I'm sure someone has already thought of this somewhere, but can a wormhole violate the conservation of energy IF the object takes two paths? I mean like this: Imagine a wormhole near a charged object, one closer, the other farther. Then imagine the particle being attracted to the charged object(Path 1) It has now lost potential energy. BUT then it goes back away through the wormhole, a shortcut in space(Path 2). The object travels the full distance relative to an outside observer. The object travels a shrunken distance relative to itself(Doesn't it?) Would it not take less energy through the wormhole to gain back the initial potential energy? This is also not relying on gravitational acceleration, which is technically nonexistent and only curves paths. The only thing I can reasonably think of is that either wormholes are impossible, thus negating this paradox entirely, or the field is kind of 'condensed' into the wormhole passage thingie. On an unrelated note, how has noone ever thought of an explanation in science fiction for being able to communicate with extremely distant ships and bases? The whole point of the warp drive or whatever is to get them to places that are further than light can reach in traversable time. Since em waves travel at light speed, they should probably die before the messages even reach them(depending on distance). Unless they have a permanent wormhole for the communications, or they generate one everytime they send a message and just send a communications probe thing through, this just seems completely, even in science fiction reasoning, unreasonable.
The electrostatic field is conservative. This means that the potential energy of a particle depends solely on its position. The "paradox" is resolved by realizing that if a wormhole connects a weak field region with a strong field region, the field must smoothly change from weak to strong within the wormhole. The shorter the wormhole "neck", the more abruptly must this change occur. The same "paradox" occurs for the gravitational field in presence of wormholes. You might want to check out this item in this Wormhole FAQ: "Is a wormhole whose mouths are arranged vertically in a gravitational field a source of unlimited energy? No. The argument in favor of such a wormhole being an energy source is this: An object falls from the upper mouth, gains kinetic energy as it falls, enters the lower mouth, reemerges from the upper mouth with this newly acquired kinetic energy, and repeats the cycle to gain even more kinetic energy ad infinitum. The problem with this is that general relativity does not permit discontinuities in the metric – the descriptor of the geometry of spacetime. This means that the gravitational potential of an object at the lower mouth must continuously rise within the wormhole to match the potential it had at the upper mouth. In other words, this traversal of the wormhole is “uphill” and therefore requires work. This work precisely cancels the gain in kinetic energy." The FAQ is an excerpt of this book.
I have rather strong doubts about the accuracy of the above answer, though it is in "genuine print". So far, I haven't found anything really clear that resolves this issue in the literature. The argument that GR "doesn't allow discontinuities in the metric doesn't seem to me to prove the point in question. I do suspect that the author of the book believes his remarks to be true, but I'm not convinced that the statements made in this FAQ follow from Einstein's field equations. The reasons why I think the approach is wrong are a bit technical, but boil down to the fact that the only way to "thread" an electric field, or by analogy a gravitational one, is by passing a charge, or mass through it. This is incompatible with the author's idea that it would "always be uphill" if the wormhole connected two regions of differing Newtonian gravitational potential - the author would have us believe that there must always be a field threading the wormhle in this case, it's not consistent with the idea that the only way to generate such a field is to pass an object with charge (or mass) through it.
I suspect that there is an assumption in how much energy it takes to keep a wormhole both stable and at a discrete separation. I suspect this energy will be non-trivial. This is where the energy will come from. When the net energy of the whole system, including a working wormhole, is taken into account, it will surely respect the LCE.
I'd be willing to bet that the total ADM mass of the pair of wormhole mouths plus the "energy at infinity" of the moving object was constant. Assuming we can separate the ADM mass of the pair of wormholes into separate parts. But it's possible for one of the wormhole mouths to acquire negative mass. So it's possilbe (and perhaps even expected_ that wormhole A gains +1 unit of mass/energy, B gains -1.1 unit of mass/energy, and the moving object C gains .1 unit for instance. For pure story-writing purposes, having the energy come from the stabilizer is a good idea. We are a ways away yet from being able to do any experiments :-).
I'll let people smarter than I answer the question about wormholes (although my knee-jerk reaction is no) but as for interstellar communication, I have seen some forms of sifi use typically two methods. The first is what most people would think. They just beam em waves through an active wormhole. Although this would draw a ridiculous amount of power for a simple message, it should work, at least in theory. The second idea relies a little less on science and a little more on fiction. The writers presume that there is some sort of underlying fabric of the universe we have yet to discover dubbed subspace. Their hope in this is that by accessing this subspace, we should be able to send messages down into this area, to the predetermined point relative to space, and back into space in little to no time at all relative to both observers. I apologize if I botched any of this, but those are the only two ways I've seen of rapid interstellar communications and, as I said, the first seems to hold more potential than the second even though the amount of energy needed to convert it to widespread use borders on the need for access to zero point energy, cold fusion, or both.
Happy to. :) Although, many advances in science did begin in, and are first explained through, fiction. I'll try to keep it a little more on the science side though.
The answer you are referring to is completely accurate. The reason this doesn't appear much in literature is because if you understand enough geometry and topology to know what a wormhole is, it's absolutely obvious how any conservative field will behave in its vicinity.
I'm afraid I don't find your answer very convincing, and I suspect that you are most likely making the error of applying ideas based on Newtonian energy conservation to the conservation of energy to general relativity, most especially if you think the answer is "obvious", and that if you think that gravity is a "conservative field". (I'm assuming that's what you meant, it'd be a bit snippy to say that the answer was obvious for any conservative field if you knew already that gravity wasn't a conservative field in the usual sense, the only way your remark would avoid being a non sequitur is if you did think gravity was a conservative field). If you do think the answer is obvious, perhaps you can explain how it's consistent with the total flux through the wormhole is altered by a charge passing through it - to wit: I.e. http://www.npl.washington.edu/AV/altvw69.html How do you reconcile this with the idea that one direction must always be "uphill", based on some Newtonian idea of local gravitational potential (which appears to be the somewhat naive argument given in the so-called FAQ) when the argument above shows that you can change which way is "uphill" only by passing a mass through the wormhole - .i.e you can thread electric field lines through the wormhole by sending a charge through it, similarly you can thread gravitational field lines through the wormhole by sending a mass through it, changing which direction is "uphill". The charge case is somewhat easier to deal with,so let me recap it in those terms: Specifically, imagine a wormhole pair far far away from any other charge, set up so that there are no fields threading it. Now, bring the wormhole pair close to a charge, so that the two ends are at different potentials. The field line argument shows that bringing a wormhole close to a charge won't change the field in the throat, it won't make one direction "uphill", because no charge has threaded the wormhole, and that's the only way to get field lines in the throat threading the wormhole. The field lines have to stay "hooked up", they must be continuous - if you drag a charge th rough the wormhole, the field lines come with it, and that's the one and only way to change the flux through the wormhole. I'm using the charge argument here rather than the mass one, because it's slightly more rigorous and because it's really easy to deal with, given that electromagnetism is a two-form and we can talk rigorously about "field lines". But you can ask the same question about bringing a field-less wormhole close to a large mass. BTW, there's lots of FAQ's and literature about the conservation of energy in GR out there, as well as some more technical expositions. I could list a few, but it seems premature, and I suspect that if you get over thinking that the answer is "obvious", you can track them down well enough yourself and you'll find it more convicing if you do the legwork. I will mention one, however, that's helpful on a historical note rather than a technical one. The issues with the conservation of energy in General relativity have been known since Hilbert's time, when he talked Emily Noether into investigating the problem for him. See for instance http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html.
Wow, I feel proud of myself. I thought of a question on my own that was important enough to warrant controversies of its own accord, even if it was originally thought of decades ago. I just hope I'm not acting as a sponge sobconsciously. It really sucked when I realized a tune I thought I made by myself was actually a part of Gilgamesh's theme from Final Fantasy. So it seems the main arguments are, the field goes through the wormhole as it does the outside, or the field is brought about by the presence of the charged particle already inside the wormhole. Too bad we don't have a wormhole to experiment with(or do they...(they probably don't)). Either way, it would be neat to one day attempt to find a method to create an artificial wormhole, even if it's untraversable, resulting in ridiculous deadliness, because it would be the first. After all, an opportunity to fail is an opportunity to succeed! Ironic, being created by me, a pessimist, isn't it(the sentence, that is, I made up the saying).
Pervect, you are making an elephant out of a fly. [tex]\partial^\mu \partial_\nu A^\rho = \mu J^\rho[/tex] So long as J is uniquely defined, A is uniquely defined. You can substitute any differential geometry you want, the potential is still unique for given charge distribution and boundary conditions. That's it. That's the whole discussion right there.
Actually in curved space-time, this formula isn't quite right. See for instance Wald, "General relativity", pg 71 eq 4.3.15 [tex]\nabla^a \nabla_a A_b = R^{d}{}_{b} A_{d} = -4 \pi J_{b} [/tex] where R is the Ricci tensor. Note that we've replaced the partial derivatives [itex]\partial_a[/itex] with the covariant derivatives [itex]\nabla_a[/itex] The important thing is that [itex]\nabla_a J^a = 0 [/itex]. (Which is true for the expression above, and not true for your expression, see Wald's discussion. Furthermore this equation, [itex]\nabla_a J^a = 0 [/itex], sometimes called the continuity equation, is what prevents you from changing the flux through a wormhole without passing a charge through it. The argument is clearer in the intergal form, rather than the differential form. Consider a wormhole that has no charge on either mouth, and no charge inside it. Then if you draw a sphere, around either wormhole mouth, or anywhere in the throat, there is no flux through the sphere, no electric field. This remains true even if you bring the wormhole close to another charge. You can't induce a flux through the throat of the But without a flux going through the wormhole, you can't have a net change in the energy by passing through the wormhole. (You can change the flux through the throat by passing a charge through it, but that's the only way to change the flux). So the explanation for how energy gets conserved in a wormhole is a bit trickier than postulating some sort of field threading the throat - which just can't happen according to the laws of physics. If we assume that the wormhole is static, it doesn't get that much more complicated for the gravitational case than it does for the electric case described above, you just need to replace charge with Komar mass.
Hm... Alright. Maybe I need to take a closer look at this. Covariant derivatives taking place of ordinary derivatives does make sense intuitively, but I've never considered the consequences. It still looks like it should reduce to something simpler, but maybe I have to add constraints, like no time-dependence and maybe no frame-dragging in the metric. Thank you for pointing this out. And hey, I have Wald's book right here. Going to go read.
As I understand it, a discontinuity in the metric would imply an infinitesimally thin, infinitely dense surface of mass-energy at the site of the discontinuity. [That is, if you put a discontinuous metric into left side of Einstein's equations, the unphysical "delta function" surface pops out of the right side.] In the absence of such an unphysical surface, a discontinuous metric could not exist. No surface ==> no discontinuity. "'thread' an electric field"?
Bold added for emphasis. A gravitational field does exist before any object passes through the wormhole. The wormhole meteric is the gravitational field in my understanding. [I seem to remember from Kip Thorne's book that the Newtonian scalar field is normally identified with the logarithm of g_{00}(x).]
The red lines in the diagram below are electric field lines threading a wormhole - it's a 2d wormhole, as that's the only sort I can draw. The electric field lines on the diagram are red. The wormhole geometry is sketched in black. Electric field lines point in the direction of the electric field. The magnitude of the force is given by the density of the lines - double the lines per inch is twice the force. Note that field lines threading the wormhole are the only way to have a potential difference when traveling down the wormhole. With no field lines, there is no electric force. Remember Gauss's law. The integral of the normal electric field over a surface is equal to the enclosed charge. Let's look at and end view of the left wormhole mouth. Applying Gauss's law, we can see that the circle must enclose a negative charge. Gravity isn't a simple two-form like the electromagnetic case. But if you have a static wormhole, the Komar mass is a conserved quantity, like the charge is, expressible as a surface integral (or as a volume integral) so you can use the same argument, as long as everything is static (or at least stationary).
Pervect, The gravitational field of the wormhole itself is completely distinct from the gravitational field of a massive test particle that passes through the wormhole. Yes, Gauss' Law does tell us that if a wormhole connecting Universe A and Universe B swallows a massive particle in Universe A, then observers in Universe A will see the wormhole acquire this mass, even after the mass has traveled light years away in Universe B from the wormhole's mouth. This has absolutely nothing to do with the continuity of the gravitational potential within the wormhole. Continuity persists, gravity remains conservative. Travel through the wormhole from a low potential to a high potential would still be "uphill". That's my understanding, anyway.
I would certainly agree that the metric coefficients are continuous. I would disagree that this implies that there would be an "uphill" direction if you moved one end of a wormhole to a region of lower newtonian potential. A few things should be clarified, I think. 1) You mention "the potential". What potential are you talking about? Currently, I'm assuming you are basically importing the Newtonian potential into General relativity - and assuming that gravity in GR can be expressed in the same way that you are used to as the gradient of a scalar potential function. It's also possible that you mean "the metric" coefficeints by "the potential", but I don't quite see how your argument follows if we assume this is what you mean by "potential". 2) You mention that gravity "is conservative". What are you basing this on? Are you assuming that the usual Newtonian ideas, or have you read some of the FAQ's or the technical literature on how and when energy is conserved in General Relativity? For instance the sci.physics.faq: http://www.desy.de/user/projects/Physics/Relativity/GR/energy_gr.html The somewhat related faq here on PF https://www.physicsforums.com/showthread.php?t=506985 (the focus is on cosmology though) Also the paper about Noether's theorem I mentioend earlier: http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html Or for a textbook source, Wald's general relativitiy, chapter 11 section 2 3) How do you explain away Cramer's argument given in http://www.npl.washington.edu/av/altvw69.html It should be clear that passing charges through the wormhole can change the direction of "uphill" and "downhill" at will, which will play havoc with any attempt to say that "one direction must always be uphill".