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In general relativity, the **Ehrenfest–Tolman effect** (also known as the **Tolman–Ehrenfest effect**), created by Richard C. Tolman and Paul Ehrenfest, argues that temperature is not constant in space at thermal equilibrium, but varies with the spacetime curvature. Specifically, it depends on the spacetime metric. In a stationary spacetime with timelike Killing vector field , the temperature satisfies instead the Tolman-Ehrenfest relation: , where 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 to the concept that temperature is the rate of thermal time with respect to proper time.^{ [1] }

In physics, **spacetime** is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur.

The **world line** of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In physics and mathematics, the **Lorentz group** is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

The **Unruh effect** is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe none. In other words, the background appears to be warm from an accelerating reference frame; in layman's terms, an accelerating thermometer in empty space, removing any other contribution to its temperature, will record a non-zero temperature, just from its acceleration. Heuristically, for a uniformly accelerating observer, the ground state of an inertial observer is seen as a mixed state in thermodynamic equilibrium with a non-zero temperature bath.

**Teleparallelism**, was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field.

In physics, a **Killing horizon** is a geometrical construct used in general relativity and its generalizations to delineate spacetime boundaries without reference to the dynamic Einstein field equations. Mathematically a Killing horizon is a null hypersurface defined by the vanishing of the norm of a Killing vector field. It can also be defined as a null hypersurface generated by a Killing vector, which in turn is null at that surface.

When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are used. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

In general relativity, the **monochromatic electromagnetic plane wave spacetime** is the analog of the monochromatic plane waves known from Maxwell's theory. The precise definition of the solution is quite complicated, but very instructive.

In general relativity, specifically in the Einstein field equations, a spacetime is said to be **stationary** if it admits a Killing vector that is asymptotically timelike.

The **Gödel metric** is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant. It is also known as the **Gödel solution** or **Gödel universe**.

In relativistic classical field theories of gravitation, particularly general relativity, an **energy condition** is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically-phrased mathematical formulation. There are multiple possible alternative ways to express such a condition such that can be applied to the matter content of the theory. The hope is then that any reasonable matter theory will satisfy this condition or at least will preserve the condition if it is satisfied by the starting conditions.

In general relativity, the **van Stockum dust** is an exact solution of the Einstein field equation in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is *increasing* with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example.

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of *nested round spheres*. There are several different types of coordinate chart which are *adapted* to this family of nested spheres; the best known is the Schwarzschild chart, but the **isotropic chart** is also often useful. The defining characteristic of an isotropic chart is that its radial coordinate is defined so that light cones appear *round*. This means that, the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances. On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart.

**Born rigidity** is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics.

In physics the **Einstein aether theory**, also called **aetheory**, is a generally covariant modification of general relativity which describes a spacetime endowed with both a metric and a unit timelike vector field named the aether. The theory has a preferred reference frame and hence violates Lorentz invariance.

The **Komar mass** of a system is one of several formal concepts of mass that are used in general relativity. The Komar mass can be defined in any stationary spacetime, which is a spacetime in which all the metric components can be written so that they are independent of time. Alternatively, a stationary spacetime can be defined as a spacetime which possesses a timelike Killing vector field.

Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the **construction of a complex null tetrad**, where is a pair of *real* null vectors and is a pair of *complex* null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature

**Symmetries in quantum mechanics** describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

A **proper reference frame** in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity.

- ↑ Rovelli, Carlo; Smerlak, Matteo (2011). "Thermal time and Tolman–Ehrenfest effect: 'temperature as the speed of time'".
*Classical and Quantum Gravity*.**28**(7): 075007. arXiv: 1005.2985 . Bibcode:2011CQGra..28g5007R. doi:10.1088/0264-9381/28/7/075007. S2CID 119250231.

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