# What do they do in numerical GR ?

## Main Question or Discussion Point

I am trying to understand how they find an exact solution of Einstein equation and the field equations for the matter fields. My problem is that in GR the coordinate system cannot be chosen independently from matter from the very begining like in Newtonian mechanics. For example, in Schwarzschild's solution they 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.

I wonder how the procedure goes in numerical solution of Einstein eq + matter field equations. That will help me probably understand how they find the exact solutions.

As a first naive guess, I can imagine I write a completely arbitrary smooth metric in some arbitrary coordinates (which is 10 arbitrary smooth functions of 4 coordinates). Then I can calculate the Einstein tensor (the left side of einstein equation) and that will give me the corresponding energy momentum tensor (for all spacetime points) of the matter fields creating that metric. That will be a valid solution, only in general the obtainted energy mom. tensor won't resemble any known matter even remotely.

So what kind of matter (what type of energy mom. tensor) they assume in numerical GR (my guess is pressureless ideal fluid i. e. 'dust')? How do they choose the coordinates of the metric when there won't be any symmetry to exploit like the Shwartzschild case? 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?

Please explain or refer me to article/book. I want to understand the procedure so that I could program a computer do it if I wanted to.

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Well, if you really want, there is a big book called "Exact solutions to Einstein's Field Equation".. Note that "numerical" normally refers to something other than "exact solutions".

the real problem is the non-linear dependence of the curvature tensor upon the metric tensor.

while i'm not familiar with this area, i believe that there are very clever numerical methods that have been worked out..the diversity and numerical stability of approaches vary greatly.

here is a good starting point:

http://www.tacc.utexas.edu/research/users/features/ripples.php [Broken]

since the papers at the bottom are all recent computational studies, i would imagine that reading their methods will yield other sources that are more comprehensive.

i suspect that you will have to rewrite the main field equation in iterative form, approximate the stress-energy tensor (as you say), make a trial guess of the metric (probably starting with a Minkowski metric as a guess) and then iteratively correct it until the field equations are satisfied to some degree of precision (the tricky part). Once the answer has converged, compute your christoffel symbols and construct the geodesic propogator to advance the system through spacetime.

implicit in all of this is also the performance scaling of the problem. what i have suggested above is probably completely unworkable for anything but the simplest of systems (im only guessing this based upon my computational expertise in quantum chemistry and numerical methods thereof).

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Chris Hillman
How to solve the nonvacuum EFE

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 [Broken] 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,
$$G^{\hat{m}\hat{n}} = \left[ \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$$
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
$$G^{\hat{m}\hat{n}} = \left[ \begin{array}{cccc} a & a & 0 & 0 \\ a & a & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$$
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
$$ds^2 = -dt^2 + \exp(2\,a) \, dx^2 + \exp(2\,b) \, \left( dy - x \, dz \right)^2 + \exp(2\,c) \, dz^2,$$
$$t_0 < t < \infty, -\infty < x,y,z < \infty$$
where $a,b,c$ are functions of t, such that
$$8 ( \ddot{a}+\dot{a}^2 ) + 4 \, (\dot{a} \dot{b} + \dot{a} \dot{c} - \dot{a} \dot{c} ) = 3 \exp(2(b-a-c))$$
$$8 ( \ddot{b}+\dot{b}^2 ) + 4 \, (\dot{a} \dot{b} - \dot{a} \dot{c} + \dot{a} \dot{c} ) = -5 \exp(2(b-a-c))$$
$$8 ( \ddot{c}+\dot{c}^2 ) + 4 \, ( -\dot{a} \dot{b} + \dot{a} \dot{c} + \dot{a} \dot{c} ) = 3 \exp(2(b-a-c))$$
which admits the Killing vector fields
$$\partial_x, \; \partial_y, \; x \, \partial_y + \partial_z$$
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.

My problem is that in GR the coordinate system cannot be chosen independently from matter from the very begining 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.

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!

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):
$$ds^2 = -dt^2 + t^{4/3} \, \left(dx^2 + dy^2 + \left( a \, x + b \, y + c + \frac{5}{9} \, k \, (x^2+y^2) + k \, t^{2/3} \right) \, dz^2 \right),$$
$$0 < t < \infty, \; -\infty < x,y,z < \infty$$
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.

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 [Broken] for a list of textbooks which discuss this.

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Stingray
Please explain or refer me to article/book. I want to understand the procedure so that I could program a computer do it if I wanted to.
You won't be able to program a computer to solve Einstein's equations. The problem is extremely difficult, and was actually unsolved until a couple of years ago. There are still many remaining issues even with so many people contributing to these huge programs running on supercomputers.

Anyway, the basic idea is that you split up spacetime into a "3+1" form. You first introduce a foliation of spacelike hypersurfaces. Then specify initial data on your initial hypersurface and finally evolve it into the future.

There are many problems with this. Einstein's equations split into "constraint" and "evolution" portions. The constraint equations must be satisfied on each constant-time hypersurface, while the evolution equations propagate data from one slice to the next. If your initial slice satisfies the constraints, the evolution equations guarantee that all other slices will as well. Obtaining good initial data is a very complicated problem.

Even after solving that, you have to choose how to "stack" the hypersurfaces on top of each other. This is a gauge (coordinate) choice. It's nontrivial to choose coordinates which don't become singular or do other undesirable things. Again, there's been a lot of work on this problem.

Finally, you have to decide how to evolve the system. There is no unique way to decompose Einstein's equations in to 3+1 form. The simplest possibility is the ADM formulation referenced by Chris. If you try to put this into your program, it will crash. The equations have terrible numerical properties, which is not completely understood. So there's a whole cottage industry that (used to) churn out different form of Einstein's equations which might be more stable.

At this point, things are kind of a mess. They mainly work, though. Most simulations are in vacuum (the sources are black holes), although some people have done detailed simulations of neutron star collisions using various models. You can look up lots of review articles on the "Living Reviews in Relativity" site.

Chris Hillman

You won't be able to program a computer to solve Einstein's equations.
We should clarify something. quetzalcoatl9 and Stingray offered summaries of the status of initial value reformulations of the EFE; these do indeed form the theoretical foundation for most work in numerical relativity (but see "Regge calculus" for an alternative foundation), although as Stingray said, the ADM reformulation turned out not to be up the task when it came to black hole merger simulations, so nowadays numerical simulations tend to use more sophisticated reformulations which give better behaved numerical integration schemes.

However, the OP appears to have used "numerical computation" when he meant "symbolic computation". He clearly stated that he is interested in learning how to find exact solutions to the nonvacuum field equations. This is a quite different question from the one which Stingray and quetzalcoatl9 answered! Thus, the immediate computer help he seeks is much more likely to be given by GRTensorII running under Maple than by papers on numerical relativity. (Not to mention, as well-informed commentators suspect http://math.ucr.edu/home/baez/week248.html this poster.)

(EDIT: well, OK, the OP did go on to express his expectation that ADM reformulation would help him find nonvacuum solutions--- maybe the real point is that if this is his immediate goal, there are probably easier and more direct attacks available to him!)

However, it is probably fair to say that at present more work is being done on initial value formulations and numerical simulations of black hole mergers than on exact solutions in the sense of the Stephani et al. monograph, so one could argue that merger simulations and so on are more important. For example, one could point out that exact solutions are "nongeneric", whereas numerical simulations are expected to give (if they are stable) a good picture of the evolution of the gravitational field under "generic" initial conditions of some kind.

Nonetheless, I think it is clear that newbies should learn some elementary methods of finding exact solutions of the EFE before they study ADM! Certainly before they study state of the art numerical integration schemes. And exact solutions remain important even for researchers--- among other things, they can provide convenient "testbeds" for numerical schemes, when suitable solutions are known.

I don't want to leave the impression that initial value formulations are of interest only as the foundation for most numerical simulations, however--- this is not at all the case. To the contrary, the fact that initial value formulations exist at all is of fundamental importance to anyone trying to understand the physical significance of the EFE. To mention one other aspect which has recently come up in my ruminations, one might ask "how many axisymmetric perfect fluid solutions exist"? The expected answer would have the form "so many functions of so many variables, chosen freely and independently". Here, comparing different formulations of the EFE provides an invaluable reality check that a proposed method for "solution counting" yields consistent results.

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Stingray
I wasn't sure exactly what the OP was asking. He now has answers to both possible interpretations

Chris Hillman
Let's hope he's happy now

I asked exactly what StingRay and CH answered. I wasn't trying to solve Einstein eq. on my laptop lol but to understand the weird way of choosing coordinates and metric at the same time and how the physical system represented by the solution, influences the process of obtaining the solution.

In real life problem you want to describe a certain physical system like a black hole or two black holes or galaxy of stars etc. I was trying to understand how the given physical description of the system leads to choice of coordinate system and metric that have to be compatible with each other through Einstein eq. In exact solutions they try an anzats that contains the necessary symmetries of the physical system and then interpret what the obtained solution, if any, represents physically. It's less clear what they do in numerical GR. Suppose I specify I want to simmulate a galaxy of stars. It is not clear how that information is exploited in choices of coordinates (foliation), form of metric, initial hypersurface, how the physical description leads to initial conditions on that hypersuface etc. There are no symmetries to exploit in that case.

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Chris Hillman
A vector space analogy

but to understand the weird way of choosing coordinates and metric at the same time
Uh oh! It just occured to me that you might be being misled by the common practice of defining an exact solution by writing down a line element. This does mean in a sense defining not only the spacetime model but a particular coordinate chart, but you shouldn't confuse the levels of structure: every spacetime admits infinitely many coordinate charts.

Consider this: how do we specify a linear operator? In all likelihood, we write down a matrix. But this matrix only has meaning with respect to a particular basis for our vector space. The operator itself has "geometric significance" independent of any basis. So here the analogy is operator:spacetime as matrix:chart on spacetime.

and how the physical system represented by the solution, influences the process of obtaining the solution.
Again, its really no different from the fact that we have to choosing a basis to make computations on vector spaces. Choosing a coordinate chart adapted to say a cylindrically symmetric spacetime is really no different from choosing to adopt a basis in which a symmetric operator is diagonalized--- it makes the computations a lot easier.

In real life problem you want to describe a certain physical system like a black hole or two black holes or galaxy of stars etc. I was trying to understand how the given physical description of the system leads to choice of coordinate system and metric that have to be compatible with each other through Einstein eq.

In exact solutions they try an anzats...
smallphi, you're making by head ache! Whatever happened to the passive tense? "In searching for exact solutions, one often starts by writing down a frame field Ansatz..."

that contains the necessary symmetries of the physical system
"... adapted to the assumed metrical symmetries of the spacetime...""

and then interpret what the obtained solution, if any, represents physically.
Good, that fragment is word perfect!

It's less clear what they do in numerical GR.
Because the symmetry Ansatz belongs to elementary gtr; numerical relativity is an advanced topic! Hence, at best only the foundations are sketched in most gtr textbooks.

Suppose I specify I want to simmulate a galaxy of stars. It is not clear how that information is exploited in choices of coordinates (foliation), form of metric, initial hypersurface, how the physical description leads to initial conditions on that hypersuface etc. There are no symmetries to exploit in that case.
Nobody said that's neccessarily the first thing astrophysicists try when they want to simulate, in some sense, a galaxy of stars. First, as you probably guessed direct simulation is a highly challenging problem even in Newtonian gravity. So physicists cleverly adopted the practice of changing the problem, and searching for a statistical description of the evolving cluster, which leads to an integro-differential equation called the Vlasov equation. Now, except for very dense clusters or a cluster with a supermassive black hole at the center, gtr is probably of minimal relevance. But advanced students can see http://relativity.livingreviews.org/Articles/lrr-2005-2/index.html [Broken] for a currently popular relativistic version of the Vlasov equation.

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My problem is that in GR the coordinate system cannot be chosen independently from matter from the very begining like in Newtonian mechanics. For example, in Schwarzschild's solution they 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.
This is not correct.

(a) GR possesses an important, essential property: diffeomorphism invariance. No coordinate system is special, you can choose whatever coordinate system you wish.

(b) Schwarzchild spacetimes have nothing to do with choosing coordinate systems based on the matter field-- it's a vacuum solution. And bringing it up points out convenient choices of coordinate systems-- those that exploit symmetry. The coordinate systems are not part of the solution however-- they are like choosing a frame for a picture.

1. Yes all coordinate systems are equal in GR but if you start with a form of the metric that doesn't respect the symmetries in the modeled physical system you won't get any solution. Part of the discussion in this thread was how to choose the form of the metric appropriately.

2. Schwarzschild solution is not only vacuum solution, but also the solution outside a spherical distribution of matter. For that case the spherical symmetry is induced by the matter distribution.

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Chris Hillman
Pedantic correction

all coordinate systems are equal in GR but if you start with a form of the metric that doesn't respect the symmetries in the modeled physical system you won't get any solution.
You mean: choosing a chart which doesn't respect the symmetries will probably make solving the field equations much harder. Indeed, for practical purposes, it may prove impossible to solve in closed form unless you choose an appropriate "symmetry adapted chart". Even then, of course, nothing is guaranteed.

I agree with your point (2).

2. Schwarzschild solution is not only vacuum solution, but also the solution outside a spherical distribution of matter. For that case the spherical symmetry is induced by the matter distribution.

True but misses the point because that's still referring to a vacuum solution, which is independent of the internal structure of your matter distribution. If you want to talk about coordinates being dependent on matter fields as you put it then you have to look *inside* a matter distribution, such a star, where T is not 0. Your example simply did not logically follow from what you were saying. I guess that's alright because what you were trying to illustrate (coordinate choice must depend on matter) was wrong.

And your first point is simply wrong. It's easier to solve Einstein's equation by choosing coordinate systems with cyclic coordinates, but they are not preferred coordinate systems. You can choose anything, it just makes life harder for you.

What I meant was that in GR the simmetries in the assumed matter distribution (if any) cut down the degrees of freedom of the metric. The most general metric in arbitrary coordinates has 10 functions of 4 variables of which only 6 functions are independend due to freedom to perform coordinate transformations. A general spherically symmetric metric depends only on two functions of two coordinates (see proof of Birkhoff theorem). So if for example your mass distribution is spherical the possible choices of metric are cut down. You can't just write down any possible metric like Minkowski or a metric with one unspecified function and get away with it.

Once the possible choices of the abstract metric are cut down, you can express it in any coordinate system but of course you will choose the one that exhibits the symmetry. In that sense symmetries in the matter pick up 'best' coordinates.

I have a physical type of thinking not mathematical, and to me Schwarzschield is the only static metric outside a spherical mass distribution. It doesn't exist physically if there isn't something at the center. Thinking of it as a 'vacuum solution' is mathematically correct and physically a nonsense like a spherical electrostatic field without the charge at the center.

:rofl:

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Chris Hillman
The most general metric in arbitrary coordinates has 10 functions of 4 variables of which only 6 functions are independend due to freedom to perform coordinate transformations.
Einstein himself suggested an interesting way to be more precise: consider the Hilbert series (think OGF=ordinary generating function) counting the number of partial derivatives of order n which cannot be eliminated using all available information.

A general spherically symmetric metric depends only on two functions of two coordinates (see proof of Birkhoff theorem).
Careful, you forgot something!

So how come you started by asking elementary questions and now you're suddenly lecturing mahlerfan? Unless you know something I don't know, since he has only two posts, maybe you should try to give him the benefit of the doubt.

("He": effectively, the closest thing to a neuter pronoun in English.)

I have a physical type of thinking not mathematical, and to me Schwarzschield is the only static metric outside a spherical mass distribution.
You forgot something. Something important.

(Don't tell him, anyone, he'll learn more if he figures it out himself.)

It doesn't exist physically if there isn't something at the center. Thinking of it as a 'vacuum solution' is mathematically correct and physically a nonsense like a spherical electrostatic field without the charge at the center.

:rofl:
OK, I guess you didn't forget.

So, what, is this whole thread a "joke", then?

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With respect to the difficulties in modeling black hole binaries I recommend http://arxiv.org/abs/astro-ph/0609172" [Broken].

The Goddard team is getting pretty successful in their attempts of modeling BHB's. Keep an eye on John Baker!