The initial value formulation of general relativity is a reformulation of
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's theory of
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. ...
that describes a
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. A ...
evolving over
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
.
Each solution of the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole
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 diffe ...
: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify
evolution equation
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
s for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields: they are self-interacting (that is,
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
even in the absence of other fields); they are
diffeomorphism invariant, so to obtain a unique solution, a fixed background metric and gauge conditions need to be introduced; finally, the metric determines the spacetime structure, and thus the domain of dependence for any set of initial data, so the region on which a specific solution will be defined is not, a priori, defined.
There is, however, a way to re-formulate Einstein's equations that overcomes these problems. First of all, there are ways of rewriting spacetime as the evolution of "space" in time; an earlier version of this is due to
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Unive ...
, while a simpler way is known after its inventors
Richard Arnowitt
Richard Lewis Arnowitt (May 3, 1928 – June 12, 2014) was an American physicist known for his contributions to theoretical particle physics and to general relativity.
Arnowitt was a Distinguished Professor (Emeritus) at Texas A&M University, whe ...
,
Stanley Deser
Stanley Deser (born 1931) is an American physicist known for his contributions to general relativity. Currently, he is emeritus Ancell Professor of Physics at Brandeis University in Waltham, Massachusetts and a senior research associate at Califo ...
and
Charles Misner
Charles W. Misner (; born June 13, 1932) is an American physicist and one of the authors of ''Gravitation''. His specialties include general relativity and cosmology. His work has also provided early foundations for studies of quantum gravity ...
as
ADM formalism
The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was fir ...
. In these formulations, also known as "3+1" approaches, spacetime is split into a three-dimensional hypersurface with
interior metric
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
and an embedding into spacetime with
exterior curvature
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement o ...
; these two quantities are the dynamical variables in a
Hamiltonian formulation tracing the hypersurface's evolution over time. With such a split, it is possible to state the ''initial value formulation of general relativity''. It involves initial data which cannot be specified arbitrarily but needs to satisfy specific
constraint
Constraint may refer to:
* Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies
* Constraint (mathematics), a condition of an optimization problem that the solution ...
equations, and which is defined on some suitably smooth three-manifold
; just as for other differential equations, it is then possible to prove
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistenti ...
and
uniqueness
Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison, or is remarkable, or unusual. When used in relation to humans, it is often in relation to a person's personality, or some specific characterist ...
theorems, namely that there exists a unique spacetime which is a solution of Einstein equations, which is
globally hyperbolic
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...
, for which
is a
Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...
(i.e. all past events influence what happens on
, and all future events are influenced by what happens on it), and has the specified internal metric and extrinsic curvature; all spacetimes that satisfy these conditions are related by
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
.
The initial value formulation with its 3+1 split is the basis of
numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and ...
; attempts to simulate the evolution of relativistic spacetimes (notably merging
black hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can defo ...
s or
gravitational collapse
Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formatio ...
) using computers. However, there are significant differences to the simulation of other physical evolution equations which make numerical relativity especially challenging, notably the fact that the dynamical objects that are evolving include space and time itself (so there is no fixed background against which to evaluate, for instance, perturbations representing gravitational waves) and the occurrence of singularities (which, when they are allowed to occur within the simulated portion of spacetime, lead to arbitrarily large numbers that would have to be represented in the computer model).
[For a review of the basics of numerical relativity, including the problems alluded to here and further difficulties, see .]
See also
*
ADM formalism
The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was fir ...
Notes
References
*
*
*
*
*
*Kalvakota, Vaibhav R. (July 1, 2021).
A brief account of the Cauchy problem in General Relativity.
*
*
*
* {{cite book , author-link1=Robert M. Wald, last1=Wald , first1= Robert M. , year=1984 , title=
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. ...
, location= Chicago , publisher= University of Chicago Press , isbn=0-226-87033-2
General relativity