Erratum to: An Entropy Functional for Riemann-Cartan Space-Times

Erratum to: Int J Theor Phys (2012) 51:362–373 DOI 10.1007/s10773-011-0913-9

The original version of this article unfortunately contained a mistake. The authors rectified the errors and are shown below:

We correct the entropy functional constructed in Int. J. Theor. Phys. 51:362 (2012). The ‘on-shell’ functional one obtains from this correct functional possesses a holographic structure without imposing any constraint on the spin-angular momentum tensor of matter, in contrast to the conclusion made in the above paper.

The error made in [1] was the missing torsion trace Q _i =Q _ij ^j in the evaluation of the divergence terms. Indeed, the integral of a divergence in Riemann-Cartan space-times is of the form ∫ _M ∇ _i V ⁱ=∫ _∂M n _i V ⁱ−∫ _M 2Q _i V ⁱ. Taking this into account, the entropy functional that should be considered is actually the following more ‘economical’ one

$$\begin{aligned} S =& \int \mathrm{d}^{4}x\sqrt{ - g} \biggl[ \alpha \bigl( \bigl( \nabla_{i}u^{j} \bigr) \bigl( \nabla_{j}u^{i} \bigr) - \bigl( \nabla_{i}u^{i} \bigr)^{2} \bigr) + ( \lambda g_{ij} + T_{ij} )u^{i}u^{j} \\ &{} + ( \varSigma_{ijk} + \varSigma _{ikj} + \varSigma_{kji} )u^{i}\nabla^{j}u^{k} \biggr]. \end{aligned}$$

(1)

The additional terms introduced in [1] arise automatically when extracting total divergences. The variation of this functional with respect to the field u ⁱ reads

$$\begin{aligned} \delta S =& \int \mathrm{d}^{4}x\sqrt{ - g} \biggl[ 2\alpha \bigl( \bigl( \nabla_{i}\delta u^{j} \bigr) \bigl( \nabla_{j}u^{i} \bigr) - \bigl( \nabla_{i}\delta u^{i} \bigr) \bigl( \nabla_{j}u^{j} \bigr) \bigr) + 2 ( \lambda g_{ij} + T_{ij} )\delta u^{i}u^{j} \\ &{} + ( \varSigma_{ijk} + \varSigma _{ikj} + \varSigma_{kji} )\delta u^{i}\nabla^{j}u^{k} + ( \varSigma _{ijk} + \varSigma_{ikj} + \varSigma _{kji} )u^{i}\nabla^{j}\delta u^{k} \biggr]. \end{aligned}$$

(2)

Extracting from the total divergences the boundary integrals which do not contribute to the variation, the condition δS=0 reads

$$\begin{aligned} &\int \mathrm{d}^{4}x\sqrt{ - g} \biggl( 2\alpha \nabla_{[ i }\nabla_{ j ]}u^{j} + ( 2\alpha Q_{i}g_{jk} - 2\alpha Q_{k}g_{ij} + \varSigma _{ikj} )\nabla^{j}u^{k} \\ &\quad{}+ \biggl[ \lambda g_{ij} + T_{ij} + \frac{1}{2} \bigl( \nabla^{k} + 2Q^{k} \bigr) ( \varSigma _{ijk} + \varSigma_{kij} + \varSigma _{kji} ) \biggr]u^{j} \biggr)\delta u^{i} = 0. \end{aligned}$$

(3)

From the identity 2∇_[i ∇ _j] u ^j=−R _ij u ^j−2Q _ij ^k∇ _k u ^j, it follows that Eq. (3) is satisfied for arbitrary variations of u ⁱ if

$$\begin{aligned} & ( 2\alpha Q_{kij} + 2\alpha Q_{i}g_{jk} - 2\alpha Q_{k}g_{ij} + \varSigma_{ikj} )\nabla^{j}u^{k} \\ &\quad{} - \biggl[ \alpha R_{ij} - \lambda g_{ij} - T_{ij} - \frac{1}{2} \bigl( \nabla^{k} + 2Q^{k} \bigr) ( \varSigma_{ijk} + \varSigma _{kij} + \varSigma_{kji} ) \biggr]u^{j} = 0. \end{aligned}$$

(4)

This in turn is satisfied for all u ⁱ if and only if the content of each of the two square brackets vanishes identically, leading straightforwardly as in [1] to the Cartan-Sciama-Kibble field equations, but containing more correctly than in [1] the metric g _ij instead of the Kronecker-delta δ _ij in front of the torsion traces Q _i g _jk and Q _k g _ij . The arguments used in [1] to prove the uniqueness of the functional Eq. (1) remain unchanged. When the field equations are substituted in Eq. (1) after integration by parts, however, the ‘on-shell’ entropy functional becomes simply

$$ S = \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{3}x \sqrt{\vert h \vert } n_{i} \bigl( u^{j} \nabla_{j}u^{i} - u^{i}\nabla_{j}u^{j} \bigr). $$

(5)

So contrary to the conclusion made in [1], the holographic structure emerges without imposing any constraint on the spin-angular momentum tensor of matter. Using Eq. (5), the application and conclusions made in [1] for the case of Dirac fields and black holes with intrinsic spin remain unchanged. For spin fluids obeying the Frenkel condition, however, the affine connection ∇ in Eq. (5) reduces as in [1] to the Levi-Civita connection ${\circ} \atop \nabla$ only for an isotropic deformation $u^{i}u^{j} = \frac{1}{4}u^{2}g^{ij}$.