Exact interior solutions for perfect fluids in Einstein–Gauss–Bonnet gravity

Since the pioneering mathematical work of Lovelock (1971) no exact interior solution for static spherically symmetric perfect fluid matter has been explicitly  found in the case of the Einstein–Gauss–Bonnet theory. We investigate this

problem and report classes of new exact solutions for this configuration. One of

the main motivations for the study of alternate gravity theories is the inability

of the highly successful theory of general relativity, obtained from the Einstein–

Hilbert action principle, to explain phenomena such as the late time accelerated

expansion of the universe without resorting to the introduction of concepts such

as dark matter. The expanding universe phenomenon has been demonstrated by

observational evidence such as in WMAP. Lovelock gravity neatly avoids calling

on such matter and instead involves a modification of geometry which generalises

that of general relativity. Unlike other competing theories, such as f(r) theory,

the Lovelock polynomials used in the action generate second order equations of

motions for all orders of the polynomial. To second order the Lovelock polyno-
mial is referred to as the Einstein–Gauss–Bonnet (EGB) quadratic polynomial

and is constructed from quadratic forms of the Ricci tensor, Ricci scalar and

Riemann tensor. This EGB term only contributes to the dynamics for metrics

with dimension more than 4. Therefore we are interested in studying the dimen-
sional order 5 as a first tangible case in order to examine the behaviour of the

EGB term and its effect on the physics when compared to the general relativity

case of dimension 4. The EGB field equations for n, n ≥ 5 have been obtained

however we confine this investigation to the 5 dimensional case only. We work

in a coordinate frame and write the associated EGB field equations for perfect

fluid matter. By introducing a coordinate transformation we rewrite the master

pressure isotropy condition in a form that allows us to locate exact solutions by

prescribing a form for one of the gravitational potentials. We impose the usual

tests of physical plausibility on the resulting solutions. In particular we exhibit a

model that is causal, satisfies the weak, strong and dominant energy conditions

and which represents a fluid with a vanishing pressure hypersurface identifying

the boundary of the fluid - the simplified Israel-Darmois junction conditions

have been used. The exact solution is matched to the exterior Boulware–Deser

(1985) exterior metric across the pressure–free boundary.