Piecewise Functions in the Einstein Field Equations

[Sciencemaster]

In terms of defining the stress energy tensor of the conducting material, I ended up just setting everything but T_μν to be 0, since one of the assumptions being made is that everything is stationary, meaning that all net forces are 0 and momentum is 0. This is valid for the Stress Energy Tensor even if the forces acting apon an object from one side are nonzero, just as long as the net forces are 0, right? I'm just not 100% positive that's how a component of the tensor works.

 

[Sciencemaster]

No, this is not correct. The conditions on the metric are indirect. From the metric on one side of boundary, you define a family of spacelike surfaces that approaches the boundary (foliation). Similarly for the metric for the other side of the boundary. The overall metric induces a 3-metric in each of these surfaces. Then, the first requirement is that the limit of induced metric from one side is the same as the limit of induced metric from the other side. Thus, the same 3-geometry is implied on surface where the overall combined metric is not differentiable.

Similarly, for each surface of the families, there is an extrinsic curvature tensor, K. These must also have well defined limits from each side. However, we don't require equality for these between the limits from both sides. Instead, the formula involving K equates to difference in limits from 2 sides to a stress energy expression, involving the "surface stress tensor". I am not really familiar with the details on this part, so I can't help you further. References are given in the paper I linked that should have more details.

Alright, that does help. Thank you for the help! I'll look through the paper further, but your description really is helpful!

 

[Sciencemaster]

Alright, so if we construct a piecewise Stress-Energy Tensor (Let's not worry about what it actually is anymore, I think I've got that part), it seems to be able to be used to find the Metric Tensor and Ricci Tensor. However, I've been looking at derivations of known exact solutions such as the FRW solution and the Schwarzschild solution, but so far, they just seem to use continuous functions and variables. For a piecewise function, to find the metric, would one start by solving for the Metric and Ricci Tensors for each piece of the function and apply boundary conditions later? Or is it something else?

 

[PAllen]

Alright, so if we construct a piecewise Stress-Energy Tensor (Let's not worry about what it actually is anymore, I think I've got that part), it seems to be able to be used to find the Metric Tensor and Ricci Tensor. However, I've been looking at derivations of known exact solutions such as the FRW solution and the Schwarzschild solution, but so far, they just seem to use continuous functions and variables. For a piecewise function, to find the metric, would one start by solving for the Metric and Ricci Tensors for each piece of the function and apply boundary conditions later? Or is it something else?

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.

 

[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.

Oh? My thought was that you could construct a SET using arbitrary constant values over different intervals and go from there? Is that not something you can do? Do you have to meet some requirements at the boundaries (perhaps some 'fuzziness' where we have a function sharply change value to approximate a discontinuous change)?

 

[PAllen]

Oh? My thought was that you could construct a SET using arbitrary constant values over different intervals and go from there? Is that not something you can do? Do you have to meet some requirements at the boundaries (perhaps some 'fuzziness' where we have a function sharply change value to approximate a discontinuous change)?

What I am getting at is that the SET via the EFE substantially metric tensor. Technically, it determines it up to Weyl curvature. Meanwhile, the requirement that the metric from both sides of the boundary induce the same intrinsic 3 geometry in the boundary, is another constraint on the metric. While I have not analyzed the strength of these constraints, in general, my guess is that you would not be able to freely choose the SET on both sides to the boundary. That is, having picked it on one side freely, you would have restricted choice on the other side.

 

[PeterDonis]

My thought was that you could construct a SET using arbitrary constant values over different intervals and go from there?

You can't just arbitrarily construct any SET you like. It has to be (up to a constant factor $8 \pi$ and a possible additional factor depending on your choice of units) equal to the Einstein tensor of some metric.

Normally when trying to construct solutions, either a form is chosen for the SET that is already known to meet the above requirement (for example, vacuum, perfect fluid, EM field, etc.), or the solution is constructed in reverse, so to speak, by first choosing a metric ansatz (possibly including some undetermined functions), then computing its Einstein tensor and seeing what that implies about the form of the SET.

 

[Sciencemaster]

You can't just arbitrarily construct any SET you like. It has to be (up to a constant factor $8 \pi$ and a possible additional factor depending on your choice of units) equal to the Einstein tensor of some metric.

Normally when trying to construct solutions, either a form is chosen for the SET that is already known to meet the above requirement (for example, vacuum, perfect fluid, EM field, etc.), or the solution is constructed in reverse, so to speak, by first choosing a metric ansatz (possibly including some undetermined functions), then computing its Einstein tensor and seeing what that implies about the form of the SET.

Alright, but would it be possible to construct an Einstein Tensor that satisfies the requirement of being equal to the SET over $8 \pi$ and such just from a SET? After all, it's not just finding an arbitrary ET, it has to meet the constraint. Finding such a metric from a metric ansatz is similar to what I was thinking but the problem is that without symmetries in spacetime or rotation, there's not a lot about the form we can deduce, I imagine, although there are still a few constraints. Since we can find the SET from the ET and such, would it be possible to do it the other way around, even if just numerically (i.e. computationally)? Or do the constraints on a valid ET make this not possible for some SET's.

Essentially, if we know the matter distribution of a system at t=0 from measurement, and we want to determine the curvature of spacetime around it, and predict how matter would move, would it be possible to determine a metric/curvature from our measured matter distrubution?

 

[PeterDonis]

would it be possible to construct an Einstein Tensor that satisfies the requirement of being equal to the SET over and such just from a SET?

You don't "construct" an Einstein tensor. The Einstein tensor is determined by the metric. You can't construct a metric from a stress-energy tensor.

if we know the matter distribution of a system at t=0 from measurement, and we want to determine the curvature of spacetime around it, and predict how matter would move, would it be possible to determine a metric/curvature from our measured matter distrubution?

If all you know at t=0 is the matter distribution, and you don't know the metric, no, that's not enough information to determine a solution. You need to know the metric and the matter distribution at t=0. Or, as I said, you can try a metric ansatz and compute its Einstein tensor and see what that implies about the stress-energy tensor.

 

[PeterDonis]

it's not just finding an arbitrary ET, it has to meet the constraint

What constraint? The only constraint on the Einstein tensor is that it is determined by the metric.

 

[PAllen]

What constraint? The only constraint on the Einstein tensor is that it is determined by the metric.

I think the OP is referring to the junction conditions for a boundary where the metric is allowed to be non-differentiable.

 

[PAllen]

You don't "construct" an Einstein tensor. The Einstein tensor is determined by the metric. You can't construct a metric from a stress-energy tensor.If all you know at t=0 is the matter distribution, and you don't know the metric, no, that's not enough information to determine a solution. You need to know the metric and the matter distribution at t=0. Or, as I said, you can try a metric ansatz and compute its Einstein tensor and see what that implies about the stress-energy tensor.

Of course, if you assume a static situation, you can try any time independent SET form you want and try to see if the pdiffs relating this to the metric are solvable.

 

[PeterDonis]

if you assume a static situation, you can try any time independent SET form you want and try to see if the pdiffs relating this to the metric are solvable.

You can only do this if you know the metric, or at least have an ansatz for it (probably containing undetermined functions), so you can compute its Einstein tensor.

 

[PAllen]

You can only do this if you know the metric, or at least have an ansatz for it (probably containing undetermined functions), so you can compute its Einstein tensor.

No, you can do the reverse - guess an an SET form with none of the components depending on the time coordinate. Then, the EFE are a system of pdiffs which some metric must satisfy. You can try to see if they have a solution. There may be none, or it may be far from unique. It all depends how good your SET guess was. Of course, solving such a pdiff system is totally non-trivial undertaking.

 

[Sciencemaster]

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)?

 

[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.