Gravity and Strings

124 Action principles for gravity
Furthermore, we are not going to assume in our derivation of the equations of motion either
the symmetry of the connection or the symmetry of the “metric,” which we will impose
at the very end. In this way, we can obtain with a minimum extra work the equations of
the Einstein–Straus–Kaufman [358, 364, 367, 369] non-symmetric gravity theory (NGT)
which was (unsuccessfully) proposed as a unified relativistic theory of gravitation and electromagnetism
in which the antisymmetric part of the “metric” g [µν] should be identified
with the electromagnetic field strength tensor 4 F µν .
In the NGT the inverse “metric” is also denoted by g µν and satisfies
g µν gνρ = δ µ ρ, gαβ g βγ = δα γ , (4.60)
but g µνgµρ = δν ρ. Also, we cannot use it to lower or raise indices.
Let us now vary the above action with respect to the metric and connection. By using
Palatini’s identity Eq. (3.286), we find
δS = dd x δgαβ Rαβ(Ɣ) + gαβ∇αδƔρβ ρ −∇ρδƔαβ ρ − Tαρ σ δƔσβ ρ
=
dd x δgαβ
ρβ Rαβ(Ɣ) +∇ρ g δσ α − gαβδσ ρ
σ
δƔαβ
+ ∇σ g αβ −∇ρg ρβ δσ α − g λβ Tλσ α δƔαβ σ .
Using now the identity for vector densities
(4.61)
∇µv µ = ∂µv µ + v µ Tµρ ρ , (4.62)
and integrating by parts, we obtain, up to a total derivative
δS = dd x δgαβRαβ(Ɣ) + Tρδ δgρβδσ α − gαβδσ ρ
+∇σ g αβ −∇ρg ρβ δσ α − g λβ Tλσ α δƔαβ σ .
(4.63)
Since the metric and the connection are independent, we obtain two equations from the
minimal action principle:
δS
δgαβ = Rαβ(Ɣ) = 0,
δS
δƔαβ γ =∇γgαβ −∇ρgρβδγ α − gλβ Tλγ α + gρβδγ αTρδ δ − gαβ Tγδ δ (4.64)
= 0.
The first equation would be the Einstein equation if the connection were the Levi-Cività
connection. Observe that, if we couple bosonic (scalar or vector) matter minimally to this
4 See also [654, 837, 838, 895]. A more recent NGT that reinterprets Einstein’s theory was proposed in
[699]. In it the antisymmetric part of the metric is also considered as a sort of new gravitational interaction.
Clearly, the weak-field limit cannot be the Fierz–Pauli theory but contains another field corresponding to
the antisymmetric part of the metric. While this suggests a relation with string theory, which also contains
a rank-2 antisymmetric tensor (the Kalb–Ramond field), these two fields appear in quite different ways: the
Kalb–Ramond field has an extra gauge symmetry, which allows it to be consistently quantized, whereas the
antisymmetric part of the NGT “metric” transforms only under GCTs. See [253, 285, 286, 616].