I Piecewise Functions in the Einstein Field Equations

[Sciencemaster]

TL;DR Summary

Can piecewise functions be used to find solutions to the Einstein field equations?

Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having components be nonzero on the interval $-a<X_\mu<a$ and 0 everywhere else?) Can piecewise functions be used to create valid solutions to the EFE's in our universe, or is this invalid??

 

[Dale]

Yes, this is absolutely a valid approach. For example the Oppenheimer Snyder collapse solution is modeled by a radial function which is zero outside the dust cloud and non-zero inside.

 

[Sciencemaster]

Alright, sounds good! The piecewise-ness is concerning me a bit though, would the area in the piecewise function still affect the geometry of the area outside?

 

[PeroK]

Alright, sounds good! The piecewise-ness is concerning me a bit though, would the area in the piecewise function still affect the geometry of the area outside?

"Piecewise-ness" is a not a property of a function; it simply refers to how you choose to define the function. All functions may be defined piecewise or not (if you try hard enough).

You can have piecewise polynominals, or piecewise continuous or piecewise differentiable functions. In these cases, the piecewiseness applies to a property of the function and is significant.

But, simply defining a function in a piecewise manner is not significant in terms of functional properties or the lack of any particular functional property.

 

[Ibix]

You require that the metric at the boundary be continuous and sufficiently differentiable for your purposes (you'll need finite second derivatives for the Riemann be finite, at least). So you can see the value and derivative(s) of one at the joints between two pieces as sort of boundary conditions on the other, which is how they affect each other.

 

[George Jones]

Summary:: Can piecewise functions be used to find solutions to the Einstein field equations?

As an analogy, consider electrostatics. Suppose there is a uniform charge density on a spherical surface, and a vacuum inside and outside. What is the electric field inside the spherical surface? Outside? What is an expression for the charge density in terms of 3-dimensional space?

 

[PAllen]

You require that the metric at the boundary be continuous and sufficiently differentiable for your purposes (you'll need finite second derivatives for the Riemann be finite, at least). So you can see the value and derivative(s) of one at the joints between two pieces as sort of boundary conditions on the other, which is how they affect each other.

Actually, there is a well established approach for junctions in GR which only requires continuity. They are called the Israel Junction Conditions. I can’t find a concise reference at the moment, but the following discusses them:

https://arxiv.org/abs/gr-qc/0505048

 

[Ibix]

Actually, there is a well established approach for junctions in GR which only requires continuity.

I didn't know that. It seems surprising to me because I think you'd want the Riemann to be well defined and continuity only guarantees that the first derivatives of the metric are finite everywhere. Presumably either (a) I'm wrong or (b) it comes out in the wash somewhere. I'll have a look at your reference.

 

[PAllen]

I didn't know that. It seems surprising to me because I think you'd want the Riemann to be well defined and continuity only guarantees that the first derivatives of the metric are finite everywhere. Presumably either (a) I'm wrong or (b) it comes out in the wash somewhere. I'll have a look at your reference.

You are not wrong. The conditions allow a discontinuity in curvature. Two common cases where any simple model must tolerate this are the Oppenheimer-Snyder solution @Dale mentioned earlier, and a hollow shell of matter. In the former, you have Schwarzschild outside the dust surface. It is too complicated to require the dust density to smoothly vanish at the surface, so the solution involves a curvature discontinuity at the surface. Similarly, for a matter shell you have discontinuity from shell to schwarzschild outside, and shell to Minkowski space inside.

 

[PeterDonis]

The conditions allow a discontinuity in curvature.

As a clarification, the discontinuous solutions are, I think, best viewed as useful approximations to a more physically realistic but also much more mathematically intractable solution that interpolates continuously between the two patches. This statement...

It is too complicated to require the dust density to smoothly vanish at the surface

...is an example of what I mean.

 

[Ibix]

The conditions allow a discontinuity in curvature.

That I expected (although as you and @PeterDonis note, only in an approximate/idealised case). But it seems to me that continuous metric allows discontinuous first derivatives and hence undefined second derivatives and Riemann tensors. Presumably that doesn't happen (or isn't a problem) for some reason - and that's where I need to read your citation. Too sleepy now, but will look in the morning.

 

[vanhees71]

As a clarification, the discontinuous solutions are, I think, best viewed as useful approximations to a more physically realistic but also much more mathematically intractable solution that interpolates continuously between the two patches. This statement......is an example of what I mean.

This is pretty common in (classical) field theory. E.g., in electrostatics we consider a problem like a conducting charged sphere and just use a boundary condition at its idealized boundary, leading to a jump in the normal component of the electric field. Of course in reality there is not really a sharp surface but a "fuzzy cloud" of charges, but the solution with the idealized conditions is good enough for all practical purposes.

 

[Sciencemaster]

Alright, so the piecewise function would still need to be continuous to get a valid equation, if I'm understanding what you all are saying. For a box floating around in space, would that just mean I should use a 'fuzzy' approximation of a change in the stress energy tensor (i.e. description of matter) representing the edge of the box? Such as a^(cx) where c is very large?
Also, would anything special need to be done at the boundaries to account for there being surfaces or can I just define the inside of the box, the outside of the box, and make the piecewise functions continuous?

 

[Sciencemaster]

Also, I was going to post this in its own thread but since it's related to this question and is set up with the physical system I have already described, I figured i'd post it here. If this isn't the right place for it, please feel free to let me know.

So for an uncharged conducting box as described initially, in space and at rest in its own reference frame, I'd like to try and construct a stress energy tensor that can describe the matter and such in this situation, i.e. the box and the space around it. For clarity, when I say space, I mean the kind of space between galaxies in the real universe, where it's mostly empty, but the cosmological constant is nonzero and there is some small amount of ambient matter.

As for the box itself, the energy density is just going to be mc² if I'm not mistaken, and the energy flux/momentum density will be zero since we are describing it in its own reference frame. However, I'm having trouble figuring out how to make valid shear stress/momentum flux indices, and figuring out what the pressure of a solid in outer space would be (especially considering that it's not expanding by it's construction, it's just sitting stationary in this reference frame). How could I describe/find these indices in such a system. I don't imagine I can just describe a conducting solid as a perfect fluid or anything even with approximating...

 

[pervect]

Summary:: Can piecewise functions be used to find solutions to the Einstein field equations?

Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having components be nonzero on the interval $-a<X_\mu<a$ and 0 everywhere else?) Can piecewise functions be used to create valid solutions to the EFE's in our universe, or is this invalid??

The usual approach would be to use a weak-field approximation. Then cartesian coordinates then make sense in the flat background assumed by the approximation. If you're not imagining a flat background approximation, you might have to say a few more words about what you really want to accomplish, there's not an exact equivalent to Cartesian coordinates in general space-times. Depending on what you want to acacomplish, you might be able to come up with something that does what you want in spite of this lack, though.

If you are thinking of something that is somehow not weak field, though, it's also be best to give a few more details of what you want to do, exactly. I will note that it will probably be hard to find a case where you need anything other than the weak field approximation. You'll also find that large enough boxes with the strengths of materials for matter we know about will pretty much have to be spherical, as materials are simply not strong enough to create a cubical planet or star, for instance.

The case of a large spherical object has of course already been analyzed with appropriate approximations. One of the wrinkles in this sort of analysis that you haven't mentioned will be the "equation of state", which relates the pressure to the density of the material composing the object.

 

[Sciencemaster]

Alright, so I'm actually considering a small box on the stellar scale, the kind we might see in our experience. I'm trying to construct a collection of regions that have different substances in them so that I can see how a path through them is affected by the warping of spacetime, however small it would be. An example of what I'm thinking of is a cube filled with air and the path of a neutrino moving through it, where we may be able to approximate the oxygen as an ideal gas. On this scale however, I imagine we should take into account the ambient matter of the universe, especially if using substances that reach that density.

 

[Dale]

An example of what I'm thinking of is a cube filled with air

Using a sphere will simplify things from the additional symmetry.

 

[Sciencemaster]

Using a sphere will simplify things from the additional symmetry.

I'm sure it will, but it doesn't really satisfy the geometry I had in mind...

 

[Sciencemaster]

At any rate, would I just be able to use a continuous, piecewise geometry to create a boundary (as in, not having to make it continuously differentiable or meet any other prerequisites)? I want to make sure this is something I can actually do.

 

[Dale]

I'm sure it will, but it doesn't really satisfy the geometry I had in mind...

Is non-spherical geometry essential? This is the first time you mentioned it.

Make sure that the non-spherical geometry is truly essential before going down that route. The difference in mathematical complexity is vast. If you don’t have a truly solid reason for a different geometry then you should reconsider. Certainly you have not expressed such a reason in this thread.

At any rate, would I just be able to use a continuous, piecewise geometry to create a boundary (as in, not having to make it continuously differentiable or meet any other prerequisites)? I want to make sure this is something I can actually do.

Has anyone been ambiguous in any way? The answer is, yes. Why do you need it repeated?

 

[Sciencemaster]

Is non-spherical geometry essential? This is the first time you mentioned it.

Make sure that the non-spherical geometry is truly essential before going down that route. The difference in mathematical complexity is vast. If you don’t have a truly solid reason for a different geometry then you should reconsider. Certainly you have not expressed such a reason in this thread.

You're right, I apologize. Yes, a non-spherical geometry is essential (even if it is tedious) for what I currently have in mind. The reason is that I wanted to analyze a particle traveling between two conducting metal plates, which surely seems to require them not to be spherical. This being the case, I'd like to construct a Stress-Energy Tensor describing the plates and go from there. I apologize for the lack of clarity.

 

[PAllen]

Please see post #7. There is more to it if you want physically sensible results. The Israel junction conditions are what you apply to a boundary.

 

[Sciencemaster]

Ah, okay.
So for a (perfectly) conducting surface, are there known values or equations for the Stress Energy Tensor indices in question (pressure and Shear Stress/Momentum Flux)? Perhaps just a close approximation?

 

[PAllen]

Ah, okay.
So for a (perfectly) conducting surface, are there known values or equations for the Stress Energy Tensor indices in question (pressure and Shear Stress/Momentum Flux)? Perhaps just a close approximation?

Presumably someone has done something like this, but I don't know the answer or a reference for it.

 

[Sciencemaster]

That's fine, I don't know either, of course. Thanks for your help though, I really appreciate it! Does anyone know what values we could use for these indices?

 

[pervect]

Well, the electric field inside the conductor is going to be zero. The field at the surface of the conductor, where the charges are, can be found by Gauss' law. The force on the surface charge will be the electric field at the surface multiplied by the charge, so the force density will be the charge density times the electric field. This will put the conductor under stress.

You'd need some material properties of the conductor to solve the for the distribution of stress. You might do using a non GR approach, but you could also do it using the equation $\nabla_a T^{ab} = 0$. In general I would expect the division of the stress terms between radial tension and circumfrential tension would depend on the details of the material properties of the conductor, though I haven't attempted any serious analysis.

Weak field approximations would ignore the stress terms anyway, so there's no real reason to calculate them if you're doing a weak field analysis. But it's still not clear if that's what you're doing.

As I mentioned before, known materials can probably not support gravitationally significant tensions and pressures, but for such a case, I see no real need to use GR to analyze your problem. Newtonian physics should be good enough.

You'll also have to consider the exterior electric field as a source of the stress energy tensor, and hence gravitation. This will be a small effect, but it might be a reason to do a full GR analysis. Wiki has the formula for the stress-energy tensor of an electrromagnetic field , see https://en.wikipedia.org/w/index.php?title=Electromagnetic_stress–energy_tensor&oldid=1028734167. But it'll probably take some specialized knowledge to solve it.

 

[Sciencemaster]

I know that the gravitational effects probably won't be very significant in this problem (unless we choose to make the materials either very large or very massive), but I'd like to calculate them anyway even if they are very small. This being the case, I don't feel a weak-field approximation would be quite adequate.

If the material within the solid is an uncharged conductor, then I imagine the electromagnetic stress energy tensor goes to zero from inside, leaving just the description of the material itself. Since this is floating around in outer space, one could consider background radiation, however that's probably VERY negligible and so let's just say that the plate is uncharged, and as a result the electromagnetic stress energy tensor is 0 (I imagine). The bigger problem here is finding the stress energy tensor of an uncharged conductor.

 

[Sciencemaster]

Actually, there is a well established approach for junctions in GR which only requires continuity. They are called the Israel Junction Conditions. I can’t find a concise reference at the moment, but the following discusses them:

https://arxiv.org/abs/gr-qc/0505048

So in order to create a metric out of two different regions, say, inside of the box and outside of the box, you have to make sure that K_+ij (inside of box)-K_-ij(outside of box) equals $$8\pi\cdot f(T_ij)$$, where K_+ij is a piecewise metric and T_ij?

Essentially, this is just a constraint on the metric at the boundaries of the piecewise stress energy tensor?

I could be (and probably am) very wrong but that's what the paper (and some other sources) seemed to imply. Is this even remotely correct?

 

[Dale]

The reason is that I wanted to analyze a particle traveling between two conducting metal plates, which surely seems to require them not to be spherical.

Why not? You don’t think particles can travel between spherical shells? That doesn’t make sense to me

 

[Sciencemaster]

Why not? You don’t think particles can travel between spherical shells? That doesn’t make sense to me

Well I believe they can, but I believe that it would be more akin to what happens in laboratories, and it would be more like, well, particles moving between plates, rather than just motion between planets but on a very small scale. I imagine they are at least a tad different, metric-wise. Basically, I'd like to try something different.

 

[Sciencemaster]

Sorry, that last message was rather confusing. What I meant to say was that a particle moving between two spheres seems very similar to its motion between two planets but on a much smaller scale. This is all well and good, but I'm more curious about the metric of a particle moving between two plates (in terms of the geodesics) since it is more akin to what one could visualize in a laboratory environment, and it isn't as commonly solved for.

 

[Dale]

it isn't as commonly solved for.

Yes, because the math is much more difficult. Anyway, good luck. I think you are “biting off more than you can chew”.

 

[Sciencemaster]

Yes, because the math is much more difficult. Anyway, good luck. I think you are “biting off more than you can chew”.

I think you might very well be right, but i'd like to at least attempt it regardless.

 

[Sciencemaster]

At any rate, am I thinking of the Israel Boundary Conditions correctly? Is there anything else special I have to do if I have a piecewise Stress-Energy Tensor (compared to one that is a single function)? Is there an easier way of dealing with the piecewise-ness?

 

[PAllen]

So in order to create a metric out of two different regions, say, inside of the box and outside of the box, you have to make sure that K_+ij (inside of box)-K_-ij(outside of box) equals $$8\pi\cdot f(T_ij)$$, where K_+ij is a piecewise metric and T_ij?

Essentially, this is just a constraint on the metric at the boundaries of the piecewise stress energy tensor?

I could be (and probably am) very wrong but that's what the paper (and some other sources) seemed to imply. Is this even remotely correct?

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 intrinsic 3-geometry is implied on the 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 the 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.

 

[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)?