Black hole entropy as a Noether Charge

[haushofer]

Hi folks, I have a question about a paper by Thomas Mohaupt, called
"Black hole entropy, special geometry and strings". It's available here:

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Ahep-th%2F0007195

My question concerns part 2.2.5 , page 18. I find it quite difficult to get a nice feeling for doing calculations like the ones in that part. Here the author defines entropy as a surface charge ( a method due to Wald ). He assumes that the Lagrangian depends on the Riemanntensor, the energy-momentum (E-M) tensor and the derivative of the E-M tensor ( the covariant, I presume? ). He then considers a variation of the E-M tensor and the metric, which I do understand ( they're defined via a general coordinate transformation, so you end up with a Lie-derivative ) So up to that it's okay.

First, I want to do the variation of the Lagrangian with respect to the Riemanntensor, and here I get some troubles. How do I write down such a variation ? It looks like

$$\delta S = \frac{\partial L }{\partial R_{\mu\nu\rho\sigma} } \delta R_{\mu\nu\rho\sigma}$$

Here L is the Lagrangian density.
If I want to rewrite this in terms of the variation with respect to the metric, can I use the usual chain rules for differentiation? Or should I do the differentiation explicitly with respect to the metric, and rewrite this in terms of the Riemanntensor?

Then they define a current J. Via Noethers theorem I know that one can define such a current as

$$J^{\mu} = \frac{\partial L}{\partial \phi ,\mu} \delta \phi$$

where we sum over all fields phi. So my guess would be that the current which is written down there in section 2.2.5 is acquired via

$$J^{\mu} = \frac{\partial L}{\partial R_{\nu\lambda\rho\sigma},\mu} \delta R_{\nu\lambda\rho\sigma} + \frac{\partial L}{\partial \psi_{\nu\lambda},\mu} \delta \psi_{\nu\lambda}$$

Is this going into the right direction? In the variation of the metric and the E-M tensor they use a test function epsilon which is a function of the coordinates, but I don't see it in the current back. That's strange, because you can't divide the function out of the current ( we get derivatices of the test function ). Or can we put those terms to 0? Also I have some doubts about when to use partial derivatives in such calculations, and when to use covariant derivatives. Should I just replace al partial derivatives in such calculations by covariant derivatives?

And does any-one know a good text(book) in which all this is nicely explained? I've been searching on the internet, but without good results. A lot of questions, I hope some-one can help me. Many thanks in forward,

Haushofer.

 

[haushofer]

Nobody ? :(

 

[Entropia]

Black hole entropy is a fundamental concept in the study of black holes and their thermodynamics. It is closely related to the Noether charge, which is a conserved quantity associated with a symmetry of a system. In the case of black holes, the Noether charge is related to the symmetries of the black hole's horizon.

The paper by Thomas Mohaupt, "Black hole entropy, special geometry and strings", discusses the calculation of black hole entropy using the Noether charge approach. In particular, part 2.2.5 focuses on the variation of the Lagrangian with respect to the Riemanntensor and the metric, and the definition of the current J.

To begin, the variation of the Lagrangian with respect to the Riemanntensor can be written as:

\delta S = \frac{\partial L}{\partial R_{\mu\nu\rho\sigma}} \delta R_{\mu\nu\rho\sigma}

where L is the Lagrangian density. To rewrite this in terms of the metric, you can use the chain rule for differentiation:

\frac{\partial L}{\partial R_{\mu\nu\rho\sigma}} = \frac{\partial L}{\partial g^{\alpha\beta}} \frac{\partial g^{\alpha\beta}}{\partial R_{\mu\nu\rho\sigma}}

Note that the metric is a function of the Riemanntensor, so you will need to use the inverse metric to find the second term in the above equation.

Next, the current J can be defined through Noether's theorem as:

J^{\mu} = \frac{\partial L}{\partial \phi ,\mu} \delta \phi

where \phi represents all the fields in the system. In the context of black hole entropy, the current is defined as:

J^{\mu} = \frac{\partial L}{\partial R_{\nu\lambda\rho\sigma},\mu} \delta R_{\nu\lambda\rho\sigma} + \frac{\partial L}{\partial \psi_{\nu\lambda},\mu} \delta \psi_{\nu\lambda}

where \psi_{\nu\lambda} represents the E-M tensor. This is the correct direction for the current, but note that the test function epsilon is not explicitly included. This is because the test function is taken to be