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Static Einstein–Maxwell space-time invariant by translation

Abstract

In this paper we study the static Einstein–Maxwell space when it is conformal to an n-dimensional pseudo-Euclidean space, which is invariant under the action of an \((n-1)\)-dimensional translation group. We also provide a complete classification of such space.

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Correspondence to Benedito Leandro.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Ana Paula de Melo was partially supported by PROPG-CAPES [Finance Code 001].

Ilton Menezes was partially supported by PROPG-CAPES [Finance Code 001].

Romildo Pina was partially supported by CNPq/Brazil [Grant number: 305410/2018-0]

Appendix

Appendix

In this section, we will provide the formula for the sectional curvature for a conformal metric. Consider the metric \(g_{ij}= \frac{\delta _{ij}\varepsilon _{i}}{\varphi ^2}\) in \(\Omega \subseteq {\mathbb {R}}^n\) a open subset, where \(\varphi \) is a positive smooth function. We can write \(g^{ij}= \varphi ^2\delta _{ij}\varepsilon _{i}\) to indicate the inverse of the metric \(g_{ij}\). In these conditions, we have:

$$\begin{aligned} \frac{\partial g_{ik}}{\partial x_j} = \frac{-2 \delta _{ik} \varepsilon _i}{\varphi ^3} \varphi _{,j}. \end{aligned}$$

For this metric g, the curvature coefficients are

$$\begin{aligned}&R_{ijij}= \sum _{l} R_{iji}^l \, g_{lj} = R_{iji}^j \, \frac{\varepsilon _j}{\varphi ^2} \\&\quad = \frac{\varepsilon _j}{\varphi ^2} \left( \sum _l \Gamma _{ii}^{l} \Gamma _{jl}^{j} - \sum _l \Gamma _{ji}^{l} \Gamma _{il}^{j} + \frac{\partial }{\partial x_j}\Gamma _{ii}^{j} - \frac{\partial }{\partial x_i}\Gamma _{ji}^{j} \right) . \end{aligned}$$

Then, from (3.3) we can calculate the derivative of \(\Gamma _{ij}^{k}\). That is,

$$\begin{aligned} \frac{\partial }{\partial x_j}\Gamma _{ii}^{j}= \varepsilon _i\varepsilon _j \left( \frac{\varphi _{,jj}}{\varphi } - \frac{\varphi _{,j} ^2 }{\varphi ^2}\right) \quad \mathrm {and} \quad \frac{\partial }{\partial x_i}\Gamma _{ji}^{j}= \left( \frac{\varphi _{,i}}{\varphi } \right) ^2 - \frac{\varphi _{,ii}}{\varphi }. \end{aligned}$$

Combining the above identities with (3.3) we get

$$\begin{aligned} \varphi ^2\varepsilon _j R_{ijij}= - \sum _l \varepsilon _i\varepsilon _l \left( \frac{\varphi _{,l}}{\varphi } \right) ^2 + \varepsilon _i\varepsilon _j \frac{\varphi _{,jj}}{\varphi } + \frac{\varphi _{,ii}}{\varphi }. \end{aligned}$$

Now, if the four indices are distinct

$$\begin{aligned} R_{ijkl}= \sum _{s} R_{ijk}^s \, g_{sl} = R_{ijk}^l \, g_{ll} = 0. \end{aligned}$$

Finally, consider the case in which we have three distinct indices:

$$\begin{aligned} R_{ijk}^i= - \frac{\varphi _{,kj}}{\varphi }, \quad R_{ijk}^j= \frac{\varphi _{,ki}}{\varphi } \quad \mathrm {and} \quad R_{ijk}^k= 0. \end{aligned}$$

Hence, the sectional curvature generated by \(\partial _{x_i}\), \(\partial _{x_j}\) is

$$\begin{aligned} K_{ij} = \varphi ^2\left( -\sum _l \varepsilon _i\varepsilon _l \left( \frac{\varphi _{,l}}{\varphi } \right) ^2 + \varepsilon _i\varepsilon _j \frac{\varphi _{,jj}}{\varphi } + \frac{\varphi _{,ii}}{\varphi }\right) \varepsilon _i. \end{aligned}$$
(4.1)

Let \(\xi =\sum _{i=1}^{n}\alpha _ix_i\), consider \(\varphi (\xi )\) a function of \(\xi \). Since

$$\begin{aligned} \varphi _{,i}=\varphi '\alpha _i, \quad \varphi _{,ii}=\varphi ''\alpha _i^2, \quad \varphi _{,ij}=\varphi ''\alpha _i\alpha _j, \end{aligned}$$

from (4.1) we get

$$\begin{aligned} K_{ij} = -\left( \varphi '\right) ^2\varepsilon _{i_0} + \varphi \varphi ''\varepsilon _j \alpha _j^2 + \varphi \varphi ''\varepsilon _i\alpha _i^2, \end{aligned}$$

where \(\displaystyle \sum \nolimits _{{i = 1}}^{n} \varepsilon _{l}\alpha _{l}^{2}=\varepsilon _{i_0}\in \{-1,\,0,\,1\}\), which depends on the direction of the tangent vector field \(\alpha =\displaystyle \sum \nolimits _{{i = 1}}^{n}\alpha _{l}\partial _{x_l}\).

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Leandro, B., de Melo, A.P., Menezes, I. et al. Static Einstein–Maxwell space-time invariant by translation. Gen Relativ Gravit 53, 92 (2021). https://doi.org/10.1007/s10714-021-02867-3

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Keywords

  • Electrostatic system
  • Conformal metric
  • Einstein–Maxwell equations

Mathematics Subject Classification

  • 53C20
  • 83C22
  • 53C21
  • 30F45