Self-similar static solutions admitting a 2-space of constant curvature

Abstract

The plupose of thii letter is to give all the static, splierically symmehic perfect fluid solutions admitting a homothety. This family can be completely characterized by means of a real parameter y (arising quite naturally from the equation of state for these fluids, see below), which must be in the interval [ I , 21 in order to satisfy energy conditions. The two limiting values of y. namely y = 1 and y = 2 correspond to Minkowski flat spacetime and to the HHB solution, respectively.A few remarks concerning the similarity group and its action are in order here. It is a well known fact that an r-parameter group of homotheties H, (in which at least one proper homothety exists) always admits an (r -l)-parameter subgroup of isomehies G,-I. Now, the maximal dimension of the group of homotheties that a perfect fluid spacetime may admit is r = 7, in which case it is one of the special Robertson-Walker spacetimes [2], and therefore they are all known. The case r = 6 is not compatible with an energy-momentum tensor of the perfect-fluid type; thus, apart from the special Robertson-Waker solutions mentioned above, the highest dimension of the group of homotheties that a perfect fluid spacetime may admit is r = 5. In such case, the associated isometry subgroup Gq has necessarily three-dimensional non-null orbits

Spherically symmetric spacetimes and kinematic self-similarity

“…It is also important to note that the static solution is the only subcase in which this solution can have an equation of state of the form p = p(µ), since this condition implies (from equations (2.23)) that ¨ = 0. Static spherically symmetric spacetimes admitting a homothetic vector were studied by Ibanez and Sanz (1982) (see also Carot and Sintes 1994).…”

Kinematic self-similar locally rotationally symmetric models

“…They have σ = θ = ω ab = 0, whereas ua = [(1 − n)/t]δ t a . The case k = 1 is further discussed in [20][21][22] corresponding to a static spherically symmetric solution. No physically realistic solutions exist for k = −1 since the energy density is then negative necessarily.…”

General solution and classification of conformal motions in static spherical spacetimes

“…These are the self-similar Tolman-Bondi metrics [5,14,15], and are the most general selfsimilar metrics admitted by (1). This follows from (30), which implies Äi = 0 and thus Gi = 0 (by (38)), leading to = 0 (by (49)), so that the metric is in class I q [0] (including its conformally flat sub-class I I [0] ( q = ±1)), and we arrive at (59).…”

“…We use the general solution to show that the class I q [0] contains the self-similar metrics (for which ψ, α = 0 = ψ in (3)) given in [14] (see also [5,15]). Furthermore, we show that these self-similar metrics contain the only non-flat static spherically symmetric spacetime with a special CKV, for which [4,17] ψ ;αβ = 0 = ψ, α .…”