---
abstract: 'Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation.'
author:
- |
    Alicia M. Sintes\
    [Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), ]{}\
    [Schlaatzweg 1, 14473 Potsdam, Germany]{}\
    \
    Alan A. Coley and Des J. McManus\
    [Department of Mathematics, Statistics and Computing Science.]{}\
    [Dalhousie University. Halifax, NS. Canada B3H 3J5]{}
title: ' On space-times admitting shear-free, irrotational, geodesic null congruences'
---

22.0 true cm 15.24 true cm 1true cm 1 true cm -0.8cm

Ł[[L]{}]{}

Introduction
============

In this article we wish to extend earlier work on shear-free, irrotational and geodesic (SIG) timelike and spacelike congruences [@des; @des2] to SIG [*null*]{} congruences. The fact that we are dealing with null congruences means that we have to approach the problem in a completely different way; we must make extensive use of the Newman-Penrose formalism.

Thus, we wish to study a congruence of curves whose tangent vector ${\bf k}$ is null and geodesic. Hence, we have a family of null geodesics $x^a=x^a(y^{\alpha},v)$, where $y^{\alpha}$ distinguishes the different geodesics, and $v$ is the affine parameter along a fixed geodesic. The null tangent vector is $k^a={\partial x^a \over \partial v} $, and satisfies ${k^a}_{;b}k^b=0$. The spin coefficients are defined in [@Kramer], where $\rho=-(\theta + i\omega)$ is called the complex divergence and $\sigma$ is the complex shear. The geodesic condition implies that the spin coefficient $\kappa$ vanishes and $\epsilon +\bar\epsilon =0$ follows from the choice of an affine parameter along the congruence. The congruence is said to be shear-free if $\sigma=0$. Also, from the relation $k_{[a;b}k_{c]}=(\bar \rho-\rho)\bar m_{[a}m_bk_{c]}$ [@Pirani], it follows that $w=0$ (i.e., zero twist) is a necessary and sufficient condition for ${\bf k}$ to be hypersurface orthogonal.

First we shall briefly review some of the results of relevance to this work. Goldberg and Sachs [@Gold] proved that if a gravitational field contains a shear-free, geodesic, null congruence ${\bf k}$, then $\kappa=\sigma=0$, and if R\_[ab]{}k\^ak\^b=R\_[ab]{}k\^am\^b=R\_[ab]{}m\^am\^b=0  ,\[s1\] then the field is algebraically special (i.e., $\Psi_0=\Psi_1=0$), and ${\bf k}$ is a degenerate eigendirection. In addition, a vacuum metric is algebraically special if and only if it contains a shear-free geodesic null congruence.

A space-time admits a geodesic, shear-free, twist-free ($\kappa=\sigma=\omega=0$) and diverging ($\rho=\bar\rho=\theta=-1/r$) null congruence ${\bf k}$, and satisfies (\[s1\]), if and only if the metric can be written in the form ds\^2=2r\^2 P\^[-2]{}(z,|z,u)dzd|z -2dudr -2H(z,|z,r,u)du\^2  . Robinson-Trautman models [@Robi] with this metric have been found for vacuum, Einstein-Maxwell and pure radiation fields with or without a cosmological constant [@Kramer].

For geodesic null vector fields we have that $(\theta +i\omega)_{,a}k^a+ (\theta +i\omega)^2+\sigma\bar\sigma=
-R_{ab}k^ak^b/2$. Therefore, in the non-diverging case (i.e., $\rho=-(\theta +i\omega)=0$), if the energy condition $T_{ab}k^ak^b\ge 0$ is satisfied, it follows that $\sigma=0=R_{ab}k^ak^b$. Thus, non-twisting (and therefore geodesic) and non-expanding null congruences must be shear-free. Hence, the space-time is algebraically special, and it corresponds to vacuum, Einstein-Maxwell, and pure radiation field. Perfect fluid solutions violate $R_{ab}k^ak^b=0$ unless $\mu+p=0$. This class of solutions has been studied by Kundt [@Kundt].

Another important case corresponds to the Kerr-Schild metric, which is given by $g_{ab}=\eta_{ab}-2\phi k_ak_b$. The null vector ${\bf k}$ of a Kerr-Schild metric is geodesic if and only if the energy-momentum tensor obeys the condition $T_{ab}k^ak^b=0$, and then ${\bf k}$ is a multiple principal null direction of the Weyl tensor and the space-time is algebraically special. The general properties of the Kerr-Schild metrics and their applications to vacuum, Einstein-Maxwell, and pure radiation space-times can be found in [@Kramer].

Finally, we note the algebraically special perfect fluid space-times corresponding to the generalized Robinson-Trautman solutions investigated by Wainwright [@Wain]. They are characterized by a multiple null eigenvector ${\bf k}$ of the Weyl tensor which is geodesic, shear-free, and twist-free but expanding (i.e., $\Psi_o=\Psi_1=0$, $\kappa=\sigma=\omega=0$, $\rho=\bar\rho\not= 0$), and the four-velocity obeys $u_{[a;b}u_{c]}=0$, $k_{[c}k_{a];b}u^b=0$. The line-element of the space-time can be written in the form ds\^2= -[12]{}\^2(r,u)P\^[-2]{}(z,|z, u)dzd|z +2du(dr-Udu)  , \[11\] with U=r(P)\_[,u]{}+ U\^0(z,|z, u)+ S(r,u)  , \_[,r]{}=0  ,0  . In this case no dust solutions nor solutions of Petrov types $III$ and $N$ are possible.

Analysis
========

Let us consider space-times admitting a shear-free, irrotational, geodesic null congruence in which the source of the gravitational field is a [*combination of a perfect fluid and null radiation*]{}, so that the energy-momentum tensor has the form T\_[ab]{}=(+p)u\_au\_b -p g\_[ab]{} +\^2k\_ak\_b  , \[13\] where $u^a$ is the four-velocity of the fluid, $\mu$ and $p$ are the density and the pressure of the fluid, respectively, and ${\bf k}$ is a null vector. The null radiation is geodesic, twist-free, and shear-free, and defines the null congruence. Wainwright [@Wain] proved that for a space-time in which there exists a SIG null congruence, coordinates can be chosen so that the metric takes on the simplified form (\[11\]) with $u=x^1$, $r=x^2$, $z=x^3+i x^4$, the tangent field of the null congruence is given by $k^a=\delta^a_2$, $k_a=\delta^1_a$, and we can introduce the null tetrad k\^a= \^a\_r  , & l\^a= \^a\_u+ U\^a\_r  , & m\^a=P\^[-1]{}( \^a\_3+ i\^a\_4 )  ,\
k\_a=\^u\_a  , & l\_a= -U \^u\_a + \^r\_a  ,& m\_a=P\^[-1]{}(\^3\_a + i \^4\_a)/2  . With the sign convention used here we have that $u^au_a=k^al_a=1=-m^a\bar m_a$. Note that the null radiation is everywhere tangent to the repeated null congruence of the space-time.

First, since $\Phi_{01}\equiv-{1\over 2}R_{ab}k^am^b=0$, we conclude that the four-velocity satisfies $u^am_a=0$, and hence it can be expressed in terms of the null tetrad by u\^a=[1 B]{}(B\^2 k\^a +l\^a) u\_a=[1 B]{}\[(B\^2-U)\^u\_a + \^r\_a\]  , \[36\] for some function $B$. The conditions $\Phi_{02}\equiv-{1\over 2}R_{ab}m^am^b=0$ and $\Phi_{12}\equiv-{1\over 2}R_{ab}m^al^b=0$ are satisfied identically. The non-zero components of the Ricci tensor are & &\_[00]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})k\^ak\^b= [12]{}(+p)([**ku**]{})\^2  ,\
& &\_[11]{}-[14]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})(k\^al\^b+m\^a|m\^b)= [14]{}(+p)([**ku**]{}) ([**lu**]{}) ,\
& &\_[22]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})l\^al\^b= [12]{}(+p)([**lu**]{})\^2 +[12]{}\^2  . In addition, since ${\bf k\cdot u}={1\over \sqrt{2} B}$ and ${\bf l\cdot u}={1\over \sqrt{2}} B$ implies ${\bf l\cdot u}=B^2({\bf k\cdot u})$, we obtain B\^2\_[00]{}&=& 2\_[11]{}   , \[34\]\
B\^4\_[00]{}&=&\_[22]{}- [12]{}\^2  . \[35\]

If we now assume that the fluid is non-rotating, then $ B^2=U+F(r,u)$, and the compatibility condition (\[34\]) can be written as (U+F)\_[00]{}=2\_[11]{} .\[39\] On differentiating this equation successively with respect to $z$ and $r$, we obtain the restriction (\^2)\_[,rrr]{}\[[U\^0]{}\_[,z]{}+r(P)\_[,uz]{}\]=0  . There are consequently two different cases to consider.

In the first case ${U^0}_{,z}+r(\ln P)_{,uz}=0$, which is equivalent to ${U^0}_{,z}=(\ln P)_{,uz}=0$, so that $P=P(z,\bar z)$ and $U^0=U^0(u)$. Obviously, the solutions admit a multiply transitive group of motions, $G_3$, acting on the 2-spaces $r=$const, $u=$const, of constant curvature, and belong to class $II$ of Stewart and Ellis [@Stew]. The metric (\[11\]) can then be rewritten as ds\^2=-\^2(r,u)[2dzd|z(1+[k2]{}z|z)\^2 ]{}+ 2du(dr-U(r,u)du)  . \[44\] The non-zero Ricci components are given by & & \_[00]{}=-[\_[,rr]{}]{}  ,\
& & \_[11]{}=[\_[,r]{}\_[,u]{}2\^2]{} + [(\_[,r]{})\^2U2\^2]{}- [U\_[,rr]{}4]{} + [k4 \^2]{}  ,\
& & \_[22]{}=[\_[,u]{}U\_[,r]{}]{}- [\_[,uu]{}]{} -2[\_[,ur]{}U ]{}-[\_[,r]{}U\_[,u]{}]{} -[\_[,rr]{}U\^2 ]{}  , and the Ricci scalar is given by =12= 4[\_[,r]{}U\_[,r]{}]{}+ 2[\_[,r]{}\_[,u]{}\^2]{} +2 [(\_[,r]{})\^2U\^2]{} + 4 [\_[,ur]{}]{}+ U\_[,rr]{} +4 [\_[,rr]{}U ]{} +[k\^2]{}  . Hence, the metric (\[44\]) can be interpreted as pure radiation plus a perfect fluid where $\mu$ and $p$ are given by =[R 4]{} +6\_[11]{} , p=-[R 4]{}+2\_[11]{} , \[45\] $u_a$ is determined by (\[36\]) with $B^2=2 \Phi_{11}/\Phi_{00}$, and $\phi^2$ is given by \^2=2( \_[22]{}-4[\_[11]{}\^2\_[00]{}]{})  . \[47\]

In the second case (i.e., ${\chi^2}_{,rrr}=0$) two possibilities arise: (i) & \^2=r,& = 1 \[52\]\
(ii) & \^2=(r\^2-k\^2), & k=const  .\[53\] In both subcases $\chi=\chi(r)$, and they can be written together as $\chi^2=ar^2+2br+c $, with $a$, $b$, $c$ taken to be appropriate constants. From equation (\[39\]) we obtain aU\^0 -b(P)\_[,u]{} + K =G(u)  , \[55\] and \[\^2 S\_[,r]{} -S(\^2)\_[,r]{}\]\_[,r]{} + [F \^2]{}= G(u)  , \[56\] where $K\equiv 4P^2 (\ln P)_{z\bar z}$, $\Sigma\equiv b^2-ac$, and $G(u)$ is an arbitrary function of $u$.

Subcase $(i)$: $a=c=0$, $b=\epsilon/2$. Integrating equation (\[56\]) we see that $S$ can be written in the form S=rh(u)+2G(u)rr -f(u) -[12]{}r\^r [drr]{}F(r,u)  , \[58\] where $h(u)$ and $f(u)$ are arbitrary functions of $u$.

Subcase $(ii)$: $a=\epsilon$, $b=0$, $c=-\epsilon k^2$, $\Sigma=k^2$. We obtain S=-G(u) + f(u) \^2 + h(u)\^2 -2k\^2\^2\^r [dr\^2(r)]{}F(r,u)  .\[59\]

Therefore, the metric (\[11\]) with $\chi(r)$ given by (\[52\]) or (\[53\]), $S(r,u)$ given by (\[58\]) or (\[59\]), and $P(z, \bar z, u)$ satisfying (\[55\]) can be interpreted as pure radiation plus a perfect fluid, in which the four-velocity is determined by (\[36\]) and $\phi^2$, $\mu$ and $p$ are determined by (\[45\]) and (\[47\]), respectively.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work was supported by the European Union, TMR Contract No. ERBFMBICT961479 (AMS), the Natural Sciences and Engineering Research Council of Canada (AAC) and the Canadian Institute for Theoretical Astrophysics (DJM).

[99]{} A.A. Coley and D.J. McManus, Class. Quantum Grav., [**11**]{}(1994)1261. D.J. McManus and A.A. Coley, Class. Quantum Grav., [**11**]{}(1994)2045.

D. Kramer, H. Stephani, M.A.H. MacCallum and E. Herlt, [*Exact Solutions of Einstein’s Field Equations*]{}, Deutscher Verlag der Wissenschaften, Berlin (1980). F.A.E. Pirani, in [*Lectures on General Relativity, 1964 Brandeis Summer Institute in Theoretical Physics*]{}, Vol. 1. Prentice-Hall, Englewood Cliffs, NJ (1965).

J.N. Goldberg and R.K. Sachs, Acta. Phys. Polon., Suppl. [**22**]{}(1962)13. I. Robinson and A. Trautman, Proc. Roy. Soc. Lond., [**A265**]{}(1962)463. W. Kundt, Z. Phys., [**163**]{}(1961)77. J. Wainwright, Int. J. Theor. Phys., [**10**]{}(1974)39. J.M. Stewart and G.F.R. Ellis, J. Math. Phys., [**9**]{}(1968)1072.
