View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Spherically symmetric static solution for colliding null dust Laszlo A. Gergely KFKI Research Institute for Particle and Nuclear Physics, Budapest 114, P.O.Box 49, H-1525 Hungary The Einstein equations are completely integrated in the presence of two (incoming and outgoing) streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is also written in double null and in radiation co ordinates and it is reinterpreted as an anisotropic uid. Interior matching with a static uid and exterior matching with the Vaidya solution along null hyp ersurfaces is discussed. The connection with two-dimensional dilaton gravity is established. I. INTRODUCTION Null dust represents the high frequency (geometrical optics) approximation to the unidirectional radial ow of unp olarized radiation. This is a reasonable approximation whenever the wavelength of the radiation is negligible compared to the curvature radius of the background. Various exact solutions of the Einstein eld equations were found in the presence of pure null dust (for reviews see [1], and more recently [2] and [3]). In some scenarios even gravitation b ehaves as null dust. Price [4,5] has shown that a collapsing spheroid radiates away all of its initial characteristics excepting its mass, angular momentum and charge. (This result is known as the no hair theorem.) The escaping radiation then interacts with the curvature of the background b eing partially backscattered. Both the escaping and the backscattered radiation can b e mo deled bynull dust [6] as the curvature radius of the background is larger than the wavelength of the radiation. Letelier has shown that the matter source comp osed of twonull dust clouds can b e interpreted as an anisotropic uid [7], giving also the general solution for plane-symmetric anisotropic uid with twonull dust comp onents. Later Letelier and Wang [8] have discussed the collision of cylindrical null dust clouds. The collision of spherical null dust streams was discussed byPoisson and Israel [6]. Their analysis yielded the phenomenon of mass in ation. However, no exact solution in the presence of two colliding null dust streams with spherical symmetry was known until now. Recently Date [9] tackled this problem under the assumption of staticity,integrating part of the Einstein equations. It is the purp ose of the present pap er to present the exact solution for the case of two colliding spherically symmetric null dust streams in equilibrium, in a completely integrated form. The plan of the pap er is as follows. In Sec. II we derive the eld equations and weintegrate them. The emerging exact solution is written explicitly in suitable co ordinates adapted to spherical symmetry and staticity. The metric in radiation co ordinates and in double null co ordinates is also given. We analyze the metric b oth analytically and by numeric plots. In Sec. III we presentvarious p ossible interpretations of the solution, including the anisotropic uid picture, and dilatonic gravity. Finally Sec. IV contains the analysis of the interior and exterior matching conditions on junctions along timelike and null hyp ersurfaces, resp ectively. We employ the matching pro cedure of Barrabes and Israel [10]. The interior junction xes the parameters of the solution in terms of physical characteristics of the central star: the mass and energy density on the junction. The exterior matching with incoming and outgoing Vaidya solutions [11] leads to the conclusion that no distributional matter is present at the junction. This feature is in contrast with the exterior matching with the Schwarzschild solution [9]. I I. SOLUTION OF THE EINSTEIN EQUATIONS 2 2 The general form of a spherically symmetric, static metric in a spacetime with top ology R S is 2 2 1 2 2 2 ds = h(r )dt + f (r ) dr + r d : (2.1) 2 2 2 Here t is time, r is the curvature co ordinate (e.g. the radius of the sphere t=const with area 4r ), d = d + 2 2 sin ( )d is the square of the solid angle element. The functions f (r ) and h(r ) are p ositivevalued. Weintro duce the lo cal mass function m(r ), related to the gravitational energy within the sphere of radius r [12]: 2m(r) f (r )=1 : (2.2) r The energy-momentum tensor in the region of the cross- owing null dust is a sup erp osition of the energy-momentum tensors of the incoming and outgoing comp onents: 1 (r ) ab a b a b T = (v v + u u ): (2.3) 2 8r All energy conditions are satis ed for (r ) 0. The same linear mass density function was chosen for b oth comp onents as staticity requires no net ow in either of the null directions. The vector elds p p 1 1 1 1 a a p p p p v = ; f; 0; 0 ; u = ; f; 0; 0 (2.4) 2 2 h h are the tangents to the (future-oriented) outgoing and incoming null congruences, partially normalized such that a v u = 1. Similarly as for the one-comp onentnull dust, here b oth null congruences are geo desics [9]. a After eliminating second time derivatives from the nontrivial Einstein equations wehave the system: df r = f +1 dr dh = h( f +1) (2.5) rf dr d rf = ( + f 1) : dr The solutions with =const all reduce to = 0, meaning vacuum. They are either the Schwarzschild or the at solution, in accordance with the Birkho theorem. For 6=const one of the equations is a relation b etween the mass density (r ) and the metric function h (r ): A h = ; (2.6) where A > 0 is a constant. This relation can be deduced also from the energy-momentum conservation [9]. The remaining equations do not contain the constant A: d ln f f = f +1 (2.7) d ln r d ln f = + f 1 : (2.8) d ln r Inserting (2.2) in (2.7) the mass is found to increase with the radius: dm=dr = =2 > 0. Nowwe solve the system (2.7),(2.8). Following Date [9] , we eliminate f from (2.7) by its expression taken from (2.8): 1+ f = : (2.9) dln 1 d ln r The resulting second order ordinary di erential equation in 1= can b e integrated, nding: 1 1 d 1 = D 2ln : (2.10) d ln r r 1+ Here D is an integration constant. Inserting this expression in (2.9), an algebraic relation b etween the metric comp o- nents emerges: 2 (1 + ) Cr =ln ; (2.11) 2f where C = exp((1 + D )=2) > 0. Then we complete the integration of the system (2.7),(2.8). The key remark is that none of the equations in this system contains explicitly the indep endent variable ln r . Thus one can pass to the new indep endent variable , in terms of which an ordinary rst order equation can b e written: f ( + f 1) df = : (2.12) d ( f +1) 2 Intro ducing the new p ositivevariables 1+ 1=2 P =(2f ) ; L = (2.13) 1=2 (2f ) the equation (2.12) takes the form of a rst order linear (inhomogeneous) ordinary di erential equation: dP =2(1PL) : (2.14) dL The solution is found byintegrating rst the homogeneous equation, then varying the constant. It is Z L 2 2 L x P =2e (L) ; (L)=B+ e dx > 0; (2.15) B B where B is a third integration constant. The function (L) in (2.15) can b e expressed either in terms of the error B function or in terms of the Dawson function: p 2 i L (L) B = erf (iL)=e Dawson(L): (2.16) B 2 For prop erties of these transcendental functions see [13]. From Eqs. (2.11),(2.13) and (2.15) b oth the curvature co ordinate r and the metric functions and f are found as functions of the radial variable L: 2 L +2L (L) (2.17) Cr(L)=e B 2 L (L)=1+ 2Le (L) (2.18) B 2 2 (L) B f (L)= : (2.19) 2 2 L L e e +2L (L) B Then the mass function m = m (L) is obtained from (2.2): 2 2 L L 2 2Cm(L)=e +2L (L)2e (L): (2.20) B B It is easy to check that b oth r and m are monotonously increasing functions of L: d (Cr) =2 (L) > 0 (2.21) B dL h i 2 2 d (Cm) L L = e (L) e +2L (L) 0: (2.22) B B dL In the last relation the equality holds for r =0. Nowwehaveeverything together to write the metric in terms of the new radial co ordinate L: 2 h i h i L 2 2 2 2 2 2 2 Ae dt d dL 2 L L L ds = +2e + e +2L (L) : (2.23) e +2L (L) B B 2 2 2 L C C e +2L (L) B There are three parameters in the solution, two of them restricted to be p ositive: A and C . Without loss of generalitywe can cho ose A =1 by rescaling the time co ordinate. The parameter C provides some distance scale. We comment on the third parameter in what follows. Both Cr 0 and the energy conditions 0 imply Z 2 L L 2 e x e dx (2.24) B (L) ; (L)= 2L valid for all admissible values of L. The equality B = (L ) holds for L = L corresp onding to r = 0.