Title: Coordinate system for numerical simulation of general relativity
Tags: general-relativity;computational-physics;coordinate-systems
Question: Lets say i want to simulate the differential equations of GR
with some numerical method. I can express the Einstein tensor
in terms of the christoffel symbols which in turn can be expressed
in terms of the metric and its first and second derivative.
Now i can impose a set of coordinates $[t, x, y, z]$ and set up
a big cartesian grid. Each point contains information about the
metric of that point, as seen by an davidbrown@example.org. Initially
the space is empty so the metric will reduce to the Minkowski-metric.
Now i place some mass at the origin, but with finite density. Assume
that the mass encapsulates many grind points and the grid extends to
a far distance so the metric at the end of the grid is approximatly
Now i want to simulate what happens. To do this i rearrange the equations
and solve for $\frac{\partial^2}{\partial t^2}g_{\mu\nu}$ which should govern
the evolution of the system. Since i know the initial conditions of the
metric and $T_{\mu\nu}$ i should be able to simulate the dynamics of what
happens.
(I assume the metric outside would converge to the outer schwarzschild metric
while the parts inside the mass would converge to the inner schwarzschild metric. Additionally a gravitational wave should radiate away because of the sudden appearance of a mass).
However, by doing so i have placed the spacetime itself on a background grid,
which seems fishy to me.
Question 1:
How does the coordinate system influences the equation? For example i could
have choosen $[t, r, \phi, \theta]$ and would have gotten the same equations
since it involves only ordinary derivatives. Do i assume correctly that the
coordinate system properties only appear during numerical integration?
Question 2:
What physical significance does this "cartesian grid" system have? If i look
at a point near the surface of the mass after a long time, where is this point
in the spacetime? A stationary observer would follow the curvature and may already have fallen into the mass. Does this mean my coordinate system itself
"moves" along? How can i get a "stationary" (constant proper radial distance) coordinate system?
Question 3:
Since i have the metric at every grid point i could calculate (numerically) the geodesic through this spacetime and find the path of an infalling observer, right?
Question 4:
Would this work for simulating non-singular spacetimes? Or is this approach too easy?
edit1:
To clarify question 1, different coordinate systems have different artefacts on their own. Instead of a grid i could use schwarzschild-coordinates. If i now expand the whole equation i would get the same form, because it is coordinate independent. However the resulting metric would look different (same metric, but expressed in another coordinate system). I'm a bit confused because the metric and the coordinate system are tied together. If i'm solving for the metric itself i still need to provide the coordinate system. And i dont understand how i introduce the coordinate system.
Comment: @WetSavannaAnimalakaRodVance is absolutely right; numerical relativity is *nasty* and setting up the Cauchy problem is difficult let alone doing the numerics. There are also numerous constraints to consider, such as those related to the extrinsic curvature not in the Einstein equations. I recommend the book 'Numerical Relativity.'

Comment: It sounds as though you're doing this from first principles: have you read any numerical GR texts or background yet? It sounds like you are thinking about this intelligently, but GR is not trivial as far as numerical simulation goes - it can be as nasty as the most advanced fluid dynamics simulations and there are, as I understand it, a great many tricks of the trade which have emerged experimentally over quite a long time.

Comment: Could you say a bit more about (1): I don't think I follow. The equations are most assuredly not the same for different co-ordinate systems (when you expand them out as partial derivatives rather than in tensor form) - but this sounds too obvious a mistake from your level - hence I don't think I follow. (2) is a good conceptual question, to answer theoretically whether or not the answer is useful numerically, and probably worth asking as a conceptual question, framing the numerical simulation as simply a thought experiment. (3) seems trivial: once you have the metric, well, of course yes!

Comment: Have you looked at any papers on the subject of numerical relativity?

Comment: Now I don't want to seem rude, but you probably are in over your head at this point. It might be better to take a few steps backwards. Perhaps solve a 1+1D problem first (plenty of examples for that, from my understanding) while you learn more diff geom

Comment: @KyleKanos I have looked at several and i get the splitting into 3D space and time-slices, but other than that i have trouble understanding these papers as my knowledge of differential geometry is very limited.

Comment: @WetSavannaAnimalakaRodVance i have not read a textbook on this subject, this is out of curiosity. I have done some numerical raytracing on the schwarzschild metric and i have some background on numerics. I have added some information on question 1.



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