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  Extrinsic curvature for generalised entropy

+ 1 like - 0 dislike

So I am reading this paper. The metric is Eq. (3.1) reads: 


The definition of extrinsic curvature can be written as: 

$$K^{a}_{b}=\nabla_b n^{a}=\partial_{b}n^{a}+\Gamma^{a}_{\ bc}n^{c}$$

where Latin indices are running from 1 to d-1. My understanding tells me this is something standard, by which I mean extrinsic curvature is being defined on the hypersurface, hence it is d-1. Yet below Eq. (3.3), the paper calls for $K^{\mu}_{\nu}$ where greek indices are running 1 to d. However, later on above Eq. (4.6), the paper mentions $K^{a}_{b}$. Is this a typo? Also how does one calculate the normal vector n's? Specifically can anyone guide how to obtain $K^{\tau}_{\tau}=1/n\epsilon$ as mentioned below Eq. (3.3).

asked Dec 16, 2016 in Theoretical Physics by Wiliam (65 points) [ revision history ]
recategorized Jun 11, 2019 by Arnold Neumaier

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