Covariantized Nœther identities and conservation laws for perturbations in metric theories of gravity

Abstract

A construction of conservation laws and conserved quantities for perturbations in arbitrary metric theories of gravity is developed. In an arbitrary field theory, with the use of incorporating an auxiliary metric into the initial Lagrangian covariantized Nœther identities are carried out. Identically conserved currents with corresponding superpotentials are united into a family. Such a generalized formalism of the covariantized identities gives a natural basis for constructing conserved quantities for perturbations. A new family of conserved currents and correspondent superpotentials for perturbations on arbitrary curved backgrounds in metric theories is suggested. The conserved quantities are both of pure canonical Nœther and of Belinfante corrected types. To test the results each of the superpotentials of the family is applied to calculate the mass of the Schwarzschild-anti-de Sitter black hole in the Einstein–Gauss–Bonnet gravity. Using all the superpotentials of the family gives the standard accepted mass.

Appendices

Appendix A: Auxiliary algebraic expressions

Here, we give useful for calculations algebraic properties of the operator presented by the notation ${\left. Q^A \right|}^\alpha _\beta$ included in (3):

$$\begin{aligned} \delta Q^A= {\pounds }_\xi Q^A =-\xi ^\alpha \partial _\alpha Q^A + {\left. Q^A \right|}^\alpha _\beta \partial _\alpha \xi ^\beta . \end{aligned}$$

(150)

In general, we follow to [24], however our treating ${\left. Q^A \right|}^\alpha _\beta$ is more simple and more effective, as we imagine. We define the operator for covariant quantities only: tensor densities or sets of tensor densities. For example, for a tensor density of the weight $+$ $n$ one has

$$\begin{aligned} \left. Q^{\alpha \beta \ldots \gamma }_{\pi \rho \ldots \sigma } \right|^\mu _\nu&= -n\delta ^\mu _\nu Q^{\alpha \beta \ldots \gamma }_{\pi \rho \ldots \sigma } +\delta ^\alpha _\nu Q^{\mu \beta \ldots \gamma }_{\pi \rho \ldots \sigma }+\delta ^\beta _\nu Q^{\alpha \mu \ldots \gamma }_{\pi \rho \ldots \sigma } +\cdots +\delta ^\gamma _\nu Q^{\alpha \beta \ldots \mu }_{\pi \rho \ldots \sigma }\nonumber \\&- \delta ^\mu _\pi Q^{\alpha \beta \ldots \gamma }_{\nu \rho \ldots \sigma }- \delta ^\mu _\rho Q^{\alpha \beta \ldots \gamma }_{\pi \nu \ldots \sigma }-\cdots - \delta ^\mu _\sigma Q^{\alpha \beta \ldots \gamma }_{\pi \rho \ldots \nu }. \end{aligned}$$

(151)

Thus, for the metric one has $\left. g_{\mu \nu } \right|^\alpha _\beta = -\delta ^\alpha _\mu g_{\beta \nu }-\delta ^\alpha _\nu g_{\mu \beta }$, or for the scalar density: $\left. {\hat{\mathcal{{L}}}}\right|^\alpha _\beta =-\delta ^\alpha _\beta {\hat{\mathcal{{L}}}}$. One can see that calculations with expressions, like (151), could be very cumbersome. Whereas the use of the abstract form ${\left. Q^A \right|}^\alpha _\beta$ is, indeed, more economical, this is a main reason why we suggest to apply it here. Only to show a final result after calculations the form (151) could be represented. Below we describe properties of the definition $\left. Q^A \right|^\alpha _\beta$, which are necessary in the present paper.

The right hand side of the Eq. (151) shows that the quantities ${\left. Q^A \right|}^\alpha _\beta$ are tensor densities of the same weight as the tensor densities $Q^A$. Therefore, the covariant derivative $\left({\left. Q^A \right|}^\alpha _\beta \right)_{;\gamma }$ of tensor densities ${\left. Q^A \right|}^\alpha _\beta$ is defined in a usual manner as applied to a covariant quantity. Thus, the evident property

$$\begin{aligned} \left. \left(Q^A{}_{;\alpha }\right)\right|^\tau _\rho =\left. \left(Q^A{}\right|^\tau _\rho \right)_{;\alpha } - \delta ^\tau _\alpha Q^A{}_{;\rho } \end{aligned}$$

(152)

follows after covariant differentiation of (151). Also an action of a double vertical line is defined by a natural way:

$$\begin{aligned} \left.\left.Q^A\right|^\alpha _\beta \right|^\mu _\nu \equiv \left.\left(\left.Q^A\right|^\alpha _\beta \right)\right|^\mu _\nu \equiv {\left. Q^B \right|}^\mu _\nu \end{aligned}$$

(153)

where ‘${}^B$’ is a new generalized index. More important properties follow from the usual properties of the Lie derivative. Thus from

$$\begin{aligned} {\pounds }_\xi \delta ^\rho _\tau&= 0;\end{aligned}$$

(154)

$$\begin{aligned} {\pounds }_\xi (Q^AP^B)&= P^B{\pounds }_\xi (Q^A) + Q^A{\pounds }_\xi (P^B);\end{aligned}$$

(155)

$$\begin{aligned} {\pounds }_\zeta {\pounds }_\xi (Q^A)- {\pounds }_\xi {\pounds }_\zeta (Q^A)&= {\pounds }_{[\zeta \xi ]} (Q^A), \end{aligned}$$

(156)

where $[\zeta \xi ]=\xi ^\rho \zeta ^\alpha {}_{,\rho }-\zeta ^\rho \xi ^\alpha {}_{,\rho }$, the next relations are derived:

$$\begin{aligned} \left.\delta ^\rho _\tau \right|^\alpha _\beta&= 0; \end{aligned}$$

(157)

$$\begin{aligned} \left.\left(Q^AP^B\right)\right|^\alpha _\beta&= \left.\left(P^B\right)\right|^\alpha _\beta Q^A +\left.\left(Q^A\right)\right|^\alpha _\beta P^B; \end{aligned}$$

(158)

$$\begin{aligned} \left. \left.Q^A \right|^\beta _\rho \right|^\tau _\alpha - \left. \left.Q^A \right|^\tau _\alpha \right|^\beta _\rho&= \delta ^\beta _\alpha \left.Q^A\right|^\tau _\rho -\delta ^\tau _\rho \left.Q^A\right|^\beta _\alpha . \end{aligned}$$

(159)

Among these the second property (158) is more useful. Thus, in our calculations frequently we use the transformations, like this

$$\begin{aligned} \frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}\left.Q_B\right|^\rho _\tau = \left. \left(\frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}Q_B\right)\right|^\rho _\tau -\left. \left(\frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}\right)\right|^\rho _\tau Q_B. \end{aligned}$$

(160)

The first term at the right hand side, to which the vertical line is applied, is a vector density, therefore it is useful also

$$\begin{aligned} \left. \left(\frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}Q_B\right)\right|^\rho _\tau = - \delta ^\rho _\tau \frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}Q_B + \delta ^\alpha _\tau \frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\rho }}Q_B. \end{aligned}$$

(161)

The above properties are enough to derive the useful relation:

$$\begin{aligned} \frac{\partial \left(\left.Q^A\right|^\alpha _\beta \right)}{\partial Q^B}\left.Q^B\right|^\rho _\tau = \left.\left.Q^A\right|^\alpha _\beta \right|^\rho _\tau . \end{aligned}$$

(162)

The definition $\left. Q^A \right|^\alpha _\beta$ in (150) corresponds to the presentation of the covariant derivative by the way:

$$\begin{aligned} \overline{D}_\alpha Q^A = \partial _\alpha Q^A + {\left. Q^A \right|}^\rho _\tau \overline{\Gamma }_{\alpha \rho }^\tau . \end{aligned}$$

(163)

Namely this presentation is used to represent a partial derivative through a covariant one. Recall that for the vector density $\hat{Q}^\alpha$: $\overline{D}_\alpha \hat{Q}^\alpha = \partial _\alpha \hat{Q}^\alpha$, and for the antisymmetric tensor density $\hat{Q}^{\alpha \beta }$: $\overline{D}_\beta \hat{Q}^{\alpha \beta } = \partial _\beta \hat{Q}^{\alpha \beta }$. Then, the definition (163) evidently gives

$$\begin{aligned} {\left. \hat{Q}^\alpha \right|}^\rho _\tau \overline{\Gamma }_{\alpha \rho }^\tau&= 0, \end{aligned}$$

(164)

$$\begin{aligned} {\left. \hat{Q}^{\alpha \beta } \right|}^\rho _\tau \overline{\Gamma }_{\beta \rho }^\tau&= 0. \end{aligned}$$

(165)

Then (161), as an example of the vector density, gives:

$$\begin{aligned} \left. \left(\frac{\partial {\hat{\mathcal{{L}}}}}{\partial Q_{B;\alpha }}Q_B\right)\right|^\rho _\tau \overline{\Gamma }_{\alpha \rho }^\tau =0. \end{aligned}$$

(166)

At last, we note that the notation $\left. Q^A \right|^\alpha _\beta$ can be used also in antisymmetrization of the covariant derivatives:

$$\begin{aligned} Q^A{}_{;\mu \nu }-Q^A{}_{;\nu \mu } =\left. Q^A \right|^\alpha _\beta \overline{R}_\alpha {}^\beta {}_{\mu \nu }. \end{aligned}$$

(167)

Calculations in the text of the paper are very prolonged and it is impossible to note in each the case what formulae in this “Appendix” have been used. Therefore, we do not do it, explaining only a general direction of calculations. The information in this “Appendix” is quite enough to repeat our calculations without principal obstacles.

Appendix B: Necessary formulae in the Einstein–Gauss–Bonnet gravity

The action of the Einstein $D$-dimensional theory with a bare cosmological term $\Lambda _0$ corrected by the Gauss–Bonnet term (see, for example, [14]) is

$$\begin{aligned} S&= -\frac{1}{2\kappa _D}\int d^D x\hat{L}_{EGB} +\int d^D x{\hat{\mathcal{{L}}}}_{m} \nonumber \\&= -\frac{1}{2\kappa _D}\int d^D x \sqrt{-g} \left[R - 2\Lambda _0 + \alpha (RR)_{GB}\right] +\int d^D x{\hat{\mathcal{{L}}}}_{m}, \end{aligned}$$

(168)

$$\begin{aligned} (RR)_{GB}&\equiv R_{\alpha \beta \gamma \delta } R^{\alpha \beta \gamma \delta } - 4 R_{\alpha \beta } R^{\alpha \beta }+ R^2 , \end{aligned}$$

(169)

where $\kappa _D = 2\varOmega _{D-2}G_D> 0$ and $\alpha >0$; $G_D$ is the $D$-dimension Newton’s constant, $\varOmega _{D-2}$ is the area of a unit $(D-2)$-dimensional sphere, and we restrict ourselves by $\Lambda _0 \le 0$. The subscript ‘${}_{E}$’ is related to the pure Einstein part of the action (168), and the subscript ‘${}_{GB}$’ is related to the Gauss-Bonnet part connected with $\alpha$-coefficient.

To present the metric Lagrangian $\hat{L}_{EGB}$ in an explicitly covariant form ${\hat{\mathcal{{L}}}}_{EGB}$ one has to change partial derivatives $\partial _\mu$ of the dynamic metric $g_{\mu \nu }$ in the Riemannian tensor by the covariant derivatives $\overline{D}_\mu$. We use next useful formulae:

$$\begin{aligned} R^\lambda {}_{\tau \rho \sigma } = \overline{D}_\rho \Delta ^\lambda _{\tau \sigma } - \overline{D}_\sigma \Delta ^\lambda _{\tau \rho } + \Delta ^\lambda _{\rho \eta } \Delta ^\eta _{\tau \sigma } - \Delta ^\lambda _{\eta \sigma } \Delta ^\eta _{\tau \rho } + \overline{R}^\lambda {}_{\tau \rho \sigma } = \delta R^\lambda {}_{\tau \rho \sigma } + \overline{R}^\lambda {}_{\tau \rho \sigma }\,\nonumber \\ \end{aligned}$$

(170)

where

$$\begin{aligned} \Delta ^\alpha _{\mu \nu } = \Gamma ^\alpha _{\mu \nu } - \overline{\Gamma }^\alpha _{\mu \nu } = {\textstyle {1 \over 2}}g^{\alpha \rho }\left( \overline{D}_\mu g_{\rho \nu } + \overline{D}_\nu g_{\rho \mu } - \overline{D}_\rho g_{\mu \nu }\right)\, \end{aligned}$$

(171)

is the difference between the Christoffel symbols related to the dynamic $g_{\mu \nu }$ and the background $\overline{g}_{\mu \nu }$ metrics. It is useful the next relations also. Analogously to (163) one defines the covariant derivative related to the dynamic metric:

$$\begin{aligned} D_\alpha Q^A = \partial _\alpha Q^A + {\left. Q^A \right|}^\rho _\tau \Gamma _{\alpha \rho }^\tau . \end{aligned}$$

(172)

Then, comparing (172) and (163) one obtains

$$\begin{aligned} D_\alpha Q^A = \overline{D}_\alpha Q^A + {\left. Q^A \right|}^\rho _\tau \Delta _{\alpha \rho }^\tau . \end{aligned}$$

(173)

The coefficients ${n^*}$ and ${m^*}$ [see formulae (176) and (177) below], corresponding to the Lagrangian ${\hat{\mathcal{{L}}}}_{EGB}$ in (168), are necessary for calculating superpotentials. However, at the first it is useful to present the next derivatives:

$$\begin{aligned}&\frac{-2\kappa _D}{\sqrt{-g}}\frac{\partial {\hat{\mathcal{{L}}}}_{EGB}}{\partial g_{\mu \nu ;\alpha }} = \frac{-2\kappa _D}{\sqrt{-g}}\left(\frac{\partial {\hat{\mathcal{{L}}}}_{E}}{\partial g_{\mu \nu ;\alpha }}+ \frac{\partial {\hat{\mathcal{{L}}}}_{GB}}{\partial g_{\mu \nu ;\alpha }} \right)\nonumber \\&\quad = 2 \left[\Delta ^{\alpha }_{\sigma \rho }g^{\sigma [\rho }g^{\mu ]\nu } + g^{\alpha \sigma }\Delta ^{(\mu }_{\sigma \rho }g^{\nu )\rho } - g^{\alpha (\mu }\Delta ^{\nu )}_{\sigma \rho }g^{\sigma \rho } \right]\nonumber \\&\qquad \, +\ 4\alpha \left[2R^{\alpha \sigma \rho (\mu }\Delta ^{\nu )}_{\sigma \rho } -\Delta ^{\alpha }_{\sigma \rho }R^{\sigma \mu \nu \rho }\right]\nonumber \\&\qquad \,-\ 4\alpha \left[2 R^{\alpha \sigma }\Delta ^{(\mu }_{\sigma \rho }g^{\nu )\rho } - 2g^{\alpha (\mu }\Delta ^{\nu )}_{\sigma \rho }R^{\sigma \rho } + 2 g^{\alpha \sigma }\Delta ^{(\mu }_{\sigma \rho }R^{\nu )\rho } -2R^{\alpha (\mu }\Delta ^{\nu )}_{\sigma \rho }g^{\sigma \rho }\right.\nonumber \\&\qquad \,+\left. \Delta ^{\alpha }_{\sigma \rho }R^{\sigma \rho }g^{\mu \nu }+ \Delta ^{\alpha }_{\sigma \rho }g^{\sigma \rho }R^{\mu \nu } - 2\Delta ^{\alpha }_{\sigma \rho }R^{\sigma (\mu }g^{\nu )\rho }\right] \nonumber \\&\qquad \,+\ 4\alpha R \left[\Delta ^{\alpha }_{\sigma \rho }g^{\sigma [\rho }g^{\mu ]\nu } + g^{\alpha \sigma }\Delta ^{(\mu }_{\sigma \rho }g^{\nu )\rho } - g^{\alpha (\mu }\Delta ^{\nu )}_{\sigma \rho }g^{\sigma \rho } \right]\,; \end{aligned}$$

(174)

$$\begin{aligned}&\qquad \quad \; \frac{-2\kappa _D}{\sqrt{-g}}\frac{\partial {\hat{\mathcal{{L}}}}_{EGB}}{\partial g_{\mu \nu ;\alpha \beta }} = \frac{-2\kappa _D}{\sqrt{-g}}\left(\frac{\partial {\hat{\mathcal{{L}}}}_{E}}{\partial g_{\mu \nu ;\alpha \beta }}+ \frac{\partial {\hat{\mathcal{{L}}}}_{GB}}{\partial g_{\mu \nu ;\alpha \beta }} \right)\nonumber \\&\quad = \left[g^{\alpha (\mu }g^{\nu )\beta }- g^{\alpha \beta }g^{\mu \nu }\right]\nonumber \\&\qquad \,+\ 2\alpha \left[2R^{\alpha (\mu \nu )\beta } \!- \!4 R^{\alpha (\mu }g^{\nu )\beta } \!+\! 2g^{\mu \nu }R^{\alpha \beta } \!+\! 2g^{\alpha \beta }R^{\mu \nu } \!+\! R \left(g^{\alpha (\mu }g^{\nu )\beta }- g^{\alpha \beta }g^{\mu \nu } \right)\right].\nonumber \\ \end{aligned}$$

(175)

Substituting (174) and (175) into (61) and (62) one obtains

$$\begin{aligned} {\hat{n}^*}_{\sigma }{}^{\lambda \alpha \beta }&\!=\!&{}_{(E)}{\hat{n}^*}_{\sigma }{}^{\lambda \alpha \beta } \!+\! {}_{(GB)}{\hat{n}^*}_{\sigma }{}^{\lambda \alpha \beta } \nonumber \\&\!=\!&\frac{\sqrt{- g}}{2\kappa _D}\left\{ g^{\alpha \beta }\delta ^\lambda _\sigma \!-\!g^{\lambda (\alpha }\delta ^{\beta )}_\sigma \right\} \nonumber \\&\quad \!+\! \frac{\alpha \sqrt{-g}}{\kappa _D} \left\{ -2R_\sigma {}^{(\alpha \beta )\lambda } \!-\! 4 R_\sigma ^{\lambda }g^{\alpha \beta } \!+\! 4 R_\sigma ^{(\alpha }g^{\beta )\lambda } \!+\! R\left(g^{\alpha \beta }\delta ^\lambda _\sigma -g^{\lambda (\alpha }\delta ^{\beta )}_\sigma \right)\right\} . \nonumber \\&\end{aligned}$$

(176)

$$\begin{aligned} {\hat{m}^*}_{\sigma }{}^{\alpha \beta }&\!=\!&{}_{(E)}{\hat{m}^*}_{\sigma }{}^{\alpha \beta } +{}_{(GB)}{\hat{m}^*}_{\sigma }{}^{\alpha \beta } \nonumber \\&= -\frac{\sqrt{- g}}{2\kappa _D}\left[\delta ^\alpha _\sigma \Delta ^{\beta }_{\rho \tau }g^{\rho \tau } - 2 \Delta ^{\alpha }_{\sigma \rho }g^{\beta \rho } + \Delta ^{\rho }_{\rho \sigma } g^{\alpha \beta } \right] \nonumber \\&\quad + \frac{2\alpha \sqrt{-g}}{\kappa _D} \left[\ R^{\alpha \tau \rho }{}_\sigma \Delta ^{\beta }_{\tau \rho } -2R^{\alpha (\tau \beta )}{}_\rho \Delta ^{\rho }_{\tau \sigma } \right] \nonumber \\&\quad + \frac{4\alpha \sqrt{-g}}{\kappa _D}\left[4 g^{\rho [\alpha } R^{\beta ]}_{\tau }\Delta ^\tau _{\rho \sigma } + 2 R^{[\alpha }_\sigma g^{\tau ]\rho }\Delta ^{\beta }_{\tau \rho }+ 2 g^{\alpha [\beta } R^{\tau ]}_{\rho }\Delta ^{\rho }_{\tau \sigma } \right. \nonumber \\&\quad - \left. g^{\tau \beta }\left(\overline{D}_{(\tau } R^\alpha _{\sigma )} + R^{\rho }_{(\tau } \Delta ^{\alpha }_{\sigma )\rho } - R^{\alpha }_\rho \Delta ^{\rho }_{\tau \sigma }\right) \right] \nonumber \\&\quad - \frac{\alpha \sqrt{- g}}{\kappa _D}\left[\left(\delta ^\alpha _\sigma \Delta ^{\beta }_{\rho \tau }g^{\rho \tau } - 2 \Delta ^{\alpha }_{\sigma \rho }g^{\beta \rho } + \Delta ^{\rho }_{\rho \sigma } g^{\alpha \beta }\right)R - 2\delta ^{(\alpha }_\sigma g^{\tau )\beta } \partial _\tau R \right].\nonumber \\ \end{aligned}$$

(177)

Using the coefficients (176) and (177) in (71) one obtains for a pure Nœther canonical starred superpotential in EGB gravity

$$\begin{aligned} \hat{\imath }^{*\alpha \beta }&= {1\over \kappa _D}\big ({\hat{g}^{\rho [\alpha }\overline{D}_{\rho } \xi ^{\beta ]}} + \hat{g}^{\rho [\alpha }\Delta ^{\beta ]}_{\rho \sigma }\xi ^\sigma \big ) \nonumber \\&\quad - \frac{2\alpha \sqrt{-g}}{\kappa _D} \left\{ \Delta ^{\rho }_{\lambda \sigma }R_\rho {}^{\lambda \alpha \beta } + 4\Delta ^{\rho }_{\lambda \sigma }g^{\lambda [\alpha }R^{\beta ]}_\rho + \Delta ^{[\alpha }_{\rho \sigma }g^{\beta ]\rho }R\right\} \xi ^\sigma \nonumber \\&\quad - \frac{2\alpha \sqrt{-g}}{\kappa _D}\left\{ R_\sigma {}^{\lambda \alpha \beta } + 4 g^{\lambda [\alpha }R^{\beta ]}_\sigma + \delta _\sigma ^{[\alpha }g^{\beta ]\lambda }R \right\} \overline{D}_\lambda \xi ^\sigma , \end{aligned}$$

(178)

$$\begin{aligned} \overline{\hat{\imath }^{*\alpha \beta }}&= {1\over \kappa _D} \overline{D^{[\alpha } \hat{\xi }^{\beta ]}} - \frac{2\alpha }{\kappa _D}\left\{ \overline{\hat{R}}_\sigma {}^{\lambda \alpha \beta } + 4 \overline{g^{\lambda [\alpha }\hat{R}^{\beta ]}_\sigma } + \delta _\sigma ^{[\alpha } \overline{g^{\beta ]\lambda }\hat{R}} \right\} \overline{D}_\lambda \xi ^\sigma . \end{aligned}$$

(179)

To finalize forming a pure canonical superpotential one needs to choose the divergence. We prefer the choice induced by the Katz-Lifshits approach [19] instead of the choice in [17] (see discussions in [20–22]). Thus, we choose in (115)–(117)

$$\begin{aligned} \hat{d}^\alpha = {}_{(E)}\hat{d}^\alpha + {}_{(GB)}\hat{d}^\alpha = {2} \Delta ^{[\tau }_{\tau \beta }\hat{g}^{\alpha ]\beta } + 4 \alpha \left( \hat{R}_\rho {}^{\beta \tau \alpha } - 2\hat{R}^{[\tau }_\rho g^{\alpha ]\beta }- 2\delta ^{[\tau }_\rho \hat{R}^{\alpha ]\beta }+ \delta ^{[\tau }_\rho g^{\alpha ]\beta } \hat{R}\right)\Delta ^\rho _{\tau \beta }.\nonumber \\ \end{aligned}$$

(180)

Using the coefficients (176) and (177) in (95) one obtains the Belinfante corrected starred superpotential in EGB gravity

$$\begin{aligned} \hat{\imath }^{*\alpha \beta }_B&= {1\over \kappa _D}\left[\left( \delta ^{[\alpha }_\sigma \overline{D}_\lambda \hat{g}^{\beta ]\lambda } -\overline{D}^{[\alpha }\hat{g}^{\beta ]\rho }\overline{g}_{\rho \sigma } \right)\xi ^\sigma +\hat{g}^{\lambda [\alpha }\overline{D}_\lambda \xi ^{\beta ]}\right]\nonumber \\&+{\alpha \over \kappa _D}\overline{D}_\lambda \left\{ \hat{R}_\sigma {}^{\lambda \alpha \beta } + 4 g^{\lambda [\alpha }\hat{R}^{\beta ]}_\sigma +\left[2\hat{R}_\tau {}^{\rho \lambda [\alpha } -2\hat{R}^{\rho \lambda }{}_\tau {}^{[\alpha } - 8\hat{R}^\lambda _\tau g^{\rho [\alpha }\right.\right. \nonumber \\&+ \left.\left.4 \hat{R}^\rho _\tau g^{\lambda [\alpha } +4 g^{\rho \lambda } \hat{R}^{[\alpha }_\tau + 2\hat{R}\left( \delta ^\lambda _\tau g^{\rho [\alpha }- \delta ^\rho _\tau g^{\lambda [\alpha }\right)\right]\overline{g}^{\beta ]\tau }\overline{g}_{\rho \sigma } \right\} \xi ^\sigma \nonumber \\&-{2\alpha \over \kappa _D}\left\{ {\hat{R}}_\sigma {}^{\lambda \alpha \beta } + 4 {g^{\lambda [\alpha }\hat{R}^{\beta ]}_\sigma } + \delta _\sigma ^{[\alpha } g^{\beta ]\lambda }\hat{R} \right\} \overline{D}_\lambda \xi ^\sigma , \end{aligned}$$

(181)

$$\begin{aligned} \overline{\hat{\imath }^{*\alpha \beta }_{B}}&= {1\over \kappa _D}\overline{\hat{g}^{\lambda [\alpha }D_\lambda \xi ^{\beta ]}}-{2\alpha \over \kappa _D}\left\{ \overline{\hat{R}}_\sigma {}^{\lambda \alpha \beta } + 4 \overline{g^{\lambda [\alpha }\hat{R}^{\beta ]}_\sigma } + \delta _\sigma ^{[\alpha } \overline{g^{\beta ]\lambda }\hat{R}} \right\} \overline{D}_\lambda \xi ^\sigma . \end{aligned}$$

(182)