Conserved charges and first law of thermodynamics for Kerr–de Sitter black holes

Abstract

Recently, a general method for calculating conserved charges for (black hole) solutions to generally covariant gravitational theories, in any dimensions and with arbitrary asymptotic behaviors has been introduced. Equipped with this method, which can be dubbed as “solution phase space method,” we calculate mass and angular momentum for the Kerr–dS black holes. Furthermore, for any choice of horizons, associated entropy and the first law of thermodynamics are derived. Interestingly, according to insensitivity of the analysis to the chosen cosmological constant, the analysis unifies the thermodynamics of rotating stationary black holes in 4 (and other) dimensions with either AdS, flat or dS asymptotics. We extend the analysis to include electric charge, i.e. to the Kerr–Newman–dS black holes.

Appendix: A deeper review on solution phase space method

The goal of this appendix is to provide a conceptual review on SPSM, although reference to the original paper [40] is recommended. Before reviewing SPSM, we need to recap a standard phase space construction, dubbed as covariant phase space formulation [10, 11, 42, 49–51].

Covariant phase space formulation: Phase space $\mathcal {F}(\mathcal {M},\Omega )$ is a manifold $\mathcal {M}$ equipped with a closed nondegenerate symplectic form $\Omega$. In classical field mechanics, it is usual to build the phase space canonically, i.e. building the $\mathcal {M}$ from a subset of field configurations $\Phi (\mathbf {x})$ and their momentum conjugates defined on some privileged time foliation of spacetime. In this construction, solutions to the equation of motion are some curves on $\mathcal {M}$ parametrized by the time. Interestingly, in the context of generally covariant gravitational theories, there is a more suitable construction which does not break general covariance by specifying a time foliation. In this construction, $\mathcal {M}$ is composed of dynamical field configurations all over the spacetime $\Phi (x^\mu )$. On the other hand, there would not be any field conjugate present. As a result, any solution to the equation of motion in the phase space would be a point on $\mathcal {M}$, instead of a curve. The tangent space of the manifold is also constituted from a subset of perturbations $\delta \Phi (x^\mu )$. The symplectic 2-form which makes $\mathcal {M}$ to be a phase space is constructed from the Lagrangian d-form $\mathbf {L}$. To this end, picking up the Lee-Wald $(d-1)$-form $\varvec{\Theta }$ from the variation of Lagrangian

$$\begin{aligned} \delta \mathbf {L}=\mathbf {E}_{{\Phi }}\delta \Phi +\mathrm {d}\varvec{\Theta }_{\text {LW}}(\delta \Phi ,\Phi ), \end{aligned}$$

(6.1)

the symplectic form would be [10, 11, 42]

$$\begin{aligned} \Omega _{\text {LW}}(\delta _1\Phi ,\delta _2\Phi ,\Phi )\equiv \int _\Sigma \varvec{\omega }_{\text {LW}}(\delta _1\Phi ,\delta _2\Phi ,\Phi )\, \end{aligned}$$

(6.2)

where

$$\begin{aligned} \varvec{\omega }_{\text {LW}}(\delta _1\Phi ,\delta _2\Phi ,\Phi )=\delta _1\varvec{\Theta }_{\text {LW}}(\delta _2\Phi ,\Phi )-\delta _2\varvec{\Theta }_{\text {LW}}(\delta _1\Phi ,\Phi ). \end{aligned}$$

(6.3)

The $\mathbf {E}_{{\Phi }}$ denotes equation of motion for the field $\Phi$, the $\Sigma$ is some codimension-1 (Cauchy) surface and $\delta _{1,2}\Phi$ are some members of the tangent space. The $\varvec{\omega }_{\text {LW}}$ is called (pre)symplectic current. Closed-ness of $\Omega$ is guaranteed by the definition (6.3). In order to make $\Omega _{\text {LW}}$ independent of the choice of $\Sigma$, one needs $\mathrm {d}\varvec{\omega }_{\text {LW}}=0$ and flow of $\varvec{\omega }_{\text {LW}}$ out of the boundaries $\partial \Sigma$ vanish. The former is achieved if $\Phi$ and $\delta \Phi$ satisfy e.o.m and linearized e.o.m respectively. So, it is standard to request them from the beginning. But achievement of the latter needs extra conditions, usually some boundary conditions on perturbations. An important thing to be mentioned in covariant phase space formulation is the ambiguity of addition an exact $(d-1)$-form $\mathrm {d}\mathbf {Y}(\delta \Phi ,\Phi )$ to the $\varvec{\Theta }_{\text {LW}}(\delta \Phi ,\Phi )$, i.e.

$$\begin{aligned} \varvec{\Theta }_{\text {LW}}(\delta \Phi ,\Phi )\rightarrow \varvec{\Theta }(\delta \Phi ,\Phi )=\varvec{\Theta }_{\text {LW}}(\delta \Phi ,\Phi )+\mathrm {d}\mathbf {Y}(\delta \Phi ,\Phi ) \end{aligned}$$

(6.4)

This ambiguity entails corresponding ambiguities in the $\Omega$ defined above, through

$$\begin{aligned} \varvec{\omega }(\delta _1\Phi ,\delta _2\Phi ,\Phi )\rightarrow \varvec{\omega }(\delta _1\Phi ,\delta _2\Phi ,\Phi )+ \mathrm {d}\big (\delta _2 \mathbf {Y}(\delta _1 \Phi ,\Phi )-\delta _1 \mathbf {Y}(\delta _2 \Phi ,\Phi )\big ). \end{aligned}$$

(6.5)

Using the symplectic form, one can associate a Hamiltonian generator (interchangeably called conserved charge) to a diffeomorphism+gauge transformation $\epsilon =\{\xi ,\lambda ^a\}$ as

$$\begin{aligned} \delta H_{\epsilon }(\Phi )&\!\equiv \! \int _\Sigma \big (\delta ^{[\Phi ]}\varvec{\Theta }(\delta _\epsilon \Phi ,\Phi )-\delta _\epsilon \varvec{\Theta }(\delta \Phi ,\Phi )\big )\!=\!\int _{\Sigma }\mathrm {d}\varvec{k}_{\epsilon }(\delta \Phi ,\Phi )\!=\!\oint _{\partial \Sigma }\varvec{k}_{\epsilon }(\delta \Phi ,\Phi ). \end{aligned}$$

(6.6)

The $\delta ^{[\Phi ]}$ emphasizes that $\delta$ acts on dynamical fields, not the $\epsilon$. Moreover, $\delta _\epsilon \Phi \equiv \mathcal {L}_\xi \Phi +\delta _{\lambda ^a}A^a$ where $A^a$ are some probable Abelian gauge fields. In the equation above, the integrand in the first integration has been replaced by an exact $(d-1)$-form $\mathrm {d}\varvec{k}_\epsilon$. So, the last equation follows from the Stokes theorem. The $(d-2)$-form $\varvec{k}_\epsilon$ is explicitly as (see Appendix A in Ref. [40] for detailed derivation)

$$\begin{aligned} \varvec{k}_\epsilon (\delta \Phi ,\Phi )=\delta \mathbf {Q}_\epsilon -\xi \cdot \varvec{\Theta }(\delta \Phi ,\Phi ), \end{aligned}$$

(6.7)

in which $\mathbf {Q}_\epsilon$ is the Noether–Wald charge density, defined by the relation

$$\begin{aligned} \mathrm {d}\mathbf {Q}_\epsilon \equiv \varvec{\Theta }(\delta _\epsilon \Phi ,\Phi )-\xi \cdot \mathbf {L}. \end{aligned}$$

(6.8)

Hence, by the Eq. (6.7), $\varvec{k}_\epsilon$ can be found for different theories straightforwardly. Putting it into Eq. (6.6), if the last integral would be finite and nonvanishing, $\delta H_{\epsilon }(\Phi )$ then corresponds to a conserved charge variation. In order to find the finite conserved charge $H_{\epsilon }$, integrability over the phase space is needed. This condition is basically $(\delta _1\delta _2-\delta _2\delta _1)H_\epsilon (\Phi )=0$, in which $\Phi$s are any field configuration in the presumed phase space $\mathcal {F}$, and $\delta _{1,2}\Phi$ are any arbitrary chosen member of its tangent space. Then, it follows that the integrability condition can be explained as [42, 43, 52]

$$\begin{aligned} \oint _{\partial \Sigma } \Big (\xi \cdot \varvec{\omega }(\delta _1\Phi ,\delta _2\Phi ,\Phi )+\varvec{k}_{\delta _1\epsilon }(\delta _2\Phi ,\Phi ) -\varvec{k}_{\delta _2\epsilon }(\delta _1\Phi ,\Phi )\Big )=0. \end{aligned}$$

(6.9)

As far as calculation of conserved charges are concerned, conservation of $\delta H_\epsilon$ can be guaranteed if $\epsilon$ is chosen such that $\varvec{\omega }(\delta \Phi ,\delta _\epsilon \Phi ,\Phi )=0$ on-shell. It is because there would not be any flow out of the boundaries locally, and hence globally. The family of $\epsilon$’s with this property, which has been dubbed “symplectic symmetry generators” [53], can be divided to two sets: (1) the ones for which $\delta _\epsilon \Phi \ne 0$ at least on one of the points of the phase space, (2) the ones for which $\delta _\epsilon \Phi =0$ all over the phase space. The former set, dubbed as “nonexact symmetry generators”, constitute a closed algebraic structure, and are considered to be responsible for generating the phase space of a solution at given constant thermodynamical variables. We can dub the generated phase space as “statistical phase space”. Hence, they open a road towards understanding microstates of the system (see [52–54] for works in this direction). The latter set are dubbed “exact symmetry generators” and are considered as generators of the set of solutions in different thermodynamical variables [40]. The generated phase space has been called “solution phase space” which we describe below. It has been conjectured that the phase space associated with the geometries without propagating degrees of freedom are composed of the combination of statistical and solution phase spaces [40].

Solution phase space method: This method is specification of the covariant phase space formulation to some specific manifolds and their tangent spaces which endows that method the power of calculability. Consider a family of (black hole) solutions to a generally covariant gravitational theory. Usually, such a family is identified by some isometries and some parameters $p_j$. The parameters are some arbitrary (but with constrained domain) real numbers appearing in the field configuration of the mentioned solutions. The parameters can be reparametrized, but can not be removed by coordinate transformations. The manifold $\hat{\mathcal {M}}$ can be chosen to be composed of the members of the family, up to unphysical coordinate/gauge transformations. The symplectic 2-form $\hat{\Omega }$ would be simply the Lee-Wald symplectic form confined to $\hat{\mathcal {M}}$. Then, the $\mathcal {F}_p=(\hat{\mathcal {M}},\hat{\Omega })$ would be a phase space, the “solution phase space”. Hence, any point of the manifold can be identified by $\hat{\Phi }(x^\mu ,p_j)$. Tangent space of the $\hat{\mathcal {M}}$ is spanned (up to infinitesimal pure gauge transformations) by “parametric variations” which are found simply by [45]

$$\begin{aligned} \hat{\delta }\Phi \equiv \frac{\partial \hat{\Phi }}{\partial p_j}\delta p_j. \end{aligned}$$

(6.10)

These variations, which are infinitesimal difference of two solutions, satisfy linearized equation of motion. Hence, they respect $\mathrm {d}\varvec{\omega }_{\text {LW}}(\hat{\delta }_1\Phi ,\hat{\delta }_2\Phi ,\hat{\Phi })=0$.

As it was advertised above, conservation of $\hat{\delta } H_\epsilon$ is guaranteed if $\epsilon$ is chosen to be an exact symmetry generators $\eta$ defined in Eq. (2.1). This results is because of $\varvec{\omega }_{\text {LW}}(\hat{\delta }\Phi ,\delta _\eta \hat{\Phi },\hat{\Phi })=0$, (which itself is a result of linearity of $\varvec{\omega }_{\text {LW}}$ in $\delta _\eta \hat{\Phi }=0$), preventing flow of $\varvec{\omega }_{\text {LW}}$ out of the boundaries $\partial \Sigma$. Along with guaranteeing the conservation, the relation $\varvec{\omega }_{\text {LW}}(\hat{\delta }\Phi ,\delta _\eta \hat{\Phi },\hat{\Phi })=0$ yields an additional interesting and unexpected result: $\hat{\delta } H_\eta$ would also be independent of the chosen $\partial \Sigma$. It is because of vanishing of $\varvec{\omega }_{\text {LW}}$ all over the $\Sigma$, and hence, vanishing of $\varvec{\omega }_{\text {LW}}$ in the region enclosed between two different integrating surfaces $\partial \Sigma _1$ and $\partial \Sigma _2$. Then, by the Stokes theorem, and noticing the Eq. (6.6), the claim is proved. Explaining this result in another way, although the integration in calculating $\hat{\delta } H_\eta$ is over codimension-2 surface $\partial \Sigma$, but the result would be independent of all coordinates, including the two coordinates which are not integrated on.

Focusing on exact symmetries results in another nice feature for calculation of their conserved charges; discarding the ambiguity $\mathbf {Y}$. This is because of $\delta \mathbf {Y}(\delta _\eta \Phi ,\Phi )-\delta _\eta \mathbf {Y}(\delta \Phi ,\Phi )=0$, which is a result of the linearity of the left hand side in $\delta _\eta \Phi =0$. Using this identity together with Eq. (6.5) in the (6.6), then there would not be any ambiguity in the definition of conserved charges as far as exact symmetries are concerned. Summarizing the last two paragraphs, the charges associated with exact symmetries are conserved, unambiguous, and independent of the chosen described surfaces of integration $\partial \Sigma$.

So far, the SPSM has provided all materials needed to calculate $\hat{\delta } H_\eta (p_j)$. The final tasks are checking integrability over $\hat{\mathcal {M}}$, and (if integrable) performing the integration. The former is feasible simply by replacing $\delta \Phi$ and $\epsilon$ in Eq. (6.9) by $\hat{\delta } \Phi$ and $\eta$. The latter is abstractly the integration in Eq. (2.4), and pragmatically integrating $\hat{\delta } H_\eta (p_j)$ over the parameters $p_j$.

$\varvec{k}_{\varvec{\xi }}$ for EH- $\varvec{\Lambda }$ theory: To make the paper self-contained, here we provide the derivation of $\varvec{k}_\xi$ for the EH-$\Lambda$ theory, which is described by the Lagrangian density $\mathcal {L}=\frac{1}{16\pi G} (R-2\Lambda )$. Beginning from the Eq. (6.1), one finds

$$\begin{aligned} \varvec{\Theta }(\delta \Phi ,\Phi )=\star \Big (\frac{1}{16\pi G}(\nabla _\alpha \delta g_{\,\,\mu }^{\alpha }-\nabla _\mu \delta g^\alpha _{\,\,\alpha })\,\mathrm {d}x^\mu \Big ). \end{aligned}$$

(6.11)

In order to find the explicit form of the $\varvec{k}_\xi$ through Eq. (6.7), in addition to the equation above, the calculation of $\delta \mathbf {Q}_\xi$ is also needed. To this end, by the definition (6.8) and using the equations of motion,

$$\begin{aligned} \mathbf {Q}_\xi&=\star \Big ( \frac{-1}{16\pi G }\frac{1}{2!} (\nabla _\mu \xi _\nu -\nabla _\nu \xi _\mu )\,\mathrm {d}x^\mu \wedge \mathrm {d}x^\nu \Big ) \end{aligned}$$

(6.12)

$$\begin{aligned}&=\frac{-1}{16\pi G}\frac{\sqrt{-g}}{(2!(d-2)!)}\epsilon _{\mu \nu \alpha _1\dots \alpha _{d-2}}(\nabla ^\mu \xi ^\nu -\nabla ^\nu \xi ^\mu )\,\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-2}}. \end{aligned}$$

(6.13)

Now by the relations

$$\begin{aligned} \delta \sqrt{-g}=\frac{\sqrt{-g}}{2}\delta g^\alpha _{\,\,\alpha }, \quad \delta \Gamma ^\lambda _{\mu \nu }&= \frac{1}{2}[g^{\lambda \sigma }\big (\nabla _\mu \delta g_{\sigma \nu }+\nabla _\nu \delta g_{\sigma \mu }-\nabla _\sigma \delta g_{\mu \nu }\big )], \end{aligned}$$

(6.14)

one finds

$$\begin{aligned} \delta \mathbf {Q}_\xi =&\frac{-1}{16\pi G}\frac{\sqrt{-g}}{(2!(d-2)!)}\epsilon _{\mu \nu \alpha _1\dots \alpha _{d-2}}\Big (\frac{1}{2}\delta g^\alpha _{\,\,\alpha }(\nabla ^\mu \xi ^\nu )-\delta g^{\mu \beta }(\nabla _\beta \xi ^\nu )\nonumber \\&+\xi ^\alpha \nabla ^\mu \delta g^\nu _{\,\,\alpha }\Big )\,\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-2}}-[\mu \leftrightarrow \nu ]. \end{aligned}$$

(6.15)

in which the notation $\delta g^{\mu \nu }\equiv g^{\mu \alpha }g^{\nu \beta }\delta g_{\alpha \beta }=-\delta (g^{\mu \nu })$ has been used. Notice that by $\delta (g^{\mu \nu })$ we meant the direct action of $\delta$ on $g^{\mu \nu }$. The next step in calculating the $\varvec{k}_\xi$ would be finding the second term in (6.7), which is

$$\begin{aligned}&-\xi \cdot \varvec{\Theta }(\delta \Phi ,\Phi )\nonumber \\&\quad =-\xi \cdot \Big (\frac{1}{16\pi G}\frac{\sqrt{-g}}{(d-1)!}\epsilon _{\mu \alpha _{1}\dots \alpha _{d-1}}(\nabla _\alpha \delta g^{\alpha \mu }-\nabla ^\mu \delta g^\alpha _{\,\,\alpha })\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-1}}\Big )\nonumber \\&\quad =\frac{-1}{16\pi G}\frac{\sqrt{-g}}{(d-2)!}\epsilon _{\mu \nu \alpha _{1}\dots \alpha _{d-2}}(\nabla _\alpha \delta g^{\alpha \mu }-\nabla ^\mu \delta g^\alpha _{\,\,\alpha })\xi ^\nu \,\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-2}}\quad \end{aligned}$$

(6.16)

$$\begin{aligned}&\quad =\frac{-1}{16\pi G}\frac{\sqrt{-g}}{2(d-2)!}\epsilon _{\mu \nu \alpha _{1}\dots \alpha _{d-2}}(\nabla _\alpha \delta g^{\alpha \mu }-\nabla ^\mu \delta g^\alpha _{\,\,\alpha })\xi ^\nu \,\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-2}}\nonumber \\&\quad \qquad -[\mu \leftrightarrow \nu ]. \end{aligned}$$

(6.17)

Finally, having found the (6.15) and (6.17), the $\varvec{k}_\xi ^{\text {EH}}$ can be read as

$$\begin{aligned} \varvec{k}_\xi ^{\text {EH}}&=\frac{-1}{16\pi G}\frac{\sqrt{-g}}{(2!(d-2)!)}\epsilon _{\mu \nu \alpha _1\dots \alpha _{d-2}}\Big (\frac{1}{2}\delta g^\alpha _{\,\,\alpha }(\nabla ^\mu \xi ^\nu )-\delta g^{\mu \beta }(\nabla _\beta \xi ^\nu )+\xi ^\alpha \nabla ^\mu \delta g^\nu _{\,\,\alpha }\nonumber \\&\quad +(\nabla _\alpha \delta g^{\alpha \mu }-\nabla ^\mu \delta g^\alpha _{\,\,\alpha })\xi ^\nu \Big )\,\mathrm {d}x^{\alpha _1}\wedge \dots \wedge \mathrm {d}x^{\alpha _{d-2}}-[\mu \leftrightarrow \nu ]. \end{aligned}$$

(6.18)

By the Hodge duality, we would have $\varvec{k}_\xi ^{\text {EH}}=\star k_\xi ^{\text {EH}}$, where

$$\begin{aligned} k_\xi ^{\text {EH}\mu \nu }= & {} \frac{-1}{16\pi G}\Big (\frac{1}{2}\delta g^\alpha _{\,\,\alpha }(\nabla ^\mu \xi ^\nu )-\delta g^{\mu \beta }(\nabla _\beta \xi ^\nu )+\xi ^\alpha \nabla ^\mu \delta g^\nu _{\,\,\alpha }\nonumber \\&+\,(\nabla _\alpha \delta g^{\alpha \mu }-\nabla ^\mu \delta g^\alpha _{\,\,\alpha })\xi ^\nu \Big )-[\mu \leftrightarrow \nu ]. \end{aligned}$$

(6.19)

Notice that this result is independent of the cosmological constant $\Lambda$.