ADM formulation Initial Value Problem data per spacepoint

[TerryW]

Homework Statement

See attached screenshot of MTW pp 532/533

Relevant Equations

N/A

I'm having a bit of trouble getting a clear picture of what is going on here, so if anyone can shed any light, it will be greatly appreciated.

1. I can see how the metric coefficients provide the six numbers per spacepoint, but it can't always be possible to transform the metric into a diagonal form can it?

2. The forward motion in time demands one number per spacepoint, which I assume is the value $\bar t$. How is this 'already willy-nilly' present in the three data per spacepoint? Is it because the various $g_{ij}$ are functions of t, x, y and z so knowing a $g_{ij}$ and its co-ordinates, t follows?

4. Why 'In conclusion...' do we now have TWO data per spacepoint telling us about the $^{(4)}\mathcal G$ and what are they?

5. What has happened to the six $\frac {\partial g_{ij}}{\partial t}$?

Regards
TerryW

 

[member 553083]

1. It is not always possible to transform the metric into a diagonal form, as there are some metrics that are already in a diagonal form and some which are not. However, the metric coefficients can still provide the six numbers per space point. 2. The forward motion in time is already present in the three data per spacepoint, as each of the metric coefficients is a function of t, x, y and z. Knowing the values of the metric coefficients and their associated coordinates will allow you to work out the value of t. 4. In conclusion, we now have two data per spacepoint telling us about the $^{(4)}\mathcal G$ - these are the metric coefficients and their associated coordinates.5. The six $\frac {\partial g_{ij}}{\partial t}$ are still present, but they are now implicit in the two data per spacepoint - each of the coordinates and the associated metric coefficients can be used to calculate the partial derivatives.