# Geometrothermodynamics for black holes and de Sitter space

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## Abstract

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.

## Keywords

General relativity Thermodynamics Black holes## 1 Introduction

*M*is the mass of a black hole, \(\kappa \) is the surface gravity, \(A_\mathrm{H}\) is the area of the event horizon,

*G*is Newton’s gravitational constant, \(\Omega _\mathrm{H}\) is the surface angular velocity,

*J*is the angular momentum, \(\Phi \) is the electromagnetic potential, and

*Q*is a charge. For (1) to be the first law of thermodynamics for a black hole, the relationship between black hole quantities, \(\kappa ,A_\mathrm{H}\), and the thermodynamic quantities, temperature and entropy, must be clarified. The temperature of a black hole can be equated with the Hawking temperature [2], \(T_\mathrm{H}=\hbar \kappa /2\pi c\), which is obtained using a semi-classical treatment of the particle radiation from the surface of a black hole. The entropy can be obtained from the area theorem [2] for black holes: \( S_\mathrm{H}={A_\mathrm{H}}/{4l_p^2} \), where \(S_\mathrm{H}\) is the entropy of a black hole and \(l_p=\sqrt{G\hbar /c^3}\) is the Planck length. Using these relations, the first term of the r.h.s. of (1) can be rewritten as \( {\kappa c}/{(8\pi G)}dA_\mathrm{H}=T_\mathrm{H}dS_\mathrm{H} \), which allows a thermodynamic interpretation of (1). The area theorem is obtained in multiple ways for various types of black holes [6, 7, 8, 9]. Wald proposed equating the Noether charge with the entropy of black holes [7]. Wald’s method is applicable to a wide variety of Einsteinian and post-Einsteinian gravitational theories [10]. The black hole entropy has been discussed from both classical and quantum approaches [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], and also string theoretically [27, 28, 29, 30, 31]. A nice overall review article can also be found [32]. Gour and Mayo demonstrated that the entropy of a black hole is (nearly) linear in the area of the horizon [33]. The relation between black hole entropy and the Noether charge is discussed by many authors [34, 35, 36, 37, 38, 39, 40, 41]. In addition to those studies, the importance of boundary terms plays the essential role of the black hole entropy, that is first pointed out by Yoke [42]. Especially, the necessity of null boundary is discussed by Parattu [43, 44]. Interesting discussions related the boundary terms can be found in Refs. [45, 46, 47, 48, 49, 50]. In spite of these intensive studies, the area theorem for a black hole entropy has not yet been proven generally.

We propose a new method to extract entropy from the general relativistic theory. Our method is sufficiently general to discuss the thermodynamics of a wide variety of solutions of the Einstein equation with an event horizon. The Lagrangian and Hamiltonian formalisms are given in a completely covariant way in Sect. 2. In Sect. 3, the abstract thermodynamic system is rewritten using the terminology for a geometry that is parallel to the treatment for general relativity given in Sect. 2. The new method is applied to black holes and the de Sitter space in Sect. 4, and then the area theorem is examined for these examples. Finally, a summary is presented in Sect. 5. Physical constants are set to unity, e.g., \(4\pi G=\hbar =c=1\), in this work unless otherwise noted.

## 2 Vierbein formalism of general relativity

First, classical general relativity is geometrically re-formulated in terms of a vierbein formalism. The formalism and terminology in this section primarily (but not completely) follow Frè [51]. The vierbein formalism of general relativity can be found in other textbooks [52, 53, 54, 55, 56, 57] too.

*SO*(1, 3) covariant derivative \(d_{\mathfrak w}\). In this work, the Fraktur letters are used for differential forms, and Greek- and Roman-indices, both run from zero to three, are used for a coordinate on the global and local space time manifold, respectively. A curvature two-form and a volume from are introduced as \(\mathfrak {R}^{a_1a_2}=d{\mathfrak {w}}^{a_1a_2}+{\mathfrak {w}}^{a_1}_{~~b}\wedge {\mathfrak {w}}^{ba_2}\) and \(\mathfrak {v}=\epsilon _{a_1a_2a_3a_4}{\mathfrak e}^{a_1}\wedge {\mathfrak e}^{a_2}\wedge {\mathfrak e}^{a_3}\wedge {\mathfrak e}^{a_4}/4!\), respectively. The Einstein–Hilbert gravitational Lagrangian is expressed using these forms as;

*conjugate operator*, is introduced for shorter representations including Levi-Civita tensors. This is a map that maps a rank-

*p*tensor field to a rank-\((n-p)\) tensor field defined on an

*n*-dimensional space-time manifold such that

*p*tensor and \(\epsilon \) is a completely anti-symmetric tensor in

*n*-dimensions. Note that \(\overline{\overline{{\mathfrak a}}}={\mathfrak a}\) when \({\mathfrak a}\) is anti-symmetric with respect to all indices. The surface form can be expressed using the \(\epsilon \)-conjugate operator as \({\mathfrak S}_{ab}=\overline{{\mathfrak e}^a\wedge {\mathfrak e}^b}\). Therefore, the \(\epsilon \)-conjugate of the surface form, which appears frequently in later parts of this report, is \(\overline{{\mathfrak S}}^{ab}=\overline{{\mathfrak S}_{ab}}={{\mathfrak e}^a\wedge {\mathfrak e}^b}\). In the four-dimensional manifold, the identities

## 3 Geometrothermodynamic formalism

In this section, before discussing the thermodynamic aspects of the gravitation, the thermodynamic system is rewritten using geometrical terminologies. The formalization is based on that of Ref. [59]; however, it is rewritten according to the conventions of this work. Here, we confine our attention to the formal relationships between the thermodynamic variables and ignore their physical meanings.

*S*is the entropy,

*T*is the temperature,

*p*is the pressure,

*V*is the volume,

*N*is the number of molecules, and \(\mu \) is the chemical potential. These variables can be categorized into two types: extensive and intensive variables. According to the natural connotation given above, \(\xi ^a=(S,V,N)\) are categorized as extensive variables and \(\zeta ^a=(T,p,\mu )\) are intensive variables, where \(a=0,1,2\). We assume the existence of a smooth three-dimensional base-manifold \(\mathcal{M}_\mathrm{TD}\) and \(\xi \) and \(\zeta \) are vectors belonging to the tangent and cotangent bundles on the manifold, respectively.

*S*is considered to be the order parameter of the system and \(T=\partial f_{\mathrm{TD}}/\partial S\) is considered as the “Hamiltonian” (with a narrower meaning compared with the discussions in previous sections that induce a “time” evolution in the system). The intensive variables are not independent of each other; however,

*T*is assumed to be independent of

*S*. Therefore, we can write the intensive variables as

*SO*(1, 2) symmetry and is consistent with an interpretation that the variable

*S*plays the role of “time” (i.e., an order parameter) and “T” plays the role of an energy (i.e., the Hamiltonian). The characteristic two-form (the Lagrangian-density form) can also be written as

*V*or

*N*. The above equations represent the canonical equation of motion with the following interpretations: \(S \leftrightarrow t\) (time) and \(T\leftrightarrow \mathcal{H}\) (Hamiltonian). These equations can also be written as the Euler–Lagrange equation of motion in a coordinate-independent manner as \(\delta _\mathfrak {x}\mathscr {I}_\mathrm{TD}=0\Rightarrow \mathfrak {z}^{a}=0\), which is, of course, equivalent to the canonical equations. The surface term \( \mathfrak {O}_\delta =\zeta _a~\delta \mathfrak {x}^a \) can be considered as another type of the symplectic potential.

Symplectic potential and the Noether current/charge in thermodynamics

Name | Type | Definition |
---|---|---|

Symplectic potential | 1-form | \(\mathfrak {O}_\mathrm{TD}=~{\mathfrak w}_\mathrm{TD}=\zeta _a\mathfrak {x}^a\) |

Noether current | 1-form | \({\mathfrak J}_\upsilon =\mathfrak {O}_\upsilon -\iota _\upsilon {\mathfrak L}_\mathrm{TD}\) |

Noether charge | 0-form | \(\mathfrak {Q}_\upsilon =\zeta _a\iota _\upsilon \mathfrak {x}^a\) |

## 4 The symplectic formalism of general relativity

A general formalization of the thermodynamics of space-time is formalized in this section. The method proposed in Wald [7] and Iyer and Wald [8, 9] are carefully rewritten using our terminology.

### 4.1 General formalism

Let us start from the Lagrangian density of (4). This Lagrangian form has two candidates for the first canonical variables. According to the geometrothermodynamic formulation introduced in the previous section, we would like to start from the intensive variables. The two-dimensional surface form \({\mathfrak S}_{ab}\) can be easily recognized as an extensive variable. Conversely, the spin form \({\mathfrak w}^{ab}\) may be intensive because it is defined locally, independent of the total volume under consideration. Therefore, we identify the spin form as the first canonical variable, as discussed in Sect. 2.

*symplectic potential*”. Here, we assume the existence of an inverse of the spin form. Compared with the geometrothermodynamics given in the previous section, it is identified as the symplectic potential. Using this symplectic potential, the surface term can be expressed as \(\mathfrak {O}_\delta =\delta _{\mathfrak w}\mathfrak {O}=\delta {\mathfrak w}^{a_1a_2}\wedge {\mathfrak S}_{a_1a_2}\).

*X*defined on the global manifold. The Lie derivative along the vectors \(\xi ^\mu \in X\) of the Lagrangian density becomes

Next, let us introduce a generating function, \(\mathcal{H}_\xi \), and its density form, \({\mathfrak H}_\xi \), such that \(\mathcal{H}_\xi =\int {\mathfrak H}_\xi \), where the integration region is the appropriate three-dimensional manifold embedded in the global space-time manifold. \({\mathfrak H}_\xi \) can be understood as a generalization of the Hamiltonian introduced in the previous section, which induces a flow in the system along the general vector fields on the local manifold. For example, when a time-like vector field can be defined globally in the local manifold and is chosen as the “time coordinate”, the generating function density form \({\mathfrak H}_t\) is just the standard Hamiltonian with its narrow meaning, which results in the “time evolution” of the system.

*X*is defined globally on the entire global manifold, the generating function with respect to the vector \(\xi ^\mu \in X\) can be defined in the follow manner. Take the variation of the Noether current with respect to the spin form to be

*symplectic current*”. The generating function density form is introduced using the symplectic current so that \(\delta _{\mathfrak w}{\mathfrak H}_\xi =\mathfrak {o}\). Therefore, the Noether current can be written as

Symplectic potential and the Noether current/charge in general relativity

Name | Type | Definition |
---|---|---|

Symplectic potential | 3-form | \(\mathfrak {O}={\mathfrak w}^{a_1a_2}\wedge {\mathfrak S}_{a_1a_2}/2\) |

Noether current | 3-form | \({\mathfrak J}_\xi =\mathfrak {O}_\xi -\iota _\xi {\mathfrak L}_G\) |

Noether charge | 2-form | \(\mathfrak {Q}_\xi =(\iota _\xi {\mathfrak w}^{a_1a_2}){\mathfrak S}_{a_1a_2}/2\) |

Based on the above geometrical formalization, one can now single the “*entropy*” out from the general relativity system by comparing the results in Sects. 3 and 4. In general relativity (*GR*), the first canonical variable is taken to be \({\mathfrak w}^{{ab}}\), which may correspond to \(\xi ^{a}=(S,V,N)\) in thermodynamics (*TD*). The symplectic structure can be discussed in parallel for *TD* and *GR*. For example, using the surface term \(\mathfrak {O}_\mathrm{TD}\leftrightarrow \mathfrak {O}_\delta \), the conserved Noether current can be introduced as \({\mathfrak J}_\upsilon \leftrightarrow {\mathfrak J}_\xi \). The *TD* analysis suggests that the entropy can be extracted from the Noether charge for \(Q_\mathrm{TD}\leftrightarrow \mathfrak {Q}_\xi \).

Before closing this section, we would like to emphasize the origin to ensure the conservation of the Noether current/charge. In above discussions, any specific properties of the vector fields are not used. Though the Killing vector-fields will be takes as the vector filed *X* in a next section, the conserved current/charge can be obtained for any vector filed. The continuous symmetry to induce the Noethe’s theorem in this case is the variational operation with respect to the first canonical form \({\mathfrak w}^{{ab}}\), which caused at most a term of total derivative as shown in (27). Some examples of the Noether charge are summarized in “Appendix”. Moreover concrete representations for the Lie derivative \(\pounds _\xi \) and contraction \(\iota _\xi \) are not important. Algebraic rules for differential forms play a essential role.

### 4.2 Application to a Schwarzschild black hole

*M*is the mass of a black hole. The vierbein form can be extracted as

*t*and \(\phi \). The third and fourth vectors satisfy the Killing equation such that

*t*can be interpreted as the time coordinate in this region. It is expected that the thermodynamic quantities are related to this Killing vector, because, for example, the entropy can only be well defined for the thermal-equilibrium state, which is static with respect to

*t*. A dual vector of the Killing vector is given by

*G*,

*c*and \(\hbar \) explicitly. By comparing this result with the thermodynamic result given in the previous section (26), this quantity can be described using the thermodynamic variables as

The differences between Wald’s original method [7] and the method given here is as follows: in Wald’s method, the Killing vector was normalized such that the Killing field had a unit surface gravity. This normalization results in a factor of \(4GM/c^4\) in the Noether charge. (Wald explained in [7] that this normalization makes the Noether charge local.) In addition, Wald’s definition of the entropy has a factor of \(2\pi (c/\hbar ) \)^{1} in front of the surface integration of the Noether charge. Therefore, Wald’s entropy has an additional factor of \(8\pi GM/\hbar c^3\) compared with our definition, which corresponds to the inverse temperature of a black hole. This is why the integration of the Noether charge simply gives the entropy in [7]. From a purely mathematical point of view, the two methods are equivalent to each other.

### 4.3 Application to a Kerr–Newman black hole

*M*, an angular momentum \(J=\alpha M\), and an electric charge

*q*can be expressed using the Boyer–Lindquist metric [60] as [61, 62]

*c*(the speed of light) is written explicitly in (49) for future convenience. In the other parts, the physical constants are still set to unity. Details of the treatment of the physical constants are given later in this section. Even though each case corresponds to different thermodynamic processes, the calculations of the Noether charge are the same. Therefore, in the following, the \(\beta \)’s are not immediately specified.

*a*” can be

*t*or \(\phi \). Compared with the case of a Schwarzschild black hole, an additional factor of 1 / 2 is placed in front of \((\iota _\xi {\mathfrak w}^{{ab}})\) to avoid double counting owing to the two surface forms that contribute to the results. The other surface forms are zero at \(\theta =\pi /2\) and therefore do not contribute to the final results. These two terms have the following integrands:

#### 4.3.1 A Kerr black hole

First, a Kerr black hole without an electric charge is treated, which can be obtained by setting \(q\rightarrow 0\). For the first case of (49), the Killing vector is null at the black hole surface (“event horizon” = ”Killing horizon”). When \(\alpha \) is zero, which corresponds to non-rotating black holes, the normalization is the same as that for a Schwarzschild black hole. Therefore, in this case, the Noether charge is expected to be \(T_\mathrm{KN}S_\mathrm{KN}\), where \(T_\mathrm{KN}\) and \(S_\mathrm{KN}\) are the temperature and the entropy of a Kerr–Newman black hole, respectively

*t*-integration, as before, the Noether charge becomes \( Q_\xi =J_\mathrm{K}. \) The Noether charge for this Killing vector consists of \(J_\mathrm{K}\) and \(\Omega _\mathrm{K}\), both of which are defined in classical physics, in contrast to the temperature, which can only be defined via the quantum effect. Because the angular momentum is correctly obtained as the Noether charge, the method proposed here is ensured to work correctly. Compared with the thermodynamics developed in Sect. 3, this may correspond to the isobaric process, which has a “Killing vector” of \(\upsilon ^a=(0,1,0)\). The Noether charge induced by this Killing vector becomes \(Q_\mathrm{TD}=pV\). This result supports the correspondence

#### 4.3.2 A Kerr–Newman black hole

*q*is defined to have a length dimension. The entropy of a Kerr–Newman black hole can be singled out using the Hawking temperature (53) as

### 4.4 Application to de Sitter space

*length*\(dimension)^{-2}\) in this convention. According to this change, the spin form can be obtained as

## 5 Concluding remarks

The method proposed in this work, called the “*geometrothermodynamic method* (GT-method)”, is sufficiently general to treat a large variety of gravitational objects that have event horizons, such as black holes and de Sitter space. The Noether charge associated with an appropriate Killing vector shows a clear relationship with thermodynamical objects, such as the temperature and entropy. One advantage of the GT-method is that it is a purely classical and thermodynamic method that does not assume any (Euclidean) ensembles. Moreover, its formalism is highly geometrical and can be described in a coordinate independent way. The same formalism can also be applied to thermodynamics, and therefore the relationships between the thermodynamics and gravitational theory are apparent. The resultant entropies obtained using the GT-method are consistent with previous results and confirm the area theorem for black holes and the de Sitter space.

*M*,

*J*, and

*q*, and has a variation with respect to these variables:

*F*and the temperature/entropy. For any black hole, the Hawking temperature and the area of the horizon have the relationship of \( TA={(r_+^2-\alpha ^2-q^2)}/{r_+}. \) Therefore, one can easily confirm the relation \( F={T}/{4} \). Therefore, the first term of the r.h.s. of (57) can be written as

Conversely, the GT-method does not assume any relationship between the entropy and the area. The area of the event horizon appears in non-zero components of the surface form \({\mathfrak S}_{{ab}}\) in the Noether charge. A second term on the r.h.s. of (57) is given by the same procedure with a different Killing vector. The reason why the Noether charge is proportional to the surface form is very simple. In diffeomorphic theory for a *d*-dimensional manifold, the Lagrangian form and the Noether current must be the *d*-form and the \((d-1)\)-form, respectively. Therefore, the Noether charge, which is obtained by integrating the Noether current, must be a \((d-2)\)-form. In four-dimensional space-time, the Noether charge is a two-form, which can be expanded by the bases \({\mathfrak e}^{a_1}\wedge {\mathfrak e}^{a_2}=\overline{{\mathfrak S}}^{a_1a_2}\). Therefore, the GT-method provides independent evidence that the entropy can be represented using only the area.

In summary, the GT-method, which can extract thermodynamic quantities from a large variety of solutions for the Einstein equation with an event horizon, is proposed in this work. This method is mathematically consistent with Wald’s method. However, the relationships between general relativity and thermodynamics are clearer because abstract thermodynamics can be reconstructed using geometrical terminologies that are parallel to general relativity.

The discussions given in this work are purely thermodynamic without any statistical (Euclidean) ensembles. Therefore, the conclusion, for example, that \(S_\mathrm{KN}T_\mathrm{KN}\) of a Kerr–Newman black hole, such as (55), must be independent of the microscopic details is expected to be correct, even if the expression of the Hawking temperature will receive some corrections from the (still-unknown) quantum gravity.

## Footnotes

## Notes

### Acknowledgements

We wish to thank Dr. Y. Sugiyama for their continuous encouragement and fruitful discussions.

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