MaxPlanckInstitut fur Mathematik in den Naturwissenschaften Leipzig Electromagnetic resolution of curvature and gravitational instantons by N K Dadhich K B Marathe and G Martucci PreprintNr IL NUOVO CIMENTO VolN Decemb er c So cieta Italiana di Fisica N K DADHICH ETC Electromagnetic Resolution of Curvature and Gravitational In stantons N K Dadhich K B Marathe andG Martucci InterUniversity Center for Astronomy Astrophysics Pune India City University of New York Brooklyn Col lege Department of Mathematics Brooklyn NY USA Dipartimento di Fisica Universita di Firenze Largo E Fermi Firenze Italy ricevuto approvato Summary We study the electromagnetic resolution of the Riemann curvature on a spacetime manifold M with metric g into its electric and magnetic parts relative to a unit timelikevector with resp ect to g There exists a duality transformation between the active and passive electric parts whichleaves invariant a sub class of eld equations which corresp ond to the generalized gravitational instanton equations We also discuss various geometric formulations of the equations of gravitational instantons and their generalization which includes as a sp ecial case the classical vacuum Einstein equations Some sp ecial solutions and their physical signicance are also considered PACS Classical general Relativity PACS Dierential geometry Riemannian geometry emailnkdiucaaernetin emailkbmscibro oklyncunyedu emailmartucciinfnit ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS Intro duction It is well known since the b eginning of this century that the electric and magnetic elds can b e combined in a natural way to obtain the electromagnetic eld tensor over the Minkowski space and that Maxwells equations can b e expressed as dierential equa tions for the comp onents of this tensor Conversely the electric and magnetic parts of the eld tensor can be recovered once a co ordinate system is xed However the fact that the electromagnetic eld tensor can b e regarded as the curvature form or a gauge eld of a connection or a gauge p otential in a principal bundle with gauge group U and that Maxwells equations can be expressed as gauge eld equations b ecame clear only recently This identication of curvature and gauge eld extends to bundles over arbitrary Riemannian manifolds The study of the space of solutions of gauge eld equa tions has led to new results and a b etter understanding of the top ology and geometry of low dimensional manifolds For background material on geometry and top ology with applications to gauge theory see for example Marathe and Martucci and It is natural to ask ab out the signicance of electric and magnetic parts of an arbitrary gauge eld regarded as the curvature of a gauge p otential There do es not seem to b e any discussion of this asp ect of gauge elds in the literature However a decomp osition of the Weyl and the Riemann curvature of the LeviCivita connection on a spacetime manifold is known see for example and references therein In section we con sider the decomp osition of the full Riemann curvature of a spacetime manifold M g into its electric and magnetic parts relative to a unit timelikevector with resp ect to g We shall refer to this decomp osition as the electromagnetic resolution of the curvature In Dadhich has dened a duality transformation which exchanges the active and passive electric parts and leaves the antisymmetric magnetic part unchanged This du ality transformation implies interchanging the Ricci and the Einstein tensors Duality transformation leaves the Einstein vacuum equations invariantbut changes the sign of the Weyl tensor Duality transformation would then imply that the mass M of the Schwarzschild solution go es over to M Furthermore it turns out that the Einstein vacuum equations can b e mo died so that the mo died equations are not dualityinvari ant and yet admit the same solution while the dual set of equations admit a solution distinct from the Schwarzschild solution Similarly we can construct spacetimes dual to the ReissnerNordstrom NUT and the Kerr solutions The dual spacetimes describ e the original spacetimes with global monop ole charge and global texture The duality so dened is thus intimately related to the top ological defects Weshow that the gravitational instanton Lagrangian R and the instanton equations are invariant under the duality transformation The main aim of this pap er is to study the role played by the duality transformation in the case of gravitational instantons with or without their coupling to matter elds In section we consider a gravitational instanton coupled to classical elds The resulting eld equations include the usual gravitational equations with source term and with or without the cosmological constant Its relation to duality is considered in section In the concluding section wemention some op en problems Electromagnetic Resolution of Curvature Let M g b e a spacetime manifold with metric g of signature We identify the Riemann curvature with the symmetric op erator it induces on M for x each x M and denote the Ho dge star op erator on M by J Let u b e a unit timelike a x N K DADHICH ETC vector We dene the electromagnetic resolution of the Riemann curvature relativeto the unit timelikevector u as follows a b d b d E R u u E JRJ u u ac abcd ac abcd b d b d H JR u u H H H RJ u u ac abcd abcd ac ac ac where b d e b d H JC u u and H R u u abcd abce ac ac d are the symmetric and the skew symmetric parts of the tensor H Here C is the ac abcd Weyl conformal curvature and is the dimensional volume element The pair of abcd tensors E E constitutes the electric part of curvature The tensor E is called the ac ac ac active part and the tensor E is called the passivepart The tensor H is called the ac ac magnetic part of the curvature H H and hence is dropp ed from further considera ac ca a a a tion WewriteE E H H and E E The active and the passive electric parts a a a are symmetric tensors The electric and the magnetic parts have the following additional prop erties b b d a E u E R u u ab bd b b d E u E R b Rg u u ab bd bd b c H u H ab We note that the electric and magnetic parts are orthogonal to the timelike vector u a and thus b ehave at least lo cally as tensors The electric parts are symmetric and account for comp onents while the magnetic part is tracefree and accounts for comp onents of the Riemann curvature Thus the entire gravitational eld is decomp osed into electromagnetic parts The resolution is not unique and dep ends on the choice of a unit timelike vector Using the ab ove decomp osition Dadhich has dened the duality transformation as follows E E H H H H ab ab ab ab ab ab Thus the duality transformation interchanges the activeandpassive electric parts and changes the sign of the symmetric magnetic part while leaving the antisymmetric mag netic part unchanged From equation it is clear that the duality transformation would map the Ricci tensor into the Einstein tensor and viceversa This is b ecause contraction of Riemann is Ricci while that of its double dual is Einstein The Ricci tensor can b e written in terms of the electric and magnetic parts as follows mn c H u u u R E E E E u u Eg acmn b bcmn a ab ab ab a b ab The scalar curvature R is given by R E E ELECTROMAGNETIC RESOLUTION OF CURVATURE AND GRAVITATIONAL INSTANTONS In section weshow that the vacuum Einstein equations with the cosmological constant are equivalent to the equations b b E E and H ab a a This together with equation imply that the vacuum equation R is in general ab equivalentto E E H and E or equivalently E ab ab ab Thus the vacuum equation is symmetric in E and E and hence is invariant under ab ab the duality transformation For obtaining the Schwarzschild solution wefollow and consider the spherically symmetric metric ds C rtdt A rtdr r d sin d The natural choice for the resolving vector is in this case p ointing along the tline whichishyp ersurface orthogonal From equation H and E E leadto ab AC for this no b oundary condition of asymptotic atness is needed Now E gives A Mr which is the Schwarzschild solution Note that we did not need to use the remaining equation E E which is hence free and is implied bythe rest Without aecting the Schwarzschild solution we can intro duce some distribution in the direction We consider the following equation which is not invariant under the duality transfor mation H E E E w w ab ab a b ab where is a scalar and w u ju j is a spacelikeunitvector parallel to the acceleration a a a It is clear that it will also admit the Schwarzschild solution as the general solution and it determines For spherical symmetry the ab ove alternate equation also characterizes vacuum b ecause the Schwarzschild solution is unique Now applying the duality transformation to the ab ove equation we obtain the equation H E E E w w ab ab a b ab Again we shall have AC and E will then yield C k Mr and kr Thus the general solution of equation for the metric is given by M C A k r This is the BarriolaVilenkin solution for the Schwarzschild particle with global monop ole charge This has nonzero stresses given by k T T r N K DADHICH ETC A global monop ole is describ ed by a triplet of scalars a a a a r f r x r where x x r Through the usual Lagrangian it generates the energymomentum distribution at large distance