A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field

Abstract

We prove that, in a space-time of dimension $n>3$ with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if and only if the contraction of the Weyl tensor with the velocity is zero. This extends a property found in generalised Robertson–Walker spacetimes, where the velocity is also eigenvector of the Ricci tensor. Despite the simplicity of the statement, the proof is involved. As a product of the same calculation, we introduce a curvature tensor with an interesting recurrence property.

Appendix

Proposition 4.4

In a twisted space the following identities hold among the Weyl tensor and the contracted Weyl tensor:

$$\begin{aligned} \nabla _p C_{ikm}{}^p= & {} (n-3)(\nabla _i E_{km} - \nabla _k E_{im})\nonumber \\&+ (n-2) [ u^p\nabla _p (u_i E_{km}-u_k E_{im} )\nonumber \\&+2 \varphi (u_iE_{km} - u_k E_{im})] \nonumber \\&+ (2u_k u_m +g_{km}) \nabla _p E_i{}^p - (2u_i u_m +g_{im}) \nabla _p E_k{}^p \quad \end{aligned}$$

(19)

$$\begin{aligned} (n-3)( u^p\nabla _p C_{iklm} + 2\varphi C_{iklm} )= & {} (n-2) [ u^p\nabla _p (u_i u_mE_{kl}\nonumber \\&-u_k u_mE_{il} -u_iu_l E_{km}+u_k u_l E_{im} ) \nonumber \\&+2\varphi (u_iu_mE_{kl} - u_k u_mE_{il} - u_iu_lE_{km} + u_k u_l E_{im}) ] \nonumber \\&+ [ u^p\nabla _p ( g_{im} E_{kl} - g_{km} E_{il} - g_{il} C_{km} + g_{kl} E_{im} ) \nonumber \\&+2\varphi (g_{im} E_{kl} - g_{km} E_{il} - g_{il} E_{km} + g_{kl} E_{im} )] \end{aligned}$$

(20)

Proof

Contraction of (13) with $u^j$ is:

$$\begin{aligned} u^j \nabla _i C_{jklm} + u^j\nabla _j C_{kilm} + u^j \nabla _k C_{ijlm}= & {} \tfrac{1}{n-3} (u_m \nabla _p C_{kil}{}^p + u_l \nabla _p C_{ikm}{}^p) \\&+ \tfrac{1}{n-3} \nabla _p [u^j ( g_{km} C_{ijl}{}^p\\&+ g_{im} C_{jkl}{}^p+ g_{kl} C_{jim}{}^p + g_{il} C_{kjm}{}^p )] \\&-\tfrac{1}{n-3}\varphi u_p u^j (g_{km} C_{ijl}{}^p\\&+ g_{im} C_{jkl}{}^p + g_{kl} C_{jim}{}^p + g_{il} C_{kjm}{}^p ) \end{aligned}$$

Where possible, the vector $u^k$ is taken inside covariant derivatives to take advantage of property (8)

$$\begin{aligned}&\nabla _i (u^jC_{jklm})-\varphi h_i^j C_{jklm} + u^j\nabla _j C_{kilm} + \nabla _k (u^jC_{ijlm})\\&\quad \quad -\varphi h^j_k C_{ijlm} = \tfrac{1}{n-3} (u_m \nabla _p C_{kil}{}^p \\&\quad \quad + u_l \nabla _p C_{ikm}{}^p) + \tfrac{1}{n-3} \nabla ^p [ g_{km}(u_p E_{li}-u_lE_{pi}) \\&\quad \quad + g_{im} (u_lE_{pk} - u_p E_{lk} ) + g_{kl} (u_m E_{pi} -u_p E_{mi}) + g_{il} (u_p C_{mk} -u_m E_{pk} ] \\&\quad \quad +\tfrac{1}{n-3}\varphi [g_{km} E_{il} - g_{im} E_{kl} - g_{kl} E_{im} + g_{il} E_{km} ] \\&\nabla _i (u_l E_{mk} -u_m E_{lk})-\varphi C_{iklm} - \varphi u_i (u_l E_{mk} -u_m E_{lk})+ u^j\nabla _j C_{kilm} \\&\quad \quad + \nabla _k ( u_m E_{li}-u_l \mathsf{}C_{mi}) -\varphi C_{iklm}-\varphi u_k ( u_m E_{li}-u_l \mathsf{}C_{mi})\\&\quad = \tfrac{1}{n-3} (u_m \nabla _p C_{kil}{}^p + u_l \nabla _p C_{ikm}{}^p) \\&\quad \quad + \tfrac{1}{n-3} u^p\nabla _p [ g_{km} E_{li} - g_{im} E_{lk} - g_{kl} E_{mi} + g_{il} C_{mk} ] \\&\quad \quad + \tfrac{1}{n-3} \nabla ^p [- g_{km} u_lE_{pi} + g_{im} u_lE_{pk} + g_{kl} u_m E_{pi} - g_{il} u_m E_{pk}] \\&\quad \quad +\tfrac{n}{n-3}\varphi [g_{km} E_{il} - g_{im} E_{kl} - g_{kl} E_{im} + g_{il} E_{km} ] \\&(n-3)[u_l (\nabla _i E_{mk} - \nabla _k E_{mi}) -u_m (\nabla _i E_{lk} - \nabla _kE_{li}) -2\varphi C_{iklm} + u^j\nabla _j C_{kilm} ]\\&\quad = (u_m \nabla _p C_{kil}{}^p + u_l \nabla _p C_{ikm}{}^p) + u^p\nabla _p [ g_{km} E_{li} - g_{im} E_{lk} - g_{kl} E_{mi} + g_{il} C_{mk} ] \\&\quad \quad - g_{km} u_l \nabla ^pE_{pi} + g_{im} u_l\nabla ^pE_{pk} + g_{kl} u_m \nabla ^pE_{pi} - g_{il} u_m \nabla ^pE_{pk} \\&\quad \quad +2\varphi [g_{km} E_{il} - g_{im} E_{kl} - g_{kl} E_{im} + g_{il} E_{km} ] \end{aligned}$$

Contraction with $u^l$ yields the first result, (19):

$$\begin{aligned} \nabla _p C_{ikm}{}^p= & {} (n-3)(\nabla _i E_{km} - \nabla _k E_{im})\\&+ (n-2) [ u^p\nabla _p (u_i E_{km}-u_k E_{im} )\\&+2 \varphi (u_iE_{km} - u_k E_{im})]\\&+ (2u_k u_m +g_{km}) \nabla _p E_i{}^p \\&- (2u_i u_m +g_{im}) \nabla _p E_k{}^p \end{aligned}$$

which is used to replace the covariant divergences $\nabla _p C_{jkl}{}^p$ in the previous expression

$$\begin{aligned}&(n-3)[u_l (\nabla _i E_{mk} - \nabla _k E_{mi}) -u_m (\nabla _i E_{lk} - \nabla _kE_{li}) -2\varphi C_{iklm} + u^j\nabla _j C_{kilm} ]\\&\quad = -u_m \{(n-3)(\nabla _i E_{kl}\\&\quad \quad - \nabla _k E_{il}) + (n-2) [ u^p\nabla _p (u_i E_{kl}-u_k E_{il} ) +2 \varphi (u_iE_{kl} - u_k E_{il})]\\&\quad \quad + (2u_k u_l +g_{kl}) \nabla _p E_i{}^p - (2u_i u_l +g_{il}) \nabla _p E_k{}^p\}\\&\quad \quad + u_l \{(n-3)(\nabla _i E_{km} - \nabla _k E_{im}) + (n-2) [ u^p\nabla _p (u_i E_{km}-u_k E_{im} )\\&\quad \quad +2 \varphi (u_iE_{km} - u_k E_{im})]\\&\quad \quad + (2u_k u_m +g_{km}) \nabla _p E_i{}^p - (2u_i u_m +g_{im}) \nabla _p E_k{}^p \}\\&\quad \quad + u^p\nabla _p [ g_{km} E_{li} - g_{im} E_{lk} - g_{kl} E_{mi} + g_{il} C_{mk}] \\&\quad \quad - g_{km} u_l \nabla ^pE_{pi} + g_{im} u_l\nabla ^pE_{pk} + g_{kl} u_m \nabla ^pE_{pi} - g_{il} u_m \nabla ^pE_{pk} \\&\quad \quad +2\varphi [g_{km} E_{il} - g_{im} E_{kl} - g_{kl} E_{im} + g_{il} E_{km} ] \end{aligned}$$

Some derivatives cancel, and we are left with

$$\begin{aligned} (n-3)[ -2\varphi C_{iklm} - u^p\nabla _p C_{iklm} ]= & {} -u_m \{ (n-2) [ u^p\nabla _p (u_i E_{kl}-u_k E_{il} ) \\&+2 \varphi (u_iE_{kl} - u_k E_{il})]\}\\&+ u_l \{ (n-2) [ u^p\nabla _p (u_i E_{km}-u_k E_{im} ) \\&+2 \varphi (u_iE_{km} - u_k E_{im})] \}\\&+ u^p\nabla _p [ g_{km} E_{li} - g_{im} E_{lk} - g_{kl} E_{mi} + g_{il} C_{mk} ] \\&+2\varphi [g_{km} E_{il} - g_{im} E_{kl} - g_{kl} E_{im} + g_{il} E_{km} ] \end{aligned}$$

The final equation is obtained. $\square$