Einstein Product Metrics in Diverse Dimensions K. R. Koehler ∗ University of Cincinnati / Raymond Walters College, Cincinnati, OH 45236 Abstract We use direct products of Einstein Metrics to construct new solutions to Einstein’s Equations with cosmological constant. We illustrate the technique with three families of solutions having the 6 geometries Kerr/de Sitter ⊗ de Sitter, Kerr/anti-de Sitter ⊗ anti-de Sitter and Kerr ⊗ Kerr. 0 0 2 n a J 7 2 1 v 2 2 1 1 0 6 0 / c q - r g : v i X r a ∗ Electronic address: [email protected] 1 Shortly after Einstein introduced the cosmological constant to General Relativity, Kasner [1] and Schouten and Struik [2] explored the geometry of Einstein Manifolds: manifolds admitting a metric whose Ricci Tensor is a constant multiple of the metric. These are manifoldsofconstant curvatureandsolutionstoEinstein’s Equationswithzerostress-energy tensor but possibly nonzero cosmological constant. With Hubble’s discovery in 1929 of the expanding universe, Einstein’s motivation for introducing the cosmological constant was lost and the canonical Einstein Manifolds, de Sitter and anti-de Sitter spacetimes, were largely relegated to serve as textbook examples. Then in 1998, measurements of high red-shift Type Ia supernovae [3] indicated that the expansion of the universe was accelerating, possibly due to a positive cosmological constant. Interest in manifolds with dimension greater than 4 has paralleled that of interest in the cosmological constant. As early as1914 [4], a fifthdimension was postulated as a mechanism for unifying the electromagnetic and gravitational forces. Kaluza and Klien independently applied this mechanism in the context of General Relativity, but for nearly half a century it was seen as unphysical for reasons which are still obvious. Since then, however, the progress in understanding supergravity and string theories has carried interest in manifolds of diverse dimension throughout the academic community and beyond into the public consciousness. In this note we construct direct products of Einstein Metrics which are solutions to Einstein’s Equations with cosmological constant in diverse dimensions. For two or more metrics defined on disjoint manifolds, the product manifold carries a metric which is the simple sum 2 2 2 ds = ds (x)+ds (y)+... (1) 1 2 For such metrics, the Christoffel Connection and Riemann and Ricci Tensor components are simple sums of the corresponding components on the submanifolds. In addition, the scalar curvature and the Kretschmann Invariant are simple sums of the corresponding invariants. The utility of these relationships has been known and exploited for over half a century to simplify manual computations of connection and tensor components. Using them we see that for the product metric, Einstein’s Equations almost uncouple into a set of disjoint equations: 1 R +(Λ− R )g = αT (2) i,ab j i,ab i,ab 2 j X 2 It is clear that if the component metrics are independently solutions to Einstein’s Equations with vanishing scalar curvatures, the product metric is also a solution. This implies, for instance, that any product of Ricci-flat metrics (ie., circles, flat tori, Minkowski and Kerr spacetimes, or their Euclidean counterparts) is a solution. If the component metrics are Einstein Metrics, we have R = χ g i,ab i i,ab R = D χ i i i D −2 i Λ = χ (3) i i 2 where χ is a constant and D is the dimension of the ith metric. Note that for D < 3, i i i Λ = 0. i For a product of n Einstein Metrics g , Einstein’s Equations reduce to a set of n algebraic i relations 1 χ +Λ− R = 0. (4) i j 2 j X Obviously the χ must all be equal for the product metric to be a solution, which fixes Λ i and the Λ in terms of χ: i D −2 j j Λ = χ 2 DP−2 i Λ = χ (5) i 2 It is clear that the cosmological constants must all have the same sign. The most commonly discussed Einstein Manifolds which are not Ricci-flat are de Sitter, anti-deSitterandKerrwithnonzerocosmologicalconstant. AEuclideandeSitter(spherical) metric for D > 2 is [5] S t 2 2 2 2 2 ds = β cosh dΩ +dt (6) S β DS−1 where dΩ2 is the standard metric on Si. For either this metric or its Lorentzian counterpart i (or for the standard metric on S2,ds2 = β2dΩ2), we find S 2 D −1 S χ = (7) S β2 A Euclidean anti-de Sitter metric for D > 2 is [5] A 2 2 2 2 2 2 2 ds = α (dr +sinh rdΩ +cosh rdt ) (8) A n−2 3 We include the constant factor α which in the product metric becomes a nontrivial param- eter. For either this metric or its Lorentzian counterpart, D −1 χ = − A (9) A α2 Kerr solutions with nonzero cosmological constant [6] exist in any dimension D > 3. K Let 2 2 2 2 ρ = r +a cos θ 2Λ r2 µ ∆(D ) = (1− K )(r2 +a2)− K (D −1)(D −2) rDK−5 K K 2a2Λ cos2θ K ψ(D ) = 1+ K (D −1)(D −2) K K 2a2Λ K Σ(D ) = 1+ (10) K (D −1)(D −2) K K where µ is proportional to the mass, a is proportional to the angular momentum and Λ is K the cosmological constant. The corresponding Kerr metric is ρ2 ρ2 2 2 2 2 2 2 ds = r cos θdΩ + dr + dθ + K ∆(D ) ψ(D ) K K (ψ(D )(r2 +a2)2 −∆(D )a2sin2θ)sin2θ K K 2 dφ + Σ2(D )ρ2 K 2asin2θ(ψ(D )(r2 +a2)−∆(D )) K K dφdt− Σ2(D )ρ2 K ∆(D )−a2ψ(D )sin2θ K K 2 dt (11) Σ2(D )ρ2 K where dΩ2 is the standard metric on SDK−4. These solutions possess a single rotation axis; more general solutions have been discussed [7] which have the maximal number of rotation parameters. For these metrics or their Euclidean counterparts, 2Λ K χ = (12) K D −2 K Forming the product metric ds2 = ds2 +ds2, we set χ equal to χ and find the product K S K S is a solution for (D +D −2)Λ K S K Λ = D −2 K (D −2)(D −1) K S β = (13) s 2ΛK Physically, an observer on the Kerr submanifold measures a smaller value for the cosmo- logical constant than an observer living in the entire product spacetime, and the ratio of 4 the two is purely a function of the dimension of the spherical submanifold. In addition, the cosmological constant measured by the Kerr observer is inversely proportional to the square of the radius parameter for the sphere. Assuming that D = 4 and Λ corresponds to data K K from current observations [8], we find that β is of the order of the radius of the observable universe. Since the geodesic equations for a simple product manifold are not coupled, each submanifold is a geodesic hypersurface. But without a mechanism to prevent energy and momentum transfer between submanifolds, these manifolds are clearly of purely academic interest. This is obviously true as well when more than one submanifold is noncompact. For the anti-de Sitter case, ds2 = ds2 +ds2 is a solution if K A (D +D −2)Λ K A K Λ = D −2 K −(D −1)(D −2) A K α = (14) s 2ΛK Finally, we form a product of two Kerr metrics with cosmological constant and confirm that the solution requires (D1 +D2 −2)Λ1 Λ = D1 −2 (D2 −2)Λ1 Λ2 = (15) D1 −2 We noted previously that the scalar curvature and Kretschmann Invariant for a simple product metric are simply the sums of those invariants for the metrics on the submanifolds. Therefore if either metric possesses curvature singularities, the product metric possesses those same singularities. This means that our example product metrics possess the well- known ring singularity forD (D ) equal to 4 or 5, andthe singularity atr = 0 forD (D ) > K i K i 5 [9].) 5 Acknowledgements We would like to thank Cenalo Vaz and Louis Witten for stimulating discussions on these matters. [1] Kasner, E., Geometrical Theorems on Einstein’s Cosmological Equations, American Journal of Mathematics 43, 217-221 (1921) [2] Schouten, J. A. and Struik, D. J., On Some Properties of General Manifolds Relating to Ein- stein’s Theory of Gravitation, American Journal of Mathematics 43, 213-216 (1921) [3] Riess, A. G. et al, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astrophys. J. 509 74-79 (1998) (arXiv:astro-ph/9805201) [4] Nordstrom, G., On the Possibility of a Unification of the Electromagnetic and Gravitational Fields, Physik Zeitchr. 15 504-506 (1914); reprinted in Modern Kaluza-Klein Theories, Ap- pelquist, T., Chodos, A. and Freund, P. G. O., eds. (Addison Wesley, 1987) [5] Hawking, S. W. and Ellis, G. F. R., The Large Scale Structure of Space-time (Cambridge University Press, 1973) [6] Z. Stuchlik and M. Calvani, Null Geodesics in Black Hole metrics with non-zero Cosmological Constant, General Relativity and Gravitation 23, 507-519 (1991) [7] G. W. Gibbons, H. Lu, Don N. Page and C. N. Pope, The General Kerr-de Sitter Metrics in All Dimensions, J. Geom. Phys. 53 49-73 (2005) (arXiv:hep-th/0404008) [8] Filippenko, A. V., The Accelerating Universe and Dark Energy: Evidence from Type Ia Su- pernovae, Lect.Notes Phys. 646 191-221 (2004) (arXiv:astro-ph/0309739) [9] K. R. Koehler, Computations in Riemann Geometry, http://www.rwc.uc.edu/koehler/crg/blackhole.html 6