Slowly rotating charged fluid balls in the presenc

Authors: Matthew Wright

Publish Date: 2016/09/10

Volume: 48, Issue: 10, Pages: 134-

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Abstract

We examine charged slowly rotating perfect fluids in the presence of a cosmological constant The asymptotic form of the vacuum solutions to the linearised Einstein–Maxwell field equations is found and the possibility of matching this vacuum to the slow rotating García metric is considered We show that contrary to the case of zero cosmological constant this García metric can be matched to an asymptotically de Sitter vacuum in the slow rotation limit We conclude the García metric may potentially be suitable for describing a charged isolated rotating body in a cosmological backgroundIt is remarkable that 100 years after the general theory of relativity was first formulated that there are still no known exact analytical solutions to Einstein’s equations describing an isolated rotating body There are of course highly accurate numerical solutions of rotating stars 1 2 but mathematically it is of interest to obtain analytic solutions In particular and of main interest in this paper there are no exact solutions describing an interior rotating charged perfect fluid which can be matched to an asymptotically flat vacuum exterior One reason for this is the sparsity of known analytic charged rotating fluid interiors One such interior is the García solution 3 also known as the WahlquistNewman metric 4 which is the charged generalisation of the rotating Wahlquist solution 5 The Wahlquist solution is in turn the rotating generalisation of the static Whittaker metric These metrics all have unphysical equations of state in the Wahlquist and Whittaker case the simple relation mu +3p=text const holds where mu is the energy density and p is the pressure Nonetheless due to the lack of explicit solutions the possibility of matching these solutions to external vacuum domains should be investigatedThe slow rotation formalism developed by Hartle  6 has been important in deriving results about the possibility of matching these interior solutions to exterior vacuum regions If the matching is not possible to a particular exterior in the slow rotation limit then it will not be possible for a more rapid rotation This argument has been used to show that the Wahlquist metric cannot be matched to an asymptotically flat vacuum 7 and one proves this by expanding the Wahlquist metric to second order in the fluid’s angular velocity If one drops the requirement that the vacuum is asymptotically flat the matching becomes possible 8 however this means the Wahlquist metric cannot serve as a model for an isolated body as a far away quadrupole mass distribution is required to keep the body in equilibriumThe matching conditions become more restrictive once charge is included In the absence of charge if one linearises the Einstein equations in the angular velocity of the metric one can show that the general slowly rotating exterior vacuum solution of a perfect fluid sphere coincides with the Kerr solution and one can match any slow rotating perfect fluid interior to this However in the presence of charge the general exterior vacuum solution to the linearised Einstein Maxwell equations does not in general coincide with the Kerr–Newman solution and in fact is in general not asymptotically flat 9 Thus if one requires asymptotic flatness as necessary for describing an isolated rotating body in a flat background then the matching conditions become overdetermined in general In the particular case of the García metric it was shown in 9 that to first order in angular velocity this metric cannot be matched to an asymptotically flat exterior metric It is found the fluid needs to be embedded in an external magnetic field parallel to the axes of rotation and thus the authors conclude that the García metric cannot be suitable for describing an isolated charged bodyIt is now consensus that the universe is accelerating in expansion 10 11 One explanation of this expansion is the presence of a small nonzero positive cosmological constant although there are a plethora of scalar field and modified gravity models which also can account for this accelerated expansion If this is the case the universe is not asymptotically flat rather asymptotically it behaves like de Sitter space This observation has resurrected interest in studying such spacetimes which are either asymptotically de Sitter or antide Sitter Of course the small observed value of the cosmological constant means that its effects on astrophysical objects are negligible However the presence of even a tiny cosmological constant completely changes the asymptotic structure of spacetime 13 14 Moreover a given perfect fluid solution to Einstein’s equation in the absence of a cosmological constant is also a solution to Einstein’s equation with cosmological constant achieved by making the substitution p rightarrow p Lambda /8pi G for the pressure p and mu rightarrow mu + Lambda /8pi G for the energy density This of course changes the equation of state of the fluid but for the Wahlquist metric the equation of state retains the same form mu +3p=text const Therefore the possibility of matching rotating fluids to asymptotically antide Sitter spaces should be investigated Recently this approach was considered in 12 where it was shown that to second order in the angular velocity the Wahlquist metric can be matched to an asymptotically antide Sitter external vacuum The purpose of this paper is to investigate this possibility for the García metric in the slow rotation limitThis paper is organised as follows We begin in Sect 2 by reviewing the García solution and its form in the slow rotation limit In Sect 3 we examine the general slow rotating electrolambdavacuum exterior metric which we show is always asymptotically antide Sitter with the electromagnetic field tensor likewise asymptotically decaying Finally in Sect 4 we describe the matching procedure and we show that up to first order in the angular velocity one can in fact match the García metric to an asymptotically antide Sitter vacuum Thus we conclude by noting that the García metric may be suitable for describing an isolated rotating charged body in antide Sitter spacesThe set of constants in the original metric have now been transformed to a new set of five constants beta kappa gamma barg and r 0 The number of constants has been reduced from the original eight independent parameters modulo diffeomorphisms by fixing a coordinate system and ensuring that the metric is completely regular at the centerTo confirm these results we can also integrate 316 numerically We set K=0 in the equation so we ignore the particular solution corresponding to the Kerr–Newman–de Sitter contribution and find simply the form of the omega 1 and omega 2 contribution We also numerically integrate the solution for omega r since we are only interested in how the solution decays for large r and this will allow us to ignore the constant contribution which arises when numerically integrating omega

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