On the energy-momentum and spin tensors in the Riemann–Cartan space

Abstract

General classical theories of material fields in an arbitrary Riemann–Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann–Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.

Appendices

Appendix A: The condition for a Lagrangian to be a scalar

Let a Lagrangian

$$\begin{aligned} \fancyscript{L}= \fancyscript{L}(\mathbf {g}, \mathbf {R};\; \mathbf {T}, {\varvec{\nabla }}\mathbf {T},{\varvec{\nabla }}{\varvec{\nabla }}\mathbf {T};\; \varvec{\varphi }, {\varvec{\nabla }}\varvec{\varphi }, {\varvec{\nabla }}{\varvec{\nabla }}\varvec{\varphi }) \end{aligned}$$

(81)

be a generally covariant scalar. To reduce the formulae let us unite temporarily the fields $\mathbf {T}= \{ T^{\alpha }{}_{\beta \gamma }\}$ and $\varvec{\varphi }= \{ \varphi ^{a}\}$ into the unique set $\varvec{\phi }= \{ \phi ^{a}\}$:

$$\begin{aligned} \mathbf {T},\; \varvec{\varphi }\quad \rightarrow \quad \varvec{\phi }= \{ \mathbf {T}, \varvec{\varphi }\}. \end{aligned}$$

Then

$$\begin{aligned} \fancyscript{L}= \fancyscript{L}(\mathbf {g}, \mathbf {R};\; \varvec{\phi }, {\varvec{\nabla }}\varvec{\phi }, {\varvec{\nabla }}{\varvec{\nabla }}\varvec{\phi }). \end{aligned}$$

In accordance with the definition of the scalar its total variation $\bar{\delta }\fancyscript{L}$ induced by an infinitesimal diffeomorphism

$$\begin{aligned} x^\mu \quad \rightarrow \quad x'^\mu = x^\mu + \delta x^{\mu }(x), \end{aligned}$$

(82)

is equal to zero:

$$\begin{aligned} \bar{\delta }\fancyscript{L}\mathop {=}\limits ^{def}\fancyscript{L}'(x') - \fancyscript{L}(x) = 0. \end{aligned}$$

Taking into account the connection between the total $\bar{\delta }$ and the functional $\delta$ variations

$$\begin{aligned} \bar{\delta }= \delta x^{\lambda }{\partial _{\lambda }}{}+ \delta , \end{aligned}$$

(83)

we find

$$\begin{aligned} \delta x^{\lambda }{\partial _{\lambda }}{}\fancyscript{L}+ \delta \fancyscript{L}= 0. \end{aligned}$$

(84)

Let us compute $\delta \fancyscript{L}$. It is evidently

$$\begin{aligned}&\delta \fancyscript{L}= \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial g_{\alpha \beta }}\delta g_{\alpha \beta }+ \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}\delta R^{\alpha }{}_{\beta \gamma \delta }\right\} + \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \phi ^{a}}\delta \phi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\phi ^{a})}\delta \left( {\nabla _{\kappa }}{}\phi ^{a}\right) \right. \nonumber \\&\quad \qquad \;\, \left. + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a})}\delta \left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right) \right\} = \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial g_{\alpha \beta }}\left[ -\delta x^{\lambda }{\partial _{\lambda }}{}g_{\alpha \beta }+ \bar{\delta }g_{\alpha \beta }\right] + \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}\right. \nonumber \\&\quad \qquad \;\left. \times \left[ -\delta x^{\lambda }{\partial _{\lambda }}{}R^{\alpha }{}_{\beta \gamma \delta }+ \bar{\delta }R^{\alpha }{}_{\beta \gamma \delta }\right] \right\} + \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \phi ^{a}}\left[ -\delta x^{\lambda }{\partial _{\lambda }}{}\phi ^{a}+ \bar{\delta }\phi ^{a}\right] + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\phi ^{a})}\right. \nonumber \\&\quad \qquad \,\; \times \left[ -\delta x^{\lambda }{\partial _{\lambda }}{}\left( {\nabla _{\kappa }}{}\phi ^{a}\right) + \bar{\delta }\left( {\nabla _{\kappa }}{}\phi ^{a}\right) \right] \nonumber \\&\quad \qquad \;\,\left. + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a})}\left[ -\delta x^{\lambda }{\partial _{\lambda }}{}\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right) + \bar{\delta }\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right) \right] \right\} , \end{aligned}$$

(85)

where at the second step we used the connection between the total and the functional variations (83). In the last formula ${\partial _{}}{}^*\fancyscript{L}/{\partial _{}}{}g_{\beta \gamma }$ means explicit derivative with respect to $g_{\beta \gamma }$, that is the differentiation is provided only with respect $g_{\beta \gamma }$, which are not included in $\mathbf {R}$ and ${\varvec{\nabla }}$; analogously, ${\partial _{}}{}^*\fancyscript{L}/{\partial _{}}{}\phi ^{a}$ means differentiation only with respect to $\phi ^{a}$, which is not included in ${\varvec{\nabla }}\varvec{\phi }$ and ${\varvec{\nabla }}{\varvec{\nabla }}\varvec{\phi }$. Note now that

$$\begin{aligned}&\left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial g_{\alpha \beta }}{\partial _{\lambda }}{}g_{\alpha \beta }+ \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}{\partial _{\lambda }}{}R^{\alpha }{}_{\beta \gamma \delta }\right\} + \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \phi ^{a}}{\partial _{\lambda }}{}\phi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\phi ^{a})}{\partial _{\lambda }}{}\left( {\nabla _{\kappa }}{}\phi ^{a}\right) \right. \nonumber \\&\quad \left. + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a})}{\partial _{\lambda }}{}\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right) \right\} = {\partial _{\lambda }}{}\fancyscript{L}.\quad \end{aligned}$$

(86)

Next, take into account tensorial nature of the quantities $\{ g_{\alpha \beta }\}$, $\{ R^{\alpha }{}_{\beta \gamma \delta }\}$, $\{ \phi ^{a}\}$, $\{ {\nabla _{\kappa }}{}\phi ^{a}\}$, $\{ {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\}$. Then, by definition, for the infinitesimal diffeomorphisms (82) we have:

$$\begin{aligned} \bar{\delta }g_{\alpha \beta }&= (\Delta ^{\sigma }{}_{\rho })\left. _{\alpha \beta }\right| ^{\eta \zeta }g_{\eta \zeta }\times {\partial _{\sigma }}{}\delta x^{\rho };\end{aligned}$$

(87)

$$\begin{aligned} \bar{\delta }R^{\alpha }{}_{\beta \gamma \delta }&= (\Delta ^{\sigma }{}_{\rho })\left. ^{\alpha }{}_{\beta \gamma \delta }\right| _{\eta }{}^{\zeta \varphi }{}^\xi R^{\eta }{}_{\zeta \varphi \xi }\times {\partial _{\sigma }}{}\delta x^{\rho };\end{aligned}$$

(88)

$$\begin{aligned} \bar{\delta }\phi ^{a}&= (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}\phi ^{b}\times {\partial _{\sigma }}{}\delta x^{\rho };\end{aligned}$$

(89)

$$\begin{aligned} \bar{\delta }\left( {\nabla _{\kappa }}{}\phi ^{a}\right)&= \left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}\phi ^{b}- \left( \delta ^{\sigma }_{\kappa }\delta ^{\pi }_{\rho }\right) {\nabla _{\pi }}{}\phi ^{a}\right] \times {\partial _{\sigma }}{}\delta x^{\rho };\end{aligned}$$

(90)

$$\begin{aligned} \bar{\delta }\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right)&= \left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{b}- \left( \delta ^{\sigma }_{\kappa }\delta ^{\pi }_{\rho }\right) {\nabla _{\pi }}{}{\nabla _{\varepsilon }}{}\phi ^{a}- \left( \delta ^{\sigma }_{\varepsilon }\delta ^{\pi }_{\rho }\right) {\nabla _{\kappa }}{}{\nabla _{\pi }}{}\phi ^{a}\right] {\partial _{\sigma }}{}\delta x^{\rho },\nonumber \\ \end{aligned}$$

(91)

where $\{ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}\}$ are the Belinfante-Rosenfeld symbols (see, for example, Ref. [14]). Using in the right hand side of the formula (85) the formulae (86)–(91), we find

$$\begin{aligned}&\delta \fancyscript{L}= - \left( {\partial _{\lambda }}{}\fancyscript{L}\right) \times \delta x^{\lambda }\\&\quad + \left( \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial g_{\alpha \beta }}(\Delta ^{\sigma }{}_{\rho })\left. _{\alpha \beta }\right| ^{\eta \xi }g_{\eta \xi }+ \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}(\Delta ^{\sigma }{}_{\rho })\left. ^{\alpha }{}_{\beta \gamma \delta }\right| _{\eta }{}^{\zeta \varphi }{}^\xi R^{\eta }{}_{\zeta \varphi \xi }\right\} \right. \nonumber \\&\quad \left. + \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \phi ^{a}}(\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}\phi ^{b}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\phi ^{a})}\left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}\phi ^{b}- \delta ^{\sigma }_{\kappa }{\nabla _{\rho }}{}\phi ^{a}\right] \right. \right. \nonumber \\&\quad \left. \left. + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a})}\left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{b}- \delta ^{\sigma }_{\kappa }{\nabla _{\rho }}{}{\nabla _{\varepsilon }}{}\phi ^{a}- \delta ^{\sigma }_{\varepsilon }{\nabla _{\kappa }}{}{\nabla _{\rho }}{}\phi ^{a}\right] \right\} \right) \times {\partial _{\sigma }}{}\delta x^{\rho }. \end{aligned}$$

Substituting this expression into the formula (84) and taking into account the arbitrariness of the vector field $\{ \delta x^{\mu }(x) \}$, we obtain

(92)

Appendix B: The calculation of ${\nabla _{\nu }}{}\fancyscript{L}$

For the transformations presented in the main text of the paper one needs the explicit expression for the ${\nabla _{\nu }}{}\fancyscript{L}$. Let us calculate it. Because $\fancyscript{L}$ is a generally covariant scalar of the type (81) one has

(93)

Using the expressions for the covariant derivatives $\{ {\nabla _{\nu }}{}g_{\alpha \beta }\}$, $\{ {\nabla _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }\}$, $\{ {\nabla _{\nu }}{}\phi ^{a}\}$, $\{ {\nabla _{\nu }}{}{\nabla _{\kappa }}{}\phi ^{a}\}$, $\{ {\nabla _{\nu }}{}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\}$

$$\begin{aligned} {\nabla _{\nu }}{}g_{\alpha \beta }&= {\partial _{\nu }}{}g_{\alpha \beta }+ \Gamma ^{\rho }{}_{\sigma \nu }(\Delta ^{\sigma }{}_{\rho })\left. _{\alpha \beta }\right| ^{\eta \zeta }g_{\eta \zeta };\end{aligned}$$

(94)

$$\begin{aligned} {\nabla _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }&= {\partial _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }+ \Gamma ^{\rho }{}_{\sigma \nu }(\Delta ^{\sigma }{}_{\rho })\left. ^{\alpha }{}_{\beta \gamma \delta }\right| _{\eta }{}^{\zeta \varphi }{}^\xi \; R^{\eta }{}_{\zeta \varphi \xi };\end{aligned}$$

(95)

$$\begin{aligned} {\nabla _{\nu }}{}\phi ^{a}&= {\partial _{\nu }}{}\phi ^{a}+ \Gamma ^{\rho }{}_{\sigma \nu }(\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}\phi ^{b};\end{aligned}$$

(96)

$$\begin{aligned} {\nabla _{\nu }}{}\left( {\nabla _{\kappa }}{}\phi ^{a}\right)&= {\partial _{\nu }}{}\left( {\nabla _{\kappa }}{}\phi ^{a}\right) + \Gamma ^{\rho }{}_{\sigma \nu }\left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}\phi ^{b}- \delta ^{\sigma }_{\kappa }{\nabla _{\rho }}{}\phi ^{a}\right] ;\end{aligned}$$

(97)

$$\begin{aligned} {\nabla _{\nu }}{}\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right)&= {\partial _{\nu }}{}\left( {\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{a}\right) \nonumber \\&+ \Gamma ^{\rho }{}_{\sigma \nu }\left[ (\Delta ^{\sigma }{}_{\rho })\left. ^{a}\right| _{b}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\phi ^{b}- \delta ^{\sigma }_{\kappa }{\nabla _{\rho }}{}{\nabla _{\varepsilon }}{}\phi ^{a}- \delta ^{\sigma }_{\varepsilon }{\nabla _{\kappa }}{}{\nabla _{\rho }}{}\phi ^{a}\right] ,\qquad \end{aligned}$$

(98)

we find the partial derivatives $\{ {\partial _{\nu }}{}g_{\alpha \beta }\}$, $\{ {\partial _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }\}$, $\{ {\partial _{\nu }}{}\phi ^{a}\}$, $\{ {\partial _{\nu }}{}({\nabla _{\kappa }}{}\phi ^{a}) \}$, $\{ {\partial _{\nu }}{}({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{} \phi ^{a}) \}$ and substitute them into the formula (93). After a rearrangement of items we obtain

(99)

Taking into account in this relationship the identity (92) and metric-compatible condition ${\nabla _{\nu }}{}g_{\alpha \beta }= 0$, we find the search expression:

(100)

For the cases of interest this expression takes the following forms:

• 1. The case of minimal coupling, $\fancyscript{L}= \fancyscript{L}(\mathbf {g};\; \varvec{\varphi }, {\varvec{\nabla }}\varvec{\varphi }, {\varvec{\nabla }}{\varvec{\nabla }}\varvec{\varphi })$,

  $$\begin{aligned} {\nabla _{\nu }}{}\fancyscript{L}= \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \varphi ^{a}}{\nabla _{\nu }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a}\right\} ;\nonumber \\ \end{aligned}$$

  (101)

• 2. The case of non-minimal $\mathbf {g}$-coupling, $\fancyscript{L}= \fancyscript{L}(\mathbf {g}, \mathbf {R};\; \varvec{\varphi }, {\varvec{\nabla }}\varvec{\varphi }, {\varvec{\nabla }}{\varvec{\nabla }}\varvec{\varphi })$,

  $$\begin{aligned}&{\nabla _{\nu }}{}\fancyscript{L}= \left\{ \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}{\nabla _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }\right\} \nonumber \\&\quad + \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \varphi ^{a}}{\nabla _{\nu }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a}\right\} ;\qquad \end{aligned}$$

  (102)

• 3. The case of non-minimal $\mathbf {g}$- and $\mathbf {T}$-coupling, $\fancyscript{L}= \fancyscript{L}(\mathbf {g}, \mathbf {R};\; \mathbf {T}, {\varvec{\nabla }}\mathbf {T},{\varvec{\nabla }}{\varvec{\nabla }}\mathbf {T};\; \varvec{\varphi }, {\varvec{\nabla }}\varvec{\varphi }, {\varvec{\nabla }}{\varvec{\nabla }}\varvec{\varphi })$,

  $$\begin{aligned} {\nabla _{\nu }}{}\fancyscript{L}&= \left\{ \displaystyle \frac{\partial \fancyscript{L}}{\partial R^{\alpha }{}_{\beta \gamma \delta }}{\nabla _{\nu }}{}R^{\alpha }{}_{\beta \gamma \delta }\right\} \nonumber \\&+ \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial \varphi ^{a}}{\nabla _{\nu }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}\varphi ^{a}+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a})}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}\varphi ^{a}\right\} \nonumber \\&+ \left\{ \displaystyle \frac{\partial ^*\fancyscript{L}}{\partial T^{\alpha }{}_{\beta \gamma }}{\nabla _{\nu }}{}T^{\alpha }{}_{\beta \gamma }+ \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{} T^{\alpha }{}_{\beta \gamma })}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}T^{\alpha }{}_{\beta \gamma }\right. \nonumber \\&\left. + \displaystyle \frac{\partial \fancyscript{L}}{\partial ({\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{} T^{\alpha }{}_{\beta \gamma })}{\nabla _{\nu }}{}{\nabla _{\kappa }}{}{\nabla _{\varepsilon }}{}T^{\alpha }{}_{\beta \gamma }\right\} . \end{aligned}$$

  (103)

Appendix C: The transformation of the expression $\left( \frac{1}{2} G^{\alpha \beta \gamma \delta }{\nabla _{\nu }}{}R_{\alpha \beta \gamma \delta }\right)$

Transform the expression $\frac{1}{2} G^{\alpha \beta \gamma \delta }{\nabla _{\nu }}{}R_{\alpha \beta \gamma \delta }$ as follows. Substituting the Ricci identity in the form

$$\begin{aligned} {\nabla _{\nu }}{}R_{\alpha \beta \gamma \delta }\equiv - \left( {\nabla _{\gamma }}{}R_{\alpha \beta \delta \nu }+ {\nabla _{\delta }}{}R_{\alpha \beta \nu \gamma }+ R_{\alpha \beta \varepsilon \nu }T^{\varepsilon }{}_{\gamma \delta }+ R_{\alpha \beta \varepsilon \gamma }T^{\varepsilon }{}_{\delta \nu }+ R_{\alpha \beta \varepsilon \delta }T^{\varepsilon }{}_{\nu \gamma }\right) , \end{aligned}$$

one obtains

$$\begin{aligned} \frac{1}{2} G^{\alpha \beta \gamma \delta }{\nabla _{\nu }}{}R_{\alpha \beta \gamma \delta }&= -G^{\alpha \beta \gamma \delta }{\nabla _{\gamma }}{}R_{\alpha \beta \delta \nu }- \left( G^{\alpha \beta \gamma \delta }R_{\alpha \beta \varepsilon \gamma }\right) T^{\varepsilon }{}_{\delta \nu }\\&- \frac{1}{2} \left( G^{\alpha \beta \gamma \delta }T^{\varepsilon }{}_{\gamma \delta }\right) R_{\alpha \beta \varepsilon \nu }. \end{aligned}$$

Differentiating by parts the first item in the right hand side one finds

Appendix D: The transformation of the expression $\left( \frac{-1}{2} b^{\gamma \beta }{}_{\alpha }{\nabla _{\nu }}{}T^{\alpha }{}_{\beta \gamma }\right)$

Let $\{ b^{\gamma \beta \alpha }\} \mathop {=}\limits ^{def}\{ \Delta ^{\overline{\gamma \beta \alpha }}_{\pi \rho \sigma }s^{\pi ,\, \rho \sigma }\}$, where $\{ s^{\pi ,\, [\rho \sigma ]} = s^{\pi ,\, \rho \sigma }\}$, be an arbitrary tensor with such a symmetry. Then $b^{[\gamma \beta ] \alpha }= b^{\gamma \beta \alpha }$. Based on this, transform the expression $\left( \frac{-1}{2} b^{\gamma \beta }{}_{\alpha }{\nabla _{\nu }}{}T^{\alpha }{}_{\beta \gamma }\right)$ as follows.

• 1. Substituting the Ricci identity in the form

  $$\begin{aligned}&{\nabla _{\nu }}{}T^{\alpha }{}_{\beta \gamma }\equiv R^{\alpha }{}_{\nu \beta \gamma }+ R^{\alpha }{}_{\beta \gamma \nu }+ R^{\alpha }{}_{\gamma \nu \beta }\nonumber \\&\qquad - \left( {\nabla _{\beta }}{}T^{\alpha }{}_{\gamma \nu }+ {\nabla _{\gamma }}{}T^{\alpha }{}_{\nu \beta }+ T^{\alpha }{}_{\lambda \nu }T^{\lambda }{}_{\beta \gamma }+ T^{\alpha }{}_{\lambda \beta }T^{\lambda }{}_{\gamma \nu }+ T^{\alpha }{}_{\lambda \gamma }T^{\lambda }{}_{\nu \beta }\right) \nonumber \\&\quad = R^{\alpha }{}_{\nu \beta \gamma }+ 2R^{\alpha }{}_{[\beta \gamma ] \nu }- 2{\nabla _{[\beta }}{} T^{\alpha }{}_{\gamma ] \nu }- T^{\alpha }{}_{\lambda \nu }T^{\lambda }{}_{\beta \gamma }- 2T^{\alpha }{}_{\lambda [\beta } T^{\lambda }{}_{\gamma ] \nu }, \end{aligned}$$

  one obtains

  $$\begin{aligned}&-\displaystyle \frac{1}{2} b^{\gamma \beta }{}_{\alpha }{\nabla _{\nu }}{}T^{\alpha }{}_{\beta \gamma }\equiv -\displaystyle \frac{1}{2} b^{\gamma \beta }{}_{\alpha }R^{\alpha }{}_{\nu \beta \gamma }- b^{\gamma \beta }{}_{\alpha }R^{\alpha }{}_{\beta \gamma \nu }+ b^{\gamma \beta }{}_{\alpha }{\nabla _{\beta }}{}T^{\alpha }{}_{\gamma \nu }\nonumber \\&\quad + \displaystyle \frac{1}{2} b^{\gamma \beta }{}_{\alpha }T^{\alpha }{}_{\lambda \nu }T^{\lambda }{}_{\beta \gamma }+ b^{\gamma \beta }{}_{\alpha }T^{\alpha }{}_{\lambda \beta }T^{\lambda }{}_{\gamma \nu }. \end{aligned}$$

  (104)

• 2. Turn to the first term on the right hand side of (104). Then, recall the identity (C2) in the Appendix C.1 of the Ref. [15]:

  $$\begin{aligned} {\mathop {\nabla }\limits ^{*}_{\mu }}{}\left[ {\mathop {\nabla }\limits ^{*}_{\eta }}{}\theta _{\nu }{}^{\mu \eta }+ \displaystyle \frac{1}{2}\theta _{\nu }{}^{\rho \sigma }T^{\mu }{}_{\rho \sigma }\right] \equiv -\displaystyle \frac{1}{2} R^{\lambda }{}_{\nu \rho \sigma }\theta _{\lambda }{}^{\rho \sigma }, \end{aligned}$$

  change here $\theta _{\nu }{}^{\mu \eta }= b^{\mu \eta }{}_{\nu }$ and obtain for this term:

  $$\begin{aligned}-\displaystyle \frac{1}{2} b^{\gamma \beta }{}_{\alpha }R^{\alpha }{}_{\nu \beta \gamma }= -{\mathop {\nabla }\limits ^{*}_{\mu }}{}\left[ {\mathop {\nabla }\limits ^{*}_{\eta }}{}b^{\mu \eta }{}_{\nu }+ \displaystyle \frac{1}{2} b^{\varepsilon \eta }{}_{\nu }T^{\mu }{}_{\varepsilon \eta }\right] . \end{aligned}$$

• 3. The second term on the right hand side of (104) is equal to

  $$\begin{aligned}&-b^{\gamma \beta }{}_{\alpha }R^{\alpha }{}_{\beta \gamma \nu }= -b^{\gamma \beta \alpha }R_{\alpha \beta \gamma \nu }= -\Delta ^{\overline{\gamma \beta \alpha }}_{\pi \rho \sigma }s^{\pi ,\, \rho \sigma }R_{\alpha \beta \gamma \nu }\nonumber \\&\quad \quad = -\displaystyle \frac{1}{2} \left( s^{\beta ,\, \gamma \alpha }+ s^{\alpha ,\, \gamma \beta }- s^{\gamma ,\, \beta \alpha }\right) R_{\alpha \beta \gamma \nu }\nonumber \\&\quad \quad = \left( s^{(\alpha ,\, \beta ) \gamma }- \frac{1}{2} s^{\gamma ,\, \alpha \beta }\right) R_{\alpha \beta \gamma \nu }= -\displaystyle \frac{1}{2} s^{\pi ,\, \rho \sigma }R_{\rho \sigma \pi \nu }; \end{aligned}$$

• 4. Using the differentiation by part in the third term on the right hand side of (104), one finds

  $$\begin{aligned}b^{\gamma \beta }{}_{\alpha }{\nabla _{\beta }}{}T^{\alpha }{}_{\gamma \nu }= -{\mathop {\nabla }\limits ^{*}_{\mu }}{}\left( b^{\mu \beta }{}_{\alpha }T^{\alpha }{}_{\beta \nu }\right) - \left( {\mathop {\nabla }\limits ^{*}_{\eta }}{}b^{\mu \eta }{}_{\lambda }\right) T^{\lambda }{}_{\mu \nu }; \end{aligned}$$

• 5. At last, one rewrites fourth and fifth terms on the right hand side of (104), respectively, as

  $$\begin{aligned}\frac{1}{2} b^{\gamma \beta }{}_{\alpha }T^{\alpha }{}_{\lambda \nu }T^{\lambda }{}_{\beta \gamma }= -\frac{1}{2} \left( b^{\varepsilon \eta }{}_{\lambda }T^{\mu }{}_{\varepsilon \eta }\right) T^{\lambda }{}_{\mu \nu }\end{aligned}$$

  and

  $$\begin{aligned}b^{\gamma \beta }{}_{\alpha }T^{\alpha }{}_{\lambda \beta }T^{\lambda }{}_{\gamma \nu }= - \left( b^{\mu \beta }{}_{\alpha }T^{\alpha }{}_{\beta \lambda }\right) T^{\lambda }{}_{\mu \nu }. \end{aligned}$$

Combining the results of the points 2–10 in the formula (104), one obtains the search identity:

(105)