1 Correction to: Gen Relativ Gravit (2018) 50:81 https://doi.org/10.1007/s10714-018-2398-9

In our paper “A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field” [1] the sentence: “ ... a contraction with \(u^i\) gives: \(0 = u^i\nabla _i E_{km} + \varphi E_{km}\)” in the end of the proof of Theorem 1.1 (page 4) is wrong (actually, it gives \(0=u^i \nabla _i E_{km}+(n-1)\varphi E_{km}\)).

The error partly changes Theorem 1.1 (stated in page 2) but does not affect Theorem 1.2 and all the other propositions in the paper, as well as the long evaluation in the Appendix.

The correct statement is:

Theorem 1.1

In a twisted space-time of dimension \(n>3\):

$$\begin{aligned} \mathrm{(i)} \quad u_m C_{jkl}{}^m =0&\,\Longrightarrow \; \nabla _m C_{jkl}{}^m =0\\ \mathrm{(ii)} \quad \nabla _m C_{jkl}{}^m =0&\,\Longrightarrow \; u^p\nabla _p (u_m C_{jkl}{}^m) = - \varphi (n-1) u_m C_{jkl}{}^m \end{aligned}$$

Proof

The proof of statement (i) remains as given in page 4 of [1]. The proof of statement (ii) is as follows.

Consider the identity (8) for the Weyl tensor \(C_{jklm}u^m = u_k E_{jl}-u_j E_{kl}\), where \(E_{kl}=u^j C_{jklm}u^m\). Then: \( u^p\nabla _p(C_{jklm}u^m )= u_k u^p\nabla _p E_{jl}-u_j u^p\nabla _p E_{kl} \).

If \(\nabla ^m C_{jklm}=0\) Eq. (15) holds, i.e. \(u^p\nabla _p E_{ij} = -\varphi (n-1) E_{ij}\). Then:

$$\begin{aligned} u^p\nabla _p(C_{jklm}u^m )= -\varphi (n-1) (u_k E_{jl}-u_j E_{kl}) = -\varphi (n-1) C_{jklm}u^m \end{aligned}$$

\(\square \)

In the special case of generalised Robertson–Walker space-times the original statement \( u_m C_{jkl}{}^m =0 \Longleftrightarrow \nabla _m C_{jkl}{}^m =0 \) remains true (Theorem 3.4 of Ref. [2]).