ds2 = f(y) ds(x)2 + ds(y)2The membranes which we examined in the last section have this form.
The warped product metric produces Christoffel Symbols of the form
Γab c = (δxac (d f / dyb) + δxab (d f / dyc)) / (2 f(y)) +The use of x and y as superscripts and subscripts indicate that the object corresponds to either ds(x)2 or ds(y)2, respectively. If we defineΓxab c + Γyab c - gxb c gya i (d f / dyi) / 2
Ψa d = (d f / dya);d - (d f / dya) (d f / dyd) / (2 f(y))and
Υ = (d f / dyi) (d f / dyj) gyi j
Ψ2 = Ψa b Ψi j gya i gyb jthe components of the Riemann Tensor have the form
Ra b c d = (- Ψb d gxa c + Ψb c gxa d + Ψa d gxb c - Ψa c gxb d) / 2 +while those of the Ricci Tensor have the formΥ ( gxa d gxb c - gxa c gxb d) / 4 + f(y) Rxa b c d + Rya b c d
Ra c = - Ψa c Dx / (2 f(y)) - Ψb d gxa c gyb d / 2 + Υ (1 - Dx) gxa c / (4 f(y)) +where Dx indicates the dimension of ds(x)2. The scalar curvature is thengxb d Rxa b c d + gyb d Rya b c d
R = Ry + Rx / f(y) + Dx (1 - Dx) Υ / (4 f(y)2) - Dx Ψa c gya c / f(y)and the Kretschmann Invariant is
K = Ky + Kx / f(y)2 + Dx Ψ2 / f(y)2 + Dx (Dx - 1) Υ2 / (8 f(y)4) - Υ Rx / f(y)3Applying this to the D = 10 5-brane from the last section, we see that ds(y)2 is conformal to Euclidean Space with conformal factor
(k + r2)3/4 / r3/2and ds(x)2 is Minkowski Spacetime, with
f(y) = r1/2 / (k + r2)1/4This gives us
Ry = 45 k5 / (8 r1/2 (k + r2)11/4)and
Ky = 27 k2 (25 k2 + 80 k r2 + 128 r4) / (64 r (k + r2)11/2)
Rx = 0
Kx = 0
Ψa c gya c = 3 k2 / (8 (k + r2)3)
Υ = k2 r1/2 / (4 (k + r2)13/4)
Ψ2 = 3 k2 (k2 + 8 k r2 + 64 r4) / (64 (k + r2)6)This in turn results in
R = 3 k2 / (2 r1/2 (k + r2)11/4)and
K = 3 k2 (59 k2 + 192 k r2 + 384 r4) / (16 r (k + r2)11/2)as we computed in the last section.
This technique for computing invariants is somewhat limited in that general expressions for more
complicated invariants will be much more difficult to derive and much less structurally informative.
While the scalar curvatures and Kretschmann Invariants for all of the metrics which we have analyzed
have been easily computable without this
technique, it does serve as an alternate algorithm for verification purposes, and enables some
computations to be done manually which otherwise would not be possible. There is, however, an
obvious useful special case:
Simple Product Metrics
When the warp factor is a constant, several simplifications occur:
the Christoffel Connection and Riemann and Ricci Tensor components
are simple sums of the corresponding components on the two submanifolds,
and the scalar curvature and the Kretschmann Invariant are simple sums of the corresponding
invariants. The simple structure of these quantities has a number of immediate consequences
So if both metrics are independently solutions to Einstein's Equations with vanishing scalar curvatures, the product metric is also a solution. In addition, if both metrics are Einstein Metrics, with Ra b = χ ga b, Einstein's equations become
- The Geodesic Equation becomes a pair of uncoupled equations, so that geodesics tangent to either submanifold are tangent to that submanifold for all values of the affine parameter: each submanifold is a geodesic hypersurface.
- Einstein's Equations almost uncouple to take the following form:Rxa b + (Λ - (Rx + Ry) / 2) gxa b = α Txa bandRya b + (Λ - (Rx + Ry) / 2) gya b = α Tya b
- If either gx or gy possess curvature singularities, the product metric possesses those same singularities.
χi + Λ - Σj Rj / 2 = 0Obviously the χi must be equal, which fixes Λ and the Λi in terms of χ:
Λ = χ (Σj Dj - 2) / 2It is clear that the cosmological constants must have the same sign. Note that they are zero for Di < 3.
Λi = (Di - 2) χ / 2
As an example, the Kerr metric with cosmological constant is an Einstein Metric with
χ = 2 ΛK / (DK - 2)Hence it's simple product with any sphere is a solution with the following relations between parameters:
rS = ((DK - 2) (DS - 1) / (2 ΛK))1/2Here, rS is the radius parameter of the sphere, and the subscripts "K" and "S" indicate the Kerr and sphere, respectively. Similar results hold for Kerr/anti-de Sitter x (Euclidean) anti-de Sitter and Kerr x Kerr.
Λ = (DK + DS - 2) ΛK / (DK - 2)
The first appendix is about geodesics.
Please send comments or suggestions to the author.