Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Dilaton gravity from Chern–Simons gravity 137
respect to the coordinates of the usual space–time, we introduce as
usual the cylinder condition
Then, the metric tensor in a non–diagonal basis is usually written as
[8, 9]
where is the four–dimensional metric tensor, the electromagnetic
potentials and is the dilaton field, is the compactification
radius [8] of the fifth dimension It is much practical to
work in a horizontal lift basis (HLB), were the vector potential does not
appear explicitly in the metric
the is a 1–form basis. The basis vectors which are dual to the
are given by
It is important to notice that the HLB is an anholonomic basis, which
means that some of the commutators among the vectors of the dual basis
Eq. (17) are different from zero. It is easy to show that
where Given a set of basis vectors the
commutation coefficients are defined as follows
The only non–vanishing commutators are
In the next section we compute the action (10) for the ansatz (15)
and compactify it to four dimensions.
4.
EFFECTIVE ACTION
The 5–dimensional connection reads