Torsion/nonmetricity duality in f(R) gravity

Abstract

Torsion and nonmetricity are inherent ingredients in modifications of Eintein’s gravity that are based on affine spacetime geometries. In the context of pure f(R) gravity we discuss here, in some detail, the relatively unnoticed duality between torsion and nonmetricity. Our novel suggestion is that torsion and nonmetricity are physically equivalent properties of spacetimes having nontrivial Weyl structure. Our main example is $R^2$ gravity where torsion and nonmetricity are related by projective transformations corresponding to a redefinition of the affine parameters for autoparallels, As a simple consequence we show that both torsion and nonmetricity can act as geometric sources of accelerated expansion in a spatially homogenous cosmological model within $R^2$ gravity. We briefly discuss possible implications of our results.

Appendix A

Denoting with $\Gamma ^{\rho }_{\;\beta \mu }$ the affine connection, covariant derivatives (of e.g. a mixed tensor) are defined as

$$\begin{aligned} \nabla _{\mu }T^{\alpha }_{\,\,\beta }=\partial _{\mu }T^{\alpha }_{\,\,\beta }+\Gamma ^{\alpha }_{\,\,\rho \mu }T^{\rho }_{\,\,\beta }-\Gamma ^{\rho }_{\,\,\beta \mu }T^{\alpha }_{\,\,\rho } \end{aligned}$$

(A.1)

Notice the position of the various indices in (A.1). The Riemann $R^{\mu }_{\,\,\nu \alpha \beta }$ and torsion tensors $S_{\alpha \beta }^{\,\,\nu }$ are defined via the commutator of two covariant derivatives acting on a vector $u^{\mu }$ as⁶

$$\begin{aligned}&[\nabla _{\alpha },\nabla _{\beta }]u^{\mu }=2\nabla _{[\alpha } \nabla _{\beta ]}u^{\mu }=R^{\mu }_{\,\,\nu \alpha \beta } u^{\nu }+2 S_{\alpha \beta }^{\,\,\nu }\nabla _{\nu }u^{\mu } \end{aligned}$$

(A.2)

$$\begin{aligned}&R^{\mu }_{\,\,\nu \alpha \beta }:=2\partial _{[\alpha }\Gamma ^{\mu }_{\,\,|\nu |\beta ]}+2\Gamma ^{\mu }_{\,\,\rho [\alpha }\Gamma ^{\rho }_{\,\,|\nu |\beta ]},\quad S_{\alpha \beta }^{\,\,\nu }:=\Gamma ^{\nu }_{\,\,[\alpha \beta ]} \end{aligned}$$

(A.3)

With a generic connection the Riemann tensor is antisymmetric in its last two indices. So, we can generically define the independent contractions giving the Ricci tensor

$$\begin{aligned} R_{\nu \beta }:= R^{\mu }_{\,\,\nu \mu \beta } = 2\partial _{[\mu }\Gamma ^{\mu }_{\,\,|\nu |\beta ]}+2\Gamma ^{\mu }_{\,\,\rho [\mu }\Gamma ^{\rho }_{\,\,|\nu |\beta ]} \end{aligned}$$

(A.4)

which, is not symmetric in $\nu ,\beta$ in general, and the homothetic curvature tensor

$$\begin{aligned} \hat{R}_{\alpha \beta }:=R^{\mu }_{\,\,\mu \alpha \beta }=2\partial _{[\alpha }\Gamma ^{\mu }_{\,\,|\mu |\beta ]}=\partial _{\alpha }\Gamma ^{\mu }_{\,\,\mu \beta }-\partial _{\beta }\Gamma ^{\mu }_{\,\,\mu \alpha } \end{aligned}$$

(A.5)

The above discussion did not require a metric. If a symmetric metric $g_{\mu \nu }$ is present we can also define a third independent contraction of the Riemann tensor as

$$\begin{aligned} \check{R}^{\mu }_{\;\;\beta } =g^{\nu \alpha }R^{\mu }_{\,\,\nu \alpha \beta }:=2 g^{\nu \alpha }\partial _{[\alpha }\Gamma ^{\mu }_{\,\,|\nu |\beta ]}+2 g^{\nu \alpha } \Gamma ^{\mu }_{\,\,\rho [\alpha }\Gamma ^{\rho }_{\,\,|\nu |\beta ]} \end{aligned}$$

(A.6)

Moreover, the Ricci scalar is still uniquely defined since

$$\begin{aligned} \check{R}=\check{R}^{\alpha }_{\,\,\alpha }=R^{\alpha }_{\,\,\beta \mu \alpha }g^{\beta \mu }=-R^{\alpha }_{\,\,\beta \alpha \mu }g^{\beta \mu }=-R_{\beta \mu }g^{\beta \nu }=-R \end{aligned}$$

(A.7)

There are two independent contractions of the torsion tensor giving the torsion vector $S_\mu$ and the torsion pseudovector $\tilde{S}_\mu$ respectively as

$$\begin{aligned} S_{\mu } \equiv S_{\mu \lambda }^{\lambda },\quad \tilde{S}^{\mu } \equiv \epsilon ^{\mu \nu \rho \sigma }S_{\nu \rho \sigma } \end{aligned}$$

(A.8)

The non-metricity tensor is defined as

$$\begin{aligned} Q_{\alpha \mu \nu } :=-\nabla _{\alpha }g_{\mu \nu } \end{aligned}$$

(A.9)

It depends both on the metric tensor and the connection i.e. using the definition of the covariant derivative we have

$$\begin{aligned} Q_{\alpha \mu \nu } :=-\nabla _{\alpha }g_{\mu \nu } =-\partial _{\alpha }g_{\mu \nu }+\Gamma ^{\rho }_{\mu \alpha }g_{\rho \nu }+\Gamma ^{\rho }_{\nu \alpha }g_{\mu \rho } \end{aligned}$$

(A.10)

from which, the dependence on $\Gamma ^{\lambda }_{\mu \nu }$ and $g_{\mu \nu }$ is apparent. Raising the last two indices we obtain

$$\begin{aligned} Q_{\rho }^{,\alpha \beta }=\nabla _{\rho }g^{\alpha \beta } \end{aligned}$$

(A.11)

From the non-metricity tensor we can construct two independent vectors. The Weyl vector is defined as

$$\begin{aligned} Q_{\alpha }:= g^{\mu \nu }Q_{\alpha \mu \nu }=Q_{\alpha \,\mu }^{\mu } \end{aligned}$$

(A.12)

A second nonmetricity vector vector can also be defined and it is given by

$$\begin{aligned} \tilde{Q}_{\nu }:= g^{\mu \alpha }Q_{\alpha \mu \nu }=Q^{\mu }_{\,\,\mu \nu }=-g^{\mu \alpha }\nabla _{\alpha }g_{\mu \nu } \end{aligned}$$

(A.13)

Finally, using the results above one can decompose the general affine connection as

$$\begin{aligned} \Gamma ^{\lambda }_{\,\,\mu \nu }=\tilde{\Gamma }^{\lambda }_{\,\,\mu \nu }+N^\lambda _{\,\,\mu \nu }, \end{aligned}$$

(A.14)

where the distortion tensor $N^\lambda _{\,\,\mu \nu }$ is given by

$$\begin{aligned} N^\lambda _{\,\,\mu \nu }&=\underbrace{\frac{1}{2}g^{\alpha \lambda }(Q_{\mu \nu \alpha }+Q_{\nu \alpha \mu }-Q_{\alpha \mu \nu })}_\text {deflection} -\underbrace{g^{\alpha \lambda }(S_{\alpha \mu \nu }+S_{\alpha \nu \mu }-S_{\mu \nu \alpha })}_\text {contorsion} \end{aligned}$$

(A.15)

$$\begin{aligned}&=\frac{1}{2}g^{\alpha \lambda }\left[ (Q_{\mu \nu \alpha }+2S_{\mu \nu \alpha })+(Q_{\nu \alpha \mu }+2S_{\nu \alpha \mu })-(Q_{\alpha \mu \nu }+2S_{\alpha \mu \nu })\right] \end{aligned}$$

(A.16)

where the Levi-Civita connection is given by the usual Christoffel symbols

$$\begin{aligned} \tilde{\Gamma }^{\lambda }_{\,\,\mu \nu }:=\frac{1}{2}g^{\alpha \lambda }(\partial _{\mu }g_{\nu \alpha }+\partial _{\nu }g_{\alpha \mu }-\partial _{\alpha }g_{\mu \nu }) \end{aligned}$$

(A.17)

Some useful identities are

$$\begin{aligned} Q_{\nu \alpha \mu }=N_{(\alpha \mu )\nu },\quad S_{\mu \nu \alpha }=N_{\alpha [\mu \nu ]},\quad N_{[\alpha \mu \nu ]}=S_{[\mu \nu \alpha ]}=S_{[\alpha \mu \nu ]} \end{aligned}$$

(A.18)

where the latter refers to the totally antisymmetric part of the distortion. as can be easily checked. Notice also only the symmetric part $N^{\lambda }_{\,\,(\mu \nu )}$ contributes to the autoparallel equation $u^\nu \nabla _\nu u^\mu =0$. This is equal to

$$\begin{aligned} N^{\lambda }_{\,\,(\mu \nu )}=g^{\alpha \lambda }\left( Q_{(\mu \nu )\alpha }-\frac{1}{2}Q_{\alpha \mu \nu }\right) -2g^{\alpha \lambda } S_{\alpha (\mu \nu )}. \end{aligned}$$

(A.19)

For vector torsion and nonmetricity as in (10) this coincides with (9). Notice that a completely antisymmetric torsion $(S_{\alpha \mu \nu }=S_{[\alpha \mu \nu ]})$ has no effect on autoparallels.