Petrov types 0 and II perfect-fluid solutions in generalized Kerr-Schild form

Petrov types D and II perfect-fluid solutions are obtained starting from conformally flat perfect-fluid metrics and by using a generalized Kerr-Schild ansatz. Most of the Petrov type D metrics obtained have the property that the velocity of the fluid does not lie in the two-space defined by the principal null directions of the Weyl tensor. The properties of the perfect-fluid sources are studied. Finally, a detailed analysis of a new class of spherically symmetric static perfect-fluid metrics is given.


I. INTRODUCTION
In a previous paper,1 the first results concerning perfectfluid solutions of Einstein's equations in generalized Kerr-Schild form were given.The generalized Kerr-Schild metrics have the following form: where gaP is the metric of any space-time, H is a scalar field, and la is a null geodesic vector field for both metrics g and g.
As is well known, the Kerr-Schild ansatz has been a powerful tool in searching for solutions of Einstein's equations in either the vacuum case or the Einstein-Maxwell case. 2 -6 In contrast, excepting a metric due to Vaidya, 7 no perfect-fluid solutions in generalized Kerr-Schild form were known up to the appearance of Ref. 1.In our opinion, there were two reasons for that lack.First, it was usual to start from metricsg which were solutions of the vacuum Einstein equations.But, if TaP and TaP are the energy-momentum tensors for the metrics g and g, respectively, it was shown in Ref. 1 that the following interesting relation holds 8 : (2) where f is a scalar field.Therefore, if TaP vanishes then la is a null eigenvector of TaP' so that TaP cannot be the energymomentum tensor of a perfect fluid.The same happens when la is an eigenvector of Tap.Thus in order to obtain perfectfluid metrics g we must consider only the case in which la is not a null eigenvector of Tap.Of course, the most interesting case arises when TaP itself is an energy-momentum tensor for a perfect fluid.
The second reason emerges from the fact that it is necessary to allow great freedom in choosing the vector field I a.For example, in the classical Kerr-Schild metrics (Ref.2), the great variety of shear-free null geodesic vector fields in flat space-time was used.For any metric g, there will be, in general, a great number of null geodesic vector fields. 9But, in order to solve the Einstein equations, it is also very useful to know an explicit expression of the general solution for vector fields of this kind.In the classical Kerr-Schild metrics it was very useful that the Kerr theorem 10 provides the general solution for the shear-free geodesic null vector fields in flat space-time explicitly.This is not the case for an arbi-aJ Present address: Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary College, Mile End Road, London EI 4NS, England.
trary metric g.However, it is known that the geodesic (shear-free) null vector fields in a conformally flat spacetime are the geodesic (shear-free) null vector fields in flat space-time, and conversely.Thus if we start from a conformally flat space-time then we can use the Kerr theorem.Moreover, the conformally flat perfect-fluid metrics have an additional advantage: all metrics of this kind are known.They are either generalized interior Schwarzschild solutions or generalized Friedmann solutions.11 Therefore, we shall start from conformally flat perfectfluid metrics g.We devote Sec.II to writing down the Einstein equations for this case.The case when the vector field I a is shearing was studied in Ref. 1. On the other hand, the case when I a is shear-free was solved only in a very particular subcase.In this paper, we try to solve the shear-free case in general.There are two outstanding subcases which are studied in Sec.III.
All the solutions we obtain are Petrov types D and II.We also point out that most of the type-D metrics are new since the velocity of the fluid does not lie in the two-space spanned by the two multiple null eigenvectors of the Weyl tensor.Apart from the results obtained in Ref. 1, only two metrics (Wahlquist,12 Kramer 13 ) with this property were known previously.
In Sec.IV we give some explicit examples.In Ref. 1 two explicit examples of how the method works and their respective metrics were given.Although the method always works in the same way, in this paper we present some explicit solutions again.In particular, a class of spherically symmetric static perfect-fluid metrics is obtained.The properties of the perfect-fluid sources themselves are discussed in Sec.V. Finally, Sec.VI is devoted to the study of the new class of spherically symmetric static perfect-fluid space-times.

II. THE EINSTEIN EQUATIONS
Hereafter, we choose the metric g and the null vector field lof ( 1) with the following properties.First, g is a solution of Einstein's equations for a perfect-fluid energy-momentum tensor, Second, g is conformally flat, that is to say where CapAI'is the Weyl tensor for the metric g.Or, equivalently, g can be transformed into gap dx a d:xf3 = 2qi( -du dv + dzdz), (6) where l/J2 is a positive function of the coordinates (the conformal factor) and where suitable coordinates {u,v,z,z} have been chosen for the flat metric.Finally, we choose the null geodesic vector field la to be shear-free.The general solution for vector fields of this kind in the metric ( 6) is known and is given by 10. 11 (Kerr theorem) la dxa=du+ Ydz+ Ydz+ YYdv, (7) where Y is a complex function of the coordinates defined implicitly by and where F is an arbitrary analytic function of three complex variables.
the Einstein equations become finally I (X is the gravitational constant) where we have put The expressions ( 17)-( 19) are not equations, but they define q, p, and u a as functions of q, p, u a , and the unknown H. Thus we need only solve Eqs. ( 20) and ( 21), which are differential equations for H.However, in addition to Eqs. ( 20) and ( 21) we have the compatibility conditions for U and V, that iS,I,15 Also, from the definition (16) of V it follows that V -V= 2H(p -pl.
The Weyl tensor for the metric g is given byl6 Therefore, all the metrics g are algebraically special and the vector field la is a multiple null eigenvector of the Weyl tensor.
When V=O, from ( 23), ( 25),andtheNewman-Penrose equations, it follows that U = a and p = p.Then, from (26) it is evident that g is conformally flat.Thus we shall only consider the case V #0.
It has become clear to the authors that cases which have a function W nonlinear in H do not have solutions in general.This is because W is always of greater order in H than Vand U.And then, Eqs. ( 16) and ( 21) are not compatible in general.I ,I5 Perhaps, this fact has something to do with a theorem due to Xanthopoulos (see Refs. 5 and 6).Moreover, it is convenient to assume that p=p. ( This assumption simplifies the calculations substantially.Therefore, we shall treat the following two cases: (29) Of course, there are more cases in which Wbecomes linear in H, for example, when DV + 2Vp -8l/Jll = a and DV + 2Vp -4Hl/Joo -8l/Jll = O.These two cases would provide different resulting metrics, which is a proof of the variety of possibilities in our Kerr-Schild transformation.However, all the cases are formally equivalent to either the case A or the case B, and then the calculations are just a pure repetition in other cases.We shall study both cases in the following section.
Because of ( 13) and ( 27) we can use the Bianchi identities for the metric g as given in the Appendix of Ref. 1.These identities and the Newman-Penrose equations for the metric g (when they are conveniently restricted to each case) will be used repeatedly (but not explicitly) in Sec.III.Anyway, we shall omit the details.
(Ala) pD4>oo = -24>00(p2 + 4>00)' From the Newman-Penrose equations and the Bianchi identities it is easily shown that this condition is equivalent to where C is an arbitrary positive real constant.In order to distinguish this case from the case (A2) we must assume that (39) (Alb) r = O.In this case ( 37) is automatically satisfied.Furthermore, this case is different from the case (A2) since now we havep2 -4>00#0.
The subcase (Ala): From the above considerations, Eqs. ( 30) and ( 28) are already compatible and Eq. ( 32) is satisfied.Also, Eq. ( 31) now becomes The compatibilities of this equation with Vand U give us, respectively, 939
( 48) Equations ( 47) and ( 48) are compatible and the integrability of ( 47) and (28) gives us the following condition: (49) It is very difficult to know if the expression ( 49) is possible in general.In fact, Eq. ( 49) should be interpreted as an equation from which V is obtained as a function of H. Then we shouldputthis VinEqs.( 28), (47), and (48) and we should obtain an (or more!) expression for Hwhich is not, in general, a solution of Eqs. ( 16) and ( 46).This procedure is useless in general.However, we can assume /LP + 4>11 + A = 0 (50) so that Eq. ( 49) becomes 2"" + 4>00(4)11 -3A) This is a condition on only g and therefore we only have to check it.In fact, it may be shown that ( 51) is possible.
The subcase (A2): This case is defined by the assumptions 7=0, ¢OO=p2 (52) so that Eqs. ( 28) and ( 30) are compatible.Furthermore, Eq. ( 31) may be written The compatibility of this equation with V gives us In order to make this expression compatible with (28) we must have

PP+¢l1+A=O
(55) and then, Eqs. ( 53) and ( 54) become, respectively, The compatibility of (56) with U is Keeping this equation in mind, Eqs. ( 57) and ( 30) are compatible.Now, the Weyl tensor is given by (59) Since V does not vanish, we only can obtain solutions of Petrov types D and II.For Petrov type-D solutions we must have Otherwise, the solutions are Petrov type II.By using the condition (33), and after a little computation, it is easily shown that Eqs. ( 60), (58), and (32) are compatible.Now, we are going to solve case B. Therefore, we assume conditions ( 29) and ( 27) so that Eqs. ( 20), ( 23), (24), and (25) become, respectively, Eqs. ( 30), ( 32), (33), and (34).Furthermore, Eq. ( 21) now may be written The compatibility of ( 30) with ( 29), making use of (32), leads us to (36) and (37).Also, we must have Consequently, as in case A, we could consider two subcases again but it turns out that the only interesting case arises when 7=0 and then, from (62) we have u=o.
(63) (64) Thus, Eqs. ( 29) and ( 30) are compatible and also Eq. ( 32) is satisfied.On the other hand, Eq. ( 61) becomes 65) which is compatible with (64).The integrability of (65) and V leads us to (66) This equation is compatible with (30).Moreover, it may be shown that under the conditions Eq. ( 66) is compatible with (29).In fact, condition (67) is not very much restrictive because it is satisfied for the metricsg which have a constant energy density.In other words, all the generalized interior Schwarzschild metrics satisfy the above-mentioned condition (67).II The results obtained in this section can be summarized as follows.
Let us choose the conformally flat perfect-fluid metric g and the shear-free geodesic null vector field f a such that they verify the possible conditions given in the first row of Table I.Then, let us define U and V by ( 16) and let us solve the integrable system of equations for U and V which appear in the second row of the table.Once this has been done, the solutions H of the compatible system of equations given by ( 16) and the third row of the table provide us generalized Kerr-Schild metrics g.These metrics are solutions of the Einstein equations for a perfect-fluid energy-momentum tensor (15), where the energy density ij, the pressure p, and the velocity ija are given, for each case, in the fourth row of the table.The Weyl tensor of the resulting metrics as well as their Petrov types are also shown in Table I.
As we can see in the table, the Petrov type-D metrics of cases Ala and A2 satisfy ¢3#-0 and ¢4#-0.Therefore, for these metrics, k a is not a multiple null eigenvector of the Weyl tensor, and then the form of ija tells us that the velocity of the fluid does not lie in the preferred two-space defined by the multiple null eigenvectors of the Weyl tensor.Thus these solutions are new.On the other hand, the metrics of cases Alb and B have u a lying in that preferred two-space so that they may be already known.II The particular case V = CHp (where C is a constant #--2) belongs to the more general case Ala and it had been solved previously by one of US. 17 Likewise, the case V = -2Hp solved in Ref. 1 (when u = 0) belongs to the general case A2.

IV. EXPLICIT EXAMPLES
In this section we give some examples of how the equations may be solved for each particular case.
Now, we know that the system of equations ( 56) and ( 16) for H is compatible.The integration of this system is standard and we obtain for H where E is an arbitrary constant.These metrics belong to the class of generalized Robinson-Trautman solutions.11.19 Unless we have M = const, the resulting metrics g are Petrov type II.
Another particular solution of Eqs. ( 69) is given by and then, the solution ofEqs.( 56) and ( 16) for In this case, Eq. ( 60) is also satisfied and therefore the resulting metrics g are Petrov type D. Since 1p3 and 1p 4 do not vanish they are new.(2) In this example we take the interior Schwarzschild metric in canonical coordinates, that is to sayll u a dx a = -Ardt, (73) and we try solve the equations for the case B. It may be shown that the only shear-free geodesic null vector field which satisfies T = 0, ( 27) and ( 68) is given by where Br=aN-b, w 2 =A2_B2= (a 2 _b 2 )IR2.( 76) A null tetrad associated with fa and such that mau a = 0 is given by20 21/2ma dx a = rei~( -dO + i sin e d</J), so that the only non-null spin coefficients are Solving the system of equations defined by ( 29), ( 30), (66), ( 16), and (65) for Vand H we obtain wheref(r) is a solution of the following differential equation: The resulting metric g is a Petrov type-D static spherically symmetric perfect-fluid solution.In the following sections, we are going to discuss the properties of the solutions we have obtained.

V. PROPERTIES OF THE SOLUTIONS
It is evident that the properties of the generalized Kerr-Schild metrics g depend, in general, on the properties of the initial metrics g themselves.However, some considerations may be made without loss of generality and then the specific properties of the explicit solutions can be deduced.
Thus, for example, in Ref. 1 it was shown that Petrov type-N metrics cannot be obtained by means of the Kerr-Schild transformation as defined by us.Also, the Petrov type of the resulting metrics has been always given in Sec.III.It is convenient to remark that this has been possible because we knew the Petrov type of the initial metrics (they are conformally flat) .
With regard to the symmetries of the solutions, one of us 15 has shown the following result: "s is a Killing vector field of the Kerr-Schild metric g if and only if where/is a function ofthe coordinates and we use standard notation for Lie derivatives."This result provides us a method to find all the Killing vector fields of the explicitly known Kerr-Schild metrics.For the first solution of the previous section, the former conditions (82) lead us to where A I, A 2 , andA 3 are arbitrary constants (A3 real).From these expressions it is evident that, in general, the only Killing vector is given by au au But also, there are some particular cases depending on the form ofthe function M(z).These are the following: vector.This is the only new Killing vector which is not a Killing vector of the initial metric g.
Similarly, the symmetries of the second solution of the previous section may be obtained.The result is that there are the following two Killing vectors: .(a a) a a 1 (a a) I az-az ' -a,;au +"2 az+ az .
Both of them were Killing vectors for the initial Robertson-Walker metric g.
In relation to the properties of the density and pressure of the solutions, first of all we must obtain the explicit expressions for these quantities, which are given by +=4Ct), (86) for the first metric obtained in Sec.IV, and by for the second metric, where From ( 85)-( 88) it is clear that the solutions do not have singularities in general.Furthermore, both metrics satisfy x(q -P) = (4113) exp (+ 2Ct). (89) Finally, we are going to study the properties of the velocity of the fluid of the Kerr-Schild metrics.By using the formulas of Ref. 21, making a change of null tetrad and after some standard and straightforward calculations we obtain for the shear, vorticity, and expansion of the fluid the following expressions: O"ap: A= _(_I_)1/2{(l+E)V+~DH (94) where we have used the notation of Ref. 21 and we have put Moreover, E = 1 for case A and E = -1 for case B. These expressions are valid in general.From (93) we see that the solutions obtained in this paper do not have vorticity.This is a direct consequence of assumption ( 27).On the other hand, they have, in general, shear, expansion, and acceleration.The explicit expressions of these quantities for the explicit metrics of Sec.IV may be easily obtained from ( 90)-( 95).However, the static and spherically symmetric solution is shear-free and expansion-free (of course!).In the next section, we are going to study this particular solution.

VI. A CLASS OF STATIC, SPHERICALLY SYMMETRIC PERFECT-FLUID METRICS
In Sec.IV, we obtained the metric the metric (96) becomes In this form, the metric is manifestly static and spherically symmetric.By using the formulas of previous sections we can get the velocity of the fluid u=Ar(l-j) l12 dT (99) and the density and pressure where Xq and XP are the density and pressure of the Schwarzschild interior solution and are given in (72).By the way, we remark that the metric (98) is a generalization of the Schwarschild interior metric.The Schwarzschild metric is the particular casef(r) =0 [as is evident from ( 98)-( 101) or directly from ( 80)].
If we want to study the properties of the solution (98) we have to solve the differential equation ( 81).This equation is linear and of second order and, in general, it has four regular singular points. 22• 23 This type of equation is called Heun's equation. 22• 23 However, there are two cases in which the equation has only three regular singular points (so that it can be reduced to the hypergeometric equation 23 ).These cases are defined by b = 0 and b = a.When b = a, it may be shown that there is not any regular solution.Then, we do not consider this case here.
Let us begin with the easier case b = O.In this case, (81) can be reduced to a hypergeometric equation and, in fact, the general solution of (81) may be expressed by means of elementary functions as follows:

(lOS)
It follows from ( 104) and ( 105) that this solution satisfies the equation of state By physical considerations, we must demand C < 0 so that i} + 3p is positive and, also, this assures the correctness of the signature for (98) because f( r) < O.This special metric is just the static limit of the Wahlquist solution 12 and it was also given by Whittaker. 24 Now, let us study the general case b =1=0.By simplicity, it is convenient to distinguish several possibilities depending on the different values of b la.Thus, for example, when b I a<! the regular solution of ( 81 where C is an arbitrary constant again.Analogously, the solution when !<b la < 1 may be given by means of the Heun's function F. We shall restrict ourselves to the case b la<~ because all the possibilities are quite similar.From (107) it may be shown Therefore, we must choose C as follows: -1 < C < (3b -a)/3(a -b) so that q(O) and tHO) are positive.
Bearing this condition in mind, and taking into account the following relation: xp(r=R) =xp(r=R) = -R -2<0, we conclude that the pressure is a decreasing function of r and that there exists a value r = ro <R such that p(r o ) = O.
Unfortunately, it is very difficult to find out the equation of state for these metrics.However, from ( 100) and ( 101) it is evident that the following relation holds in general: X(q+p) =X(q+p)(l-j) = [2a/ArR2](I-j).
This expression proves that there are no solutions in which the density and the pressure vanish at the same value of r without singularities in the metric [see ( 98) ] and, therefore, the equation of state cannot be that of a polytropic fluid. •ull•u.

TABLE I .
Integrability conditions, compatible systems of equations, and properties for Kerr-Schild metrics.