# Initial value formulation (general relativity)

﻿
Initial value formulation (general relativity)

The initial value formulation is a way of expressing the formalism of Einstein's theory of general relativity in a way that describes a universe evolving over time.

Each solution of Einstein's equation encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: 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 equations 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 even in the absence of other fields; they are diffeomorphism invariant, so two 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. [Cf. Harvnb|Hawking|Ellis|1973|loc=sec. 7.1.]

There is, however, a way to re-formulate Einstein's equations that overcomes these problems. First of all, there are ways of re-writing spacetime as the evolution of "space" in time; an earlier version of this is due to Paul Dirac, while a simpler way is known after its inventors Arnowitt, Deser and Misner as ADM formalism. In these formulations, also known as "3+1" approaches, spacetime is split into a three-dimensional hypersurface with interior metric and an embedding into spacetime with exterior curvature; these two quantities are the dynamical variables in a Hamiltonian formulation tracing the hypersurface's evolution over time. [Harvnb|Arnowitt|Deser|Misner|1962; for a pedagogical introduction, see Harvnb|Misner|Thorne|Wheeler|1973|loc=§21.4–§21.7.] 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 equations, and which is defined on some suitably smooth three-manifold $Sigma$; just as for other differential equations, it is then possible to prove existence and uniqueness theorems, namely that there exists a unique spacetime which is a solution of Einstein equations, which is globally hyperbolic, for which $Sigma$ is a Cauchy surface (i.e. all past events influence what happens on $Sigma$, 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. [Harvnb|Fourès-Bruhat|1952 and Harvnb|Bruhat|1962; for a pedagogical introduction, see Harvnb|Wald|1984|loc=ch. 10; an online review can be found in Harvnb|Reula|1998.]

The initial value formulation with its 3+1 split is the basis of numerical relativity; attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers. [See Harvnb|Gourgoulhon|2007.] 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 Harvnb|Lehner|2001.]

ee also

*Einstein's equations

Notes

References

*Arnowitt, Richard; Stanley Deser&amp; Charles W. Misner(1962),"The dynamics of general relativity", inWitten, L.,"Gravitation: An Introduction to Current Research", Wiley, pp. 227-265
*Bruhat, Yvonne(1962),"The Cauchy Problem", inWitten, Louis,"Gravitation: An Introduction to Current Research", Wiley, pp. 130
*Citation
first=Yvonne
last= Fourès-Bruhat
title= Théoréme d'existence pour certains systémes d'équations aux derivées partielles non linéaires
journal= Acta Mathematica
volume=88
year=1952
pages=141–225

*Citation
first=Eric
last=Gourgoulhon
title=3+1 Formalism and Bases of Numerical Relativity
id=arxiv|gr-qc|0703035
year=2007

*Citation
last1=Hawking
first1=Stephen W.
last2=Ellis
first2=George F. R.
title=The large scale structure of space-time
publisher=Cambridge University Press
isbn=0-521-09906-4
year=1973

*Citation
last=Lehner
first=Luis
title=Numerical Relativity: A review
journal =Class. Quant. Grav.
volume=18
year=2001
pages=R25-R86
id=arxiv|gr-qc|0106072

* Misner, Charles W.; Kip. S. Thorne&amp; John A. Wheeler(1973),"Gravitation", W. H. Freeman, ISBN 0-7167-0344-0
*Citation
first= Oscar A.
last=Reula
title=Hyperbolic Methods for Einstein's Equations
journal=Living Rev. Relativity
volume=1
year=1998
url=http://www.livingreviews.org/lrr-1998-3
accessdate=2007-08-29

* Wald, Robert M.(1984),""General Relativity"", Chicago: University of Chicago Press, ISBN 0-226-87033-2

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• General relativity — For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia

• Golden age of general relativity — The Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics. During this… …   Wikipedia

• Exact solutions in general relativity — In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field.… …   Wikipedia

• Mathematics of general relativity — For a generally accessible and less technical introduction to the topic, see Introduction to mathematics of general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia

• Contributors to general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

• Mass in general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

• 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 many other… …   Wikipedia

• History of special relativity — The History of special relativity consists of many theoretical and empirical results of physicists like Hendrik Lorentz and Henri Poincaré, which culminated in the theory of special relativity proposed by Albert Einstein, and subsequent work of… …   Wikipedia

• Time value of money — The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory. For example, \$100 of today s money invested for one year… …   Wikipedia

• Path integral formulation — This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see line integral. The path integral formulation of quantum mechanics is a description of quantum theory which… …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.