On thermodynamics of compact objects

In curved spacetime, a fluid element sufficiently small can be described by local thermodynamic variables in the local rest frame of the fluid element [21]. According to the equivalence principle, these local thermodynamic variables shall respect the fundamental laws of thermodynamics. Here, we take two basic relations as the starting point. One is the local Euler equation for thermodynamics [22], relating the proper quantities in the local frame,

$$\begin{align} s = \frac{\rho+p-\mu\, n}{T}\;, \end{align} \tag{ 3 }$$

where s is the entropy density, n is the number density, µ is the chemical potential and T is the temperature. The other relation comes from the local version of the first law (together with the second law) of thermodynamics [23]

$$\begin{align} \mathrm{d}\rho = \frac{\rho+p-s\,T}{n} \mathrm{d}n+T d s = \mu\, d n+ T\, d s\,, \end{align} \tag{ 4 }$$

where the energy density ρ is treated as a function of number density n and entropy density s.

For the canonical ensemble, the local thermodynamic potential is the Helmholtz free energy density f. It is clear from the fundamental equation that $f = \rho-s\, T$ is a function of n and T. Following equation (4), we have

$$\begin{align} \mathrm{d}f = \mu\, \mathrm{d}n-s \,\mathrm{d}T\;, \end{align} \tag{ 5 }$$

from which the chemical potential and entropy density can directly be found as

$$\begin{align} \mu = \left(\frac{\partial f}{\partial n}\right)_T\;,\quad s = -\left(\frac{\partial f}{\partial T}\right)_n\;. \end{align} \tag{ 6 }$$

Using equation (6) and the local Euler equation, given in equation (3), together with the definition of f, one obtains the energy density ρ and the pressure p in terms of f as

$$\begin{align} \rho = -T^2 \frac{\partial}{\partial T}\left(T^{-1} f \right)_n\,,\quad p = -\frac{\partial }{\partial n}\left(n\,f\,\right)_T\;. \end{align} \tag{ 7 }$$

Similarly, for the grand canonical ensemble, we consider the grand potential density $\omega = \rho-s\,T-\mu\,n = -p$. As a function of T and µ, the total differential is

$$\begin{align} \mathrm{d}\omega = -s \,\mathrm{d}T-n\,\mathrm{d}\mu\;. \end{align} \tag{ 8 }$$

In analogy with equations (6) and (7), we obtain the derived quantities as follows:

$$\begin{align} n &= -\left(\frac{\partial \omega}{\partial \mu}\right)_T\;,\quad s = -\left(\frac{\partial \omega}{\partial T}\right)_\mu\,,\quad p = -\omega\,,\quad \rho = -T^2 \frac{\partial}{\partial T}\left(T\,^{-1} \omega \right)_\mu+n\,\mu\,. \end{align} \tag{ 9 }$$

The spacetime variation of local thermodynamic quantities is governed by the conservation law of the stress tensor, $\nabla^{\mu}T_{\mu\nu} = 0$. For a static and spherically symmetric spacetime, this yields a single constraint from the momentum conservation along the radial direction. For the metric in equation (1) and stress tensor in equation (2), this constraint is obtained as

$$\begin{align} p^{^{\prime}}+(p+\rho)\frac{B^{^{\prime}}}{2B} = 0\;, \end{align} \tag{ 10 }$$

where $(^{^{\prime}})$ denotes derivative with respect to r.

As recently emphasized in [24], the momentum conservation of the stress tensor is directly related to Tolman's law for local thermal equilibrium in curved spacetime. With use of the local Euler equation, given in equations (3) and (10) can be expressed as

$$\begin{align} \frac{B^{^{\prime}}}{2B} = -\frac{1}{\rho+p}\left(-\rho^{^{\prime}}+T\,s^{^{\prime}} +\mu\,n^{^{\prime}}\right)-\frac{T^{^{\prime}}}{T}-\frac{n\,T}{\rho+p}\left(\frac{\mu}{T}\right)^{^{\prime}}\;. \end{align} \tag{ 11 }$$

The term in the parenthesis vanishes due to the first law given in equation (4), and therefore

$$\begin{align} \frac{B^{^{\prime}}}{2B}+\frac{T^{^{\prime}}}{T}& = &-\frac{n \, T}{\rho+p}\left(\frac{\mu}{T}\right)^{^{\prime}}\;. \end{align} \tag{ 12 }$$

When the chemical potential is zero, as for the photon gas, equation (12) reduces to the commonly known Tolman's law [10, 11]: $T(r)\sqrt{B(r)} = T_{\infty}$. The constant $T_{\infty}$ is the temperature of the gas at spatial infinity or the redshifted temperature for observers at spatial infinity. More generally, the quantity $\mu/T$, the exponential of which is commonly called fugacity, is position independent, i.e. $(\mu/T)^{^{\prime}} = 0$, as the condition of vanishing heat flow and diffusion for a system in equilibrium [25]. This then leads to the generalized version of Tolman's law [26]

$$\begin{align} T(r)\sqrt{B(r)} = T_{\infty}\,,\quad \mu(r)\sqrt{B(r)} = \mu_{\infty}\,. \end{align} \tag{ 13 }$$

This means that the temperature and chemical potential as intensive quantities are each uniquely specified by single numbers, $T_\infty$ and $\mu_\infty$, respectively. Since we consider only equilibrium thermodynamics, we employ the condition of constant fugacity throughout this work.

The global thermodynamic characteristics of a strongly gravitating system are of interest to observers far away from the gravitational potential. For a generic discussion here, we write down global extensive thermodynamic variables in terms of their local counterparts by assuming additivity [27]. This is expected for a variety of matter sources and will be explicitly verified for non-interacting gas in later subsections. Thermodynamic potentials are then obtained by appropriately integrating the corresponding potential energy density such as

$$\begin{align} U = \int_0^R \sqrt{A\,B}\, \rho \,d^3r\;,\quad F = \int_0^R \sqrt{A\,B}\, f \,d^3r\;,\quad \Omega = \int_0^R \sqrt{A\,B}\, \omega \,d^3r\;, \end{align} \tag{ 14 }$$

where the factor $\sqrt{A\,B}$ comes from the term $\sqrt{-g}$, with g being determinant of the metric, and R denotes the boundary of the matter distribution. U, F, and Ω are the total internal energy, Helmholtz free energy, and the grand potential, respectively. The energy densities multiplied by $\sqrt{B}$ can be interpreted as the redshifted values measured at spatial infinity. The total number of particles N and the entropy S are defined through the spatial integral of the temporal components of the conserved currents J^µ and S^µ (with $\sqrt{-g}$ in the integrand). Considering that the densities are given as $n = -J^{\mu}u_{\mu}$ and $s = -S^{\mu}u_{\mu}$, and that the four-velocity in the rest frame of fluid becomes $u^{\mu} = (1/\sqrt{B},0,0,0)$, it is obtained that

$$\begin{align} N = \int_0^R \sqrt{A}\,n \,d^3r\,,\quad S = \int_0^R \sqrt{A}\,s \,d^3r\,, \end{align} \tag{ 15 }$$

where the factor $\sqrt{B}$ drops out from each integrand and the volume element becomes the geometric one.

Let's consider the canonical ensemble to verify the conventional thermodynamics. Given the Helmholtz free energy density $f = \rho-s\,T$ and the Tolman's law equation (13), we can first obtain the fundamental equation

$$\begin{align} F = \int_0^R \sqrt{A\,B}\, (\rho-s\, T) \,d^3r = U-T_{\infty} S\;. \end{align} \tag{ 16 }$$

Then, taking the total differential of the Helmholtz free energy F and implementing equations (5) and (13), we have

$$\begin{align} \mathrm{d}F& = \int_0^R \sqrt{A\,B}\, (\mu\, \mathrm{d}n-s \,\mathrm{d}T) \,d^3r+ (\mathrm{d}F)_{f} = \mu_\infty \int_0^R \sqrt{A}\, \, \mathrm{d}n \,d^3r-\int_0^R \sqrt{A\,B}\, s \,\frac{d T_\infty}{\sqrt{B}} \,d^3r+ (\mathrm{d}F)_{f}\nonumber\\ & = \mu_\infty\, \mathrm{d}N -S\, \mathrm{d}T_\infty+ (\mathrm{d}F)_{N,T_\infty}\,. \end{align} \tag{ 17 }$$

The last term $(\mathrm{d}F)_{N,T_\infty}$ denotes the variation of the metric function or size of the system independent of N and $T_\infty$. If we attribute this change to the variation of a thermodynamic volume element

$$\begin{align} \mathrm{d}V_\mathrm{th} = -p_{\infty}^{-1} \left(\mathrm{d}F\right)_{T_{\infty}, N}\;, \end{align} \tag{ 18 }$$

the conventional equation can be recovered

$$\begin{align} \mathrm{d}F = -S \mathrm{d}T_{\infty}-p_{\infty}\mathrm{d}V_\mathrm{th}+\mu_\infty \mathrm{d}N\;. \end{align} \tag{ 19 }$$

Thus, by considering F as a function of N, $T_\infty$ and $V_\mathrm{th}$, one gets the consistent picture with the desired relations

$$\begin{align} S = -\left(\frac{\partial F}{\partial T_{\infty}}\right)_{V_\mathrm{th},N}\;,\qquad p_{\infty} = -\left(\frac{\partial F} {\partial V_\mathrm{th}}\right)_{T_{\infty},N}\;,\qquad \mu_\infty = \left(\frac{\partial F} {\partial N}\right)_{T_{\infty},V_\mathrm{th}}\,. \end{align} \tag{ 20 }$$

By using these relations and equation (16), the internal energy can be simply expressed as $U = -T_{\infty}^2\;(\partial (T_{\infty}^{-1} F)/\partial T_{\infty})_{V_\mathrm{th},N}$. Then, with equations (16) and (19), one obtains the fundamental thermodynamic relation for internal energy,

$$\begin{align} \mathrm{d}U = T_\infty \mathrm{d}S-p_\infty \mathrm{d}V_\mathrm{th}+\mu_\infty \mathrm{d}N\,, \end{align} \tag{ 21 }$$

as the manifestation of the first law of thermodynamics for global variables.

Similarly, for the grand canonical ensemble (see appendix A for the microcanonical ensemble), given Tolman's law in equation (13), the global grand potential satisfies

$$\begin{align} \Omega = \int_0^R \sqrt{A\,B}\, (\rho-s\,T-\mu\,n) \,d^3r = U-T_{\infty}S-\mu_{\infty} N\;, \end{align} \tag{ 22 }$$

where the term in parenthesis is the grand potential density $w = -p$, as defined above equation (8). Together with equation (19), the total differential of Ω is given as

$$\begin{align} \mathrm{d}\Omega = -S \mathrm{d}T_{\infty}-p_{\infty}\mathrm{d}V_\mathrm{th}-N \mathrm{d}\mu_{\infty}\;, \end{align} \tag{ 23 }$$

where the thermodynamic volume can be expressed in terms of the grand potential as

$$\begin{align} \mathrm{d}V_\mathrm{th} = -p_{\infty}^{-1}(\mathrm{d}\Omega)_{T_{\infty}, \mu_{\infty}} = \left(d \int_0^R \sqrt{A\,B}\, \frac{p}{p_{\infty}} \,d^3r\right)_{T_{\infty}, \mu_{\infty}}\,, \end{align} \tag{ 24 }$$

and consequently, the required relations are obtained as

$$\begin{align} S = -\left(\frac{\partial \Omega}{\partial T_{\infty}}\right)_{V_\mathrm{th},\mu_\infty}\;,\qquad p_{\infty} = -\left(\frac{\partial \Omega} {\partial V_\mathrm{th}}\right)_{T_{\infty},\mu_\infty}\;,\qquad N = -\left(\frac{\partial \Omega} {\partial \mu_\infty}\right)_{T_{\infty},V_\mathrm{th}}\,. \end{align} \tag{ 25 }$$

In short; we have seen that global thermodynamic variables obey conventional thermodynamics in the sense that their definitions fully encode the curved spacetime effects. The key ingredient is to appropriately identify the thermodynamic volume $V_\mathrm{th}$ as in equations (18) and (24). Finding the explicit expression for $V_\mathrm{th}$ is not in general straightforward and its attainability highly depends on the equation of state (EoS) in question. This can be conveniently seen from the definition given in equation (24). Unlike T(r), $\mu(r)$ that follow Tolman's law, the spatial variation of pressure depends on the EoS as we will see later in our examples. The difficulty is in separating the spatial integral from the $T_{\infty}$ when the latter is tangled in position-dependent non-trivial functions. If such a separation is possible, then one can have a clear expression for $V_\mathrm{th}$. For instance, in the case of a massless ideal gas with the EoS $\rho = 3p$, which yields $p\propto T^{\,4}$, we have $p(r)/p_{\infty} = B^2(r)$ from the Tolman's law, and the thermodynamic volume emerges as

$$\begin{align} V_\mathrm{th} &= \int_0^R \sqrt{\frac{A(r)}{B^3(r)}}\; d^3r\,, \end{align} \tag{ 26 }$$

with $\Omega = -p_\infty V_\mathrm{th}$. Apparently, $V_\mathrm{th}$ differs from the geometric volume

$$\begin{align} V_\mathrm{geo} = \int_0^R \sqrt{A} \;d^3r\,. \end{align} \tag{ 27 }$$

When the curved spacetime features a deep gravitational potential, i.e. $B(r)\ll 1$, $V_\mathrm{th}$ will be much larger than $V_\mathrm{geo}$. As we will show later in section 3 by explicit examples of compact objects with back-reaction taken into account, a larger $V_\mathrm{th}$ is responsible for the difference between the physical mass M and internal energy U. Thus, in comparison to previous studies where M is interpreted as the internal energy, we keep U unchanged but replace $V_\mathrm{geo}$ by $V_\mathrm{th}$, where the gravitational field contribution is more conveniently encoded.

As the first example, we consider a box of a semi-classical ideal gas in thermal equilibrium. The local thermodynamic variables can be derived from the Boltzmann distribution in the local rest frame. For later discussion, we display expressions [22] for the number density, energy density, pressure, and entropy density,

$$\begin{align} n& = &\frac{e^{\mu/T}}{(2\pi)^3}\int e^{-E/T} d^{3}\mathrm{p} = \frac{e^{\mu/T}m^2}{2\pi^2} T K_2 (b)\,, \end{align} \tag{ 28 }$$

$$\begin{align} \rho& = &n T \left( 3+b \frac{K_1(b)}{K_2 (b)}\right)\;, \end{align} \tag{ 29 }$$

$$\begin{align} p& = &nT\;, \end{align} \tag{ 30 }$$

$$\begin{align} s& = & n\left(4+b\frac{K_1 (b)}{K_2 (b)}-\frac{\mu}{T}\right)\,, \end{align} \tag{ 31 }$$

where $E = \sqrt{\mathbf{p}^2+m^2}$ is the locally measured proper energy of a particle, K₁, K₂ are the modified Bessel functions, and $b\equiv m/T$.

To complete our previous derivation of consistency between local and global pictures, we first verify the additivity of the global Helmholtz free energy in equation (14). The starting point is the fundamental equation in the canonical ensemble, namely,

$$\begin{align} F = -T_{\infty} \ln Z_{N}(T_\infty)\;, \end{align} \tag{ 32 }$$

where Z_N is the N-particle global partition function. As usual, we evaluate the one-particle partition function Z₁ first. Taking the energy eigenstates of the Hamiltonian $\hat{H}$, we have

$$\begin{align} Z_1(T_\infty) = \operatorname{Tr}[e^{-\hat{H}/T_\infty}] = \sum_\mathbf{l} e^{-E_{\mathbf{l},\infty}/T_\infty} = \frac{1}{(2\pi)^3}\int e^{-E_\infty/T_\infty} g(E_\infty) \mathrm{d}E_\infty\,, \end{align} \tag{ 33 }$$

where $\mathbf{l} = (l_x, l_y, l_z)$ labels the momentum eigenstates in the box, and the summation over l is approximated by an integral for the box sufficiently large. $E_\infty = -\xi^\mu p_\mu$ is the conserved energy, where $\xi^\mu = (1,\boldsymbol{0})$ is the timelike killing vector of the static spacetime and p^µ is the particle's four-momentum. The density of states available to one-particle $g(E_\infty)$ for a given energy $E_\infty$ can be obtained by $g(E_\infty) = \mathrm{d}P(E_\infty)/\mathrm{d}E_\infty$, with the invariant phase space volume [28, 29]

$$\begin{align} P(E_\infty) = \int d^3r \,d^3\mathrm{p}\, \Theta(E_\infty+\xi^\mu p_\mu) = \frac{4}{3}\pi \int (E_\infty^2/B-m^2)^{3/2} \sqrt{A}\, d^3r \,, \end{align} \tag{ 34 }$$

where Θ is the Heaviside step function accounting for the volume of momentum space with energy less than or equal to $E_\infty$. The one-particle partition function is then

$$\begin{align} Z_1(T_\infty) = \frac{1}{(2\pi)^3}\int e^{-E_\infty/T_\infty} \frac{\mathrm{d}P(E_\infty)}{\mathrm{d} E_\infty} \mathrm{d} E_\infty = \frac{1}{(2\pi)^3}\int e^{-E/T} \sqrt{A}\; d^3r\; d^3 \mathrm{p} = N e^{-\mu_\infty/T_\infty}\,,\nonumber\\ \end{align} \tag{ 35 }$$

where equations (15) and (28) are used in the last step to relate Z₁ and N. Note that $E = E_\infty/\sqrt{B}$ is the proper energy and so $E/T = E_\infty/T_\infty$. The physics behind the metric dependence for E and T is different. $E = E_\infty/\sqrt{B}$ is gravitational redshift due to the change of reference frames, while $T = T_\infty/\sqrt{B}$ is the Tolman's law due to thermodynamic equilibrium [29, 30].

As in the case of flat spacetime, the N-particle global partition function, Z_N , in the semi-classical limit is related to the one-particle partition function by $Z_N\approx Z_1^N/N!\approx (Z_1e/N)^N$, where the Stirling approximation [31] is used. Thus, with equations (30), (32) and (35), the global Helmholtz free energy is

$$\begin{align} F&\approx-T_\infty N\left(1+\ln\frac{Z_1}{N} \right) = N(\mu_\infty-T_\infty)\nonumber\\ & = \int_0^R\sqrt{AB}\,(n\,\mu-p)\,d^3r = \int_0^R\sqrt{AB}\,f\,d^3r\,, \end{align} \tag{ 36 }$$

where the local Euler equation, given in equation (3), and $f = \rho-s\,T$ are used in the last step. This supports the earlier definition equation (14) based on additivity.

Next, let us examine the explicit expressions for global variables. Considering the massless particle case first, the one-particle partition function is given through equation (35) as

$$\begin{align} Z_1(T_\infty) = \frac{1}{\pi^2}T_\infty^3\int_0^R \sqrt\frac{A(r)}{B^3(r)}\,d^3r \equiv \frac{1}{\pi^2}T_\infty^3 V_\mathrm{th}\,. \end{align} \tag{ 37 }$$

In the last step, the thermodynamic volume is directly identified as that in equation (26) since the $T_\infty$ dependence is fully separable from the spatial integral. This is consistent with the general definition of $\mathrm{d}V_\mathrm{th}$ in equation (18) from the global Helmholtz free energy F. To see this, we first write down F as a function of N, $T_\infty$ and $V_\mathrm{th}$,

$$\begin{align} F = -T_{\infty} \ln Z_{N}(T_\infty) = -T_{\infty} N \left(1+\ln \frac{T_{\infty}^3 V_\mathrm{th}}{\pi^2 N}\right)\,. \end{align} \tag{ 38 }$$

Equation (18) implies that the intensive quantity $p_\infty$ is the conjugate to $V_\mathrm{th}$ from equation (20),

$$\begin{align} p_\infty = -\left(\frac{\partial F} {\partial V_\mathrm{th}}\right)_{T_{\infty},N} = \frac{N T_\infty }{V_\mathrm{th}}\,. \end{align} \tag{ 39 }$$

On the other hand, in the massless limit, the local pressure $p\propto T^{\,4}$ from equations (28)–(31) and satisfies $p(r) B^2(r) = p_{\infty}$ from Tolman's law equation (13). Together with equation (30), we can find

$$\begin{align} p_\infty V_\mathrm{th} = \int_0^R \sqrt{\frac{A(r)}{B^3(r)}} B(r)^2\,n(r)\, T(r)\;d^3r = T_\infty N\,. \end{align} \tag{ 40 }$$

This agrees with the derivative definition in equation (39), and so it validates the $V_\mathrm{th}$ definition in equation (26). This shows that the thermodynamic system mimics the one in flat spacetime with the extensive global variables properly encoding the curvature effects and the intensive quantities given by the ideal gas properties at spatial infinity.

Then from equation (20), we can find the total entropy and chemical potential from F as

$$\begin{align} S& = &-\left(\frac{\partial F}{\partial T_{\infty}}\right)_{V_\mathrm{th},N} = N\left(4+\ln\left[\frac{T_{\infty}^3 V_\mathrm{th}}{\pi^2 N}\right]\right)\,, \end{align} \tag{ 41 }$$

$$\begin{align} \mu_\infty& = &\left(\frac{\partial F} {\partial N}\right)_{T_{\infty},V_\mathrm{th}} = -T_\infty \ln\left[\frac{T_{\infty}^3 V_\mathrm{th}}{\pi^2 N}\right]\,. \end{align} \tag{ 42 }$$

As expected S, derived in this way, agrees with the definition in equation (15) with the local variable $s = n(4-\mu/T)$ from equation (31). The internal energy is obtained as

$$\begin{align} U = F+T_\infty S = 3 N T_{\infty}\,, \end{align} \tag{ 43 }$$

which agrees with the local definition in equation (14) as well, with $\rho = 3nT$ from equation (29). The first law of thermodynamics, given in equation (21), follows accordingly.

For the massive particle case, the one-particle partition function is

$$\begin{align} Z_1 = \frac{m^2T_{\infty}}{2\pi^2}\int d^3r\sqrt{\frac{A}{B}}\;K_2 (m\sqrt{B}/T_{\infty})\,, \end{align} \tag{ 44 }$$

and the global Helmholtz free energy F is given as

$$\begin{align} F = -T_{\infty} \ln Z_{N}(T_\infty) = -T_{\infty} N \left(1+\ln\left[ \frac{m^2 T_{\infty}}{2\pi^2 N}\int d^3r \sqrt{\frac{A}{B}}K_2(m\sqrt{B}/T_{\infty})\right]\right)\,. \end{align} \tag{ 45 }$$

In contrast to the massless case, the thermodynamic volume $V_\mathrm{th}$ cannot be simply identified as the spatial integral part in the partition function due to the $T_\infty$ dependence in the Bessel function $K_2(m\sqrt{B}/T_\infty)$. Instead, we determine its differential form by evaluating the total derivative of F as given in equation (18). By using the pressure at infinity $p_{\infty}$, found from equations (28) and (30) as

$$\begin{align} p_{\infty} = \frac{m^2 e^{\mu_{\infty}/T_{\infty}}}{2\pi^2} T_{\infty}^2\; K_2 (m/T_{\infty})\;, \end{align} \tag{ 46 }$$

we obtain the differential form of thermodynamic volume

$$\begin{align} \mathrm{d}V_\mathrm{th} = -p_{\infty}^{-1} (\mathrm{d}F)_{N,T_{\infty}} = \frac{1}{K_2(m/T_\infty)}d\left(\int d^3r \sqrt{\frac{A}{B}}K_2(m\sqrt{B}/T_{\infty}) \right)_{N,T_\infty}\;. \end{align} \tag{ 47 }$$

In the massless limit, with $K_2(b) = b^2/2$, this agrees with equation (26). In general, $\mathrm{d}V_\mathrm{th}$ is sensitive to the particle mass m and is quite different from the universal geometric volume. As we will show later in section 3.2, the combination $p_\infty \mathrm{d}V_\mathrm{th}$ might be insensitive to the mass if ultracompact objects feature a deep gravitational potential and the spatial integral is dominated by the relativistic contribution.

The total entropy, chemical potential, and total internal energy are derived in a similar way as

$$\begin{align} S = -\left(\frac{\partial F}{\partial T_{\infty}}\right)_{V_\mathrm{th},N} = N\left(4-\frac{\mu_{\infty}}{T_{\infty}}+\frac{e^{\mu_{\infty}/T_{\infty}}m^3}{2\pi^2 N}\int K_1(m\sqrt{B}/T_{\infty})\sqrt{A} \;d^3r\right)\,, \end{align} \tag{ 48 }$$

$$\begin{align} \mu_\infty = \left(\frac{\partial F} {\partial N}\right)_{T_{\infty},V_\mathrm{th}} = -T_\infty \ln\left[ \frac{m^2T_\infty}{2\pi^2 N}\int d^3r\sqrt{\frac{A}{B}}\;K_2(m\sqrt{B}/T_{\infty})\right]\,, \end{align} \tag{ 49 }$$

$$\begin{align} U = F+T_\infty S = 3T_{\infty} N+\frac{e^{\mu_{\infty}/T_{\infty}}m^3 T_{\infty}}{2\pi^2 }\int K_1(m\sqrt{B}/T_{\infty})\sqrt{A} \;d^3r\,, \end{align} \tag{ 50 }$$

which agree with the quantities obtained from the local parameters. In comparison to the massless gas, S and U include additional terms that depend on the spatial integral of $K_1(b)$. This is also related to the difficulty in defining the full form of $V_\mathrm{th}$ from equation (47).

When the quantum nature of the source is taken into account, the particle number can fluctuate and hence is not appropriate to be treated as a state parameter in equilibrium thermodynamics. Therefore, the grand canonical ensemble is generally used to describe quantum gases where the chemical potential µ, in addition to temperature and volume, can be a state parameter that handles the change of particle number and is appropriately fixed.

In similarity with the canonical ensemble, we verify the consistency between the global and local pictures in the grand canonical ensemble by demonstrating the additivity of the global grand potential Ω, given in equation (14). In the global picture, as in the conventional thermodynamics [32], the grand potential can be derived from the fundamental equation

$$\begin{align} \Omega = -T_{\infty} \ln Z\,, \end{align} \tag{ 51 }$$

where the global partition function Z is

$$\begin{align} Z = \operatorname{Tr}[e^{-(\hat{H}-\mu_{\infty}\hat{N})/T_\infty}] = \prod_\mathbf{l}\sum_{n_\mathbf{l}}e^{-(n_\mathbf{l} E_{\mathbf{l},\infty}-\mu_\infty n_\mathbf{l} )/T_\infty}\,. \end{align} \tag{ 52 }$$

For a given quantum state labeled by l, the trace is evaluated with the number of eigenstates $n_\mathbf{l}$. For Bose–Einstein and Fermi–Dirac gas, summing over $n_\mathbf{l}$ leads to

$$\begin{align} \Omega = -T_{\infty} \begin{cases} &g\underset{l}{\sum} \ln\left[1+e^{-(E_{l,\infty}-\mu_{\infty})/T_{\infty}}\right]\;,\quad \textrm{Fermi-Dirac}\\ &-g\underset{l}{\sum} \ln\left[1-e^{-(E_{l,\infty}-\mu_{\infty})/T_{\infty}}\right]\;,\quad \textrm{Bose-Einstein}\;, \end{cases} \end{align} \tag{ 53 }$$

where $g = 2\sigma+1$ denotes the multiplicity for massives particles with spin σ and g = 2 for massless particles. As in the case of canonical ensemble, the sum can be approximated by the integral over the density of states $P(E_\infty)$. From equation (34), one obtains that

$$\begin{align} \Omega& = -\int_0^R \sqrt{AB} \,d^3r\; \dfrac{T}{(2\pi)^3}\int d^3p\; \begin{cases} &g \ln\left[1+e^{-(E-\mu)/T}\right]\;,\quad \textrm{Fermi-Dirac}\\ &-g \ln\left[1-e^{-(E-\mu)/T}\right]\;,\quad \textrm{Bose-Einstein} \end{cases} \nonumber\\ & = \int_0^R \sqrt{AB} \;w\, d^3r \end{align} \tag{ 54 }$$

where $\omega = -p$ is the grand potential density. The main difference for the grand canonical ensemble here is that the connection between local and global pictures in equation (54) is established through the logarithm of the partition function, given that the sum over states appears after taking the logarithm. For canonical ensemble, on the other hand, we take the sum over states before taking the logarithm, and the additivity emerges from the overall dependence on N. Note that from equation (9), by using the expression for w given in equation (54), one can find the other local quantities. For instance, the common case of relativistic gas of bosons and fermions for which we can assume for simplicity that $m, \mu \ll T$, one obtains the (average) quantities as

$$\begin{align} n\simeq\frac{\zeta (3)}{\pi^2}\left(g_b+\frac{3}{4}g_f \right) T^3\;,\quad s\simeq\frac{2\pi^2}{45} \left(g_b+\frac{7}{8}g_f \right) T^3\;,\quad \rho \simeq 3 p \simeq \frac{\pi^2}{30} \left(g_b+\frac{7}{8}g_f \right) T^{\,4}\;, \end{align} \tag{ 55 }$$

where $\zeta (3)\approx 1.2$ is the Riemann zeta function of argument 3 [32], g_b and g_f are the total bosonic and fermionic degrees of freedom respectively, which are the generalizations of g in equation (54). For pure radiation (photons), since $m = \mu = 0$, the expressions in equation (55) become exact rather than approximate (with $g_b = 2\;, g_f = 0$), which will be our first example below.

Now we will proceed to examine some examples and identify the thermodynamic volume in each case.

Photon gas: As massless Bose–Einstein gas, the particle number of photon gas is not conserved, and so its chemical potential vanishes, i.e. µ = 0. The local variables in this case are given as

$$\begin{align} \rho = 3p = \frac{\pi^2}{15}T^{\,4}\;,\qquad s = \frac{4\pi^2}{45}T^3\;. \end{align} \tag{ 56 }$$

Plugging in the Tolman's law, given in equation (13), we can obtain the relations

$$\begin{align} p(r)B^2(r) = p_{\infty}\,,\quad \rho(r)B^2(r) = \rho_{\infty}\,, \quad s(r)B^{3/2}(r) = s_{\infty}\,, \end{align} \tag{ 57 }$$

where $\rho_{\infty} = 3p_{\infty} = \pi^2 T^{\,4}_{\infty} /15$ and $s_{\infty} = 4\pi^2 T^3_{\infty} /45$ denote the gas properties at spatial infinity. These relations are insensitive to the numerical coefficients in equation (56) and can be directly read from the momentum conservation law in equation (10), given the EoS $\rho = 3p$ and the local Euler equation, given in equation (3).

With equations (54) and (56), the global grand potential is obtained as

$$\begin{align} \Omega = -\int \sqrt{A(r)B(r)} \;p \;d^3r = -p_\infty \int_0^R \sqrt{\frac{A(r)}{B^3(r)}} \;d^3r \equiv-p_\infty V_\mathrm{th}\,. \end{align} \tag{ 58 }$$

With vanishing chemical potential, Ω is a function of $T_\infty$ and $V_\mathrm{th}$. The thermodynamic volume $V_\mathrm{th}$ is again identified as that in equation (26), consistent with the derivative relation $p_\infty = -(\partial\Omega/\partial V_\mathrm{th})_{T_\infty}$ in equation (25). It is not surprising that the thermodynamic volume of the photon gas takes the same form as in the case of the semi-classical massless gas. From equation (24), the expression of $V_\mathrm{th}$ in equation (26) follows from the relation $p(r)B^2(r) = p_{\infty}$, as dictated by the corresponding EoS, $\rho = 3p$. From equations (54) and (25), we can obtain the total entropy and internal energy

$$\begin{align} \,\,S = -\left(\frac{\partial \Omega}{\partial T_{\infty}}\right)_{V_\mathrm{th}} = s_{\infty} V_\mathrm{th}\,, \end{align} \tag{ 59 }$$

$$\begin{align} U = \Omega+T_\infty S = \rho_{\infty} V_\mathrm{th}\,, \end{align} \tag{ 60 }$$

with $s_\infty$, $\rho_\infty$ given below equation (57). These expressions are consistent with the expected local definitions, given in equations (14) and (15).

Cold Fermi gas at T = 0: At zero temperature, all states of the cold Fermi gas below the cutoff Fermi energy are occupied. The local variables are then characterized by the corresponding Fermi momentum k_F and the fermion mass m as

$$\begin{align} p = 3g \rho_c \,h_p\left(\frac{k_F}{m}\right),\quad \rho = 3g \rho_c\, h_\rho\left(\frac{k_F}{m}\right),\quad n = g \frac{k_F^3}{6\pi^2} \,, \end{align} \tag{ 61 }$$

where g = 2 is the multiplicity in equation (53), $\rho_c = m^4/(6\pi^2)$ is the critical density, and

$$\begin{align} h_p(x)& = \frac{1}{8}\left[x\left(\frac{2}{3}x^2-1\right)\sqrt{x^2+1}+\sinh^{-1}x\right],\quad\nonumber\\ h_\rho(x)& = \frac{1}{8}\left[x(2x^2+1)\sqrt{x^2+1}-\sinh^{-1}x\right].\;\; \end{align} \tag{ 62 }$$

With vanishing temperature, the local Euler equation now becomes $\rho+p = n\mu$, where the chemical potential is given as $\mu = \sqrt{k_F^2+m^2}$ and Tolman's law in equation (13) imposes the constraint $\mu(r)\sqrt{B(r)} = \mu_\infty$.

The global grand potential is a function of the chemical potential $\mu_\infty$ and the thermodynamic volume $V_\mathrm{th}$, and is given as

$$\begin{align} \Omega = -\int \sqrt{A(r)B(r)} \;p \;d^3r = -3\rho_c \int_0^R\sqrt{A(r)\,B(r)}\,h_p\left(\frac{k_F}{m}\right) \;d^3r\,. \end{align} \tag{ 63 }$$

Similar to the massive case for a semi-classical ideal gas, the thermodynamic volume $V_\mathrm{th}$ cannot simply be identified as the spatial integral part in Ω due to the complicated $T_\infty$ dependence in $h_p(k_F/m)$. It is then found from the generic derivative definition equation (24). By using $p_\infty = 3\rho_c h_p(k_{F,\infty}/m)$ from equation (61), we obtain

$$\begin{align} \mathrm{d}V_\mathrm{th} & = & d\left(\frac{1}{h_p\left(\frac{k_{F,\infty}}{m}\right)} \int d^{3} r \sqrt{A\,B}\, h_{p}\left(\frac{k_F}{m}\right)\right)_{\mu_{\infty}}\;, \end{align} \tag{ 64 }$$

where $k_F/m = \sqrt{\mu_\infty^2/(B \,m^2)-1}$ and the demanded relation, $p_{\infty} = -(\partial \Omega/\partial V_\mathrm{th})_{\mu_\infty}$, is then recovered. Similarly, $\mathrm{d}V_\mathrm{th}$ is sensitive to the fermion mass, in general.

Self-gravitating systems in GR provide us with natural examples to see the curvature effects on statistical thermodynamics. Here, we consider two concrete examples of compact objects, self-gravitating photon gas confined to a spherical box [15] and neutron stars composed of cold Fermi gas (i.e. the Oppenheimer-Volkof model) [33].

Considering static and spherically symmetric solutions, the Einstein field equations can be simplified by the conservation of the stress-tensor and the ToV equation, given in equations (10) and (92) of appendix B, respectively. For a given EoS of the matter source, the pressure p(r) and mass profiles $\mathcal{M}(r)\equiv \int_0^r 4\pi r^{^{\prime} 2}\rho(r^{^{\prime}})\mathrm{d}r^{^{\prime}}$ can be solved simultaneously as functions of the central pressure p_c at the origin. The solutions obey a simple scaling behavior, with the following rescaled dimensionless quantities

$$\begin{align} \tilde{p}(\tilde{r}) = p(r)\,\lambda^4,\, \tilde{\mathcal{M}}(\tilde{r}) = \mathcal{M}(r) \,\lambda^{-2} {\ell_{\mathrm{Pl}}}^3\,, \tilde{r} = r\, \lambda^{-2} {\ell_{\mathrm{Pl}}},\, \end{align} \tag{ 65 }$$

uniquely determined by the field equations. A one-parameter family of solution for $p(r), \mathcal{M}(r)$ can then be obtained by scaling with an arbitrary length scale λ. In GR, the physical mass of the object takes a simple form

$$\begin{align} M = \mathcal{M}(R)\equiv \int_0^R 4\pi r^2\rho(r)\mathrm{d}r\,. \end{align} \tag{ 66 }$$

Given that the metric determinant $\sqrt{A\,B}\lt1$ in the interior, the total internal energy U, defined in equation (14), is always smaller than the mass M, and its smallness reflects the contribution from the gravitational field.

Let's first consider self-gravitating photon gas confined to a spherical box, with the EoS given in equation (56). Here, a finite box of radius R is imposed to prevent the massless gas from spreading all the way to infinity. Self-gravitating photon gas confined to a spherical box is then described by a two-parameter family of solutions, and the scaling behavior in equation (65) can be used to relate solutions of different p_c with $\lambda = p_c^{-1/4}$.

Figure 1 displays properties for self-gravitating photon gas. The global thermodynamic variables are related by $U = 3\,T_\infty S/4 = 3\,p_\infty V_\mathrm{th}$ as given in equations (59) and (60). When the mass-to-radius ratio $M/R$ is small, the solution describes the dilute photon gas in a box. Gravitational effects become strong with increasing $M/R$. The maximal $M/R\approx 0.25$ defines a turning point, beyond which the solutions become unstable. Along the way, the internal-energy-to-mass ratio $U/M$ decreases slowly from unity and reaches the minimum $U\approx 0.64M$ around the turning point. In the weak gravity regime, the thermodynamic volume $V_\mathrm{th}$ is very close to the geometric one $V_\mathrm{geo}$ and the entropy $S\propto R^3$ with $T_\infty$ roughly a constant. In the unstable branch, $V_\mathrm{th}$ becomes slightly larger and $S\propto R^{3/2}$ grows slower with R given $T_\infty\propto R^{-1/2}$.

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The numerical solutions can be used to examine the first law of thermodynamics. Since the solutions are described by the two parameters p_c and R, the entropy S (or the temperature $T_\infty$) and the thermodynamic volume $V_\mathrm{th}$ can vary independently. The conventional first law then applies, i.e. $\mathrm{d}U = T_\infty \mathrm{d}S-p_\infty \mathrm{d}V_\mathrm{th}$, as we would expect from the general derivation in section 2. On the other hand, it has been proved in GR [15] that the physical mass M satisfies the following first law,

$$\begin{align} \mathrm{d}M = T_\infty \mathrm{d}S-p(R)\sqrt{B(R)} \, \mathrm{d}V_\mathrm{geo} = T_\infty \mathrm{d}S-p_\infty\, B(R)^{-3/2} \, \mathrm{d}V_\mathrm{geo}\,, \end{align} \tag{ 67 }$$

where M is considered as a function of S and $V_\mathrm{geo}$ instead. The difference between the physical mass and total internal energy is then

$$\begin{align} \mathrm{d}M-\mathrm{d}U = p_\infty \left(\mathrm{d}V_\mathrm{th}-\mathrm{d}V_\mathrm{geo}B(R)^{-3/2}\right)\,. \end{align} \tag{ 68 }$$

Considering the gravitational potential in the object's interior, i.e. $B(R)\gt B(r)$ for r < R, we expect $V_\mathrm{th}\gt V_\mathrm{geo}B(R)^{-3/2}$ and then $\mathrm{d}M\gt \mathrm{d}U$. This reveals the U and M relation from a different perspective apart from their definitions. Interestingly, we find no discussion of such a relation between equation (67) and the conventional first law in the literature.

As the second example, we consider neutron stars composed of cold Fermi gas at zero temperature, with the EoS given in equation (61). In contrast to the self-gravitating photon gas, the radius R of neutron stars is determined with $p(R) = 0$ and is not an independent parameter. Neutron stars composed of cold Fermi gas are then described by a two-parameter family of solutions, i.e. the central pressure p_c and the Fermion mass m, and the scaling behavior in equation (93) can be used to relate solutions of different mass m, with $\lambda = 1/m$.

Figure 2 shows properties for neutron stars composed of cold Fermi gas. Similarly, the gravitational effects become stronger with increasing central pressure. In the stable branch of solutions, the physical mass M and the total number of particles N increase, and the radius R decreases. In the weak gravity regime, we find $N\propto r_H/(m\,{\ell_{\mathrm{Pl}}}^2)$, $\mu_\infty\propto r_H^0\, m$. The maximum of $M/R$ remains the turning point, but the value is slightly smaller than that of self-gravitating photon gas. The internal energy to mass ratio $U/M$ decreases from unity in a similar way and is bounded from below by $U\approx 0.72M$. For the first law of thermodynamics, since the solution for a given Fermion mass m is described by only one parameter, the particle number N and the thermodynamic volume $V_\mathrm{th}$ cannot vary independently. The conventional first law equation (21) then may not be properly defined at zero temperature limit. From the numerical solutions, we find that

$$\begin{align} \mathrm{d}M\approx \mu_\infty \mathrm{d}N\,. \end{align} \tag{ 69 }$$

This can be viewed as the first law built upon the physical mass M. The thermodynamic volume term in equation (21), which accounts solely for the contribution of the gravitational field, is incorporated into $\mathrm{d}M$. In practice, neutron stars may have a small but nonvanishing temperature. The pressure, then, would not drop to zero at a hard surface, and so a small independent $\mathrm{d}V_\mathrm{th}$ term would appear in equation (69). For the unstable solutions, the global variables vary less with the central pressure and become attracted by the infinite pressure limit [33].

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As a final remark, the compactness $M/R$ is sensitive to source matter properties. With exotic EoSs, it is possible to realize ultra-compact objects in GR with R smaller than the photon sphere radius, i.e. $M/R\gtrsim0.33$, or even black hole mimickers with $M/R\sim 0.5$ and no horizon [8]. Nonetheless, no candidate makes connections between black hole thermodynamics and the conventional one as we focus on here. In the following subsection, we will discuss a candidate in a theory of modified gravity, where such a connection appears due to super-Planckian high curvature effects.

An interesting candidate for horizonless ultracompact objects is a new type of solution, 2-2-holes [34, 35], in quadratic gravity as described by the classical action, given in equation (96) in appendix B. The existence of 2-2-holes relies on the Weyl term $C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$, which introduces a new spin-2 mode with the Compton wavelength λ₂. In the presence of a compact matter source, a typical 2-2-hole with mass M much larger than the minimum mass $M_{\mathrm{min}}\sim m_{\mathrm{Pl}}^2 \lambda_2$ resembles a black hole closely from the exterior, while a novel interior takes over right above the would-be horizon $r_H = 2M{\ell_{\mathrm{Pl}}}^2$. The interior curvature can easily reach super-Planckian values when $\lambda_2\sim {\ell_{\mathrm{Pl}}}$, and approaches infinity at the origin. In contrast to proposed ultracompact objects in GR, the 2-2-hole properties are crucially determined by the high curvature interior, while being quite insensitive to the details of EoS. Thus, 2-2-holes provide a tractable model to study the influence of high curvature effects on statistical thermodynamics. In the following, we consider concrete examples of 2-2-holes sourced by various simple forms of matter.

First, we consider thermal 2-2-holes sourced by massless particles, i.e. photon gas ⁴ and massless classical ideal gas, e.g. EoS being $\rho_{\infty} = 3p_{\infty}$. The gas profile and the metric functions can be solved from the momentum conservation of the stress-tensor equation (10), which yields $p(r)B(r)^2 = p_\infty$ for relativistic gas, and two field equations, given in equation (97) in appendix B. Figure 3 displays the 2-2-hole solutions with two different values of mass $M\gg M_{\mathrm{min}}$. Despite the complicated form of field equations in quadratic gravity, the numerical solutions of 2-2-holes turn out to be governed by simple qualitative features. The exterior ($r\gtrsim r_H$), where the curvature becomes lower for increasing size of the object, obeys the Einstein field equations to a good approximation and follows the conventional scaling in equation (65). The interior $r\lt r_H$, on the other hand, is a high curvature regime and features a huge gravitational potential. At the leading order of high curvature expansion, it is described by a novel scaling behavior, with the following rescaled dimensionless quantities [35]

$$\begin{align} \tilde{p}(\tilde{r}) = p(r) \lambda_2^2\,{\ell_{\mathrm{Pl}}}^2,\; \tilde{A}(\tilde{r}) = A(r) \frac{r_H^2}{\lambda^2_2},\; \tilde{B}(\tilde{r}) = B(r) \frac{r_H^2}{\lambda^2_2} \end{align} \tag{ 70 }$$

uniquely determined as functions of $\tilde{r} = r/r_H$. Similarly, a box has to be imposed to make the total energy finite, and the box radius R has to be considerably larger than r _H to not ruin the simple behavior.

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From the novel scaling in equation (70) and the numerical solutions, we find the rescaled gas pressure at infinity

$$\begin{align} \tilde{p}_\infty = \tilde{p}(\tilde{r})\tilde{B}(\tilde{r})^2 = p_\infty \frac{{\ell_{\mathrm{Pl}}}^2}{\lambda_2^2}r_H^4\approx 1.35\times10^{-5}\,. \end{align} \tag{ 71 }$$

The thermodynamic volume $V_\mathrm{th}$ in equation (26) can be separated into two parts, the interior contribution $V_\mathrm{th}^\mathrm{(in)}$ for $r\lesssim r_H$ and the exterior contribution $V_\mathrm{th}^{(ex)}$ for $r_H\lesssim r \lesssim R$. The interior contribution is governed by the novel scaling in equation (70), satisfying

$$\begin{align} \tilde{V}_\mathrm{th}^\mathrm{(in)} = V_\mathrm{th}^\mathrm{(in)}\frac{\lambda_2^2}{r_H^5} \approx 4.63\times 10^3\,. \end{align} \tag{ 72 }$$

In comparison to the flat spacetime scaling $V_\mathrm{th}^\mathrm{(in)}\sim r_H^3$, the high curvature effects lead to a large enhancement of order $(r_H/\lambda_2)^2$. The gain could be enormous given that λ₂ can be as small as ${\ell_{\mathrm{Pl}}}$. The exterior contribution encodes the box radius R dependence. Since the gas temperature drops drastically at $r\gt r_H$ as shown in figure 3, we expect a trivial dependence $V_\mathrm{th}^{(ex)}\propto R^3$ from the cold and dilute gas outside as in the case of GR.

Thus, for macroscopic 2-2-holes with R not significantly larger than r_H , we expect $V_\mathrm{th}$ to be dominated by the interior contribution. This is also true for the total internal energy and entropy of the gas, which are proportional to $V_\mathrm{th}$. With equations (60), (71) and (72), the total internal energy is

$$\begin{align} U\approx U^\mathrm{(in)} = 3 \, p_\infty\, V_\mathrm{th}^\mathrm{(in)}\approx \frac{3}{8} M\,. \end{align} \tag{ 73 }$$

Since the compactness of 2-2-holes is nearly maximal, it is not surprising that its internal energy-to-mass ratio is far smaller than that for compact objects in GR. The same ratio has also been obtained in the brick wall model [38], but there, the back-reaction has not been taken into account. Using equations (56), (59), (71) and (72), we can obtain the temperature at spatial infinity and the entropy

$$\begin{align} T_{\infty}\approx 1.4\,\hat{M}_{\mathrm{min}}^{1/2}\,T_\textrm{BH}\,,\quad S\approx S^\mathrm{(in)}\approx 0.71\,\hat{M}_{\mathrm{min}}^{-1/2}\,S_\textrm{BH}\,\,, \end{align} \tag{ 74 }$$

where $\hat{M}_{\mathrm{min}} = M_{\mathrm{min}}/m_{\mathrm{Pl}} = 0.63\lambda_2/{\ell_{\mathrm{Pl}}}$, the Hawking temperature $T_\textrm{BH} = m_{\mathrm{Pl}}^2/8\pi M$ and the Bekenstein–Hawking entropy $S_\textrm{BH} = \pi\, r_H^2/{\ell_{\mathrm{Pl}}}^2$. ⁵ As a result of high curvature effects, a 2-2-hole sourced by relativistic particles exhibits Hawking-like temperature and the entropy area law [35]. Given that $S = s_\infty V_\mathrm{th}$ in equation (59), we can see that the enormous amount of microscopic entropy is achieved here mainly through the huge value of $V_\mathrm{th}^\mathrm{(in)}$ given in equation (72). This is usually considered difficult for self-gravitating objects with no horizon. The numerical values of $T_{\infty}$ and S differ from the black hole counterparts due to the $M_{\mathrm{min}}$ or λ₂ dependence.

For the first law of thermodynamics, as 2-2-holes sourced by photon gas are described by two parameters r_H and R, the entropy S and the thermodynamic volume $V_\mathrm{th}$ can vary independently, and the conventional first law still applies, i.e. $\mathrm{d}U = T_\infty \mathrm{d}S-p_\infty \mathrm{d}V_\mathrm{th}$, as we would expect. Given the absence of analytical solutions, the variation of the physical mass M requires to be checked numerically. Using $T_\infty S^\mathrm{(in)} = 4\,p_\infty V_\mathrm{th}^\mathrm{(in)}\approx M/2$, as obtained from the relations given below equation (57) and in equation (73), with the relations $T_\infty\propto M^{-1}$ and $S^\mathrm{(in)}\propto M^2$ as can be seen in equation (74), we obtain

$$\begin{align} \mathrm{d}M\approx T_\infty \mathrm{d}S^\mathrm{(in)}\approx T_\infty \mathrm{d}S\,. \end{align} \tag{ 75 }$$

In comparison to equation (67), the work performed at the box radius is numerically negligible for macroscopic 2-2-holes with $V_\mathrm{geo}\sim R^3\ll V_\mathrm{th}^\mathrm{(in)}$. The major difference between the physical mass and total internal energy is then accounted for by the thermodynamic volume contribution $p_\infty \mathrm{d}V_\mathrm{th}\approx 5\mathrm{d}M/8$ with $\mathrm{d}M\approx \mathrm{d}U+p_\infty \mathrm{d}V_\mathrm{th}$.

It is instructive to make some comparisons with black hole thermodynamics. For Schwarzschild black holes, $\mathrm{d}M = T_\infty \mathrm{d}S$ applies without the volume term because of the vanishing thermodynamic pressure. For thermal 2-2-holes, equation (75) holds at the leading order of high curvature expansion since the volume term associated with variation of the box boundary has a negligible contribution. This shows that certain aspects of black hole thermodynamics could be derived from conventional thermodynamics by utilizing the non-trivial structure of curved spacetime, and thus are not too mysterious.

For 2-2-holes sourced by a semiclassical ideal gas, there is the additional dependence on the conserved number of particles N or the nonzero chemical potential $\mu_\infty$ given in equation (42). The relation between U and M remains the same as in equation (73), while the temperature and total entropy change. With equations (28), (30), and (41) for the massless ideal gas, we obtain

$$\begin{align} T_{\infty}\approx 1.7\,\hat{M}_{\mathrm{min}}^{1/2}\,e^{-\frac{1}{4}\mu_{\infty}/T_{\infty}}\,T_\textrm{BH}\,,\; S^\mathrm{(in)}\approx 0.59\,\hat{M}_{\mathrm{min}}^{-1/2}\,e^{\frac{1}{4}\mu_{\infty}/T_{\infty}}\left(1-\frac{\mu_{\infty}}{4T_{\infty}}\right)\,S_\textrm{BH}\,\,. \end{align} \tag{ 76 }$$

Given that $\exp(\mu_\infty/T_\infty)\ll1$ for a semiclassical ideal gas, the corresponding thermal 2-2-holes have higher temperature and lower entropy than holes sourced by photon gas of the same mass in equation (74). As the generalization of conventional thermodynamics, it is not surprising that thermal 2-2-hole characteristics can depend on certain aspects of matter properties.

Considering the entropy S as a function of two independent variables M and N as given in equation (41), the total differential of S for the interior contribution is

$$\begin{align} \mathrm{d}S^\mathrm{(in)}\approx \mathrm{d}N^\mathrm{(in)} \ln(e^{-\mu/T})+N^\mathrm{(in)}\left(5 \frac{\mathrm{d}M}{M}+3 \frac{\mathrm{d}M}{M}\right) \approx-\frac{\mu_\infty}{T_\infty} \mathrm{d}N^\mathrm{(in)}+\frac{1}{T_\infty}\mathrm{d}M\,, \end{align} \tag{ 77 }$$

with $V_\mathrm{th}^\mathrm{(in)}\propto M^5$ and $T_\infty N^\mathrm{(in)}\propto M$. This yields a generalization of equation (75) with the additional contribution from the conserved number of particles,

$$\begin{align} \mathrm{d}M\approx T_\infty \,\mathrm{d}S+\mu_\infty \mathrm{d}N\,. \end{align} \tag{ 78 }$$

The exterior contributions are again ignored. Since black holes do not conserve charges associated with global symmetries [39], this has no counterpart in black hole thermodynamics. For a macroscopic 2-2-hole, the particle-number-to-mass ratio $Nm_{\mathrm{Pl}}/M\approx m_{\mathrm{Pl}}/(8T_\infty)\propto M/m_{\mathrm{Pl}}$ is enormous, and it can increase indefinitely with the hole mass, unlike the local charge of black holes to be bounded by the extremal limit.

2-2-holes can also be sourced by semiclassical ideal gas with nonzero mass. As the temperature satisfies Tolman's law $T(r)\sqrt{B(r)} = T_\infty$, the gas becomes relativistic around the origin, while the density and pressure are suppressed by the Boltzmann factor when T(r) drops below the particle mass m. Although the field equations, given in equation (99) in appendix B, are more involved for this case, the interior metric functions and gas profile remain characterized by the same scaling behavior equation (70) as for relativistic thermal gas, for a given $m\sqrt{\lambda_2{\ell_{\mathrm{Pl}}}}$. Interestingly, global thermodynamic quantities such as $T_\infty$, S, N, and U are found to be quite insensitive to the particle mass due to the dominance of the relativistic contribution. Thus, the results obtained for the massless case can be used as a reliable approximation for the massive case, with the mass dependence having a negligible impact on the leading order results. This includes the intricacy of identifying the thermodynamic volume for the massive case, as discussed around equation (47).

Another example is 2-2-holes sourced by the cold Fermi gas, which may serve as the endpoint of the gravitational collapse of neutron stars after sufficient cooling. Together with Tolman's law equation (13) for chemical potential, the metric functions and gas profile are solved from the field equations (equation (99) in appendix B) with EoS given in equation (61). The cold Fermi gas becomes relativistic around the origin due to the quantum pressure. As in the case of neutron stars in GR, the pressure (and the Fermi momentum k_F ) drops to zero at some radius R and defines the object's surface. The difference is that R is within the would-be horizon r_H for 2-2-holes, while it is considerably larger than r_H for neutron stars.

For a given $m\sqrt{\lambda_2{\ell_{\mathrm{Pl}}}}$, the interior is again characterized by the novel scaling behavior in equation (70). With this scaling, the rescaled chemical potential redshifted to infinity is

$$\begin{align} \tilde{\mu}_\infty = \mu_\infty \sqrt{\frac{{\ell_{\mathrm{Pl}}}}{\lambda_2}}r_H\approx 0.22\,. \end{align} \tag{ 79 }$$

Since $R\lt r_H$, the total internal energy and (average) number of particles receive contributions only from the 2-2-hole interior. The internal energy to mass ratio $U/M$ remains roughly $3/8$. The rescaled number of particles is

$$\begin{align} \tilde{N} = N \, \frac{{\ell_{\mathrm{Pl}}}^{3/2}\lambda_2^{1/2}}{r_H^2}\approx 1.15\,. \end{align} \tag{ 80 }$$

Note that the r_H (or M) dependences of $\mu_\infty$ and N are quite different from that for neutron stars, but resemble closely that for $T_\infty$ and S for thermal 2-2-holes in equation (76). With equations (79) and (80), it is straightforward to obtain

$$\begin{align} \mathrm{d}M\approx \mu_\infty\, \mathrm{d}N\,, \end{align} \tag{ 81 }$$

the analog of equation (69) for neutron stars and of equation (78) for thermal 2-2-holes. Again, the thermodynamic volume term accounts for the difference $1-U/M$, and the contribution is the same as in the case of thermal 2-2-holes.

In summary, the global thermodynamic variables of 2-2-holes exhibit interesting universal characteristics due to the high curvature interior and its novel scaling behavior. It is intimately related to the greatly enhanced interior thermodynamic volume that dominates the contribution for macroscopic holes. This renders the EoS dependence rather weak, as opposed to the situation in GR.