I Piecewise Functions in the Einstein Field Equations

[PAllen]

Alright, so if we construct a system in a way that's static, we could try solving the EFE's as a pdiff system of equations to find a time-independent metric (and Einstein Tensor)? Would this still work if the SET is piecewise, and does the SET itself have to be continuous I know the metric does, but does the SET have to be as well)?

You don't need to solve for the Einstein tensor, since it is equal, up to constants, to the SET. You are effectively guessing an Einstein tensor when you guess an SET. Solving the pdiff system would give you a metric (or parameterized family of them) if your guess was good.

I don't think it would be safe to make any guesses about the SET across the boundary. If you could pull off the pdiff solution for the SET of the material body part, then look for a general electrovac metric ansatz, assume this for the outside, and apply the junction conditions to constrain it. If your electrovac ansatz was not general enough, this may not be solvable.

I really doubt anyone has ever pulled this off analytically except for the case of spherical conductors or spherical charged fluid balls, bounded by an electrovac solution with spherical symmetry satisfying the junction conditions, and also satisfying EM consistency conditions across the boundary.

If you can find a full treatment of a solution for a charged ball, this would be at least your starting point for treating a different shape - which is much much more complicated.

If you are really serious, one reasonable place to start is Chapter 10, on Electromagnism in GR, in Synge's 1960 GR book. Specifically, the section on electrovac universes is exactly what you are trying to do (including an interior region of matter plus EM fields, and an exterior region of vacuum plus EM fields). This is a hard to find reference. Perhaps another science advisor knows of a more accessible reference for this material.

(Note: I think Synge uses junction conditions that predate Israel's, as Israel's work came after Synge's book).

 

[PeterDonis]

Then, the EFE are a system of pdiffs which some metric must satisfy.

In the general case, a system of ten partial differential equations for ten unknown functions (one for each independent component of the metric), yes. But in the general case you won't get a unique solution. Even in highly constrained special cases, such as vacuum, there isn't a unique solution; to get a unique vacuum solution you have to impose spherical symmetry as an additional constraint, and that is a constraint on the metric, not the SET.

 

[PAllen]

In the general case, a system of ten partial differential equations for ten unknown functions (one for each independent component of the metric), yes. But in the general case you won't get a unique solution. Even in highly constrained special cases, such as vacuum, there isn't a unique solution; to get a unique vacuum solution you have to impose spherical symmetry as an additional constraint, and that is a constraint on the metric, not the SET.

I stated after this "There may be none, or it may be far from unique." so I see no disagreement.

Also, spherical symmetry, at least, would carry over to the SET.

 

[PeterDonis]

spherical symmetry, at least, would carry over to the SET.

Yes, but you can't impose it as a symmetry on the SET; it has to be imposed as a symmetry on the metric. The OP keeps asking whether it's possible to just specify an SET and get a solution that way. It isn't. In order to get a unique solution you will have to impose conditions on the metric.

 

[PAllen]

Yes, but you can't impose it as a symmetry on the SET; it has to be imposed as a symmetry on the metric. The OP keeps asking whether it's possible to just specify an SET and get a solution that way. It isn't. In order to get a unique solution you will have to impose conditions on the metric.

I don’t think the OP necessarily wants a unique solution, any meeting the requirements will do.

The spherical symmetry is a side issue, as that is definitely not what the OP wants.

 

[PeterDonis]

The spherical symmetry is a side issue, as that is definitely not what the OP wants.

Moderator's note: the discussion of spherical symmetry has been moved to a new thread:

https://www.physicsforums.com/threads/spherical-symmetry-in-the-einstein-field-equations.1009305/

 

[Sciencemaster]

Yes, your final sentence is what would typically be done. Note, though, the SET is basically equal to the Ricci tensor. Thus the boundary conditions in terms of induced metric seriously constrain the SET near a boundary.

Given this, for a static SET and time-independent metric and Ricci Tensor, could one take an arbitrary SET (including a piecewise one), set the Ricci Tensor equal to the SET, and solve for the metric tensor to find an approximate metric?

 

[PeterDonis]

could one take an arbitrary SET (including a piecewise one), set the Ricci Tensor equal to the SET

To even write down the Ricci tensor, you need to have some kind of ansatz for the metric. The Ricci tensor components are differential equations written in terms of the metric and its first and second derivatives.

solve for the metric tensor to find an approximate metric?

You could evaluate the differential equations numerically if there is no closed form exact solution (which there won't be in most cases). However, if you just write down an arbitrary SET with no consideration given to any constraints or assumptions, it is quite possible (I would say likely) that the system of differential equations you come up with will not be solvable because some of the equations will be inconsistent with others. You could spend a very long time guessing along these lines before you were lucky enough to come up with a system of equations that was solvable, even numerically.