Rainich-type Conditions for Null Electrovacuum Spacetimes I

Transcription

1 Rainich-type Conditions fo Null Electovacuum Spacetimes I Synopsis In this woksheet I descibe local Rainich-type conditions on a spacetime geomety which ae necessay and sufficient fo the existence of a null electomagnetic field. When it exists, the electomagnetic field is easily constucted. The algoithm illustated hee is based upon esults of [3, ]. In this woksheet I shall illustate this analysis with a couple of pue adiation spacetimes involving a twist-fee null conguence. A companion woksheet (Null Electovac Geomety II, to appea) will teat the twisting case, which is somewhat moe complicated. Theoy Let M, g be a spacetime a -dimensional manifold M endowed with a Loentzsignatue metic g. The Rainich conditions ae puely geometic conditions on g such that thee exists an electomagnetic field F with g, F satisfying the Einstein-Maxwell equations, G ab = F a c Fcb 1 g ab F mn Fmn, V a F bc = 0, V a F ab = 0. The Rainich conditions involve the Ricci tenso R ij, the covaiant deivative, and the volume fom e ijhk of the metic, and ae given by i h 1 R Rj h d i j R hk Rhk = 0, i R i = 0, V j a i = 0, whee a i = j e ijhk R m V h R mk R ij R ij When these conditions ae satisfied thee is a staightfowad pocedue fo constucting the electomagnetic field, which is detemined by the metic up to a global duality otation: F ab / cos q F ab sin q - F ab, q R, whee - denotes the Hodge dual on -foms detemined by the metic. If a metic

2 satisfies the Rainich conditions we say that it detemines an electovacuum spacetime. A spacetime admits a non-null electomagnetic souce if and only if it satisfies the Rainich conditions. The Rainich conditions ae not defined fo a null electovacuum spacetime, i.e., solutions of the Einstein-Maxwell equations with a null electomagnetic field, fo which the enegy-momentum tenso and Ricci tenso ae also null: F ab F ab = - F ab - F ab = R ab R ac = 0. Thus the Rainich conditions do not povide geometic citeia fo null electovacua. In this woksheet (and its companion, Null Electovac Geomety II) I descibe Rainichtype conditions on a spacetime geomety which ae necessay and sufficient fo the existence of a null electomagnetic field. When it exists, the electomagnetic field is easily constucted. The algoithm illustated hee is based upon the esults of [3, ]. The stuctue of the analysis is as follows. A metic with a null Ricci tenso a pue adiation spacetime detemines a null vecto field k a via G ab = R ab = 1 k a k b. This vecto field detemines a family of -foms, f ab = k a s b, whee s b is any spacelike unit vecto othogonal to k a. The enegy-momentum tenso of f ab satisfies the Einstein equations in the sense that its enegy-momentum tenso equals the Ricci tenso (which is the same as the Einstein tenso in the null case). Theefoe, if thee is a solution F ab to the Einstein-Maxwell equations, at each point of M it must be elated to f ab by a duality otation. Thus thee will exist a function : M / R such that the electomagnetic pat of the solution to the Einstein-Maxwell equations takes the fom F ab = cos f ab sin - f ab. The Maxwell equations fo F ab impose a numbe of conditions on k a and. Fist of all, the vecto field k a must be tangent to a shea-fee, null, geodesic conguence, with k b V b k a = V b k b k a. This last equality follows automatically fom the contacted Bianchi identity. In tems of a null tetad whose fist leg is k a, using the Newman-Penose fomalism [], these conditions on k a take the fom s = 0 = k, 1 C = e C e. (1)

3 To analyze the conditions on, one must distinguish two cases: the conguence tangent to k a is (i) twisting o (ii) is twist-fee. (The twist is minus the imaginay pat of.) These two cases have significantly diffeent solution spaces fo and hence fo the electomagnetic field. This woksheet consides the twist-fee case only. In the twist-fee case the function detemining the duality otation must satisfy 1 i d C t b = 0, 1 i d t C b = 0, 1 D e C e = 0. () i Hee the lettes t, b, e, d, D denote the standad Newman-Penose quantities [], which ae detemined once k a has been incopoated into the fist leg of a null tetad. Thee is a single non-tivial integability condition fo these equations; it is given in tems of the Newman-Penose fomalism by Re d C b a t b 1 ee µ µ = 0, = 0 (twist-fee case) (3) Conditions (1) (3) ae invaiant unde the set of local Loentz tansfomations which fix k a, theefoe they ae invaiant conditions, defined independently of the choice of null tetad adapted to k a. Conditions (1) and (3) on the null conguence detemined by the Ricci tenso povide geometic conditions on the spacetime geomety which ae necessay and sufficient fo the existence of a (null) electomagnetic souce [3]. Thus we obtain Rainich-type conditions fo null electovac spacetimes. Any two solutions to () will diffe by a solution to d = d = D = 0. Thus an abitay function of the leaves of the foliation defined by the (non-twisting) vecto field k a will appea in the electomagnetic field. Example 1: A homogeneous null electovacuum The fist spacetime we shall analyze was not known until ecently to admit an electomagnetic souce [1]. In fact, it was this Rainich-type analysis which led to the discovey that it was an electovacuum. This metic can be initially chaacteized as a homogeneous, pue adiation solution to the Einstein equations. The metic can be witten as follows. > > > with(diffeentialgeomety): with(tenso): with(tools): DGsetup([u, v, x, y], M); fame name: M (.1) g := evaldg(dx &t dx + dy &t dy + *du &s dv - *exp(*s*x)*du &t du); g := e s x du 5 du C du 5 dv C dv 5 du C dx 5 dx C dy 5 dy (.)

4 The fist condition we check is that the Ricci tenso of the metic g ab is null, R ab R ac = 0. This implies that the Ricci tenso is the squae of a null vecto []. We find the null vecto k a which puts the null Ricci tenso into a standad fom, R ab = 1 k a k b. Ricci := RicciTenso(g); Ricci := s e s x du 5 du TensoInnePoduct(g, Ricci, Ricci, tensoindices = [1]); 0 du 5 du := DGzip([1,, 3, ], [du, dv, dx, dy], "plus"); := 1 du C dv C 3 dx C dy EQTenso := evaldg(ricci - 1/* &t ); EQTenso := 1 s e s x 1 du 5 du du 5 dv 1 3 du 5 dy dv 5 dy dx 5 dy dy 5 dy dv 5 du dx 5 du dy 5 du 3 dv 5 dv DGsolve(EQTenso,, {1,, 3, }); RootOf _Z e s x s du 3 dx 5 dv 3 dy 5 dv du 5 dx dv 5 dx dx 5 dx dy 5 dx down := simplify(allvalues(dgsolve(eqtenso,, {1,, 3, }))[1], symbolic)[1]; down := e s x s du up := RaiseLoweIndices(InveseMetic(g), down, [1]); up := e s x s D_v Veify the esult. (.3) (.) (.5) (.6) (.7) (.8) (.9) evaldg(ricci - 1/*down &t down); 0 du 5 du TensoInnePoduct(g, down, down); 0 (.10) (.11) It now follows that f ab = k a s b has the coect enegy-momentum tenso, whee s a is any unit vecto othogonal to k a. Constuct a spacelike unit vecto othogonal to k a. S := DGzip([s1, s, s3, s], [du, dv, dx, dy], "plus"); S := s1 du C s dv C s3 dx C s dy eq1 := TensoInnePoduct(g, down, S); (.1)

5 eq1 := e s x s s eq := TensoInnePoduct(g, S, S) - 1; eq := e s x s C s s1 C s3 C s 1 DGsolve([eq1, eq], S, [s1, s, s3, s]); s1 du C RootOf _Z C s 1 dx C s dy (.13) (.1) (.15) A simple solution is theefoe: S := Tools:-DGsimplify(eval(S, {s1=0, s=0, s3=1, s=0})); S := dx f := SymmetizeIndices(down &t S, [1,], "SkewSymmetic"); f := e s x s du 5 dx e s x s dx 5 du (.16) (.17) Veify the esult. EnegyMomentumTenso("Electomagnetic", g, f); s e s x D_v 5 D_v EinsteinTenso(g); s e s x D_v 5 D_v (.18) (.19) We now compute the popeties of the conguence associated to the vecto field k a. ContactIndices(up, CovaiantDeivative(up, Chistoffel(g) ), [[1,]]); 0 D_u ConguencePopeties(g, up); table "Expansion" = 0, "RotationNomSquaed" = 0, "Raychaudhui" = 0, "SheaNomSquaed" = 0 (.0) (.1) Thus k a is tangent to a conguence which is geodesic, shea-fee, non-expanding, and twist-fee and satisfies satisfies k b V b k a = V b k b k a. Condition (1) fo g to be a null electovac metic is theefoe satisfied. (This can also be checked using the Newman Penose spin coefficients, computed below.) We now conside condition (3). We begin by constucting a null tetad whose fist leg is k a and then compute its Newman-Penose spin coefficients and diectional deivatives. OT := OT := DGGamSchmidt([D_v, D_u, D_x, D_y], g, signatue=[[1, -1], 1, 1]); D_u C e s x C 1 D_v, D_u e s x 1 D_v, D_x, (.) D_y

6 NT0 := NT0 := NullTetad([OT[], OT[3], OT[], OT[1]]); D_v, D_u e s x D_v, D_x C I D_y, D_x I D_y NT := NullTetadTansfomation(NT0, "boost", *exp(s*x)*s); NT := e s x s D_v, es x D_u es x s s D_v, D_x C I D_y, D_x (.3) (.) I D_y NPS:=NPSpinCoefficients(NT); NPS := table "gamma" = 0, "pi" = 0, "tau" = 0, "beta" = 1 s, "epsilon" = 0, "sigma" (.5) = 0, "lambda" = 0, "mu" = 0, "ho" = 0, "nu" = 1 16 = 0 s, "alpha" = 1 s, "kappa" NPD := NPDiectionalDeivatives(NT): We compute d C b a t b 1 ee µ µ = 0, featuing in (3). alpha :=NPS["alpha"]: beta:=nps["beta"]: epsilon:=nps ["epsilon"]: mu:=nps["mu"]: tau := NPS["tau"]: X := simplify(tau - *beta) assuming s::eal: X1 := simplify(conjugate(x)) assuming s::eal: Y := simplify(beta - conjugate(alpha)) assuming s::eal: Z := simplify((epsilon - conjugate(epsilonn))*(mu - conjugate (mu))) assuming s::eal: NPD["delta"](X1) + Y*X1 - Z; 0 (.6) To illustate the invaiance of the integability condition with espect to the choice of tetad adapted to k, we conside a new tetad obtained fom NT by a combination of a spatial and null otation. NT1 := NT1 := convet(nulltetadtansfomation (NullTetadTansfomation(NT, "null otation", u, "L"), "spatial otation", x + y), exp) assuming u::eal; e s x s D_v, es x D_u C es x 16 s u 1 D_v C u D_x, s s e I x C y u e s x s D_v C ei x C y e I x C y u e s x s D_v C ei x C y D_x C I ei x C y D_x I ei x C y D_y, D_y (.7)

7 We check that NT1 is in fact a null tetad. TensoInnePoduct(g, NT1, NT1); (.8) We veify the integability conditions exactly as befoe. NPS1:=NPSpinCoefficients(NT1); NPS1 := table "gamma" = 1 s u C 1 I u, "pi" = 0, "tau" = 0, "beta" (.9) = 1 s ei x C y C 1 C 1 I ei x C y, "epsilon" = 0, "sigma" = 0, "lambda" = 1 u e I x C y s, "mu" = 1 s u, "ho" = 0, "nu" = 1 16 e I x I y s x 16 e s x s u C e s x s, "alpha" = 1 s ei x C y C 1 C 1 I ei x C y, "kappa" = 0 NPD1 := NPDiectionalDeivatives(NT1): alpha1 :=NPS1["alpha"]: beta1:=nps1["beta"]: epsilon1:=nps1 ["epsilon"]: mu1:=nps1["mu"]: tau1 := NPS1["tau"]: X := simplify(tau1 - *beta1) assuming s::eal, x::eal, y::eal, u::eal: X1 := simplify(conjugate(x)) assuming s::eal, x::eal, y::eal, u::eal: Y := simplify(beta1 - conjugate(alpha1)) assuming s::eal, y::eal, x::eal, u::eal: Z := simplify((epsilon1 - conjugate(epsilon1))*(mu1 - conjugate(mu1))) assuming s::eal, x::eal, y::eal, u::eal: NPD1["delta"](X1) + Y*X1 - Z; 0 (.30) We now constuct the electomagnetic field. The equations detemining the duality otation ae: 1 i d C t b = 0, 1 i d t C b = 0, 1 i The solution is obtained as follows. D e C e = 0. NPSvalues := {tau = NPS["tau"], beta = NPS["tau"], epsilon =

8 NPS["epsilon"]}: Phi := phi(u,v,x,y); eq1 := I F := f u, v, x, y (.31) eq1 := eval(1/i*npd["delta"](phi) + tau - *beta, NPSvalues); 1 v (.3) vx f u, v, x, y C 1 I v vy f u, v, x, y 1 s eq := eval(-1/i*npd["badelta"](phi) + conjugate(tau) - * conjugate(beta), NPSvalues) assuming s::eal; 1 v eq := I vx f u, v, x, y 1 I v vy f u, v, x, y 1 s (.33) eq3 := eval(1/i*npd["d"](phi) - epsilon + conjugate(epsilon), NPSvalues); eq3 := I e s x v s f u, v, x, y (.3) vv phisol:=pdsolve({eq1, eq, eq3}, Phi); phisol := f u, v, x, y = s y C _F1 u (.35) The electomagnetic field is obtained fom the duality otation given by. F0 := evaldg(cos(phi) * convet(f, DGfom) - sin(phi) * HodgeSta(g,convet(f, DGfom), detmetic=-1)); F0 := cos f u, v, x, y e s x s du Y dx sin f u, v, x, y e s x s du Y dy F:=eval(F0, phisol); F := cos s y C _F1 u e s x s du Y dx sin s y C _F1 u e s x s du Y dy (.36) (.37) We veify that g, F satisfy the Einstein-Maxwell equations. The Einstein equations: G := EinsteinTenso(g): T := EnegyMomentumTenso("Electomagnetic", g, F): evaldg(g - T); 0 (.38) The Maxwell equations: MatteFieldEquations("Electomagnetic", g, F); 0 D_u, 0 du Y dv Y dx (.39) Notice that the electomagnetic field involves an abitay function of u, which labels the null hypesufaces geneated by k a. This is a geneal featue of null electovac spacetimes when k a is twist-fee. In the twisting case the electomagnetic field involves only one fee paamete (specifying a global duality otation).

9 Example : A pue adiation Robinson-Tautman spacetime > > > estat; with(diffeentialgeomety): with(tenso): with(tools): Hee we conside a class of pue adiation solutions of Robinson-Tautman type (See Theoem 8.6 of [].). We detemine unde what conditions these solutions ae in fact solutions to the Einstein-Maxwell equations. Initialize the manifold, define the metic, and check that it is of pue adiation type. DGsetup([u,, x, y], N); fame name: N g := evaldg(/*(-f(x)*diff(f(x), x, x)* + diff(f(x), x)^* + m(u))*du &t du - *du &s d + ^//f(x)^*(dx &t dx + dy &t dy)); g := f# x f$ x C m u C dx 5 dx C Ricci := RicciTenso(g); dy 5 dy du 5 du du 5 d d 5 du f$ x 3 f& x C d du m u Ricci := du 5 du TensoInnePoduct(g, Ricci, Ricci, tensoindices = [1]); 0 du 5 du This is a pue adiation spacetime only when the u-u component of Ricci is positive, R uu > 0, which we shall assume hencefoth. Define R uu = 1 y. (3.1) (3.) (3.3) (3.) psi := *sqt(hook([d_u, D_u], Ricci)); y := 3 d dx d dx d du m u (3.5) We can easily ead off the covaiant fom of the pefeed null vecto field k. down := evaldg(psi*du); down := 3 f& x f$ x d du m u evaldg(ricci - 1/*down &t down); 0 du 5 du du (3.6) (3.7)

10 The contavaiant fom of k is as follows. up := RaiseLoweIndices(InveseMetic(g), down, [1]); up := 3 f& x f$ x d du m u D_ (3.8) We now build a null tetad NT adapted to k and compute its Newman-Penose spin coefficients and diectional deivatives. OT0 := OT0 := DGGamSchmidt([D_u, D_, D_x, D_y], g, signatue = [ [-1, 1], 1, 1]); D_u f$ x f# x m u D_, D_u (3.9) NT0 := f$ x f# x m u C D_y D_, D_x, NT0 := simplify(nulltetad([ot0[1], OT0[3], OT0[], -OT0[]] ), symbolic); D_, D_u f$ x f# x m u D_, D_x C I D_y, D_x I D_y NT := NullTetadTansfomation(NT0, "boost", psi); (3.10) NT := 3 f& x f$ x d du m u D_, (3.11) 3 f& x f$ x d du m u D_u

11 C f# x f$ x C m u 3 f& x f$ x d du m u D_, D_x C I D_y, D_x I D_y Spin := NPSpinCoefficients(NT): NPdiff := NPDiectionalDeivatives(NT): We check that the conguence is indeed geodesic and sheafee: k = 0 and s = 0. In addition, the spin coefficient is eal, indicating the conguence tangent to k is twist-fee. Spin["kappa"]; Spin["sigma"]; Spin["ho"]; 3 d dx d dx 0 0 d du m u (3.1) (3.13) (3.1) We now compute and solve the geometic condition on the spacetime (in the twist-fee case), denoted int_cond, which indicates whethe the spacetime admits an electomagnetic souce. X := simplify(spin["tau"] - *Spin["beta"]): Y := simplify(spin["epsilon"] - DGconjugate(Spin["epsilon"])) : Z := simplify(spin["mu"] - DGconjugate(Spin["mu"])): W := simplify(dgconjugate(spin["beta"]) - Spin["alpha"]): eq:=npdiff["badelta"](x) + W*X - 1/*Y*Z: int_cond := DGRe(eq); int_cond := 1 1 d dx 3 d dx (3.15)

12 C 3 d 3 dx 3 d dx d dx C d3 dx d 6 dx 6 5 d 5 dx 5 3 d dx 3 d dx d dx d dx d du m u 3 d dx d dx d du m u C d3 dx 3 3 d dx C 1 d 5 dx 5 C 1 d dx 3 d dx C d dx d du m u 3 d 1 d dx d dx C 3 d 5 dx 5 dx d dx 3 d dx 3 d dx d dx 3 d du m u d dx d dx d 3 dx 3 3 d dx d 5 dx 5 d dx 3 d 3 d dx d dx dx d dx d du m u d du m u

13 We solve fo functions f and m such that the integability condition holds. This command may take a few minutes to complete. PDETools:-Solve(int_cond); = e _C1 x _C, m u = m u, = e _C1 x _C, m u = _C3 u C _C, = 0, m u = m u (3.16) Evidently, fo geneic f this pue-adiation Robinson-Tautman spacetime does not admit an electomagnetic souce. Thee exists an electomagnetic souce fo the choice f = a e bx, with m u emaining abitay. Hee is the metic and Ricci tenso in this case. g := eval(g, f(x) = a*exp(b*x)); g := m u C du 5 du du 5 d d 5 du C dy 5 dy a b x e RicciTenso(g); d du m u dx 5 dx a b x e du 5 du (3.17) (3.18) We see that this is a pue adiation solution povided dm! 0. We now constuct the du coesponding electomagnetic field fom a duality otation detemined by the equations 1 i d C t b = 0, 1 i d t C b = 0, 1 D e C e = 0. i X := simplify(spin["tau"] - *Spin["beta"]): Y := simplify(dgconjugate(spin["tau"]) - *DGconjugate(Spin ["beta"])): Z := simplify(spin["epsilon"] - DGconjugate(Spin["epsilon"]) - 1/*(Spin["ho"] - DGconjugate(Spin["ho"]))): solutions := {f(x) = a*exp(b*x)}: NP1:=simplify(1/I*NPdiff["delta"](phi(u,,x,y)) + X): NP:=simplify(-1/I*NPdiff["badelta"](phi(u,,x,y)) + Y): NP3:=simplify(1/I*NPdiff["D"](phi(u,,x,y)) - Z): simplify(eval({np1, NP, NP3}, solutions)): phisol := pdsolve(simplify(eval({np1, NP, NP3}, solutions)), {phi(u,,x,y), m(u)}); phisol := m u = m u, f u,, x, y = b y C _F1 u (3.19)

14 The only non-tivial solution is theefoe f =b y C h u, with m u emaining abitay. The electomagnetic field is constucted as follows. phi := -b*y + h(u): k := eval(raiseloweindices(g, NT[1], [1]), solutions): S := eval(eval(evaldg(1/sqt() * (exp(i*phi) * NT[3] + exp(- I*phi) * NT[]))), solutions): s := eval(raiseloweindices(g, S, [1]), solutions): F := convet(evaldg(1/*k &w s), DGfom) assuming > 0; F := d du x m u cos b y h u eb a du Y dx (3.0) d du x m u sin b y h u eb a du Y dy We veify that g and F do define a solution to the Einstein-Maxwell equations. MatteFieldEquations("Electomagnetic", g, F); 0 D_u, 0 du Y d Y dx evaldg(einsteintenso(g) - EnegyMomentumTenso ("Electomagnetic", g, F)); 0 (3.1) (3.) This solution is (isometic to) the geneal fom of the Petov type D, Robinson-Tautman null electovacuum []. Refeences 1. Toe, C.G., "All homogeneous pue adiation spacetimes satisfy the Einstein-Maxwell equations", Class. Quant. Gav. 9 (01) Stephani, H. ame, D MacCallum, M. Hoenselaes, C., and Helt, E. Exact Solutions to Einstein's Field Equations. nd ed. (Cambidge Monogaphs on Mathematical Physics, 003). 3. Toe, C.G., to appea.. Ludwig, G., Commun. Math. Phys. 17, 98 (1970). Release Notes The illustated commands ae all available in Maple 17 and subsequent eleases. Autho C. G. Toe Depatment of Physics Utah State Univesity Mach 5, 013

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