Solutions of the Einstein field equations
Solutions of the Einstein field equations are metrics of
The Einstein field equations are
where is the Einstein tensor, is the cosmological constant (sometimes taken to be zero for simplicity), is the metric tensor, is a constant, and is the stress–energy tensor.
The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten partial differential equations in which the metric tensor can be solved for.
- Where appropriate, this article will use the abstract index notation.
Solving the equations
It is important to realize that the Einstein field equations alone are not enough to determine the evolution of a gravitational system in many cases. They depend on the
To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine evolution of the stress–energy tensor):
This is clearly not enough, as there are only 14 equations (10 from the field equations and 4 from the continuity equation) for 20 unknowns (10 metric components and 10 stress–energy tensor components). Equations of state are missing. In the most general case, it's easy to see that at least 6 more equations are required, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout spacetime.
In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation. Some common approximations are:
- where
Here is the mass–energy density measured in a momentary co-moving frame, is the fluid's 4-velocity vector field, and is the pressure.
- Non-interacting dust ( a special case of perfect fluid ):
For a perfect fluid, another equation of state relating density and pressure must be added. This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected.
Next, notice that only 10 of the original 14 equations are independent, because the continuity equation is a consequence of Einstein's equations. This reflects the fact that the system is
A popular choice of gauge is the so-called "De Donder gauge", also known as the harmonic condition or harmonic gauge
In numerical relativity, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism. In this decomposition, metric is written in the form
- , where
and are functions of spacetime coordinates and can be chosen arbitrarily in each point. The remaining physical degrees of freedom are contained in , which represents the Riemannian metric on 3-hypersurfaces with constant . For example, a naive choice of , , would correspond to a so-called
Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved. Unfortunately, even in the simplest case of gravitational field in the vacuum (vanishing stress–energy tensor), the problem is too complex to be exactly solvable. To get physical results, we can either turn to numerical methods, try to find exact solutions by imposing symmetries, or try middle-ground approaches such as perturbation methods or linear approximations of the Einstein tensor.
Exact solutions
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Exact solutions are
External reference
Scholarpedia article on the subject written by Malcolm MacCallum
Non-exact solutions
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The solutions that are not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.
Applications
There are practical as well as theoretical reasons for studying solutions of the Einstein field equations.
From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters. From a physical standpoint, knowing the solutions of the Einstein Field Equations allows highly-precise modelling of astrophysical phenomena, including black holes, neutron stars, and stellar systems. Predictions can be made analytically about the system analyzed; such predictions include the perihelion precession of Mercury, the existence of a co-rotating region inside spinning black holes, and the orbits of objects around massive bodies.
See also
References
- J.A. Wheeler; C. Misner; K.S. Thorne (1973). ISBN 0-7167-0344-0.
- J.A. Wheeler; I. Ciufolini (1995). Gravitation and Inertia. ISBN 978-0-691-03323-5.
- R.J.A. Lambourne (2010). Relativity, Gravitation and Cosmology. ISBN 978-0-521-13138-4.