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While studying the 3+1 Formalism of General Relativity, the slices of constant $t$ confuse me on what the physical essence is.

For example (and I've made another question related to that, after that started studing 3+1 GR book) in simulations where a Black Hole - Neutron Star merger is happening, the $t$ parametere is refering to these slices is, but I'm lacking to see the physical relation.

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It's a time coordinate for the spacetime. It's not a unique choice, of course, because we have diffeomorphism invariance; any other scalar function $t'$ on our manifold with $(\nabla_a t') (\nabla^a t') < 0$ is just as valid of a choice. This coordinate freedom is why we have all that business with the lapse and the shift and the Hamiltonian and momentum constraints.

But as far as the equations of motion are concerned, $t$ plays the same role that $t$ does in any other field theory, allowing us to cast the equations of motion in the form $$ {\text{rate of change} \choose \text{of fields w.r.t. } t} = {\text{some expression involving the fields} \choose \text{& their derivatives at time } t}. $$

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  • $\begingroup$ I see, really really thank you for the replay first of all :). So, going back to the example I refered (say we have a collision of two massive objects, and proceed to "filiming" it with respect to the t coordinate). That metric will be "time-dependant" (wrt to t). At each 'slice' of t we can "print" the evolution of that system, but that time t will be the coordiate of our chosen metric, and therefore will have no physical meaning or "intuitive" relation with an observer at infinity. Am I somewhere wrong until now? $\endgroup$
    – user174411
    Commented Nov 22, 2020 at 21:10
  • $\begingroup$ @user174411: That's all basically correct. If you did want to reconstruct the full metric to figure out what an an observer at infinity might observe, you can in principle do so from the spatial metric $g_{ij}$, the lapse, and the shift (all given as functions of time.) $\endgroup$
    – Michael Seifert
    Commented Nov 23, 2020 at 12:31
  • $\begingroup$ Hi again and sorry for the delayed reply. Had some things to do. Hope you still recieve this :) What I wanted to point out is that the slice of "constat" t is practically a "3D" box covering the whole space, so If I get those figures, I should be able through a transformation $\tau = W(t)*t$ to find the "corresponing" time of a ZAMO in the inf right? $\endgroup$
    – user174411
    Commented Nov 27, 2020 at 0:49

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