Called general relativity, Albert Einstein’s theory of gravitation was created as a generalization of his special relativity theory. As special relativity is a theory of physical space-time (neglecting gravitational effects), general relativity is a theory of physical space-time in the presence of gravitation.
While Maxwell’s theory of electromagnetism is a relativistic theory that is covariant with respect to Lorentz transformations, Newton’s theory of gravitation is incompatible with special relativity. In 1907, two years after proposing special relativity, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. Einstein described this as “the happiest thought of my life.” He proposed the equivalence principle as “the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame” that “extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.”
In fact, Einstein’s equivalence principle is a generalization of the so-called weak equivalence principle, which dates from Galileo and Newton and states that the inertial mass and gravitational mass of any object are equal. Thus (neglecting friction), the acceleration of different bodies in a gravitational field is independent of their masses and other physical characteristics, and hence, with given initial conditions, their motions will be the same.
The next important step was made by Einstein in his 1912 papers, where he concluded that “if all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.” Further investigations by Einstein to find the correct form of equations for a gravitational field were connected with applications of Riemannian geometry and tensor analysis. The final form of equations of general relativity was given by Einstein in his paper “The Field Equations of gravitation,” submitted on 25 November 1915. At about the same time, David Hilbert submitted a paper entitled “Foundations of Physics,” which also contains the correct field equations for gravitation, introduced by applying a variation principle.
Physical Space-time and Gravitating Matter in General Relativity
According to general relativity, physical space-time in a gravitational field is non-Euclidean; so to describe the properties of space-time, Einstein applied Riemannian geometry. Without gravitation, physical space-time is a flat pseudo-Euclidean Minkowski 4-continuum, where free particles move uniformly and linearly along geodesic worldlines of Minkowski space-time. Einstein’s key idea is that in a gravitational field, particles move along geodesic lines of curved space-time, and in accordance with the Equivalence principle, their movement does not depend on the particles’ characteristics. Thus, motion for an observer is motion along curves in 3-space with variable velocity. Curvature of space-time in general relativity is created by sources of gravitational field. In general relativity, the role of sources of gravitational field is played by the energy-momentum tensor describing the distribution and motion of gravitating matter. Energy (or mass) density (the source of gravitation in Newton’s theory) is only a component of the energy-momentum tensor. Besides energy (mass), a gravitational field in general relativity is also created by momentum and other components of an energy-momentum tensor. The dependence between geometrical properties of physical space-time and gravitating matter is described by Einstein’s gravitational equations.
Einstein Gravitational Equations
The principal geometrical characteristics of a gravitational field in general relativity are given by the metric tensor gik, which determines the square of distance between two infinitesimally close points of pseudo-Riemannian 4-space-time
ds2=gikdxidxk (i, k=0, 1, 2, 3).
(1)
By means of the metric tensor, time intervals and spatial distances can be defined; thus the formula for proper time fixed by a clock at rest in some reference frame is Because the value of g00 in a gravitational field depends on location, time flow depends on gravitational field.
Einstein’s gravitational equations are nonlinear second-order differential equations with respect to the metric tensor and have the form
Rik−1/2gikR=8πG/c4Tik,
(2)
where Rik is the so-called Ricci tensor (a contraction of the curvature tensor), R is the scalar curvature, Tik is the energy-momentum tensor, G is Newton’s gravitational constant, and c is the velocity of light in a vacuum. Einstein’s equations are covariant with respect to arbitrary coordinate transformations.
Equation (2) can be changed by adding to the right-hand side, the so-called cosmological term Λgik, where Λ is a cosmological constant introduced by Einstein. This term describes energy density and pressure of the vacuum, and it plays an essential role in cosmology, leading to the effect of gravitational repulsion if Λ>0. In the case of weak gravitational fields, when the variation of metric with respect to the Minkowski metric is small, the Einstein equations lead to Newton’s law of gravitational attraction and allow one to find first relativistic corrections. In the case of strong gravitational fields, Einstein equations can give new physical results, including black holes and gravitational waves.
Experimental Verification and Bounds of Applicability
Several classical effects of general relativity have been verified observationally, including the bending of light in a gravitational field, gravitational redshift, the advance of the perihelion of the planet Mercury (43 s of arc per century), and retarding of the propagation of light in a gravitational field. The first three effects were discussed by Einstein even before the creation of general relativity.
The weak equivalence principle has been verified to high precision (10−12), and general relativity provides a basis for relativistic astrophysics and cosmology. The Hot Big Bang model was built within the framework of general relativity. Over the past two decades, the role of general relativity has grown in connection with discoveries in cosmology—in particular, acceleration of cosmological expansion, dark matter and dark energy, and other problems that need to be resolved.
General relativity is a classical theory, and a consistent quantum theory of gravitation has not yet been developed. In fact, the formulation of a quantum-gravitation theory requires a unified theory of all fundamental physical interactions. At present, the most popular candidate for such a unified theory is the superstring theory. This theory is in higher-dimensional space, and it leads to a generalization of Einstein’s gravitation theory. A second problem with general relativity is the presence of gravitational singularities (cosmological singularities, collapsing systems, etc.). According to theorems by Stephen Hawking and Roger Penrose, this problem is connected under certain conditions with internal properties of the gravitational equations of general relativity. As with classical theory, general relativity is inapplicable near singular states.
The creation and development of Einstein’s gravitation theory was a triumph of 20th-century science, providing the basis of gravitation theory, relativistic astrophysics, and cosmology. Within the frame of its applicability, general relativity will remain a great achievement of human culture.
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Because the value of g00 in a gravitational field depends on location, time flow depends on gravitational field.
