Warped Product Spaces

We define a warped product space by the (block diagonal) ansatz
ds2 = f(y) ds(x)2 + ds(y)2
The 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)) +
Γxab c + Γyab c - gxb c gya i (d f / dyi) / 2
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 define
Ψa d = (d f / dya);d - (d f / dya) (d f / dyd) / (2 f(y))

Υ = (d f / dyi) (d f / dyj) gyi j

Ψ2 = Ψa b Ψi j gya i gyb j
the 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 +
Υ ( gxa d gxb c - gxa c gxb d) / 4 + f(y) Rxa b c d + Rya b c d
while those of the Ricci Tensor have the form
Ra c = - Ψa c Dx / (2 f(y)) - Ψb d gxa c gyb d / 2 + Υ (1 - Dx) gxa c / (4 f(y)) +
gxb d Rxa b c d + gyb d Rya b c d
where Dx indicates the dimension of ds(x)2. The scalar curvature is then
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)3
Applying 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 / r3/2
and ds(x)2 is Minkowski Spacetime, with
f(y) = r1/2 / (k + r2)1/4
This gives us
Ry = 45 k5 / (8 r1/2 (k + r2)11/4)

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)
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 [Koehler1]:
  1. 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.
  2. Einstein's Equations almost uncouple to take the following form:
    Rxa b + (Λ - (Rx + Ry) / 2) gxa b = α Txa b
    Rya b + (Λ - (Rx + Ry) / 2) gya b = α Tya b
  3. If either gx or gy possess curvature singularities, the product metric possesses those same singularities.
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
χi + Λ - Σj Rj / 2 = 0
Obviously the χi must be equal, which fixes Λ and the Λi in terms of χ:
Λ = χ (Σj Dj - 2) / 2

Λi = (Di - 2) χ / 2

It is clear that the cosmological constants must have the same sign. Note that they are zero for Di < 3.

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))1/2

Λ = (DK + DS - 2) ΛK / (DK - 2)

Here, 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.

The first appendix is about geodesics.

©2005, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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