Empty Singularities in Higher-Dimensional Gravity

Abstract

We study the exact solution of Einstein’s field equations consisting of a (n+2)-dimensional static and hyperplane symmetric thick slice of matter, with constant and positive energy density ρ and thickness d, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of ρ and d, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if $\sqrt{\rho} d$ is small enough, the dominant energy condition is satisfied all over the spacetime. We find that this solution presents many interesting features. In particular, it has an empty singular boundary in one of the vacua.

Appendix: Killing Vectors and Adapted Coordinates

We want to find coordinates adapted to a hyperplane symmetric distribution of matter. That is, space-time must be invariant under n translations and under rotations in n(n−1)/2 hyperplanes.

More precisely, a n+2 dimensional space-time will be said to be n-dimensional Euclidean homogenous if it admits the r=n(n+1)/2 parameter group of isometries of the n-dimensional Euclidean space ISO(n).

Since the spacetime admits n mutually commuting independent motions, we can choose coordinates x ⁱ,z,t so that the corresponding Killing vectors are ξ _(i)=∂ _i (i=1,…,n), and so

$$ \xi_{(i)}^k = \delta^k_i \quad \left ( \begin{array}{c} k=t,1,\ldots,n,z\\ i=1,\ldots,n \end{array} \right ) . $$

(33)

The equations of Killing,

$$ \xi^k {\partial_{k}}g_{ij}+ g_{kj} { \partial_{i}}\xi^k + g_{ik} { \partial_{j}} \xi^k= 0 , $$

(34)

corresponding to these vectors become

$$ {\partial_{k}} g_{ij}= 0 \quad \left ( \begin{array}{c} i,j=t,1,\ldots,n,z\\ k=1,\ldots,n \end{array} \right ) . $$

(35)

Hence all the components of the metric tensor depend only on the coordinates t and z and the metric is unaltered by the finite transformation

$$ x^i \rightarrow x^i + a^i \quad (\mathrm{for} \ i=1,\ldots,n) . $$

(36)

We can take for the remaining n(n−1)/2 motions, the generators ξ _(ij)=x ⁱ ∂ _j −x ^j ∂ _i (i<j and i,j=1,…,n), so

$$ {\partial_{l}} \xi_{(ij)}^k = \delta^i_l \delta^k_j- \delta^j_l \delta^k_i \quad \left ( \begin{array}{c} k,l=t,1,\ldots,n,z\\ i<j\ \mathrm{and}\ i,j=1,\ldots,n \end{array} \right ) . $$

(37)

Taking into account (35) and (37) from the equations of Killing (34), we get

$$ g_{jm} \delta^i_l -g_{im} \delta^j_l+g_{lj} \delta^i_m-g_{li} \delta^j_m= 0 \quad \left ( \begin{array}{c} m,l=t,1,\ldots,n,z\\ i<j\ \mathrm{and}\ i,j=1,\ldots,n \end{array} \right ) . $$

(38)

From the last equation we readily find

$$ g_{ii}=g_{jj} \quad\mathrm{and}\quad g_{ij}=g_{it}=g_{iz}=0 \quad (\ i\neq j\ \mathrm{and}\ i,j=1,\ldots,n) . $$

(39)

Furthermore, we can take the bidimensional metric of the V ₂ spaces x ⁱ= constant (i=1,…,n) in the conformal flat form. Hence, the most general metric admitting this group of isometries may be written as

$$ ds^2= - e^{2 U(z, t)} \bigl(dt^2-dz^2 \bigr)+ e^{2V(z, t)} \bigl(\bigl(dx^{ 1}\bigr)^2+\cdots+ \bigl(dx^{ {n}}\bigr)^2 \bigr) . $$

(40)

If, in addition, we impose staticity, U and V must be time independent, and the change of variable ∫e ^U(z) dz→z brings the line element to the form (1).