general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, the Ehrenfest–Tolman effect (also known as the Tolman–Ehrenfest effect), created by Richard C. Tolman and

Paul Ehrenfest Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition an ...

, argues that

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

is not constant in space at

thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...

, but varies with the spacetime curvature. Specifically, it depends on the

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

metric. In a stationary spacetime with timelike

Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...

\xi

, the temperature

T

satisfies instead the Tolman-Ehrenfest relation:

T\,, , \xi, , =\mathrm

, where

, , \xi, , =\sqrt

is the norm of the timelike Killing vector field. This relationship leads to the concept of ''thermal time'' which has been considered as a possible basis for a fully general-relativistic thermodynamics. It has been shown that the Tolman–Ehrenfest effect can be derived by applying the

equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...

to the concept that temperature is the rate of thermal time with respect to proper time.

References

General relativity Quantum mechanics {{relativity-stub