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

Γ^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 / 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

Ψ_{a d} = (d f / dy^a);d - (d f / dy^a) (d f / dy^d) / (2 f(y))
Υ = (d f / dyⁱ) (d f / dy^j) g_y^{i j}

and

Ψ² = Ψ_{a b} Ψ_{i j} g_y^{a i} g_y^{b j}

the components of the Riemann Tensor have the form

R_{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}

while those of the Ricci Tensor have the form

R_{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}

where D_x indicates the dimension of ds(x)². The scalar curvature is then

R = R_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)

and the Kretschmann Invariant is

K = K_y + K_x / f(y)² + D_x Ψ² / f(y)² + D_x (D_x - 1) Υ² / (8 f(y)⁴) - Υ R_x / 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 / r^3/2

and ds(x)² is Minkowski Spacetime, with

f(y) = r^1/2 / (k + r²)^1/4

This gives us

R_y = 45 k⁵ / (8 r^1/2 (k + r²)^11/4)
K_y = 27 k² (25 k² + 80 k r² + 128 r⁴) / (64 r (k + r²)^11/2)
R_x = 0
K_x = 0
Ψ_{a c} g_y^{a c} = 3 k² / (8 (k + r²)³)
Υ = k² r^1/2 / (4 (k + r²)^13/4)

and

Ψ² = 3 k² (k² + 8 k r² + 64 r⁴) / (64 (k + r²)⁶)

This in turn results in

R = 3 k² / (2 r^1/2 (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 [Koehler1]:

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:
R^x_{a b} + (Λ - (R_x + R_y) / 2) g^x_{a b} = α T^x_{a b}
and
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.

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_{a b} = χ g_{a b}, Einstein's equations become

χ_i + Λ - Σ_j R_j / 2 = 0

Obviously the χ_i must be equal, which fixes Λ and the Λ_i in terms of χ:

Λ = χ (Σ_j D_j - 2) / 2
Λ_i = (D_i - 2) χ / 2

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

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

χ = 2 Λ_K / (D_K - 2)

Hence it's simple product with any sphere is a solution with the following relations between parameters:

r_S = ((D_K - 2) (D_S - 1) / (2 Λ_K))^1/2
Λ = (D_K + D_S - 2) Λ_K / (D_K - 2)

Here, r_S 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.

Table of Contents

Index:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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