"Light cones tipping over" A common phrase used to show alleged time travel solutions in GR. Even a person like Kip Thorne uses it. But my question is, is that an accurate representation of GR in strong gravitational fields? The Schwarzschild metric expressed using the Eddington-Finkelstein coordinates show those "light cones tipping over", and eventually the radial and time coordinates reverse. But this reversal, and even the tipping over seems to me a peculiarity of the choice of coordinates. It seems to me that it is assumed that there is a particular relationship between the radial and time coordinate. That seems to be a rather liberal interpretation of the metric. Afteral nothing in the Minkowskian metric of ds^{2} = dt^{2} - dr^{2} implies that if dt^{2} becomes negative it must be considered space and if dr^{2} becomes negative it must be considered time. Any comments or explanations?
Minkowski space-time, at least according to one of my textbooks, is specifically defined as a 4-D real linear space. Neither dt^{2} nor dr^{2} can ever become negative while dr and dt are real numbers. That wouldn't apply to a complex linear space though.
http://casa.colorado.edu/~ajsh/schwp.html may have useful diagrams for this question. Physically, the tipping of the light cones [which are traced out by null geodesics (a coordinate invariant idea)] is an indication that timelike observers travelling across the horizon cannot escape back out beyond the horizon.
Yeah the tipping just means the future part of the light cones point into a point in space (the black hole).
Yep, keep in mind that "r" and "t" are just coordinates, they don't necessarilly have any global meaning. In particular, t certainly isn't purely timelike for all observers. The original motivation for my current work is all the existing EFE solutions which don't yet have much physical interpretation.
But even in coordinate space, what is the compelling argument for calling this a change from one to the other? Would that not imply that t and r are both related to each others imaginary plane? So that if t^{2} becomes negative it becomes space and that if r^{2} becomes negative it becomes time? Would that not be a rather liberal interpretation of the Minkowski metric? In plain Schwarzschild the light cones get narrower the closer one gets to the event horizon and eventually becomming zero. I would think it much more logical to see them get an imaginary size past the event horizon than to assume that, magically, the time and space coordinates get switched. You say to keep in mind that coordinate space is not observer space is of course true. But, people use the tipping of light cones, which by the way is only happening when you use the Eddington-Finkelstein coordinates, to "explain" closed time curves. That again is a rather premature conclusion IMHO. But on the other hand we see that people like Kip Thorne and more surprisingly Roger Penrose use it as well. Penrose who at one point proposed the use of a complex compactified Minkowski spacetime. Obviously this spacetime does not satisfy the condition that t and r are both related to each others imaginary plane. So go figure, what's the real story here?
Normally you would leave r as simply a real valued coordinate; it's the metric components (eg. [itex]g_{rr}[/itex] vs. [itex]g_{tt}[/itex]) that may change in sign. I don't know that it makes any sense to say that (by what additional metric are you measuring the cones?).. In the plain old (ie. Boyer-Lindquist) coordinates, the light cones "tip" as you approach the event horizon, which means light rays begin to traverse greater intervals of r coordinate (note we're implicity comparing to a euclidean metric over the same chart). From points on the horizon (neglecting that the chart is actually invalid just there), the cone would be rotated such that the future cone is centered towards the negative r direction, and the past cone in the direction of increasing r coordinate. The tipping happens in general, unless you specifically choose a coordinate system to avoid it. The relation to causality violation can occur for example if null cones in a coordinate space tip such that the [itex]\phi[/itex] coordinate becomes everywhere timelike AND if there is also some kind of closed geometry such that having zero [itex]\phi[/itex] coordinate be identified with having [itex]\phi[/itex] coordinate equal to [itex]2\pi[/itex].
Using standard Schwarzschild coordinates light slows down for decreasing values of r. That is why the cones get narrower. But remember that r is nothing but a coordinate, it does not represent the physical distance from the center of gravity. Using standard Schwarzschild coordinates light cones do not tip over. They also do not tip using Kruskal coordinates. However they do tip over using the Eddington-Finkelstein coordinates. Anyway we can do a lot of juggling with coordinates and then demonstrating all kinds of effects that are completely non physical but only due to the usage of certain coordinate systems. More important is the critical review of the suggestion that time can become like space and space can become like time. I know for some that is a done deal and I simply do not "understand" it but to me it is not a done deal at all.
By saying "standard", you are referring to "Boyer-Lindquist", right? In any case, say you draw a t vs. r plot (standard coords) depicting the null cones for observers at different positions.. would you not describe them as "tipping"? Hmm. I would really avoid saying "time can become like space" or vice-versa, because I don't think it means anything. Physically, to an observer journeying across the event horizon, nothing ever happens of local significance. It's a confusing coincidence that the original Schwarzschild coordinates have undesirable features exactly where the event horizon is... Just that when we try to draw some set of coordinate axis lines (initially chosen somewhat arbitrarily), we sometimes find one of those lines is timelike in one region, and spacelike in another.
Boyer-Lindquist is used for a rotating mass not for a non rotating mass. Using Schwarzschild coordinates there is only one r coordinate. Eddington-Finkelstein introduces an additional r* called the tortoise coordinate. With this in place we get tilting cones. And if one additional coordinate is not enough we can always use the Krusal coordinates that introduces two coordinates that relate to t and r. In Krusal coordinates the cones are always straight up and also 45 degrees (so no narrowing). A very good question! For starters it would not be applicable for Schwarzschild coordinates since this is a view of the situation from the perspective of a distant observer far removed from the gravitational field. So the point is that in Schwarzschild coordinates the observer is fixed. But with regards to a observer in free fall who is approaching the event horizon and continuing towards the singularity, the speed of light will remain c, at least locally, so that implies that the angle of the cone would remain constant. However we can hardly speak of a cone from the observer's perspective, it would seem clear that since the curvature for this observer is so strong that a construction of an orthonormal coordinate system would show anything but a cone for incoming and outgoing light rays except for a very small local region. Past the event horizon the observer would still measure the speed of light at c, again only locally, but because the curvature is getting so strong here that the closure between the observer and the center of mass is faster than the width angle of the observer's light cone would allow. Now some seem to interpret this as time becoming space and vice versa, which is the whole point of this topic. To me that is not a coincidence at all. Remember that the Schwarzschild model is a view from the perspective of an observer far removed from the gravitational influence.
So your answer is no? If so I disagree. Regardless of whether you use Schwarzschild or Ingoing (Eddington-Finkelstein) coordinates, the tilting is what you see when you draw light "cones" on a t vs. r (or t* vs. r*) plot. It's a coordinate effect. Since coordinates are nothing physical, and you can't ever speak of shape or orientation of a light cone from the observer's perspective, why make a big deal of the tilting at all? As for time travel, that occurs when the tilting is such that one's future light cone intersects one's past light cone. It isn't an issue in a black hole (the future cone is directed at the singularity, closed timelike loops don't become possible), it only means that from our perspective the gravity of a black hole sucks things away too strongly for their rockets or even their radio messages to approach us.
Well I am quite sure that you would not accept from me that using Schwarzschild coordinates light cones do not tip, and that instead you need Eddington-Finkelstein coordinates for that. Perhaps from someone else you will.
I accept I may be wrong (you haven't yet motivated me to draw the plot myself) but my reasoning is that in Schwarzschild coordinates: far from the black hole, light tends to travel in the time direction (the cone spreading equally in +/- radial directions); inside the even horizon light must always travel radially inward (presumably also spreading in the t coordinate directions); if anything Ingoing coordinates would demonstrate less tilt (limited to 45 degrees). Regardless, if you believe the tilting appears only in one of two equally valid coordinate systems, do you agree that "space becoming time" is not a physical effect (in black holes)?
The whole issue IMHO with cones is that they work well with SR but miserably fail with GR. In flat spacetime we have cones, but not in curved spacetime. Yes of course if we take a very small region that we can consider flat then we have a mini cone but in curved spacetime we cannot speak about a cone at all. Sure we can insist on cones by picking the proper coordinate system, but it does not mean anything, for all intents and purposes we might as well create a coordinate system that shows cones as passa doblĂ© steps. I agree that such coordinate representations are more or less meaningless from a physical perspective. But the problem is that tipping of cones is shown as theoretical evidence for things like closed time loops. Even people like Kip Thorne, Roger Penrose and Stephen Hawking seem run away with it and write books that "clearly shows" what is going on. In my understanding at least it seem that they should know better, but clearly I just don't seem to understand why it is obvious that time and space can flip inside the event horizon. Any takers on a simple explanation? The main issue that I wanted to bring up in this topic is that suggestion, the suggestion that time becomes like space and vice versa. I don't see any indication for that, except for when we make a particular interpretation of the time and space relationship in the Minkowski metric.
The interpretation is natural though since asymptotically thats the preffered coordinate system. The choice of tipping is indeed a coordinate artifact, but it makes good sense from a physical standpoint to compare it with what we know best. Eg a rockets journey starts out with the obvious Minkowski coordinates system, and as you get close and pass through the blackholes event horizon it serves as an illustration how either you have to abandon your choice of coordinates, or you have to admit that ones notion of time/space are going to warp and switch places. And theres nothing wrong with writing down lightcones locally. Indeed we specifically choose not to pick coordinates where they are torus's or something like that, b/c no one has any intuition whatseover about that and indeed most calculational strategies vanish with such a stupid choice of local coordinates.
So demonstrate to me how they switch place! How do two separate dimensions get intertwined on a Riemann surface?
In Minkowski spacetime M, we have two ways to think of the "light cone of an event p"... It is the "set of events in M" that can only reach or be reached by the vertex event p by a light ray. It also forms the boundary between events that can be causally connected to p from those that cannot. It is also a "set of directions [set of vectors] in T_{p}M (the tangent space at p)" that are tangent to lightlike paths through event p (i.e., "set of lightlike tangent vectors at p"). It also forms the boundary between the spacelike and non-spacelike (i.e. causal) tangent vectors. T_{p}M is a vector space with a Minkowski metric. In a general curved spacetime, the notion of "light cone of an event p" usually means the "set of lightlike tangent vectors at p". Nothing I have said above depends on any choice of coordinates. To draw these light cones in the spacetime diagram of a spacetime, curved or otherwise, one must identify all of the lightlike geodesics (a coordinate invariant notion). At a particular event, its light cone is determined by the tangents to these geodesics. Depending on your choice of coordinates, the image of these geodesics in your coordinate chart may trace out all sorts of crazy looking paths (akin to the distortions one gets from various map projections of the earth). In some cases, the image of these light cones may look tipped or distorted relative to the images of other light cones in your coordinate chart. Regardless of appearances in your chart, the physics is determined by the lightlike geodesics, essentially telling you which events are in the causal future [and causal past] of events in spacetime (i.e. the causal connectivity of events). The worldlines of observers are bounded by these lightlike geodesics. It may turn out that the null geodesics tell you that you might have closed causal curves. Or it may turn out that certain sets of events have causal futures that don't extend to spatial or null infinity. Or something other feature that one doesn't see in Minkowski spacetime. Depending on your choice of chart (and thus the images of the light cones), it may be easier or harder to tell the story of what is going on physically. Depending on the particular aspect of the story you want to tell, some charts are better suited than others. More correctly, these show that closed time loops are mathematically possible, given the constraints imposed in the situation. In other words, saying that one has a 4-manifold with a Lorentzian-signature metric places some restrictions on what "physics" is possible. However, by themselves, they don't restrict the possibility of closed time loops or other pathologies. Even imposing the field equations might still allow pathologies. That is why one is led to the notion of "causality conditions" and the study of "causal structure", which were developed using "global methods" (i.e. geometric, coordinate-free methods). One may also impose other conditions like "energy conditions", "asymptotic conditions", etc... When I find the time, I'll try to address your concern about "switching".
Sure I follow what you say here. Sure in certain coordinates. Completely agree! Well at one point the time part of the geodesic has to connect to another time part of the same geodesic while the spatial parts are irrelevant. Apart from a closed spacetime or a wormhole I do not see how that can be the case. Can you? Please do, to me it makes absolutely no sense.
Ok you guys are much better than I am at this, but I thought I had a handle on it. Lets look at this in the Schwarzschild coordinates. Is it sufficient to show t becomes imaginary & that a decrease in r becomes as inevitable as going forward in time?
In case anyone's still interested in this, the clearest pedagogical explanation of this is (unsurprisingly) in MTW. Check out pp. 823-826 for a discussion that uses the Schwarzschild geometry and the surface at [tex]r=2M[/tex] as an example.