# Derivation of Wormhole Mouth Mass/Charge?

1. Jun 15, 2009

### Julian M

It is said that the mouths of a wormhole can have mass or charge; I can't follow the arguments used to support this conclusion and would be very grateful if anyone could help clarify the issue for me. In fact, not to put too fine a point on it, this question is driving me nuts: the conclusion is apparently uncontroversial, but the argument(s) seem absurd, so clearly the fundamental problem is my ignorance of something essential. What am I missing?

Apologies if this seems an over-long enquiry, though I may have failed miserably I have striven for clarity and conciseness.

What should be the technical level of the answer? The basic GR concepts are understood, but as for the math... I'm happy with manifolds, metrics, etc. but My tensor math is decrepit; I can however probably follow the steps of a mathematical exposition provided the steps are small enough. On the other hand, if you start talking about homologies etc. my eyes will probably just glaze over.

I've used Matt Visser's book "Lorentzian Wormholes – from Einstein to Hawking" (Springer Verlag, New York Inc.; 1996) ["MV"] as a convenient source for a specific argument, but I've also referenced some other papers/books as follows

MTY = M. S. Morris, K.S. Thorne, U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition (Phys. Rev. Lett. 61(13), 1446-1449; 1988) available online at http://authors.library.caltech.edu/9262/
RP = Roger Penrose The Road to Reality (Jonathan Cape, London; 2004)
KT = Kip Thorne, Black Holes and Time Warps – Einstein’s Outrageous Legacy, (Norton, [n. p.]; 1994)
LB = Leo Brewin, A simple expression for the ADM mass (Gen. Rel. Grav. 39(2007) pp.521-528) available online at http://users.monash.edu.au/~leo/research/papers/files/lcb07-01.pdf

So, here goes.

I am most particularly interested in the mass/charge of the mouths of traversable wormholes, i.e. wormholes without horizons.

MV argues [MV pp111-113] for the existence of mass by reference to ADM Mass, beginning with a wormhole between two universes and then extending the argument to an intra-universe wormhole as follows.

Consider the ADM mass of two separate universe; ADM mass is defined with reference to the asymptotically flat region where spacetime has become "sufficiently" flat; ADM mass is conserved. Join the two universes with a wormhole. "There are now two asymptotically flat regions and thus two ADM masses" [MV, p111]

Why are there now two regions? I can only see one: from any point in either "universe" one can reach any point in either universe (including infinity). Against that however, I can also see that within the wormhole throat there is a unique (circumferential) direction in which motion gets you nowhere (so to speak) - as a result of which it would seem that there is no asymptotic flatness in this direction. Is it the existence of this direction that leads to the existence of two distinct regions? (If the resultant manifold is no longer asymptotically flat in all directions, presumably the ADM Mass cannot be defined for the connected universes as a whole.)

But - if it were the existence of such a special geodesic direction that created a region boundary, it should apply in other contexts e.g. around any mass, and since extra regions are not apparently created around "ordinary" masses this argument would seem to be undermined.

[Side thought: suppose we consider the region of the wormhole in which the geodesics are closed - can we excise this region from the joined universes? Whilst each universe is asymptotically flat, it would seem that the curvature of the wormhole mouth also extends "to infinity" (being everywhere non-zero except "at infinity" itself, so it does not seem obvious that one could excise the offending region and actually have anything left. But even if one could, would the remaining region still satisfy the requirements for having a defined ADM Mass? This might be what LB was doing (in another context) with reference to the Weyl-Lewy embedding theorem, but I'm not actually wiser yet. NB I haven't read LB thoroughly - I only just stumbled across it when looking for a good definition of ADM Mass.]

MV also [p112] shows a Penrose (conformal) diagram of a Morris-Thorne wormhole which contains "two asymptotically flat regions" either side of a dashed line representing the wormhole throat; according to Penrose [RP, p725] such dashed lines represents an axis of symmetry, so whilst I can't disagree about the symmetry I can't see why the existence of the symmetry would create two regions [c.f. RP p725 Fig. 27.16 (b)]. Furthermore, Penrose [RP, p833 Fig 30.10 (b)] illustrates such a hypothetical wormhole (held open by negative energy) without the axis of symmetry... and where it seems that there is only one region.

Moving beyond the inter-universe case MV extends the argument to intra-universe wormholes with the additional qualifications [MV, p111]

i) The two wormhole mouths are sufficiently far apart that their mutual gravitational interaction can be neglected
ii) the initial and final positions of an object traversing the wormhole are sufficiently far away from both mouths

i) seems reasonable at first if the reasoning requires a flat background, but why should the argument only work in flat space? ii) probably ditto, but if infinity is always flat, what does it matter what happens in a finite region around the wormhole mouths etc.?

However, most fundamentally, since one has by definition only one ADM mass for a single universe how can the inter-universe argument be extended to the intra-universe case, unless it is again by virtue of the unique the circumferential direction, which now creates two regions instead of merely maintaining them?

With regard to charge, MV says "One can phrase the argument either in terms of the imprint at infinity or in terms of a more physical picture using flux lines" [MV, p113].

However, according to the flux line picture it would seem that charge on a wormhole mouth is an illusion. Speaking in the context of Wheeler's "charge without charge" via trapped flux lines, MTW note that an observer having constructed a "boundary" around a region that appears to be the "seat of charge" may "incorrectly apply the theorem of Gauss… it isn't a boundary" [MTW, p1200] - because there isn't an "inside", the nominal interior can be reached by another path within the same universe. Furthermore, in the absence of horizons there is nowhere for any apparent charge to reside.

Your patience and understanding are much appreciated...

Yours, confused and ignorant,

Julian

2. Jun 18, 2009

### George Jones

Staff Emeritus
There are two regions because the asymptotically flat part of spacetime is a disconnected set that consists of two path-connected components.
But any such path passes through a region of spacetime that is not asymptotically flat.

3. Jun 18, 2009

### Julian M

Many, many thanks George... I feel I'm making (slow) progress but if you can clarify a little further that would be great. Please forgive me for apparent quibblings in places... what brain there is not taken up with thoughts of hunny I have to try and keep tidy. We're on the right track and I hope you are willing to bear with me... I also hope you can forgive the inevitable mathematical errors (and a bit of rambling...)
Quoting Wikipedia (http://en.wikipedia.org/wiki/Connected_space" [Broken], the redirect from "Disconnected Set")

If I have understood you correctly, I would rephrase your statement as:

"There are two regions because the asymptotically flat path-connected parts of spacetime are disconnected." i.e. (hence confusion) that the parts are disjoint. (is that fair?)

Disjoint sets = empty intersection; in our context re connectedness a space is disconnected if it is the union of at least two disjoint non-empty open sets. It seems to me that mathematically I could carve (any?) spacetime into three parts: two open and disjoint sets abutting a single closed set, but where is the physical justification for any particular decomposition? If I were to carve up a universe in such a way total mass is still conserved and I could say that as an object moved from one region to another the mass of one decreased and the other increased but the mass won't act as though it is located at the boundary (he asserted hesitantly).

In the case under consideration I agree that the union of the two universes is originally disconnected, and that the union remains technically disconnected even after the addition of a third closed set (the wormhole) if the union is defined only wrt to those two parts. (considering a spherical wormhole mouth from the perspective its boundary, spacetime outside is an open set; spacetime inside is an open set; the boundary is a closed set). However, the union of all three still seems connected to me.

i.e. unfortunately I still don't see where/how the disconnection arises.
In other words, to be asymptotically flat one must be able to reach flatness in all directions not merely from any point? But what constitutes a valid "direction" - I can't see that it could be other than a geodesic, in which case doesn't the original objection apply? In the presence of any (spherically symmetric) massive object there are closed geodesics, so any "non-empty" spacetime (by which I mean a horizonless spacetime containing masses but not e.g. a uniform Van Stockum dust or a fluid filled spacetime) would contain at least one such direction and therefore be disjoint... wouldn't it?

In the presence of a horizon I could be persuaded there are physically reasonable disjoint sets (from outside the horizon some geodesics fall in and reach the singularity rather than flatness, and from inside none reach out) but traversable wormholes don't have horizons.

Er... </ramble> Over to you?

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4. Jun 19, 2009

### Julian M

Aha. A search for the definition of asymptotic flatness turned up something useful in General Relativity by Malcolm Ludvigsen (http://books.google.co.uk/books?id=...-kK&dq=define "Asymptotic flatness"&pg=PA115", see also attached quote JPG; copyrighted material, minimal quote) where he says that a spacetime is flat if the metric tends to Minkowskian along any spacelike or null direction. (He does however go on to say that the geometric definition of asymptotic flatness is "a whole branch of relativity in itself"... this is seriously non-trivial.)

However, keeping it simple for the moment, in the case under discussion, the wormhole joined universe(s) (i.e. universe(s) + wormhole) is/are not asymptotically flat because nowhere along the null direction around the wormhole throat (i.e. the path of a light ray that circulates around the throat) is spacetime Minkowskian.

It seems to me therefore that this suggests not that ADM Mass is separately conserved in the universe(s) to either side the wormhole but that the ADM Mass cannot be defined for this topology.

Whilst it might seem that each universe is separately asymptotically flat, to reach some null infinities one must pass through the wormhole. If there are two universes one might say either

A) that each "is" a universe - in which case along certain null directions there is no infinity (??) because the other end is not in the same universe and therefore that the ADM Mass cannot be defined, or

B) that the joined universes (less wormhole) taken together constitute a spacetime - in which case all null directions reach infinity (even those that pass through the wormhole) and the ADM Mass is defined for the union of the two, which seems to lead to a contradiction when we consider a mass traversing the wormhole: while the mass is within the wormhole it is not in the ADM Mass conserving spacetime at all, so somehow that spacetime has lost mass whilst at the same time conserving it; this would tend to suggest that the ADM Mass is not definable for the union of two spacetimes that are not joined by a mass conserving (at least ADM Mass) region

In case A, without ADM Mass mass can't be ascribed to a wormhole mouth because it isn't defined for either universe. And in case B, ADM Mass can't be ascribed to a wormhole mouth because it isn't defined for the union of the universes.

Having already dismissed the case of U+U+W above, it would seem that there isn't a meaningful ADM Mass for this topology (however, with nowhere to go common sense - which may be unreliable - suggests that U+U+W should conserve mass... which leads to the question: what is the appropriate equivalent of ADM Mass for this topology?)

The only downside to all this is that Visser et al are perfectly happy with ADM Mass in this context...

How do these arguments stack up with you... how do I reconcile myself to the POV of Visser et al?

#### Attached Files:

• ###### Asymptotic Flatness Defined (in 'General Relativity', by Malcolm Ludvigsen).JPG
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5. Jun 19, 2009

### George Jones

Staff Emeritus
I agree with Visser that a wormhole spacetime can have two asymptotically flat regions, that each region separately defines an ADM mass, and that Figure 11.3 is a Penrose diagram for a wormhole spacetime. Over the last few days I have been looking at Ludvigsen (I got my copy shortly after it was published) and several other books that are on my shelf.

Ludvigsen is an interesting character; see

Plan of action:

1) write down a specific wormhole metric;

2) use metric and (something like) Ludvigsen's preliminary definition definition of asymptotic flatness to find the two asymptotically flat regions;

3) use metric and (something like) equation (11.64) in Visser to find the ADM masses.

Even if you don't find my arguments convincing, it will be good for me to write them down.

First, though, I want to get to at least a couple of other threads. Also, I will be somewhat distracted by the Twenty20 semi-final that just started, and my weekends are taken up by family activities, so this likely won't happen until sometime next week.

6. Jun 19, 2009

### Julian M

George - thanks for making me think about the right things.

Having posted a short while ago, on the way home I suddenly thought, "I've answered my own question...!"

That having been said (I'll think it through more carefully and post longer later) I have to disagree with the statement that each universe has separately defined ADM mass: the joined spacetimes (less wormhole) do have a defined ADM Mass and so as a mass enters the wormhole it leaves an imprint behind ("the mass of the wormhole mouth"), and when it emerges from the other mouth it must imbue the exit mouth with the equivalent negative mass (but now I'm worrying about conservation of 4-momentum...)

I am however still uncomfortable with the situation; despite this making sense of the argument and apparently correctly applying the definition of ADM Mass, I can't help wondering whether mathematically the ADM definition is applicable to disjoint spacetimes. Would love to know there's an accepted proof of that!

More later. Thanks

Regards

Julian

7. Jun 19, 2009

### Julian M

No.... I retract... sort of.

I think I now understand why Visser et al say - by application of the ADM Mass definition - that wormhole mouths acquire/lose mass as mass passes through the wormhole.

However, attending carefully to the definition of asymptotic flatness in terms of null lines required for the ADM Mass to be defined there is only one physically justifiable decomposition of wormhole joined universes into regions for which ADM Mass can be defined - and that decomposition is not to "spherically cap" the wormhole mouths.

Hint: Think about where the asymptotic flatness breaks down... excise it. Two disjoint regions will then exist and taken together they will satisfy the requirements for having an ADM Mass (IFF it applies to disjoint regions), but the mass gain/loss regions will in fact be the same and they will cancel out. Wormhole mouths will not acquire/lose mass... but I can see how it might seem as if they do.

I'll write it up as carefully as I can over the weekend... then you can tear it to pieces.

PS The reason each separate universe on its own does not have a defined ADM mass is that neither the (implied/inferred) Visser decomposition (or the alternative) satisfy the asymptotic flatness condition: excise a closed region (e.g. the sphere of a wormhole mouth) and one is left with an open set whose curvature does not tend to zero as one approaches the boundary.

8. Jun 19, 2009

### George Jones

Staff Emeritus
Are you arguing that (11.66) and (11.67) in Visser are true, or that they are false? I can't tell from what you've written.
Which boundary? The boundary of the stuff that was removed? Curvature does not have to tend to zero as this boundary is approached.

9. Jun 20, 2009

### Julian M

Hmmm.... I do tend to have a "shoot first, ask questions later approach." This would be so much easier at a whiteboard where as soon as I open my mouth you could tell me I've put my foot in it. (NB - I can't work out to "multiquote" properly)

To be honest, I was saying that 11.66 and 11.67 are false. However, on a closer reading I see that Visser is clear that the appropriate criterion for ADM mass is spatial infinity. Ludvigsen's statement re asymptotic flatness "along any spacelike or null direction" refers to two different forms of flatness. With reference to spacelike infinity one has the ADM Mass to work with; with reference to null infinity one has the Bondi energy (this I have only just discovered). I was considering only null infinity... thus conflating the ADM Mass (used by Visser) and the Bondi energy (which he didn't). Whether the choice of criterion for flatness ultimately makes any difference I don't know

And to say why I was disputing 11.66/67: I thought that (imagining the wormhole to progressively narrow and grow along its length) there was only one place where the null direction never became flat: around the narrowest point of the throat. I thought that if that infinitesimally thin region were excised - so as to leave a disjoint but asymptotically flat pair of universes either side, a traversing mass would deposit mass on one side of this region and take it away on the other side, and being infinitesimally thin the effects would cancel out.

However I am no longer sure of the shape of the region within the wormhole where the asymptotic flatness condition(s) are broken, i.e. what would need to be excised. I am though still of the opinion that the "spherical cap" of either mouth is not physically justifiable as the boundary of that region: I see no problem (this could be just poor sight, I admit) reaching spatial or null infinity from within either mouth.

With regard to the "which boundary" question - boundaries being the crux of the issue - that the curvature does not have to tend to zero approaching the boundary was the point I was making.

If we divide out joined universes into three parts - universes A & B, and the wormhole - and, say, bound the wormhole with a closed region, then each universe is an open region but, as we agree, approaching the interface of the open/closed from within A (or B) the curvature does not tend to zero and thus the asymptotic flatness requirement of ADM Mass is not met for either universe on its own.

With your facility with real math you might be able to shed some more light on these questions... but I would hesitate to ask after making such a (repeated) fool of myself... though if you're already on it - don't stop!

PS I have just found a reprint of Arnowitt, Deser, Misner's original work on the Hamiltonian that led to the derivation of ADM mass... it was reprinted at arXiv; from the abstract
the pdf is http://lanl.arxiv.org/PS_cache/gr-qc/pdf/0405/0405109v1.pdf". Alas the math is orders of magnitude beyond me, but you would get a lot out of it I'm sure. Ah... you already have the book

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10. Jun 20, 2009

### Julian M

PS... I'm off to have a look at http://relativity.livingreviews.org/Articles/lrr-2004-1/" [Broken] on Conformal Infinity at Living Reviews in Relativity

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11. Jun 20, 2009

### Julian M

Just for info... at the Living Reviews in Relativity article on Conformal Infinity http://relativity.livingreviews.org/open?pubNo=lrr-2004-1&page=articlesu3.html" [Broken], esp the point about 1 screen down where it says

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12. Jun 20, 2009

### George Jones

Staff Emeritus
Condition (3) is satisfied by spherically symmetric traversable wormholes. Condition (3) is not satisfied by spherically symmetric black holes because of the null geodesics on the photon sphere,

Embedding diagrams for wormholes, like, for example, the diagram on Visser's cover, represent one instant of "time". Consequently, any curve drawn on such a diagram is necessarily spacelike (never lightlike or timelike).

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13. Jun 20, 2009

### DrGreg

Press "multiquote" in each of the messages you want to quote from. The button turns blue to indicate you have selected that message for multiquoting. When you've selected them all, press "quote" in one of them. You can even do a quote-within-a-quote with a bit of copy-&-paste inside a multiquote.

14. Jun 20, 2009

### Julian M

Thanks Dr Greg

15. Jun 20, 2009

### Julian M

You mean light would not circle around the wormhole at its narrowest point? Ah. I must confess I have great difficulty visualising wormholes so intuition has led me very astray - can you help me to understand why? (apart from the embedding diagram limitations).

Second point though - if a wormhole spacetime is properly asymptotically flat (as opposed to weakly asymptotically flat) then isn't the whole universe + wormhole + universe asymptotically flat? And why then is it described as separate regions?

I don't know whether things are becoming clearer or murkier, but the input is much appreciated George.

16. Jun 21, 2009

### Julian M

Two points about the spacelike nature of the embedding diagram... My hardcopy of Visser is in storage in England - I'm in Budapest - but the cover can be seen at Matt Visser's pages http://homepages.ecs.vuw.ac.nz/~visser/book.shtml"

1. Agreed, it is a view at an instant - but if the wormhole is static it is surely also a view at all instants, isn't it? So could I not draw a line on it and label it with t intervals to indicate movement?

2. Though I would still like to understand why there is no closed null geodesic in the wormhole throat, accepting this for the moment it occurs to me that, http://relativity.livingreviews.org/open?pubNo=lrr-2004-1&page=articlesu3.html" [Broken], this would lead to the wormhole being asymptotically null flat whereas for ADM Mass to be defined we require asymptotic spacelike flatness.

Agreeing that any interval drawn on the standard embedding diagram when viewed at an instant is spacelike, do we not have a closed spacelike curve if we cut the throat with a plane perpendicular to the axis of symmetry? If this is a closed spacelike curve, it is nowhere flat and does not reach infinity, and we would not seem to have asymptotic spacelike flatness for the combination of universe + wormhole + universe. This does not go against the ADM Mass argument for wormhole mouths per se, it merely returns us to the status quo ante of having >2 regions to consider.

Visser acknowledges that use of the ADM Mass neglects radiation to infinity,
but the argument is given in terms of ADM Mass, yet if the wormhole is null flat, and the universes either side are too, then isn't the whole is null flat? In which case, wouldn't the argument in terms of Bondi energy suggest conservation for the whole, in apparent contradiction to the ADM perspective?

I am going away now to do the promised re-write ab initio...

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17. Jun 24, 2009

### George Jones

Staff Emeritus
If done with mathematical care, yes; if done arbitrarily, maybe, no.
My intuition that wormhole spacetime is not "curved enough" to allow circular light orbits was wrong. A calculation that I did yesterday morning showed that there is exactly one (unstable) circular light orbit located at the throat minimum (and nowhere else). Not trusting my calculation, I looked through the literature and found two papers by Thomas Muller that verify this:

"Visual appearance of a Morris-Thorne-Wormhole," American Journal of Physics 72, 1045-1050 (2004);

"Exact geometric optics in a Morris-Thorne wormhole spacetime," Physical Review D 77, 044043 (2008).

I have not found any freely available on-line versions of these papers, but I haven't looked very hard.
I'm not sure what you mean here. There are two asymptotically flat regions, and I hope to soon post a mathematical demonstration of this using a simple definition of asymptotically flat.
ADM and Bondi masses are the same for all stationary spacetimes, and hence for all static spacetimes as well.

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18. Jun 24, 2009

### Julian M

You beat me to it: I had also looked again at the metric and for constant theta, l=0, phi seemed to cancel out the dt... suggesting precisely what I had "intuited" (an unreliable source at the best of times) but I was hesitant - unpacking old calculus etc. one wonders whether what seemed automatic was right. I still want to work it out/see it explicitly but it's nice to have confirmation that the throat (precisely) is a null geodesic. Yes it's unstable but we are being purely classical here... (if we went quantum then there would probably definitely be nowhere that violated the flatness conditions, so the whole system would be a single mass conserving region).

[I'll see if I can find any of those papers online... some good places are not indexed; the original MTY 1988 paper took ages to track down]

Yes, I understand that the Bondi-Sachs energy is most often used in considering radiation, so from POV of a static spacetime they would be equivalent.

I do see that there are two asymptotically flat "regions" in the broadest sense of the term "region" (for a pair of conjoined universes): in the sense that there are two values for l ($$\pm\infty$$) at which spacetime is flat, but I am hoping that your calculation would show where the boundaries are and established why the regions meet the criteria for flatness required to define ADM Mass.

I do note however on reading around further that the "location of the mouth" is not well defined, so I am beginning to suspect that "mass of the wormhole mouth" refers to the mass attributed to the wormhole per se as seen from either side.

Still working on (intermittently - more of a weekend task) the overall argument, but since the throat (r = rmin) of the wormhole is the source of non-flatness - both spacelike and null - if that alone is excised the mass ought, I feel to be attributed to the two sides of this excision... at which point I would start repeating myself, and I'm leaving that for the re-write.

We shall see.

Very glad you're on the case George.

19. Jun 26, 2009

### Julian M

I thought it was worth looking for these but after employing the "usual techniques" was unable to find them freely available.

I did however recall on seeing Muller's home page that I have a another paper by him. For anyone who is interested, this one is freely available.

"How computers can help us in creating an intuitive access to relativity" Hanns Ruder, Daniel Weiskopf, Hans-Peter Nollert and Thomas Müller

Online at http://www.njp.org/ doi:10.1088/1367-2630/10/12/125014

With regard to the visualisation of null geodesics within a wormhole Fig 15 shows null lines equally inclined either side the d$$\phi$$ direction emerging symmetrically from opposite mouths...