How to solve the nonvacuum EFE
smallphi said:
I am trying to understand how they find an exact solution of Einstein equation and the field equations for the matter fields.
Well, gtr has been studied for almost a century and by now many approaches to finding exact solutions are known. I suggest you first learn the most elementary approach, the metric symmetry Ansatz method.
You might try studying a derivation of two of the simplest and most important non-vacuum solutions of the EFE, (1) Reissner-Nordstrom electrovacuum (static spherically symmetric electric/gravitational field), (2) Schwarzschild perfect fluid (static spherically symmetric constant density perfect fluid). Almost any good gtr textbook derives at least one of these; many derive both. See http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#gtrmoderntext for a table showing which of almost a dozen standard gtr textbooks discussion which exact solutions.
You can also try
http://en.wikipedia.org/wiki/User:Hillman/Archive#Category:Exact_solutions_in_general_relativity especially
http://en.wikipedia.org/w/index.php?title=Electrovacuum_solution&oldid=43582933
(Note that I cite specific versions because I haven't read more recent versions, which for all I know are full of errors).
As you'll see from all these sources, the basic idea is to obtain relationships between the components of the Einstein tensor (which must be proportional to the stress-energy tensor). It helps to start by assuming that your solution has some metrical symmetries (self-isometries, which are usually more conveniently described, as per Lie theory, by Killing vector fields), e.g. you can assume your solution will be a spherically symmetric Lorentzian manifold. Then write down an appropriate adapted frame, compute the Einstein tensor components wrt this frame, and demand that these obey the appropriate restrictions. The resulting equations will usually form a system of coupled partial differential equations in the metric functions appearing in the Ansatz. With good judgement, luck, and skill, these can be solved; for example as I recently mentioned in another thread, it might happen that there is only one "master PDE", whose solutions then lead to a complete solution of the system, so if you can solve this master PDE, you are almost there. If not, if you can show that your system of PDEs is self-consistent (if not, it means there is no solution with the specified physical properties which has the assumed metrical symmetries), and if you can show that conditions for physical reasonableness such as the "energy conditions" are satisfied, you might be content to express your solution using your system of PDEs, without actually solving these.
Two particularly easy types of nonvacuum solutions you can cut your teeth on are dust solutions and "null dusts". The former are pressureless perfect fluids (familiar examples include the "matter dominated FRW models"), while the latter model incoherent massless radiation, such as EM radiation. In the first case, in a frame comoving with the dust particles, only one component of the Einstein tensor vanishes, namely the one representing the mass-energy density of the dust as measured by dust-riding observers,
<br />
G^{\hat{m}\hat{n}} = <br />
\left[ \begin{array}{cccc} a & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \end{array} \right]<br />
so the EFE says all the components of the Einstein tensor but G^{\hat{0}\hat{0}} (evaluated in the comoving frame) vanish. In the second case, in an adapted frame the Einstein tensor must have the form
<br />
G^{\hat{m}\hat{n}} = <br />
\left[ \begin{array}{cccc} a & a & 0 & 0 \\<br />
a & a & 0 & 0 \\<br />
0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 \end{array} \right]<br />
So all you have to do is write an adapted frame with undetermined functions, compute the Einstein tensor, and set certain components to zero and perhaps equate some others. Pretty simple, really! (But I stress that this procedure is only valid if you work with a frame field rather than a coordinate basis.)
You mentioned "numerical" several times; as others said, numerical relativity is mostly at the opposite end of the spectrum from "exact solutions". However, there are cases where researchers study numerical solutions to an "exact solution" which is given in terms of a system of PDES or ODES which are too hard to solve exactly in closed form.
Alternatively, one can try to apply the theory of qualitative characterizations of systems of differential equations. In the case of exact solutions expressed in terms of a system of ODEs, this approach is often loosely called "dynamical systems". For example, the Bianchi II analog of the classic Mixmaster (which is Bianchi IX) is
<br />
ds^2 = -dt^2 + \exp(2\,a) \, dx^2 <br />
+ \exp(2\,b) \, \left( dy - x \, dz \right)^2 + \exp(2\,c) \, dz^2, <br />
<br />
t_0 < t < \infty, -\infty < x,y,z < \infty<br />
where a,b,c are functions of t, such that
<br />
8 ( \ddot{a}+\dot{a}^2 ) + 4 \, (\dot{a} \dot{b} + \dot{a} \dot{c} <br />
- \dot{a} \dot{c} ) = 3 \exp(2(b-a-c))<br />
<br />
8 ( \ddot{b}+\dot{b}^2 ) + 4 \, (\dot{a} \dot{b} - \dot{a} \dot{c} <br />
+ \dot{a} \dot{c} ) = -5 \exp(2(b-a-c))<br />
<br />
8 ( \ddot{c}+\dot{c}^2 ) + 4 \, ( -\dot{a} \dot{b} + \dot{a} \dot{c} <br />
+ \dot{a} \dot{c} ) = 3 \exp(2(b-a-c))<br />
which admits the Killing vector fields
<br />
\partial_x, \; \partial_y, \; x \, \partial_y + \partial_z<br />
which generate a three dimensional Lie algebra belonging to the isomorphism class traditionally called Bianchi II. This Lie algebra determines a Lie group of self-isometries on the spacetime which is isometric to the group of upper triangular three by three real matrices with ones on the diagonal, called UT(3) or, sometimes, the Heisenberg group). For example, the Killing vector field \partial_x "exponentiates" to translation in x. In fact, in a sense the spatial hyperslices are
isometric to this Lie group (a Lie group is a group which is also a smooth manifold, such that the group operations are smooth, but there's a standard way of placing a Riemannian metric on this one). So this is a homogeneous but anisotropic cosmological model.
smallphi said:
My problem is that in GR the coordinate system cannot be chosen independently from matter from the very beginning like in Newtonian mechanics.
Right, this is at once one of the great beauties of gtr (since it is essential to expressing the universal character of gravitation as spacetime curvature) and a real curse, not so much because it causes technical problems as because it greatly complicates the problem of
interpreting exact solutions once you have found them. This only became clear in the second half of the 20th century.
smallphi said:
For example, in Schwarzschild's solution [the authors] kind of decide on the coordinates and the expected form of metric based on the desired symmetries of spacetime (spherically symmetric, static etc.) and then workout the Einstein equations. There are no field equations for the matter fields since there is no matter in this case.
Bleah!--- "they decide" sounds overly primitive to my ear. Anyway, you are describing the symmetry Ansatz method. Except that you meant "the field equations simply say that all the Einstein tensor components vanish" rather than "there are no field equations"! The Einstein tensor certainly does
not vanish for a generic Lorentzian manifold, so the vacuum field equations are certainly not trivial!
smallphi said:
How do they choose the coordinates of the metric when there won't be any symmetry to exploit like the Shwartzschild case?
It's Schwarzschild. Parse that
schwarz (black) +
Schild (shield).
I think you meant to ask "how does one find an exact solution without assuming some kind of metrical symmetry?" The answer is that one must in effect assume some other kind of symmetry, or combine other assumptions to render the field equations sufficiently tractable to solve. One way to do this is to start with a frame field in a known solution and to "perturb" it using one or more undetermined functions, and then impose the condition that the result be a solution of the same kind. A simple example of an exact dust solution which admits no Killing vector fields at all and which can easily be found in this way is the Szekeres dust (1975):
<br />
ds^2 = -dt^2 + t^{4/3} \, \left(dx^2 + dy^2 + <br />
\left( a \, x + b \, y + c + \frac{5}{9} \, k \, (x^2+y^2) + k \, t^{2/3} \right) <br />
\, dz^2 \right),<br />
<br />
0 < t < \infty, \; -\infty < x,y,z < \infty<br />
where a, b, c, k are (almost) arbitrary smooth functions of t, which is an inhomogeneous anisotropic perturbation of the familiar FRW dust with E^3 hyperslices orthogonal to the dust particles, which is the case a=b=k=0, c=1.
smallphi said:
The Einstein eqs. are second order in metric derivatives. I guess they specify the metric and the first derivatives on a spatial hypersurface. How do they choose the surface and the derivatives?
You'll want to read about the ADM "initial value" reformulation of the EFE (not valid for all Lorentzian manifolds). See http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#gtrmoderntext for a list of textbooks which discuss this.
