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Julian M
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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
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