have assumed and to the metric-compatibility
of the same connection.
In the Fierz–Pauli theory there is no room for the constant b. Thus, let us set it to zero for
the moment. Now we have only to fix the proportionality constant a, which can be inferred
from the linearized (Fierz–Pauli) theory. We obtain the Einstein equation
Gµν = 8πG(d)
N
c4 Tmatter µν. (3.272)