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136 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
or In what follows we assume
that is compact so that the action (5) is gauge invariant.
We identify the components of the six dimensional gauge field as
with the five dimensional connection and the five dimensional
coframe, they are related by the first Cartan equation
where the corresponding indices are lowered or raised by the local
Lorentz metric. Then, we write the action (5) in terms of the new
variables, ignoring boundary terms as follows
where
is the field strength of the SO(1,4)–valued connection and is related
to the signature of the fifth group index for SO(2,4) and
for SO(1, 5).
The first term in (10) is a Gauss–Bonnet one, the second term is
the Einstein term, and the last one is a cosmological constant term.
Decomposing the field equations (6) in terms of the component field
and we obtain
where
In what follows we will restrict ourselves to the case
3. FIVE DIMENSIONAL PRINCIPAL FIBER
BOUNDLE
The topology of the vacuum is assumed to be and thus
or equivalently The five-dimensional coordinates
are denoted by and the metric (M, N, L, … =
0, 1, 2, 3, 5) has signature (+, –, –, –, –). To account for the fact that
below Plank energy the physical quantities appear only to change with