Gravity and Strings

140 Action principles for gravity
Tcan a µ is the Dirac-spinor covariant canonical energy–momentum tensor. It has the same
form as in Eq. (4.100) but now the total covariant derivative uses the general connections
considered here. As we have already pointed out, in the first-order formalism, the covariant
canonical energy–momentum tensor is obtained by direct variation with respect to the
Vielbeins:
δSmatter
δea = eTcan a
µ
µ . (4.142)
Finally, S µ ab is the spin-angular-momentum tensor, which is totally antisymmetric and
given by Eq. (2.67).
The equations of motion are
Ga µ 2 χ
=
2 Tcan a µ , Dνeρ c =
The second equation has the solution
χ 2
2 Scνρ, i∇ ψ − mψ = i
2 Tνµ µ γ ν ψ. (4.143)
Tµν a =−χ 2 S a µν, (4.144)
as in the CSK theory. On account of the complete antisymmetry of S, this equation implies
that the r.h.s. of the third equation vanishes identically, so we are left with
i∇ ψ − mψ = 0. (4.145)
Finally, the first equation is just the Einstein equation one obtains in the CSK theory after
several manipulations. We can split it into a Riemannian part and the torsion contributions,
which we know are of quartic order in χ.
As we have stressed before, the simplicity of the first-order formalism is related to the
previously mentioned fact that this kind of action makes contact with the formulation of
gravity as the gauge theory of the Poincaré group which we are going to study next.
4.5 Gravity as a gauge theory
In [674] MacDowell and Mansouri formulated gravity as the gauge theory of the Poincaré
group and supergravity as the gauge theory of the super-Poincaré group. 16 This approach
was later extended successfully to many other situations and it is interesting enough to
review it briefly here because the similarities with and differences of gravity from the gauge
theories of internal symmetries (some of which we have already mentioned) are manifest
in this formulation. Here we will loosely follow [404, 912].
One of the differences we observed in the previous section between the first-order formalism
for gravity using Vielbeins and spin connection and a pure gauge theory is that we
did not have an interpretation of the Vielbeins as gauge fields. Furthermore, our intuition
tells us that, if gravity can be interpreted as a gauge theory at all, it cannot be a gauge theory
of the Lorentz group alone and at least gauge translations should be introduced into the
game. We should then consider the gauging of the Poincaré group. It is worth stressing here
that we are talking about the “Poincaré group of the tangent space.” That is, at each point
16 The earliest work on this subject is [918].