Tolman Effect in Gravitational Thermodynamics

Updated 6 July 2026

Tolman effect is the relativistic principle where local temperature varies with gravitational potential to maintain a constant redshifted equilibrium.
The concept is derived using gravitational redshift, stress-energy conservation, and variational methods to explain perfect fluid equilibrium in stationary spacetimes.
Applications range from astrophysical models to black hole thermodynamics, with modifications for non-ideal and quantum-regulated systems.

The Tolman effect, usually called the Tolman–Ehrenfest effect or Tolman law, is the relativistic equilibrium statement that thermal equilibrium in a gravitational field does not require a spatially uniform local temperature. In a static or stationary spacetime, the proper temperature $T$ must vary with position so that the redshifted temperature is constant, typically written as $T\chi=\text{constant}$ or $T\sqrt{-g_{00}}=\text{constant}$ , where $\chi$ is the gravitational redshift factor. The effect is the thermal analogue of gravitational redshift: local temperatures are higher deeper in a gravitational potential well, but the redshift-corrected temperature is uniform in equilibrium (Xia et al., 2023).

1. Historical origin and conceptual content

The historical development of the effect predates Tolman. A prehistory runs from Francis Guthrie’s 1873 objection to Maxwell’s claim that gravity does not make the top or bottom of a gas column hotter or colder, through Maxwell’s and Boltzmann’s defense of uniform equilibrium temperature, to Loschmidt’s disagreement. In the modern literature surveyed from a foundational perspective, Einstein is credited with deriving the core of the effect before Tolman did, using the equivalence principle and the gravitational weight of energy; Tolman and Ehrenfest later gave the exact static general-relativistic form and made it standard (Chua et al., 14 Jul 2025).

The conceptual content is that equilibrium is controlled by gravity-sensitive temporal structure rather than by bare local thermometer readings alone. In the thermal-time formulation, equilibrium in a stationary spacetime is characterized by two statements: thermal time flows geometrically along a timelike symmetry, and temperature is the rate of thermal time with respect to proper time. With a timelike Killing vector $\xi^a$ , this yields the stationary relation

$T\,\xi=\text{const},$

where $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ in the paper’s notation. In the Newtonian limit this becomes

$\frac{\nabla T}{T}=\frac{g}{c^2},$

so equilibrium in a gravitational field implies a small but definite temperature gradient (Rovelli et al., 2010).

A recurring interpretive theme is that the effect is more naturally understood in terms of clocks than in terms of energy “loss” alone. In a static spacetime, proper time and coordinate time are related by $d\tau=\sqrt{g_{00}}\,dt$ , so the Tolman factor expresses how local thermodynamic rates are compared across regions whose clocks tick at different rates. This perspective recurs in both the thermal-time derivation and recent foundational discussion (Rovelli et al., 2010).

2. Geometric formulation in stationary spacetimes

In a stationary spacetime with timelike Killing field $\xi^a$ , the redshift factor is defined by

$T\chi=\text{constant}$ 0

and the 4-velocity of stationary observers is written as

$T\chi=\text{constant}$ 1

The Tolman law then takes the covariant form

$T\chi=\text{constant}$ 2

In adapted coordinates this is equivalent to

$T\chi=\text{constant}$ 3

In flat spacetime, where $T\chi=\text{constant}$ 4, the law reduces to ordinary uniform temperature (Xia et al., 2023).

The physical meaning is straightforward but non-Newtonian. A photon climbing out of a gravitational field loses energy, so equilibrium requires the local temperature profile to compensate exactly for that redshift. For this reason, the proper temperature is higher deeper in the potential well and lower where the redshift factor is larger. The redshifted temperature, not the local temperature, is the invariant equilibrium quantity (Xia et al., 2023).

The same redshift structure applies to the chemical potential. In the variational proof for perfect fluids, the same argument that yields $T\chi=\text{constant}$ 5 also gives

$T\chi=\text{constant}$ 6

so temperature and chemical potential redshift in the same manner (Xia et al., 2023).

For matter models with explicit partition-function control, the same dependence appears at the level of exact thermodynamic functions. For an ideal gas in a time-independent background metric, pressure, energy density, and the thermally averaged energy-momentum tensor depend on temperature and chemical potential through the combinations

$T\chi=\text{constant}$ 7

which is the exact Tolman–Ehrenfest scaling in that setting (Weldon, 2024).

3. Thermodynamic derivations and the general proof

A broad proof for perfect fluids in stationary spacetimes is given in “General Proof of the Tolman law” (Xia et al., 2023). The theorem states that in a stationary spacetime containing a perfect fluid, the Tolman law holds in a region $T\chi=\text{constant}$ 8 provided three conditions are satisfied: $T\chi=\text{constant}$ 9 and $T\sqrt{-g_{00}}=\text{constant}$ 0 are determined by the spacetime metric via the field equations; given $T\sqrt{-g_{00}}=\text{constant}$ 1 and $T\sqrt{-g_{00}}=\text{constant}$ 2, the particle number density $T\sqrt{-g_{00}}=\text{constant}$ 3 and temperature $T\sqrt{-g_{00}}=\text{constant}$ 4 cannot be solved uniquely from the equation of state; and the total entropy in $T\sqrt{-g_{00}}=\text{constant}$ 5 is extremized among all matter configurations with the same metric and fixed total particle number $T\sqrt{-g_{00}}=\text{constant}$ 6 (Xia et al., 2023).

The key technical assumption is that $T\sqrt{-g_{00}}=\text{constant}$ 7 cannot be determined from $T\sqrt{-g_{00}}=\text{constant}$ 8 and $T\sqrt{-g_{00}}=\text{constant}$ 9. The paper emphasizes that many known fluids satisfy an equation of state of the form $\chi$ 0, so the metric may determine $\chi$ 1 and $\chi$ 2 without fixing $\chi$ 3 or $\chi$ 4 separately. Radiation is explicitly noted as a counterexample to this assumption, but radiation already satisfies the Tolman law by the standard argument using $\chi$ 5 and $\chi$ 6 (Xia et al., 2023).

The variational argument uses

$\chi$ 7

with fixed metric and fixed total particle number. Entropy extremization gives

$\chi$ 8

so, since only $\chi$ 9 varies independently,

$\xi^a$ 0

Using

$\xi^a$ 1

this implies that $\xi^a$ 2 is spatially constant. Combining the Gibbs–Duhem relation

$\xi^a$ 3

with its differential form

$\xi^a$ 4

one obtains

$\xi^a$ 5

Stress-energy conservation for a stationary perfect fluid yields the hydrostatic relation

$\xi^a$ 6

Equating the two gives

$\xi^a$ 7

which integrates to

$\xi^a$ 8

The proof uses only stress-energy conservation, total particle-number conservation, thermodynamic identities, and entropy extremization, and does not invoke the gravitational field equations themselves. The paper therefore argues that the Tolman law should hold for a generic perfect fluid in a stationary spacetime, even beyond general relativity (Xia et al., 2023).

4. Statistical mechanics, kinetic theory, and limits of the scalar law

Relativistic kinetic theory reproduces the classical Tolman–Ehrenfest relation for ideal isotropic equilibrium and clarifies where it fails. For a collisionless relativistic $\xi^a$ 9-body system in Schwarzschild spacetime described by the covariant Vlasov equation, a Maxwellian kinetic equilibrium of the form

$T\,\xi=\text{const},$ 0

produces an isotropic stress-energy tensor and recovers

$T\,\xi=\text{const},$ 1

In this case the statistical temperature obtained from kinetic theory and the thermodynamic temperature coincide and are isotropic (Cremaschini et al., 2023).

The same paper shows that non-ideal, non-Maxwellian equilibria alter the result substantially. If the equilibrium distribution function depends on invariants through nontrivial structure functions, or explicitly on higher powers such as $T\,\xi=\text{const},$ 2 in Schwarzschild or the Carter constant $T\,\xi=\text{const},$ 3 in Kerr, the stress-energy tensor becomes anisotropic. Directional temperatures then replace a single scalar temperature, and it is not possible to define a Tolman–Ehrenfest relation in terms of an isotropic scalar temperature. This failure is not restricted to rotating or non-diagonal metrics: sufficiently non-Maxwellian equilibria can violate the scalar law even in Schwarzschild spacetime (Cremaschini et al., 2023).

An exact canonical calculation for an ideal gas in a time-independent metric with $T\,\xi=\text{const},$ 4 sharpens the domain of validity of the standard scaling. A stationary background metric does not spoil Tolman–Ehrenfest scaling, and a magnetostatic field does not change it because the vector potential $T\,\xi=\text{const},$ 5 can be shifted out of the momentum integral. By contrast, a background electrostatic potential contributes a multiplicative factor

$T\,\xi=\text{const},$ 6

which is an exception to the strict Tolman–Ehrenfest rule because the system is open in the grand canonical ensemble (Weldon, 2024).

5. Quantum fields, black holes, and effective Tolman temperature

In black-hole thermodynamics, the conventional Tolman temperature for a static observer in Schwarzschild spacetime is

$T\,\xi=\text{const},$ 7

so $T\,\xi=\text{const},$ 8 at the horizon. This standard infinite-blueshift result is puzzling in the Hartle–Hawking state because the renormalized stress tensor of a conformal scalar field is finite everywhere, including at the horizon. The origin of the mismatch is that the usual Tolman temperature assumes a traceless stress tensor, whereas Hawking radiation is associated with a nonzero trace anomaly (Eune et al., 2015).

The anomaly-corrected resolution replaces the ordinary Stefan–Boltzmann law with a modified relation. In four dimensions, for the anisotropic thermal stress tensor, the corrected laws are

$T\,\xi=\text{const},$ 9

with $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 0 for a conformal scalar field. The resulting effective Tolman temperature is finite everywhere outside the horizon, approaches $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 1 at infinity, and vanishes at the horizon. In the Schwarzschild case the paper gives

$\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 2

and reports a maximum $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 3 at $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 4 (Eune et al., 2015).

The two-dimensional anomaly-corrected analysis leads to the same qualitative conclusion. The generalized Tolman temperature becomes

$\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 5

so the Tolman factor cancels in the final expression, the temperature is finite everywhere, and it vanishes at the horizon. This is interpreted as restoring the equivalence principle at the horizon rather than supporting a divergent thermal bath there (Gim et al., 2015).

A broader review extends this framework to different quantum states. In the Hartle–Hawking–Israel state, the effective Tolman temperature is finite and vanishes at the horizon. In the Unruh state, by contrast, the horizon divergence is associated not with outgoing Hawking quanta but with the infinitely blueshifted negative ingoing flux crossing the horizon; the outgoing Hawking radiation characterized by the effective Tolman temperature is then interpreted as originating from the quantum atmosphere rather than exactly at the horizon (Kim, 2017).

The Tolman effect enters other relativistic constructions through its redshifted equilibrium variables. In static spacetimes, the Tolman relation is used to turn local thermodynamic quantities into global bounds. One example is the quasi-local Tolman mass

$\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 6

which can be rewritten as a Gauss-like surface flux of generalized surface gravity. Combining this with the Tolman temperature relation and the Unruh effect yields an entropy bound of area type,

$\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 7

under the assumptions stated in that work (Abreu et al., 2010).

Recent work also proposes a many-channel generalization. When multiple thermodynamic channels are coupled, the naive condition that each intensive variable is constant is replaced by the invariant

$\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 8

where $\xi=\sqrt{g_{ab}\xi^a\xi^b}$ 9 is determined by the geometry of the entropy manifold through the Ruppeiner connection. This is presented as a many-channel Tolman–Ehrenfest effect: the analogue of the gravitational redshift factor is a channel-dependent geometric correction extracted from off-diagonal thermodynamic curvature (Hamblin et al., 15 May 2026).

The term “Tolman effect” is also used differently in other subfields, and those usages are distinct from the Tolman–Ehrenfest effect. In stellar modeling, a “like–Tolman IV complexity factor” refers to a generalized complexity scalar derived from the Tolman IV interior solution and used as a closure condition in gravitational decoupling (Andrade et al., 2021). In interfacial thermodynamics, the “Tolman effect” denotes the curvature dependence of surface tension for droplets and bubbles, characterized at first order by the Tolman length $\frac{\nabla T}{T}=\frac{g}{c^2},$ 0 through

$\frac{\nabla T}{T}=\frac{g}{c^2},$ 1

at the surface of tension (Lulli et al., 2021). Tolman–Oppenheimer–Volkoff equilibrium, Tolman VII stellar interiors, and related Tolman-named constructions are likewise separate from the Tolman–Ehrenfest law, even when they concern relativistic equilibrium of self-gravitating matter (Bors et al., 2024).

A recent foundational analysis argues that the effect forces a fragmentation of the classical concept of temperature. On that view, one may distinguish a local temperature $\frac{\nabla T}{T}=\frac{g}{c^2},$ 2, a global or “wahre Temperatur” $\frac{\nabla T}{T}=\frac{g}{c^2},$ 3, and a proposed “wahre-local temperature” defined only with respect to local clocks and local equilibrium frames. This suggests that the Tolman effect is not merely a correction to thermodynamics in gravity but an instance of a broader relativistic fragmentation of equilibrium variables tied to the fragmentation of time itself (Chua et al., 14 Jul 2025).