Purely magnetic vacuum solutions Norbert Van den Bergh Faculty of Applied Sciences TW16, Gent University, Galglaan, 9000 Gent, Belgium 1. Introduction Using a +1 formalism based on a timelike congruence of observers, the covariant derivative of the velocity field u being decomposed in the usual way, b u a = u a u b + D b u a (1) D b u a = σ ab + 1 θh ab + ɛ abc ω c () h ab = g ab + u a u b () ɛ abc = ɛ abcd u d, (4) reveals a close analogy between electromagnetism and gravity [Maartens et al. 1997,1998]: the source free Maxwell equations become D.E = iω.e (5) E + id E = θe + σ.e (ω + i u) E (6) E a = F ab u b H a = F ab u b (8) E = E + ih (9) (7) while the vacuum Einstein equations are given by (Bianchi identities) D.E = iω.e iσ E (10) E + id E = θe (ω + i u) E + tr(σ E) (11)
Purely magnetic vacuum solutions and E ab = C acbd u c u d (1) H ab = C acbd u c u d (1) E = E + ih (14) (Ricci identities) H = D σ + Dω + u ω (15) E = σ θσ + D u tr(σ σ) ω ω + u u (16) θ + 1 θ ω + σ u D. u Λ = 0 (17) ω + θω + 1 D u ω.σ = 0 (18) Dθ D.σ + D ω = ω u (19) D.ω = u.ω (0) Based on this analogy the (spatial and trace-free) tensors E and H are called resp. the electric and magnetic part of the Weyl curvature. While the non-vanishing of both is essential for the existence of gravitational waves, only E has a Newtonian analogue. It is the tidal tensor, making its appearance in the equation of geodesic deviation: ξ + R(ξ, u)u = 0 (1) ξ a + C acbd u c ξ b u d = 0 (in vacuum) () ξ + E.ξ = 0 () Whereas large classes of purely electric vacuum or Λ type solutions ( gravito-electric monopoles ) exist (for example all the D vacua a real Ψ in the canonical frame, the static Weyl solutions, the Gödel universe), not a single gravito-magnetic monopole is known. Some partial results concerning the latter are. Trümper (1965): no solutions exist having ω = σ = 0. McIntosh, Arianrhod, Wade and Hoenselaers (1994): the Petrov type is necessarily I; more particularly a canonical frame exists in which Ψ 1 = Ψ = 0, while Ψ 0 = Ψ 4 and Ψ are imaginary.. Haddow (1995): no solutions exist σ = 0
Purely magnetic vacuum solutions. VdB (00): no solutions exist ω = 0; this was generalised to space-times vanishing Cotton-tensor by Ferrando and Saez (00): using eq. (11) one obtains σ H = 0, acting on which the D operator results in (D σ).h σ.(d H) = 0 (4) and hence, using the Ricci identities H = 0. Maartens (1998) tries to construct stationary purely magnetic solutions ( u aligned the timelike Killing vector), which however is impossible in view of the Haddow result. He notices that, although only the Bianchi identities are invariant under the duality rotation E e iα E, the full system of linearized equations allows non-rotating gravito-electric monopoles (e.g. linearized Schwarzschild) to be converted in a non-accelerating gravito-magnetic monopole (linearized Taub-NUT). The question whether the full non-linear equations allow the existence of such a solution was left open. Existence of such vacuum models would provide possibly interesting examples of spacetimes in which the genericity condition was violated.. Non-accelerating gravito-magnetic monopoles: do they exist? The relevant dynamical equations for a purely gravito-magnetic vacuum space-time in which the timelike congruence u is geodesic are given below, using the notations and conventions of the orthonormal tetrad formalism [10]. The coefficients n aa being redefined as follows: n 11 = (n + n )/, n = (n + n 1 )/, n = (n 1 + n )/ (5) and the tetrad being specified as an eigenframe of H (each equation represents a triplet of equations, obtained by cyclic permutation of the spatial indices). The vanishing of the gravito-electric part of the Weyl-tensor can be expressed by the 9 equations 0 θ 1 θ 1 σ 1 σ 1 + ω + ω + σ 1 Ω σ 1 Ω + 1 Λ = 0 (6) 0 (σ 1 + ω ) (θ 1 + θ )(σ 1 + ω ) (σ 1 ω )(σ ω 1 ) +Ω 1 (σ 1 ω ) Ω (σ ω 1 ) + Ω (θ θ 1 ) = 0 (7) 0 (σ 1 ω ) (θ 1 + θ )(σ 1 ω ) (σ + ω 1 )(σ 1 + ω ) +Ω 1 (σ 1 + ω ) Ω (σ + ω 1 ) + Ω (θ θ 1 ) = 0 (8) while the vanishing of the off-diagonal components of H leads to 0 (n 1 + a ) 1 Ω θ 1 (n 1 + a ) (n a 1 )(σ 1 + ω ) + 1 n (σ 1 ω ) +Ω 1 (n 1 a ) Ω (n a 1 ) + 1 Ω (n 1 n ) = 0 (9) 0 (n 1 a ) Ω 1 θ (n 1 a ) (n 1 + a )(σ ω 1 ) + 1 n 1(σ 1 + ω ) +Ω 1 (n 1 + a ) Ω (n + a 1 ) + 1 Ω (n 1 n ) = 0 (0)
Purely magnetic vacuum solutions 4 Together the Ricci-identities, we obtain from these equations the evolution for θ a, σ ab, ω a, a a, n ab and n a, while the curl of the shear can be expressed in terms of the diagonal components of H: H 11 = σ 1 σ 1 + 1 ω 1 1 (θ 1(n + n ) θ n θ n ) + n σ n 1 (σ 1 + ω ) a (σ 1 + ω ) n 1 (σ 1 ω ) + a (σ 1 ω ) (1) The remaining Ricci identities contain spatial gradients of the kinematical scalars only and can be used to simplify the integrability conditions which result by considering the following commutators: [ 0, 1 ]θ [ 0, ](σ 1 ω ) and [ 0, 1 ]θ [ 0, ](σ 1 + ω ) () One obtains then three pairs of equations, (σ ω 1 + Ω 1 )H 11 (σ ω 1 Ω 1 )H = 0 () (σ + ω 1 + Ω 1 )H 11 + (σ + ω 1 + Ω 1 )H = 0 (4) in which one recognizes the previously obtained relation [, 5, 9] between σ, ω and H, σ H = ω H, (5) together a relation between σ, ω and the rotation rate Ω of the H-eigenframe respect to a Fermi-propagated triad: ω 1 + ω 1 Ω 1 = σ (6) (+ cyclic permutations). This leads to the following expressions for σ and Ω: σ 1 = ω H 11 + H H 11 H (7) Ω 1 = ω 1 (H 11 H )(H 11 H ) (H H ) (8) (note that H is not allowed to have equal eigenvalues [1]). Substituting the latter in the propagation of the shear and making use of the time evolution of the curvature (given by the Bianchi identities), there result algebraic relations between ω and H: ω (H H )(H 11 H )[θ 1 (H H ) + θ (H 11 H ) + θ (H 11 H )] +ω 1 ω (H 11 H )(5H 11 + 8H 11 H + 5H ) = 0 (9) Elimination of θ a from this equation and its cyclic permutations eventually results in ( H11 + H 11 H + 5 H ) (H 11 H ) 4 ω 1 ω + ( H + H H + 5 H ) (H H 11 ) 4 ω ω 1 + ( H + H 11 H + 5 H 11 ) (H H ) 4 ω ω = 0 (40) showing that a purely magnetic vacuum is inconsistent the assumption of the congruence being geodesic. (The analysis becomes slightly more complicated when some of the components of the vorticity vanish: higher order time derivatives are then needed to obtain an inconsistency between the signs of the involved vorticity and curvature terms.)
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