Gravity and Strings

308 The Kaluza–Klein black hole
What one would like to do now is to fix the value of this momentum, which completely
determines the dynamics in the isometry direction, and find the effective dynamics in the
remaining directions. Doing this in a general coordinate system is very complicated (if it is
possible at all) and hence we have to work in adapted coordinates as before. We will use,
then, all the machinery and notation developed in this section.
In adapted coordinates the fact that there is a conserved momentum becomes evident
since the action no longer depends on the isometric coordinate z.
To simplify the problem further, we split the ˆd-dimensional fields and coordinates in
terms of the d-dimensional ones according to Eq. (11.28), obtaining
where the combination
ˆS[X µ (ξ), Z(ξ), γ (ξ)] =− p
2
F(Z) = ˙Z + Aµ
dξ γ − 1
2 gµν ˙X µ ˙X ν − k 2 F 2 (Z) , (11.73)
˙X µ
(11.74)
that naturally appears in the action is the “field strength” of the extra worldline scalar Z,
which now does not have a coordinate interpretation.
As we explained, the original action (11.68) above is covariant under target-space diffeomorphisms
and so must the action (11.73) be, since it is a simple rewriting of the former. In
particular, it must be covariant under X µ -dependent shifts of the redundant coordinate Z,
δZ =−(X µ ), (11.75)
which do not take us out of our choice of coordinates (i.e. coordinates adapted to the isometry)
either. As discussed before, these transformations generate gauge transformations of
the U(1) gauge potential,
δ Aµ = ∂µ. (11.76)
The field strength of Z is covariant under this transformation, which justifies its definition.
Related to the constant shifts of Z (which is an invariance) is the conservation of the
momentum conjugate to Z,
Pz ≡ ∂L
∂ ˙Z = pγ − 1 2 F(Z), ˙Pz = 0. (11.77)
Now we want to eliminate Z from the action completely, using its equation of motion
( ˙Pz = 0), and thus obtain the action that governs the effective d-dimensional dynamics.
However, we cannot simply substitute into the action Eq. (11.73) Pz = pγ 1 2 F(Z) =
constant because from the resulting action one does not obtain the same equations of motion
as one would from making the substitution into the equations of motion. The reason
for this is that the equation of motion of Z is not algebraic because ˙Z occurs in the
action.
Aconsistent procedure by which to eliminate Z is to perform first the Legendre transformation
of the Lagrangian with respect to the redundant coordinate Z, just as one would
do to find the Hamiltonian if the Lagrangian depended only on Z. Weexpress ˙Z in terms