Gravity and spin with a nonsymmetric metric tensor

Abstract

It is shown the antisymmetric part of the metric tensor is the potential for the spin field. Various metricity conditions are discussed and comparisons are made to other theories, including Einstein’s. It is shown in the weak field limit the theory reduces to one with a symmetric metric tensor and totally antisymmetric torsion.

Appendices

Appendices

For completeness we derive the field equations for the case of zero metricity. In this case (49) and (50) still hold, but $E^{\lambda \omega }_{\ \ \sigma }$ is different. In fact it is,

$$\begin{aligned} E^{\lambda \omega }_{\ \ \sigma }= & {} g^{\alpha \beta }\Gamma _{\alpha \beta }^{\ \ \ \omega }\delta ^\lambda _\sigma +g^{\lambda \omega }\Gamma _{\gamma \sigma }^{\ \ \ \gamma } -g^{\lambda \beta }\Gamma _{\sigma \beta }^{\ \ \ \omega }-g^{\alpha \omega }\Gamma _{\alpha \sigma }^{\ \ \ \lambda } -\left( \frac{\sqrt{-g},_\sigma }{\sqrt{-g}}g^{\lambda \omega }+g^{\lambda \omega },_\sigma \right) \nonumber \\&+\,\delta ^\lambda _\eta \left( \frac{\sqrt{-g},_\sigma }{\sqrt{-g}}g^{\eta \omega }+g^{\eta \omega },_\eta \right) \end{aligned}$$

(89)

which, with (28) set equal to zero reduces to

$$\begin{aligned} E^{\lambda \omega }_{\ \ \sigma }=2\delta ^\lambda _\sigma g^{\eta \omega }S_\eta +2g^{\theta \omega }S_{\sigma \theta }^{\ \ \lambda } -2g^{\lambda \omega }S_\sigma \end{aligned}$$

(90)

where $S_\eta =S_{\eta \sigma }^{\ \ \sigma }$ does not vanish in this case. With this we can work out (49). From (48) we may note

$$\begin{aligned} L^{\mu \nu }\delta g_{\mu \nu }=L^{\mu \nu }(\delta \gamma _{\mu \nu }+\delta \phi _{\mu \nu }) \end{aligned}$$

(91)

so we can work out the symmetric part, $L_{S}^{\mu \nu }$ and then the antisymmetric part $L_{A}^{\mu \nu }$. Thus, assuming taking the symmetric part (in $\mu \nu$) is implied,

$$\begin{aligned} \tilde{L}_{S}^{\mu \nu }= & {} \tilde{E}^{\lambda \omega }_{\ \ \sigma }\left[ -\gamma ^{\mu \sigma }C_{\lambda \omega }^{\ \ \nu } -\gamma ^{\sigma \mu }S_{\lambda \ \ {\underset{-}{\omega }}}^{\ \ {\underset{-}{\nu }}} +\delta ^\nu _\omega S_{\lambda }^{\ {\underset{-}{\sigma }}\mu } +\delta ^\nu _\lambda S_\omega ^{\ {\underset{-}{\sigma }}\mu } - \gamma ^{\mu \sigma }S_{\omega \ {\underset{-}{\lambda }}}^{\ {\underset{-}{\nu }}} \right] \nonumber \\&-\,\frac{\partial _\eta }{2}\left[ \tilde{E}^{\lambda \omega }_{\ \ \sigma }\left( \gamma ^{\sigma \mu }\delta ^{\nu \eta }_{\lambda \omega } +\gamma ^{\sigma \mu }\delta ^{\nu \eta }_{\omega \lambda } -\gamma ^{\sigma \eta }\delta ^{\mu \nu }_{\lambda \omega } \right) \right] \end{aligned}$$

(92)

we adopt the unconventional notation $\delta ^{\nu \eta }_{\lambda \omega }=\delta ^\nu _\lambda \delta ^\eta _\omega$. Defining

$$\begin{aligned} e^{\nu \eta {\underset{-}{\mu }}}\equiv E^{\nu \eta {\underset{-}{\mu }}} +E^{\eta \nu {\underset{-}{\mu }}} -E^{\mu \nu {\underset{-}{\eta }}} \end{aligned}$$

(93)

Equation (92) may be written as

$$\begin{aligned} L_{S}^{\mu \nu }= - E^{\lambda \omega {\underset{-}{\mu }}} S_{\lambda \omega }^{\ \ \nu } +E^{\lambda \nu }_{\ \ \omega } S_\lambda ^{\ {\underset{-}{\omega }}\mu } -S_\eta e^{\nu \eta {\underset{-}{\mu }}} -\frac{1}{2}\nabla _\eta e^{\eta \nu {\underset{-}{\mu }}} .\end{aligned}$$

(94)

Now we may find the nonsymmetric part $L_{A}^{\mu \nu }$, where the antisymmetric part is implied. It is,

$$\begin{aligned} L_{A}^{\mu \nu }= & {} E^{\lambda \omega }_{\ \ \sigma }( -\phi ^{\sigma \nu } S_{\lambda \omega }^{\ \ \mu } -\gamma ^{\sigma \theta }\gamma _{\xi \omega }\phi ^{\xi \nu }S_{\lambda \theta }^{\ \ \mu } -\gamma ^{\sigma \theta }\gamma _{\xi \lambda }\phi ^{\xi \nu }S_{\omega \theta }^{\ \ \mu } )\nonumber \\&-\frac{\partial _\eta }{6}[\tilde{E}^{[\lambda \omega ]}_{\ \ \ \ \sigma } ( \phi ^{\sigma \eta }\delta ^{\mu \nu }_{\lambda \omega } +2\phi ^{\sigma \mu }\delta ^{\nu \eta }_{\lambda \omega } ) ] \nonumber \\&+\,2(\gamma ^{\sigma \nu }\gamma _{\xi \phi }\phi ^{\xi \eta }\delta ^\mu _\omega +\gamma ^{\sigma \eta }\gamma _{\xi \omega }\phi ^{\xi \mu }\delta ^\nu _\lambda +\gamma ^{\sigma \mu }\gamma _{\xi \omega }\phi ^{\xi \nu }\delta ^\eta _\lambda ) \end{aligned}$$

(95)

which may be written

$$\begin{aligned} L_{A}^{\mu \nu }= & {} E^{\lambda \omega }_{\ \ \sigma }\phi ^{\sigma \nu }S_{\lambda \omega }^{\ \ \mu } -E^{\lambda \omega {\underset{-}{\theta }}} (\gamma _{\xi \omega }\phi ^{\xi \nu }S_{\lambda \theta }^{\ \ \mu } +\gamma _{\xi \lambda }\phi ^{\xi \nu }S_{\omega \theta }^{\ \ \mu }) \nonumber \\&-\,\frac{1}{6}\left[ (\nabla _\eta +S_\eta )(E^{\mu \nu }_{\ \ \sigma }\phi ^{\sigma \eta } +E^{\nu \eta }_{\ \ \sigma }\phi ^{\sigma \mu } -E^{\eta \nu }_{\ \ \sigma }\phi ^{\sigma \mu }) +2S_{\theta \eta }^{\ \ \mu }(E^{\theta \nu }_{\ \ \sigma }\phi ^{\sigma \eta }+E^{\nu \theta }_{\ \ \sigma }\phi ^{\sigma \eta } \right] .\nonumber \\ \end{aligned}$$

(96)

These, with (50), are the field equations for minus minus metricity.