Coding the Gibbons-Hawking metric

Question

I am studying the Gibbons-Hawking metric, which is

$ g= U^{-1}(d\tau + \omega.dx)^2 + U.dx.dx$

where $U = \sum_{s=1}^n \frac{1}{|x-P_n|}$.

It is a family of metrics defined on a four-dimensional manifolds (the $P_n$ are a series of points, $\tau, x_1, x_2, x_3$ are coordinates in $R^4$ with $dx = dx_1 + dx_2 + dx_3$ and $\omega$ is a vector field on $R^3$). I was asked to "visualize" this metric with Mathematica, which means "taking a slice" of it, which would give something in $R^3$. I am not too sure how to proceed.

To simplify the metric, I changed it in spherical coordinates which gave me $ g= U^{-1}(d\tau + \phi sin\theta d\theta)^2 + U(dr^2 + r^2(d\theta^2 + \sin \theta^2 d\phi^2))$

I found some examples here "http://www.digi-area.com/DifferentialGeometryLibrary/categories.php" but I am not too sure how to write down the metric with so many parameters: how do I write down the point $P_n$ ? and how can I visualize an "image" of what is happening in $R^3$? If the question is too broad, perhaps someone knows a good tutorial for mathematica that could help ? Thanks a lot

Answer 1

In classical mechanics a system is solvable to rigid motion if the number of degrees of freedom is equal the number of contraints. So it is no good approach to go to details without any proper target for the project.

In classical gravitations the two particle problem is consider possible to be solved. The three body problem can not be in general be solved in closed form, but numerical. So good practice for example is:

how to best simulate n body systems in a functional way

The therein given animation is very impressive.

Make your relativistic approximation and drop motion or speed up with values. The mention simulation is valid for great masses and great speeds and show impressive mass dances and all together motion.

Real relativistic problem are otherwise hard to simulate in modern workstation computers. If You have the geometric part of the differential approximate the time part and you are well done.

As an example how capturing relativistics in computer have a look at this slow motion videos:

ps laser pulse bouncing through a glass plate

The speed of light is too fast for realtime simulation. Numerical simulation needs speeds down, slow motion to be viewed and relativistic effects are to preserved in the calculation solely.

To have the zeros order in time is always a good approximation.