# IBVP of general relativity when stress-energy tensor is known for “past” submanifold

I am reading the relativity texts, but meanwhile, I do get that general relativity can be formulated as IBVP (initial-boundary-value problem). But for now, it seems that I need to read much more to completely understand what is required in order for the IBVP problem to ensure existence.

So rather than keeping the question unanswered until I completely read the text, I would like to ask the question here.

Suppose that we know stress-energy tensor for "past" submanifold (depending of course according to "reference frame" or coordinate system]. By past submanifold, I mean this: pick some coordinate system, and pick time variable. Let present time be $t=0$. This allows one to pick "entire" space submanifold $S_t$ which evolves by time. Let stress-energy tensor be known for all points in $S_t$ for $t_{-1}\le t \le 0$ (or $t_{-1} < t \le 0$), where $t_{-1}$ is some chosen time.

In this case, to what time $t \geq 0$ is it possible to compute metric tensor of points at $t$? Is this only possible for $t=0$? What is the restriction required on initial stress-energy tensor to compute metric tensor at $t \geq 0$? Can metric tensor be computed for entire space submanifold?

• Comment to the post (v1): Consider to spell out acronyms. – Qmechanic Mar 2 '17 at 9:53
• Related question by OP: physics.stackexchange.com/q/315560/2451 – Qmechanic Mar 2 '17 at 9:54
• The Einstein equations are local, so if you want to know the metric at time $t > 0$ you'll certainly need to know the stress-energy tensor then. So you'll need to know the evolution equations for the matter. – gj255 Mar 2 '17 at 10:01
• What about $t=0$? – Master of Life Mar 2 '17 at 10:28

If you can pick a time function that is $\approx \Bbb R$, this is already a pretty good sign.

If you know the metric tensor and matter fields on a time interval, then you can predict the metric tensor for future $t$ if the following conditions are met :

• The spacetime is globally hyperbolic. If you can define a time function then it's probably gonna be one if you don't have any naked singularities, although you can always construct some really pathological spacetimes.
• The metric's initial data on the achronal slice selected is sufficiently regular
• The matter fields are well behaved. The conditions are rather long and involved (they are in 7.7 of Hawking and Ellis), involving solutions of matter fields belonging to Sobolev spaces of at least $W^4$ for the initial data on the achronal slice and various conditions between the metric and matter fields, as well as the stress energy tensor being polynomial in the fields.

If your matter fields are already well behaved on Minkowski space, that will probably be fine (avoid Rarita-Schwinger fields). All the basic fields obey those conditions.

• What does it mean for a function to be $\approx \Bbb R$? – Ryan Unger Mar 2 '17 at 13:31