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Relocalization of Total Conserved Charge for Schwarzschild-AdS Spacetime

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Abstract

We use a tetrad field that his associated metric gives Schwarzschild-AdS spacetime. This tetrad constructed from a diagonal tetrad, which is the square root of Schwarzschild-Ads metric and two other local Lorentz transformations. One of these transformations is a special case of Euler angles and the other is a boost transformation. We then apply the approach of invariant conserved currents to calculate the conserved quantity of Schwarzschild-Ads. Such approach needs a regularization to give the correct result. Therefore, a relocalization procedure is used to calculate the total conserved charge. This procedure leads to physical results in terms of total energy.

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Notes

  1. Latin indices i,j,… refer to local holonomic spacetime coordinates while Greek indices α,β,… label (co)frame components are used. Using local coordinates x i, we have \(\vartheta^{\alpha}=h^{\alpha}_{i} dx^{i}\) and \(e_{\alpha}=h^{i}_{\alpha}\partial_{i}\) where \(h^{\alpha}_{i}\) and \(h^{i}_{\alpha}\) are the covariant and contravariant components of the tetrad field. We define the volume 4-form by

    $$\begin{aligned} \eta \stackrel {\mathrm{def.}}{=} \vartheta^{{0}}\wedge \vartheta^{{1}}\wedge \vartheta^{{2}}\wedge\vartheta^{{3}}. \end{aligned}$$

    The line element

    $$\begin{aligned} ds^2=g_{\alpha \beta} \vartheta^\alpha \otimes \vartheta^\beta, \end{aligned}$$

    is defined by the spacetime metric g αβ .

  2. The general form of Euler′s angles in three dimensions have the form [70]

    $$\begin{aligned} \bigl( {h^\alpha}_i \bigr)_{Euler}= \left ( \begin{array}{c@{\quad}c@{\quad}c} \cos\psi \cos{\phi}_1-\sin\psi\cos{\theta}_1\sin{\phi}_1 & -\cos\psi \sin{\phi}_1-\sin\psi\cos{\theta}_1\cos{\phi}_1 &- \sin\psi\sin{\theta}_1 \\ \sin\psi \cos{\phi}_1+\cos\psi\cos{\theta}_1\sin{\phi}_1 & -\sin\psi \sin{\phi}_1+\cos\psi\cos{\theta}_1\cos{\phi}_1 & \cos\psi\sin{\theta}_1 \\ -\sin{\theta}_1\sin{\phi}_1& -\sin{\theta}_1\cos{\phi}_1 & \cos{\theta}_1 \cr \end{array} \right ). \end{aligned}$$

  3. The calculations are checked using Maple “15” software.

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Acknowledgements

This work is partially supported by the Egyptian Ministry of Scientific Research under project ID 24-2-12.

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Authors and Affiliations

  1. Center for Theoretical Physics, British University in Egypt, P.O. Box 43, Sherouk City, 11837, Egypt

    Gamal G. L. Nashed

  2. Mathematics Department, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt

    Gamal G. L. Nashed

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Nashed, G.G.L. Relocalization of Total Conserved Charge for Schwarzschild-AdS Spacetime. Int J Theor Phys 53, 1580–1589 (2014). https://doi.org/10.1007/s10773-013-1956-x

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