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

40 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
equations are equivalent to the Ernst equation
for the complex Ernst potential where the
imaginary part is related to the metric coefficient a via
and
The boundary value problem for the Ernst potential is described in detail
in[7]. These theta functions were first introduced by Rosenhain[8] and
are a generalization of the Jacobian theta functions. They depend on five
arguments: two main arguments and three moduli. The definition of the
four Jacobi theta functions and of the 16 Rosenhain functions
can be found in the appendix. The solution in
terms of ultraelliptic theta functions, is:
The solution exists and is regular everywhere outside the disk in the
range where constant
angular velocity, : radius of the
disk). The arguments of the
“theta formula” for depend via ultraelliptic line integrals on the normalized
space-time coordinates and on the parameter
µ. The functions and (also depending on are explicitly
known. In the following the arguments (using normalized coordinates
are explained. The convention for the root is
The functions and are
given by
The normalization parameters the moduli of
the theta functions and the quantities are defined via integrals
on the two sheets of the hyperelliptic Riemann surface (see Figure 1)
related to