dsThe membranes which we examined in the last section have this form.^{2}= f(y) ds(x)^{2}+ ds(y)^{2}

The warped product metric produces Christoffel Symbols of the form

ΓThe use of x and y as superscripts and subscripts indicate that the object corresponds to either ds(x)^{a}_{b c}= (δ_{x}^{a}_{c}(d f / dy^{b}) + δ_{x}^{a}_{b}(d f / dy^{c})) / (2 f(y)) +Γ_{x}^{a}_{b c}+ Γ_{y}^{a}_{b c}- g^{x}_{b c}g_{y}^{a i}(d f / dy^{i}) / 2

Ψand_{a d}= (d f / dy^{a});d - (d f / dy^{a}) (d f / dy^{d}) / (2 f(y))Υ = (d f / dy

^{i}) (d f / dy^{j}) g_{y}^{i j}

Ψthe components of the Riemann Tensor have the form^{2}= Ψ_{a b}Ψ_{i j}g_{y}^{a i}g_{y}^{b j}

Rwhile those of the Ricci Tensor have the form_{a b c d}= (- Ψ_{b d}g^{x}_{a c}+ Ψ_{b c}g^{x}_{a d}+ Ψ_{a d}g^{x}_{b c}- Ψ_{a c}g^{x}_{b d}) / 2 +Υ ( g^{x}_{a d}g^{x}_{b c}- g^{x}_{a c}g^{x}_{b d}) / 4 + f(y) R^{x}_{a b c d}+ R^{y}_{a b c d}

Rwhere D_{a c}= - Ψ_{a c}D_{x}/ (2 f(y)) - Ψ_{b d}g^{x}_{a c}g_{y}^{b d}/ 2 + Υ (1 - D_{x}) g^{x}_{a c}/ (4 f(y)) +g_{x}^{b d}R^{x}_{a b c d}+ g_{y}^{b d}R^{y}_{a b c d}

R = Rand the Kretschmann Invariant is_{y}+ R_{x}/ f(y) + D_{x}(1 - D_{x}) Υ / (4 f(y)^{2}) - D_{x}Ψ_{a c}g_{y}^{a c}/ f(y)

K = KApplying this to the D = 10 5-brane from the last section, we see that ds(y)_{y}+ K_{x}/ f(y)^{2}+ D_{x}Ψ^{2}/ f(y)^{2}+ D_{x}(D_{x}- 1) Υ^{2}/ (8 f(y)^{4}) - Υ R_{x}/ f(y)^{3}

(k + rand ds(x)^{2})^{3/4}/ r^{3/2}

f(y) = rThis gives us^{1/2}/ (k + r^{2})^{1/4}

Rand_{y}= 45 k^{5}/ (8 r^{1/2}(k + r^{2})^{11/4})K

_{y}= 27 k^{2}(25 k^{2}+ 80 k r^{2}+ 128 r^{4}) / (64 r (k + r^{2})^{11/2})R

_{x}= 0K

_{x}= 0Ψ

_{a c}g_{y}^{a c}= 3 k^{2}/ (8 (k + r^{2})^{3})Υ = k

^{2}r^{1/2}/ (4 (k + r^{2})^{13/4})

ΨThis in turn results in^{2}= 3 k^{2}(k^{2}+ 8 k r^{2}+ 64 r^{4}) / (64 (k + r^{2})^{6})

R = 3 kand^{2}/ (2 r^{1/2}(k + r^{2})^{11/4})

K = 3 kas we computed in the last section.^{2}(59 k^{2}+ 192 k r^{2}+ 384 r^{4}) / (16 r (k + r^{2})^{11/2})

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:

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 R

- 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
almostuncouple to take the following form:Rand^{x}_{a b}+ (Λ - (R_{x}+ R_{y}) / 2) g^{x}_{a b}= α T^{x}_{a b}R^{y}_{a b}+ (Λ - (R_{x}+ R_{y}) / 2) g^{y}_{a b}= α T^{y}_{a b}

- If either g
_{x}or g_{y}possess curvature singularities, the product metric possesses those same singularities.

χObviously the χ_{i}+ Λ - Σ_{j}R_{j}/ 2 = 0

Λ = χ (ΣIt is clear that the cosmological constants must have the same sign. Note that they are zero for D_{j}D_{j}- 2) / 2Λ

_{i}= (D_{i}- 2) χ / 2

As an example, the Kerr metric with cosmological constant is an Einstein Metric with

χ = 2 ΛHence it's simple product with any sphere is a solution with the following relations between parameters:_{K}/ (D_{K}- 2)

rHere, r_{S}= ((D_{K}- 2) (D_{S}- 1) / (2 Λ_{K}))^{1/2}Λ = (D

_{K}+ D_{S}- 2) Λ_{K}/ (D_{K}- 2)

The first appendix is about geodesics.

- Table of Contents
- Index:

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

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