Energy conservation and wormholes

[PWiz]

Okay, so I've had this question on wormholes which seems to have hijacked my mind from some time now, and what better place to bring it out rather than PF?

A previous thread containing lots of lengthy posts was made here a few years ago, but I'm not getting a clear-cut answer despite leafing through it. (Here is the thread: https://www.physicsforums.com/threads/wormholes-a-way-to-violate-energy-conservation.523089/ )

I'll present my doubt - how does a static wormhole connecting 2 points in space which are at different potentials (i.e. with non-identical metric tensor components at those 2 points) along a geodesic path in curved spacetime not violate local energy conservation?

Thanks for reading!

 

[A.T.]

how does a static wormhole connecting 2 points in space which are at different potentials (i.e. with non-identical metric tensor components at those 2 points) along a geodesic path in curved spacetime not violate local energy conservation?

The metric of the wormhole has to match smoothly with the metrics at the entry areas. So any difference in potential (gravitational time dilation) must also happen along the worm hole path.

 

[PWiz]

That's new for me. Does this mean that a proportion of the kinetic energy that an in-falling object on a geodesic might gain (with respect to an observer hovering at constant radius R from the gravitating mass) will be lost to "place" the object in the region of lower potential (at the other end of the wormhole)? So wormholes which lead to regions of lower gravitational potential offer "repulsive forces" which drain the corresponding amount of energy from the object to prevent violation of CoE?

 

[PeterDonis]

how does a static wormhole connecting 2 points in space which are at different potentials

Different potentials with respect to what? Your description here seems incomplete.

Does this mean that a proportion of the kinetic energy that an in-falling object on a geodesic might gain (with respect to an observer hovering at constant radius R from the gravitating mass) will be lost to "place" the object in the region of lower potential (at the other end of the wormhole)?

Kinetic energy relative to what?

You appear to be confusing yourself by focusing on things that are frame-dependent instead of things that are invariant.

 

[PWiz]

Different potentials with respect to what? Your description here seems incomplete.

Different gravitational potentials; I'm talking about the metric tensor (different points in the tensorial field)

Kinetic energy relative to what?

The static wormhole/ (a static) observer.

 

[PeterDonis]

Different gravitational potentials

Once again: different gravitational potentials relative to what? Having read through the other thread now, I'm still not clear on which of two possible scenarios you are asking about:

(1) A large gravitating body, and a wormhole placed in the vacuum region outside that mass such that the two mouths are at different altitudes;

(2) Two large gravitating bodies with different masses, and a wormhole placed with one mouth outside the first body and the other mouth outside the second.

In scenario #1, if we ignore the wormhole, "gravitational potential" has a well-defined meaning, as a function of altitude (here I am assuming the simplest case of a spherically symmetric, non-rotating body). But we can't ignore the wormhole: the wormhole has to also have mass and gravity, and what's more, it has to be threaded with exotic matter in order to hold it open. So we can't just consider the wormhole as a "test body" that doesn't affect the spacetime geometry; we can't even ignore effects of the wormhole on the spacetime geometry other than what it does to the topology (making it multiply connected). Basically, this scenario is not a one-body problem; it's a two-body problem, and it won't be static, so the concept of "gravitational potential" doesn't have a well-defined meaning.

In scenario #2, we already have a two-body problem even if we ignore the wormhole; but even if we finesse that by saying that the two large bodies are so far apart that we can ignore their effects on each other's motion, we still have all the issues of scenario #1 above, and the concept of "gravitational potential" won't have a well-defined meaning.

 

[PeterDonis]

how does a static wormhole connecting 2 points in space which are at different potentials (i.e. with non-identical metric tensor components at those 2 points) along a geodesic path in curved spacetime not violate local energy conservation?

To comment on this question directly: if the spacetime in question is a valid solution of the Einstein Field Equation, it can't violate local energy conservation. Local energy conservation is an identity (the Bianchi identity) for any valid solution of the EFE.

The interesting question here, to me, is whether there actually are any valid solutions of the EFE that can be described as either scenario #1 or scenario #2 in my previous post.

 

[pervect]

My position hasn't changed much since then. I believe the definitive reference will be in Matt Visser's "Lorentzian Wormholes", which I don't have. I gather it's out of print, too. I do recall George making some posts about this topic baed on Visser, but I didn't locate them.

Basically I believe that the differential conservation laws will all be satisfied, and that you can define a multi-valued potential function for the energy of the body. Usually, Stokes theorem says that the path integral of the gradient of a potential around a closed curve will be zero, but it only applies when the domain is simply connected, which the wormhole is not - it's multiply connected.

 

[PeterDonis]

the differential conservation laws will all be satisfied

Yes, this has to be true for any valid solution of the EFE.

you can define a multi-valued potential function for the energy of the body

Multi-valued in the sense that the "potential energy" is path dependent?

 

[pervect]

Yes, this has to be true for any valid solution of the EFE.
Multi-valued in the sense that the "potential energy" is path dependent?

Yes, basically if you have a particle traversing a close path through the wormhole and back around through normal space, it can gain or loose kinetic energy by making a circuit compared to one that "stays put". The total ADM energy (assuming asymptotic flatness) of the wormhole + particle system won't change. You can regard this as the circulating particle transferring energy to and from the wormhole geometry, you get the usual issues if you try too hard to localize exactly where the energy goes. As I recall, though you could assign energy to each wormhole mouth by taking the ADM energy in a large enough sphere encompassing each of the wormhole mouths, so you can assign the energy to three bins - one for the particle, and one for each wormhole mouth.

There aren't any suden "jumps" in the potential energy of the particle, it's continuous everywhere so that the gradient of the potential exists, but if you make a complete path, you wind up on a different branch of the potential vs position surface.

The wormhole geometry will change as the mouth masses change - though if the particle is small enough (a test particle) you can say that this effect is small. I tend to think of it as a flux through the wormhole, but that may be taking one short-cut too many. The flux idea is more rigorously defined if you consider a charge circulating through the wormhole rather than a mass.

 

[PWiz]

I'm still not clear on which of two possible scenarios you are asking about

Sorry about not being clear, Peter; I was talking about scenario 1.

(here I am assuming the simplest case of a spherically symmetric, non-rotating body)

Yes, I should've mentioned that I was talking about a vacuum solution of a gravitating mass that is spherically symmetrical with no angular momentum and net charge (basically a solution that is described by Birkhoff's theorem, except that I'm talking about a Schwarzschild solution).

But we can't ignore the wormhole: the wormhole has to also have mass and gravity, and what's more, it has to be threaded with exotic matter in order to hold it open.

This was going to be my follow up question since I had originally talked about a static wormhole, but you guessed it already

To comment on this question directly: if the spacetime in question is a valid solution of the Einstein Field Equation, it can't violate local energy conservation. Local energy conservation is an identity (the Bianchi identity) for any valid solution of the EFE.

The interesting question here, to me, is whether there actually are any valid solutions of the EFE that can be described as either scenario #1 or scenario #2 in my previous post.

Since this is a 2 body problem either way, I'm guessing an exact EFE solution is not going to exist. I want to know whether the presence of exotic matter which holds the wormhole open will allow an (non-exact ) EFE solution to exist. If we were talking about normal matter, I wouldn't give this a second thought (since a solution [exact or inexact] can always be obtained using computers in these cases), but I'm not certain how spacetime behaves in the presence of a wormhole. So will a solution which can be obtained by numerical analysis exist for these scenarios? If not, will the covariant divergence of the stress-energy tensor $∇^μ T_{μν}$ in general vanish (locally) if we talk about the closed path between the "mouths" of the wormhole in particular? Will Noether's theorem apply?

 

[PWiz]

The total ADM energy (assuming asymptotic flatness) of the wormhole + particle system won't change.

Does this imply that EFE solutions for these scenarios exist?

 

[PeterDonis]

Since this is a 2 body problem either way, I'm guessing an exact EFE solution is not going to exist.

If by "exact" you mean "expressible in closed form", I think this is correct. However, one can always solve the EFE numerically; this is what is done in other such cases, for example, binary pulsars.

I want to know whether the presence of exotic matter which holds the wormhole open will allow an (non-exact ) EFE solution to exist.

If by "non-exact" you mean "numerical" (which it looks like you do from your statement further on in your post), I don't see why not. The computer doesn't care whether the stress-energy tensor you're using satisfies the energy conditions or not.

 

[PWiz]

If by "exact" you mean "expressible in closed form", I think this is correct. However, one can always solve the EFE numerically; this is what is done in other such cases, for example, binary pulsars.
If by "non-exact" you mean "numerical" (which it looks like you do from your statement further on in your post), I don't see why not. The computer doesn't care whether the stress-energy tensor you're using satisfies the energy conditions or not.

Yes, by "exact" I meant expressible in closed forms / finite mathematical operations.

My question is if we use coordinate systems where the wormhole is described as a spacelike hypersurface, will we get a vanishing covariant divergence for the stress-energy tensor (after integration over the hypersurface of course)?

 

[pervect]

Does this imply that EFE solutions for these scenarios exist?

I believe they should, though I've never seen anything in the literature specifically about the exact scenario I'mgoing to propose, so it's unfortunately possible I'm missing something :(. My approach uses multiple metrics rather than a single metrics, metrics that are joined together by a process that I'll call "cutting and pasting".

My basic approach for constructing this solution to the EFE's would be as follows: Start with two solutions to the EFE's. One solution is a wormhole solution that connects a flat space-time to another flat space-time. The second solution is a massive hollow sphere solution, one that has flat space-time inside and asymptotically flat space-time outside. Join the two solutions together by cutting and pasting, so that the wormhole solution joins the flat interior region of the hollow sphere to the asymptotically flat outside region of the massive hollow sphere, defining a 4-d multiply connected space-time that's rather similar to a 3d torus

"Cutting and pasting" isn't something I've really done, but there's a section in MTW as to how to do it, where the authors join an exterior Schwarzschild solution to a flat interior solution, creating a thin-shelled massive sphere solution. The process was confusing enough that I never quite followed all the details even with the text to follow, but certain "junction conditions" must be satisfied for the process to give valid results. I wound up preferring to deal with a thick-shell massive sphere solution in preference to the thin-shelled solution in MTW.

Note that there isn't a really a single coordinate chart in my approach for this construction - rather, we attempt to join together different charts together, so we're covering the space-time with multiple charts.

I also need to assume that the asymptotically flat region can be successfully cut and pasted as it it were truly flat. I don't think this is an issue, but I can't guarantee anything.

 

[PeterDonis]

if we use coordinate systems where the wormhole is described as a spacelike hypersurface

Whether the wormhole is described by a spacelike hypersurface can't depend on your choice of coordinates. More precisely, whether or not there are spacelike hypersurfaces in the spacetime which have a multiply connected "wormhole" topology can't depend on your choice of coordinates. The only thing that can depend on your choice of coordinates is whether such hypersurfaces, if they exist, are surfaces of constant coordinate time. But no actual physics depends on that.

will we get a vanishing covariant divergence for the stress-energy tensor (after integration over the hypersurface of course)?

The covariant divergence is local, not global, and the identity that the EFE satisfies that makes the covariant divergence zero is also local, not global. The condition does not involve integrating over anything. It only says the local version must hold at every event in the spacetime.

Integrals over particular hypersurfaces can give quantities like the ADM energy of the spacetime, but what is integrated is not the covariant divergence of the stress-energy tensor.

 

[PeterDonis]

One solution is a wormhole solution that connects a flat space-time to another flat space-time.

Wormhole solutions don't connect two flat regions; they connect two asymptotically flat regions. They can only be asymptotically flat because the wormhole has mass.

The second solution is a massive hollow sphere solution, one that has flat space-time inside and asymptotically flat space-time outside.

This won't work as you state it because the wormhole itself has mass, so once you put it into the interior of the massive hollow sphere, spacetime inside the sphere will no longer be flat. But you should be able to match conditions at the inner edge of the hollow sphere similarly to how you would do it if you put a normal massive body inside the sphere.

Note that there isn't a really a single coordinate chart in my approach for this construction - rather, we attempt to join together different charts together, so we're covering the space-time with multiple charts.

Not necessarily; multiple charts are not required to do the "cutting and pasting" process. (Coordinate charts are just a convenience anyway; the cutting and pasting can, in principle, be described entirely in terms of invariants.) For example, in the original Oppenheimer-Snyder paper where they described their solution for a spherically symmetric collapse of dust to a black hole, IIRC they used the same chart for both the dust region and the vacuum region of the solution.

I also need to assume that the asymptotically flat region can be successfully cut and pasted as it it were truly flat.

No, you don't; in fact I don't think you want to "cut and paste" into a truly flat region, because that would require matching "truly flat" to "only approximately flat", which AFAIK can't be done. AFAIK you can only match "approximately flat" to "approximately flat" by choosing the matching surface appropriately so the junction conditions are met. The junction conditions involve quantities which depend on the spacetime curvature, so there's no way to match "exactly flat" to something that isn't exactly flat.

The main potential issue I see with this scheme is that the wormhole breaks spherical symmetry, because the mouth that exits in the "outer" asymptotically flat region has to be in some particular direction from the hollow sphere. I don't think that's a showstopper, but it complicates things.

 

[PWiz]

@PeterDonis Thanks for clarifying that. I read the "integrate over a hypesurface" part somewhere, and the integration of divergence over a region didn't make a whole lot of sense back then either!
So (just confirming) a spacelike hypersurface is an invariant (coordinate-independent) topological feature, right? (Put another way, do sets of points which have spacelike separation for constant coordinate time appear to have the same relation regardless of where these points are observed from?)

@pervect This 'cut and paste' method seems a little strange to me. Won't it introduce discontinuities in some regions of spacetime? I mean what if the regions where the wormhole mouths are placed have lots of tensorial deviations - how will this method "clean" up? If we end up with some ill-defined points (coordinate singularities) in the overlap regions of the different coordinate charts, differentiating / integrating any quantity over those regions will give weird results! Can please link the section where this process is explained? (Hopefully it won't go over my head lol)

 

[PeterDonis]

a spacelike hypersurface is an invariant (coordinate-independent) topological feature, right?

It's an invariant feature, but not a topological one, because the "spacelike" relationship requires the metric, and the metric is not a topological feature; it's additional structure over and above the topology.

sets of points which have spacelike separation for constant coordinate time

Spacelike separation doesn't depend on your choice of coordinates; the "for constant coordinate time" part is not necessary (and indeed it's not strictly correct, since there will always be coordinate charts in which any set of spacelike separated points do not all share the same coordinate time).

appear to have the same relation regardless of where these points are observed from?

If by this you mean "appear to be spacelike separated", then yes, obviously this must be true since spacelike separation is an invariant relation.

This 'cut and paste' method seems a little strange to me. Won't it introduce discontinuities in some regions of spacetime?

Not if it's done correctly; the whole point of the junction conditions that pervect mentioned is to ensure that no discontinuities are introduced.

 

[PWiz]

Not if it's done correctly; the whole point of the junction conditions that pervect mentioned is to ensure that no discontinuities are introduced.

But how can this be done mathematically? Sorry if this sounds stupid, but the approach just seems so abstract to me. Can one use "cut and paste" to describe approximate closed form EFE solutions for multiple body problems in GR as well?

 

[PeterDonis]

how can this be done mathematically?

You match the appropriate quantities at the junction. For a simple example, consider a hollow spherical shell with vacuum inside and outside (with no wormhole). If we adopt the usual Schwarzschild coordinates, then the junction conditions amount to matching the metric coefficients and their (IIRC) radial derivatives at the outer and inner surfaces of the shell, for the two solutions at each boundary (the vacuum solution and the solution using the stress-energy tensor of the shell).

Can one use "cut and paste" to describe approximate closed form EFE solutions for multiple body problems in GR as well?

In principle this should be possible, at least for certain problems, but I don't know if it's actually been done. The work I'm aware of on multi-body problems in GR involves numerical solutions of the EFE.

 

[PWiz]

Okay, the theory part seems a little more digestible now, but I would nonetheless love to see the math part of this method [if pervect would be kind enough to link it :) ].

 

[pervect]

Some further thoughts, though still short of answers:

Starting with the "cut and paste idea", we take two flat planes, cut a circle out of them, and join them by a tube. This inspires the metric:

$dr^2 + r^2 d\phi^2 - dt^2$ (r>=1 or r<= -1, $0 <= \phi < 2 \pi$)
$dr^2 + d\phi^2 - dt^2$ (-1 <= r <= 1)

Each piece of the solution has no curvature, so it should be a vacuum solution of Einstein's equations.

Then we need to ask if the junctions conditions at r=1 and r=-1 are OK. Alternately, we work around the need for doing this by defining a function f(r) which behaves as r^2 for |r|>>1, approaches a constant for |r|<1, and is differentiable at least to second order. In this case, though, we'll probably wind up with a non-vacuum solution of Einstein's equations. One that probably doesn't satisfy any of the usual energy conditions.

Somewhere along the line we also need to add in $\theta$ to make it a 4-d solution (3space + time, rather than 2space + time).

 

[PeterDonis]

we take two flat planes, cut a circle out of them, and join them by a tube

I assume that this describes a spacelike surface of constant time, correct?

This inspires the metric:

$$
dr^2 + r^2 d\phi^2 - dt^2 (r>=1 or r<= -1, 0<=ϕ<2π)

dr^2 + d\phi^2 - dt^2 (-1 <= r <= 1)
$$

Just to make sure I understand: the first part of this describes the metric on the two flat planes, correct? And the second part describes the metric on the tube, correct? And you are defining the $r$ coordinate so that it is $\pm 1$ at the junctions of the tube with the two planes, and $0$ halfway along the tube, correct?

Each piece of the solution has no curvature, so it should be a vacuum solution of Einstein's equations.

As you've written it, yes. The flat planes are just portions of Minkowski spacetime. The tube is Minkowski spacetime with a non-trivial topology.

Then we need to ask if the junction conditions at r=1 and r=-1 are OK.

I'm not sure they are, because I don't think the derivative of $g_{\phi \phi}$ is continuous at those values of $r$.

 

[pervect]

I assume that this describes a spacelike surface of constant time, correct?
Just to make sure I understand: the first part of this describes the metric on the two flat planes, correct? And the second part describes the metric on the tube, correct? And you are defining the $r$ coordinate so that it is $\pm 1$ at the junctions of the tube with the two planes, and $0$ halfway along the tube, correct?

Yes, that's it.

As you've written it, yes. The flat planes are just portions of Minkowski spacetime. The tube is Minkowski spacetime with a non-trivial topology.
I'm not sure they are, because I don't think the derivative of $g_{\phi \phi}$ is continuous at those values of $r$.

That is the part that needs work. However, discontinuities in the metric aren't necessarily fatal. Consider the hollow sphere solution. As I recall (see for instance https://www.physicsforums.com/threa...ffected-by-gravity.623866/page-4#post-4022914) see even more severe discontinuities when gluing an interior flat solution to an exterior Schwarzschild solution via a thin but massive shell.This is well short of proving that the idea has no problems, of course. Basically, I wanted to get a little bit away from the handwaving approach I was using, to something that was well defined enough to be tested, even though I haven't yet done the necessary analysis.

 

[PeterDonis]

discontinuities in the metric aren't necessarily fatal

They aren't in highly idealized scenarios like infinitely thin massive shells (see below). But in any real scenario, where anything with stress-energy has a finite thickness and changes in SET components are continuous, there shouldn't be any discontinuities in the metric, or in certain first derivatives of it (see below). Note that I specifically talked about the derivative of $g_{\phi \phi}$, not the coefficient itself.

see even more severe discontinuities when gluing an interior flat solution to an exterior Schwarzschild solution via a thin but massive shell.

Only in the highly idealized case where the shell is infinitely thin. For a shell of finite thickness, the metric coefficients will be continuous everywhere. For an idealized case where the stress-energy tensor components undergo a step discontinuity at the shell boundaries, the radial derivatives of the metric coefficients will not be continuous there; but again, in a real case, the SET components will not have discontinuities, although they may change in a very rapid but continuous way at boundaries, so the derivatives of the metric coefficients won't be discontinuous either.

In your specific scenario, you're basically asking if there is any way of concocting a stress-energy tensor that will yield the geometry you've specified. I'm not sure, because I'm not sure where the junction conditions allow discontinuities, even in highly idealized cases. It may be the same sort of thing as the idealized infinitely thin massive shell; I'm just not sure without actually reviewing a textbook like MTW to see the explicit form of the junction conditions.