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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 \(n>3\):
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:
\(\square \)
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]).
References
Molinari, L.G., Mantica, C.A.: A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field. Gen. Relativ. Gravit. 50, 81 (2018)
Mantica, C.A., Molinari, L.G.: On the Weyl and Ricci tensors of generalized Robertson–Walker space-times. J. Math. Phys. 57(10), 102502 (2016)
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Molinari, L.G., Mantica, C.A. Correction to: A simple property of the Weyl tensor for a shear, vorticity and acceleration-free velocity field. Gen Relativ Gravit 50, 133 (2018). https://doi.org/10.1007/s10714-018-2451-8
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DOI: https://doi.org/10.1007/s10714-018-2451-8