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In ADM numerical relativity, I take it that initial data is given as a 3-dimensional $g_{ij}$ configuration slice frozen in time (eg. BL, Misner solutions). The time evolution is to be calculated step by step via

$$ \frac{d}{dt}g_{ij} = \frac{2N}{\sqrt g}(\pi_{ij} - \frac{1}{2}\pi g_{ij}) + N_{i;j} + N_{j;i}$$

However, in the exposition provided here, all the quantities used for calculating $\frac{d}{dt}g_{ij}$ has to be pre-calculated using the 4-dimensional metric $^{(4)}g_{\mu\nu}$ and Christoffel connections $^{(4)}\Gamma^i_{jk}$.

This seems like a catch-22.

Since the initial data is presumably only for a 3-dimensional space frozen in time, how would we get the rest of the data to pre-construct the the full 4-dimensional $g_{\mu\nu}$ over time, in order to calculate $^{(4)}g_{\mu\nu}$, $^{(4)}\Gamma^i_{jk}$, and finally back to calculating $\frac{d}{dt}g_{ij}$?

James
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  • You are asking a physics question in a website whose topic is math. – Mariano Suárez-Álvarez Feb 08 '23 at 17:51
  • @MarianoSuárez-Álvarez yes, it is rather surprising but numerical methods usually get no answers on physics SE. – James Feb 08 '23 at 17:52
  • Your question is, more or less, why do we use initial conditions of this type instead of that type when solving this equation of physical relevance, and that is something that can only be answered in terms of the physical relevance of things. It is not a lot different to asking why we use position and speed as initial conditions in Newtonian mechanics. – Mariano Suárez-Álvarez Feb 08 '23 at 17:58
  • @MarianoSuárez-Álvarez there seems no data to calculate $^{(4)}g_{\mu\nu}$ with, if only the initial data of 3-dimensional space is given (future time is empty)? – James Feb 08 '23 at 18:02

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As you can see in the time evolution, you need the conjugate momenta $\pi^{i,j}$. Purely in terms of free parameters, $g_{\mu,\nu}$ has 10, while $g_{i,j}$ and $\pi^{i,j}$ have 6 each, giving 12. The two extra parameters are "eaten" by your lapse function and shift vector if I recall correctly (although on that part I am more fuzzy). So as usual with initial value problems, you also need the first derivative at the start time in order to get anything sensible.

Jean Du Plessis
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