On the weak field approximation of the de Sitter gauge theory of gravity

Abstract

The weak field approximation of a model of de Sitter gauge theory of gravity is studied in two cases. Without torsion and spin current, the model cannot give the right non-relativistic approximation unless the density is a constant. With small torsion, a satisfactory Newtonian approximation can be obtained.

Appendix: the process of simplification

When torsion is included, we should study the original form of the Yang-like equation, which reads

$$\begin{aligned} F_{ab\ ||\nu }^{\mu \nu }&= R^{-2}\left(16\pi G S^{\mu }_{\mathrm{M}ab}+S^{\mu }_{\mathrm{G}ab}\right). \end{aligned}$$

(6.1)

where $R$ is the radius of de Sitter spacetime. Again, suppose the spin currents of matter fields vanish. We have

$$\begin{aligned} F_{ab\ ||\nu }^{\mu \nu }&= R^{-2}(S_{\mathrm{F}ab}^{ \mu }+2S_{\mathrm{T}ab}^{\mu }) \nonumber \\&= R^{-2}\left(Y^\mu _{ \lambda \nu } e_{ab}^{\lambda \nu }+Y ^\nu _{\lambda \nu } e_{ab}^{\mu \lambda }+T_{a}^{\mu \lambda }e_{b\lambda }^{} -T_{b}^{\mu \lambda }e_{a\lambda }^{}\right). \end{aligned}$$

(6.2)

Multiply it with $e^b_\mu$, one gets

$$\begin{aligned} e^b_\mu F_{ab\ ||\nu }^{\mu \nu }=R^{-2}e^b_\mu \left(Y^\mu _{\lambda \nu } e_{ab}^{\lambda \nu }+Y ^\nu _{\lambda \nu } e_{ab}^{\mu \lambda } -T_{b}^{\mu \lambda }e_{a\lambda }^{}\right) \end{aligned}$$

(6.3)

It is

$$\begin{aligned} F_{a ||\nu }^{\nu } -\frac{1}{2} T^\lambda _{\nu \mu } e^b_\lambda F_{ab}^{\mu \nu }&= -R^{-2}\left(2Y ^\nu _{ \lambda \nu } +T_{\mu \ \, \lambda }^{~\mu }\right)e_{a}^{\lambda } \nonumber \\&= -3R^{-2} T_{\mu \lambda }^{\mu }e_{a}^{\lambda } \end{aligned}$$

(6.4)

Multiply it with $e^a_\sigma$, one gets

$$\begin{aligned}&e^a_\sigma F_{a~\, ||\nu }^{\nu } - \frac{1}{2} T^\lambda _{\nu \mu } F_{\sigma \lambda }^{\mu \nu }= -3R^{-2} T_{\mu \ \, \sigma }^{\mu }\end{aligned}$$

(6.5)

$$\begin{aligned}&F_{\sigma ~\, ;\nu }^{~\nu }= Y^\lambda _{\sigma \nu } F_{\lambda }^{\nu } +\frac{1}{2} T^\lambda _{\nu \mu } F_{\sigma \lambda }^{\mu \nu }-3R^{-2} T_{\mu \ \, \sigma }^{\mu } \approx 0 \end{aligned}$$

(6.6)

A semicolon ‘;’ represents the covariant derivative with respect to Christoffel symbols. Therefore,

$$\begin{aligned}&F_{\sigma ~\, ,\nu }^{~\nu }+\left\{ \begin{array}{l}\nu \\ \lambda \nu \end{array}\right\} F_\sigma ^{\ \lambda } -\left\{ \begin{array}{l}\lambda \\ \sigma \nu \end{array} \right\} F_\lambda ^{\ \nu } = Y^\lambda _{\ \ \sigma \nu } F_{\lambda }^{\ \nu } +\frac{1}{2} T^\lambda _{\ \ \nu \mu } F_{\sigma \lambda }^{\ \ \mu \nu }-3R^{-2} T_{\mu \ \, \sigma }^{~\mu }\qquad \quad \end{aligned}$$

(6.7)

$$\begin{aligned}&F_{\mu \nu }=\partial _\nu \Gamma ^\lambda _{\mu \lambda }-\partial _\lambda \Gamma ^\lambda _{\mu \nu } +\Gamma ^\lambda _{\mu \nu }\Gamma ^\sigma _{\lambda \sigma }-\Gamma ^\lambda _{\mu \sigma }\Gamma ^\sigma _{\lambda \nu } \nonumber \\&\qquad \quad \!\!=\mathcal{R }_{\mu \nu }+\partial _\nu Y^\lambda _{\ \mu \lambda }-\partial _\lambda Y^\lambda _{\ \mu \nu } +\left\{ \begin{array}{l}\lambda \\ \mu \nu \end{array}\right\} Y^\sigma _{\ \lambda \sigma }\nonumber \\&\qquad \qquad -\left\{ \begin{array}{l} \lambda \\ \mu \sigma \\ \end{array}\right\} Y^\sigma _{\ \lambda \nu } +Y^\lambda _{\ \mu \nu }\left\{ \begin{array}{l} \sigma \\ \lambda \sigma \\ \end{array}\right\} \nonumber \\&\qquad \qquad -Y^\lambda _{\ \mu \sigma }\left\{ \begin{array}{l} \sigma \\ \lambda \nu \\ \end{array}\right\} +Y^\lambda _{\ \mu \nu } Y^\sigma _{\ \lambda \sigma }-Y^\lambda _{\ \mu \sigma }Y^\sigma _{\ \lambda \nu } \nonumber \\&\qquad \quad \!\!\approx \mathcal{R }_{\mu \nu }+\partial _\nu Y^\lambda _{\ \mu \lambda }-\partial _\lambda Y^\lambda _{\ \mu \nu }. \end{aligned}$$

(6.8)