Generalized Komar charges and Smarr formulas for black holes and boson stars

Dear editor,

After careful reading of the two referee's reports we have made a number of
changes in the manuscript that we list in our answers to the referees.

Answers to the questions posed by referee 1:

$$\begin{enumerate} \item As we have commented in the paper, other methods have been used to derive Smarr formulae and one may say that they are, in the end, equally effective, since formulae which are identically satisfied for the relevant black hole solutions are derived. The same basic relation can be expressed in terms of different variables )as in the references mentioned by the referee), but not all variables have the same physical standing. It is well known that in General Relativity only the total (ADM) mass and angular momentum, defined by surface integrals at infinity, are conserved and that there is no invariant definition of local mass/energy or angular momentum density that allows the assignment of some amount of energy or angular momentum to a given spacetime region, like the event horizon. It seems reasonable to expect that meaningful physical expressions should be exclusively and finally written in terms of them. The method we use here (which was pioneered by Bardeen, Carter and Hawking in the new references [5,6], see also the new references [7,8]) only involves surface integrals and, therefore, establishes a clear separation between quantities which are defined asymptotically and quantities which are defined on the event horizon. The former are total total, ADM conserved charges, some of them (electric and magnetic charges) multiplied by their chemical potentials evaluated at infinity and the latter are the temperature and entropy, total electric and magnetic charges and their chemical potentials evaluated over the horizon. The reason why the total electric and magnetic charges appear in both integrals is that they satisfy Gauss laws. In contrast to this, the method used in the references mentioned by the referee and in Townsend's lectures involves volume integrals. The results of these volume integrals can be, in the end, may be empirically related to total, ADM, conserved charges but, by construction, they cannot be identified with energies/masses. When the charges are volume integrals of conserved currents associated to global symmetries, they cannot be directly associated to the black hole (they are computed on their exterior) nor to the spacetime (they are not the total charges). We believe that it is not natural to express the Smarr formula (which is nothing bu a Gibbs-Duhem-type thermodynamical relation) in terms of this kind of charges which are not thermodynamical variables. The method used in this paper leads to this final expressions in a more straightforward way. We have made several changes in the paper to address this important point: \begin{enumerate} \item We have added a 2-page discussion of the charges associated to global symmetries and computed as volume integrals in the context of black-hole physics at the beginning of the introduction. \item We have added the references [5-8] in which the method used in this paper was used. \item We have rephrased the paragraph below (1.24) to make our point clear. \end{enumerate}$$

\item We have added a long footnote (number 13 in page 8) citing the classical
references suggested by the referee and commenting on their results, in
relation with ours.

\item In this work we do cover boson stars in the sense mentioned by the
referee, although we have probably used it in an unconventional or improper
way in a few places. Readers were warned of this fact in the second
paragraph below equation (1.32). Nevertheless, we have added a footnote in
page 12 (formerly page 10) commenting our terminology.




\end{enumerate}

Answers to the questions posed by referee 2:

$$\begin{enumerate} \item We have moved the content of footnote~18 to footnote~5, combining it with the previous content of that note and extending it a bit. $\mathcal{D}$ was implicitly defined in footnote~18 but we have added a line clarifying its definition. \item We have mentioned the reference suggested by the referee as well as Heusler, Phys. Rev. D, 56, 961–973, (1997) in which the alternative method to derive Smarr formulas is introduced. \end{enumerate}$$


We have made some further changes:

$$\begin{enumerate} \item We have corrected some misprints. \item We have cited the original and earlier works in which the Komar charge and its generalizations were used to derive Smarr fomulae ([5,6,7,8] in the revised version). \end{enumerate}$$

\begin{enumerate}
\item We have added a 2-page discussion of the charges associated to global
symmetries and computed as volume integrals in the context of black-hole
physics at the beginning of the introduction.
\item We have added the references [5-8] in which the method used in this
paper was used.
\item We have rephrased the paragraph below (1.24) to make our point clear.
\item We have added a long footnote (number 13 in page 8) citing the classical
references suggested by the referee and commenting on their results, in
relation with ours.
\item We have added a footnote in page 12 (formerly page 10) commenting our terminology.
\item We have moved the content of footnote~18 to footnote~5, combining it
with the previous content of that note and extending it a bit. $\mathcal{D}$
was implicitly defined in footnote~18 but we have added a line clarifying
its definition.
\item We have mentioned the reference suggested by the referee as well as
Heusler, Phys. Rev. D, 56, 961–973, (1997) in which the alternative method
to derive Smarr formulas is introduced.
\item We have corrected some misprints.
\item We have cited the original and earlier works in which the Komar charge
and its generalizations were used to derive Smarr fomulae ([5,6,7,8] in the
revised version).

\end{enumerate}