Temperature of Gravity

Updated 31 January 2026

Temperature of Gravity is a concept uniting gravitational phenomena with thermodynamic principles, illustrated by horizon radiation, equilibrium laws, and modified gravity frameworks.
It encompasses both classical and quantum perspectives, applying Hawking/Unruh radiation and nonextensive Tsallis statistics to reveal spacetime’s thermal characteristics.
Experimental studies, such as mass–temperature coupling tests, challenge conventional predictions of General Relativity and suggest new insights into gravitational thermodynamics.

The temperature of gravity refers to a diverse set of concepts linking gravitational phenomena to thermodynamic temperature, ranging from horizon-associated temperatures (Hawking/Unruh radiation) to equilibrium properties of self-gravitating systems, modified-gravity “thermal” deviation coefficients, gravitational analogies to the Stefan–Boltzmann law, statistical mechanics of spacetime, and experimentally detectable mass–temperature coupling in terrestrial systems. These notions are united by the emergence of genuine thermodynamic or effective temperatures within gravitational or gravitating contexts, sometimes encoding deep insights about gravity’s microscopic degrees of freedom, and sometimes revealing properties of gravitational mass under changes of internal energy.

1. Temperature in Classical and Quantum Gravitation

The association of temperature with gravitational systems originated with the discovery that black-hole event horizons emit thermal (Hawking) radiation, characterized by the Hawking temperature: $T_H = \frac{\hbar\kappa}{2\pi c k_B}$ where $\kappa$ is the surface gravity. For Schwarzschild black holes, $T_H = \hbar c^3/(8\pi G k_B M)$ , establishing the identification of a physical temperature directly related to geometric invariants of the spacetime. Similarly, the Unruh effect shows that a uniformly accelerated observer in Minkowski space detects a thermal bath at $T_U = \hbar a/(2\pi c k_B)$ . In de Sitter spacetime, the cosmological horizon has Gibbons–Hawking temperature $T_{GH} = \hbar H/(2\pi k_B)$ with $H=\sqrt{\Lambda/3}$ (Moustos, 2017).

These results reveal that gravity, through the structure of spacetime and causal horizons, encodes temperature—even in the absence of conventional matter.

2. Tolman Law and Equilibrium Temperature Gradients in Gravity

In static gravitational fields, the equilibrium distribution of temperature is dictated by the Tolman relation: $T(x)\,\sqrt{-g_{00}(x)} = \text{constant}$ This formula arises from the universality of gravitational redshift and the requirement that local observers see a Planckian spectrum at position-dependent temperature $T(x)$ , so as to avoid violations of the second law of thermodynamics in closed cycles across potential gradients (Santiago et al., 2018, Majhi, 2015). Maxwell’s two-column argument generalizes to the relativistic domain: only force fields that couple universally (gravity) permit genuine equilibrium temperature gradients. The relation holds for all static, gravitationally bound systems in thermal equilibrium, with the redshift factor ensuring thermodynamic consistency.

For black holes, this implies the local temperature measured near the event horizon diverges due to infinite blueshift, aligning with Unruh radiation for accelerated observers (Majhi, 2015, Moustos, 2017). For cosmological horizons, the same logic yields the Gibbons–Hawking temperature (Moustos, 2017).

3. Gravitational Temperature in Nonextensive Self-Gravitating Systems

In astrophysical contexts, the “gravitational temperature” $T_g$ emerges from nonextensive Tsallis statistics applied to pure self-gravitating systems—multi-body gaseous spheres where gravity dominates over molecular interactions (Zheng et al., 2014). In the equilibrium (maximal Tsallis entropy) state, the system’s local kinetic temperature $T(\mathbf{r})$ and the gravitational potential $\varphi(\mathbf{r})$ combine into a spatially homogeneous gravitational temperature: $T_g(\mathbf{r}) = T(\mathbf{r}) + (1-q)\,\frac{m}{k}\,\varphi(\mathbf{r}) + T_0$ with $q<1$ the Tsallis parameter. $T_g$ is constant throughout the system at stationarity, ensuring no net heat flow despite position-dependent kinetic temperature. The associated gravitational thermal capacity $C_{V,g}$ modifies the traditional (negative) heat capacity of gravitating systems, acting as the relevant measure for thermodynamic stability in nonextensive frameworks. The stability criterion is $C_{V,g}>0$ , which, for three-dimensional monatomic gases, requires $q<2/3$ .

4. Thermodynamic Frameworks in Modified Gravity

Scalar–tensor and more general modified gravity theories support internal, effective “temperature of gravity” constructs, recasting deviations from Einstein gravity as departures from equilibrium (Faraoni et al., 2021, Karolinski et al., 2024, Faraoni et al., 2022):

In Jordan-frame scalar–tensor theory, the field equations admit an effective imperfect fluid description with heat flux obeying Eckart’s first-order thermodynamics. The gravitational temperature arises as

$K\,T_\text{grav} = \frac{\sqrt{-\nabla^c\psi\,\nabla_c\psi}}{\kappa\,\psi}$

where $K$ is a (spacetime) thermal conductivity (Karolinski et al., 2024, Faraoni et al., 2021, Faraoni et al., 2022).

General Relativity (constant scalar) is singled out as the unique zero-temperature equilibrium state; any $\nabla_a\psi \neq 0$ drives the theory into a finite-temperature, dissipative regime.
The approach to or departure from equilibrium is governed by relaxation-type equations, with new terms like $\Box \psi$ enabling non-monotonic “thermal” evolution (Karolinski et al., 2024).
Modified gravities with non-propagating scalars (Brans–Dicke with $\omega=-3/2$ , Palatini $f(R)$ , cuscuton) correspond to $T=0$ (strict equilibrium), while pathologies such as Nordström’s scalar theory feature formally negative $T_\text{grav}$ (Faraoni et al., 2022).

This thermodynamic analogy offers a unifying principle for emergent-gravity scenarios, classifying gravities by their “thermal” distance from GR and providing a diagnostic for healthy propagating degrees of freedom.

5. Temperature of Gravitons and Statistical Gravity

In quantum gravity and semiclassical graviton physics, the temperature of gravity can refer to the thermodynamic properties of the graviton field. Using Teleparallel Equivalent of General Relativity (TEGR) and Thermo Field Dynamics (TFD), one computes the Stefan–Boltzmann law for gravitons: $\rho(T) = a_g\,T^4,\quad p = \frac{1}{3}\rho,\quad a_g = \frac{\pi^2}{15}\frac{k_B^4}{\hbar^3 c^3}$ mirroring the blackbody law for photons. The equation of state enables consistent definitions of entropy and pressure for a thermal graviton bath, with gravitational Casimir energies evaluated in finite-temperature setups (Ulhoa et al., 2019, Santos et al., 2021). In de Sitter spacetime, novel phenomena such as an “anti-Unruh effect”, connecting proper acceleration $a$ to a decreasing $T_\text{grav} \propto a^{-1/3}$ , have been derived (Santos et al., 2021). This suggests nontrivial gravitational vacuum structure and links between acceleration, horizon structure, and thermodynamic response.

In recent approaches, the entire Einstein spacetime can acquire an effective temperature through statistical mechanics of the metric field, grounded in virial-theorem arguments and implemented by path-integral Monte Carlo over a lattice of metric components: $\tilde T = -\frac{\bar\upsilon}{4\tilde k_B}\langle T^\mu{}_\mu\rangle_t$ where all expectation values are evaluated in the Euclidean statistical ensemble (Fantoni, 2024). This positions gravitational dynamics within a genuine thermodynamic partition function framework.

6. Experimental Evidence: Temperature Dependence of Gravitational Mass

Terrestrial experiments investigate whether a material’s temperature alters its gravitational force. Dmitriev’s precision balance experiment demonstrated that heating a thermally isolated copper sample ( $m=28$ g) by $\Delta T \simeq 9$ K reduced its apparent weight by $\Delta W \simeq -0.7$ mg, yielding a relative temperature coefficient: $y = \frac{\Delta m}{m\,\Delta T} \approx -2.8 \times 10^{-6}\,\text{K}^{-1}$ This is negative and nine orders of magnitude larger (and of opposite sign) than the (positive, $10^{-15}\,\text{K}^{-1}$ ) prediction of General Relativity (Dmitriev, 2012). Related ultrasonic or laser-heating experiments on other materials agree in both sign and order of magnitude. Main sources of systematic error (balance resolution, air convection, vessel expansion, leakage) were shown to be an order of magnitude smaller than the observed effect. If confirmed and universal, such a negative temperature coefficient would challenge the standard tenets of General Relativity, particularly the role of internal energy in gravitational mass, and carry implications for the theory of gravitational collapse (Dmitriev, 2012).

7. Broader Theoretical Implications and Outlook

The identification of a temperature of gravity permeates several domains:

Black-Hole Mechanics and Holography: The thermodynamic view of spacetime, rooted in the analogy of gravitational entropy and temperature, underpins emergent-gravity and holographic principles. Extremization of entropy functionals over spacetime volumes recovers the Einstein equations, with surface terms encoding all the entropy in agreement with the holographic principle (Moustos, 2017, Smoot, 2010).
Modified Gravity Classifications: The thermal-language provides a unifying taxonomy for distinguishing equilibrium (zero- $T$ ) and off-equilibrium (finite- $T$ ) gravities, identifying pathologies and healthy modifications.
Experimental Signatures and Novel Phenomena: Precision measurements of mass–temperature coupling, gravitational Casimir effects, and the anti-Unruh effect in specific spacetime backgrounds broaden the phenomenology accessible to gravitational thermodynamics.

Further research directions include widening the range of materials and temperature intervals for mass–temperature coupling experiments, refining path-integral gravity simulations, exploring gravitational temperature in wider classes of modified gravity (with explicit numerical implementation), and clarifying the microscopic origin of spacetime thermodynamic quantities (Fantoni, 2024, Dmitriev, 2012, Karolinski et al., 2024).

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