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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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:
For this metric g, the curvature coefficients are
Then, from (3.3) we can calculate the derivative of \(\Gamma _{ij}^{k}\). That is,
Combining the above identities with (3.3) we get
Now, if the four indices are distinct
Finally, consider the case in which we have three distinct indices:
Hence, the sectional curvature generated by \(\partial _{x_i}\), \(\partial _{x_j}\) is
Let \(\xi =\sum _{i=1}^{n}\alpha _ix_i\), consider \(\varphi (\xi )\) a function of \(\xi \). Since
from (4.1) we get
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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DOI: https://doi.org/10.1007/s10714-021-02867-3