As part of my effort to fully understand black holes I'm trying to develop an intuition for Kruskal coordinates and how they apply to our perception of space and time. Here's the best definition I could find on the requirements of a time coordinate: What I want to know is how a mathematically valid (according to the above definition) time coordinate applies to our traditional concept of time. Are they necessarily the same thing? It bewilders me how this is possible. The Kruskal time coordinate, for example, contains parameters of both time and space; this implies to me that movement along the Kruskal time coordinate axis implies spacial movement "in our world". Is there a better definition than the one above that would help me understand this? Thanks
The definition is fine for our usual Euclidean spacetime coordinates. For example, "now" consists of the whole of space (i.e. the spacelike hypersurface) such that all events occuring "now" are happening at the same time. When you say the Kruskal coordinate system contains parameters of both time and space, I assume you're expressing it as some function of space and time coordinates in a different system, and I assume you're referring to the "mixing" of the space and time coordinates from that other system. I'm not sure if the Kruskal system has a "time coordinate" according to the definition you have given. Does it?
I'm glad you brought this up. Is there a standard definition for "now"? Is it our "past light cone"? Or is it basically our "past light cone" time-adjusted by its distance traveled (I hope not; isn't it circular to measure time by distance which is measured by time in the first place?)? Either way, SR has botched the concept up quite soundly; even if we agree on a definition all "nows" are uniquely subjective. Without a "now", how can we ever discuss the "state" of a system? According to what I've read, the Kruskal "time coordinate" V qualifies as a valid mathematical one; I just want to know what that means as it applies to the real world. Specifically, I'm discussing the conversion between Schwarzschild and Kruskal coordinates when analyzing a black hole. I'll explain what I think I've learned, restrict the discussion to the area outside of the event horizon, and probably get some stuff wrong as I'm not comfortable with heady math or LaTex. Anyway Kruskal replaces t and r in the Schwarzschild coordinates with V and U as the new time and radial coordinates, respectively. Wiki says to do the following; \(V = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\sinh\left(\frac{t}{4GM}\right)\) \(U = \left(\frac{r}{2GM} - 1\right)^{1/2}e^{r/4GM}\cosh\left(\frac{t}{4GM}\right)\) This already raises questions for me but I'll keep going. Apparently, if you want coordinates that behave well at the EH you need to utilize the "tortoise coordinate", named after Zeno's tortoise that infinitely creeps up on its target. Note that this value approaches negative infinity as it gets closer to the EH. \(R = r + 2GM\ln\left|\frac{r}{2GM} - 1\right|\) Where R + t = constant would represent an infalling photon. Using this, Kruskal coordinates (setting c = 1 and r_s = 1) can be written \(U + V = 2e^{{(R+t)}/2}\) Przyk was my original motivator to understanding Kruskal coordinates while discussing black holes in another thread. He wrote the following Specifically, the discussion was whether or not there exists a time on B's watch such that he could not 'rescue' A. Intuition (yeah, yeah) and my understanding of the Schwarzschild coordinate description tell me that this time does not exist - rather that B could begin his journey after waiting an arbitrary amount of time after A had left and, provided he moved quickly enough, he could reach A before A crossed over the EH. Przyk's Kruskal diagram, though, appears to contradict my thinking. This is why I want to understand the principle behind the math of coordinate switching. When we choose a new time coordinate is it valid to continue to refer to "a time on B's watch"? I think if I could understand the concept of what defines a time coordinate it would go a long way...
A couple of follow-up thoughts...and maybe this should be directed to pzryk. Since I'm unfamiliar with Kruskal diagrams, but I am aware that all world lines must be at an angle less than 45 degrees, how could any observer avoid crossing the EH in your picture regardless of acceleration? This statement intrigues me. Are you saying that Kruskal diagrams have the property of representing proper times for all frames?
Well a straight line on the diagram doesn't necessarily represent constant velocity, and a curved line doesn't necessarily represent acceleration. Mostly they do have to accelerate though. Someone sitting at a constant Schwarzschild radius from a black hole is accelerating away from it. If they weren't, i.e. if they were free-falling, they'd fall into the black hole. To stay out of a black hole without accelerating, you have to either orbit it or move away faster than its escape velocity. No, just that if \(\Delta V\) is finite along a curve, then \(\Delta \tau\) is also finite. The reason is simply that to work out \(\Delta \tau\) given a trajectory and initial and final V coordinates, you'd be evaluating the integral of a finite function over a finite range. To give at least a partial answer to your OP: in general relativity, a coordinate system specified on its own is pretty much meaningless. The equations you gave in post 3 tell you how V and U are related to t and r and vice-versa, but alone they can't tell you what the coordinates represent, just how they're related to one another. In GR, you use the spacetime metric to keep track of the physical significance of your coordinates. If you're familiar with how GR is constructed you can read things off the metric like which of the coordinates are space- or time-like, how stretched or compressed they are, and the angles between them. When you apply a coordinate transformation, there is an associated transformation of the metric components, which will tell you what the new coordinates represent.
It's not completely relevant, but you still didn't really answer how ANY worldline could avoid the EH in your Kruskal diagram if they are restricted to angles < 45. OK, good, I was thinking along these lines. The use of the tortoise coordinate made me...let's say...suspicious. From my layman's perspective it appeared that when we are presented with a positive-infinite parameter [the distantly observed Schwarzschild time coordinate at the EH] we define a new one which is negative-infinite at the same area (based on a separate, spacial dimension, mind you), and wrap them together so the infinities go away. This strikes me as sleight-of-hand, exploiting the fact that GR apparently treats time and space as equivalent dimensions even though we do not in our everyday experience. Here's more proof that it doesn't work, przyk: Let's say you give me a time T after which B can no longer rescue A. However, A's experience as he approaches the EH includes watching the outside world apparently speed up in complementary fashion to the outside world watching A slow down. This "speeding up" continues to infinity as the EH is approached; this means that photons representing time T+1 reach A before he crosses the EH. Simply replace those photons with an extremely powerful jetpack and B has now rescued A after time T.
Very easily, if a world line has an angle that asymptotically approaches 45 degrees but never actually reaches it. I gave you an illustration of such a curve in a later diagram, where the trajectory of an accelerating observer approaches a 45 degree line, but never touches or crosses it. No, in general it doesn't work that way. B seeing A age more slowly doesn't imply A should see B age at the reciprocal rate.
OK...I believe you are mistaken on this point, but before we discuss A's experience can we agree that IF A see's the outside world speed up to and beyond any arbitrary time on B's watch before A crosses the EH that he will "always be rescueable by B"?
Er, why? For starters, it's not even true in special relativity. If two observers move away from each other at constant velocity, each sees the other age more slowly. If they're moving toward one another, each sees the other age more rapidly. Yes, trivially. But whole point of my old Kruskal diagram was that it shows this doesn't happen: A never sees B reach point y before crossing the horizon for example, and the way GR is constructed guarantees you'll get the same result no matter which chart you use. The only point left to check is that B does indeed reach point y in finite proper time. If you're worried about that, it isn't difficult to check. Suppose you write B's trajectory as a function \(u = \gamma(v)\). On the diagram I gave you, A leaves B at about v = 0, and reaches point y at some (finite) Kruskal time we could call v[sub]y[/sub]. The Kruskal metric takes the form \( \mathrm{d}s^{2} \,=\, f^{2} \bigl( -\, \mathrm{d} v^{2} \,+\, \mathrm{d} u^{2} \bigr) \,. \)Here f is a function that varies over space-time. Its expression in terms of u and v is defined implicitly by its expression in terms of r. You can look it up (eg. on Wikipedia) if you need it, but the relevant property here is that it's finite except on the singularity. In terms of f, the proper time B accumulates in the period between v = 0 and v[sub]y[/sub] is given by \( \Delta \tau \,=\, \int^{v_{y}}_{0} f(v) \, \Bigl[ 1 - \Bigl( \frac{\mathrm{d} \gamma}{\mathrm{d} v} \Bigr) ^{2} \Bigr]^{1/2} \mathrm{d} v \,. \)(where f(v) is f evaluated on the trajectory \(\gamma\) at Kruskal time v). This is an integral of a finite function over a finite range, so \(\Delta \tau\) is finite. Since proper time differences are Lorentz scalars, you'll get the same result no matter which chart you use.
Agreed but we're subjecting one observer to an increasing gravitational field so the symmetry is broken, right? Asymptotic blueshifting while "standing on the shell" of the EH is a given (agreed?) but I thought it was also a given for the free falling observer. The asymptotic passing of the distant observers time is a function of r only, not t (i.e. not how long one remains at that distance from the EH). ...and the only reconciliation I have to offer for this currently is that the tortoise coordinate "sleight-of-hand" is what is allowing for this. This is why I want to understand time coordinate conversion, and why I started this thread. I can read what qualifies a candidate to be considered a time coordinate but I want to understand how (or rather IF) that coordinate still applies to our concept of seconds ticking on a watch.
That doesn't mean that you get "the outside world apparently speed[ing] up in complementary fashion to the outside world watching A slow down". There's no reason there should be a simple relationship between the rate each observer sees the other age, or that one should become infinite because the other becomes zero. It doesn't, directly. It's proper times, like the \(\Delta \tau\) I mentioned above, that measure the ticking of clocks. The Kruskal V coordinate isn't a proper time. The terminology I used was "time-like coordinate". It doesn't directly measure the passage of time in the sense of eg. V increasing by one unit for every second that passes on a clock. But in GR there is a sense in which we can say that V generally does parameterize the time dimension as opposed to a spatial one, in terms of the causal structure GR inherits from SR. I explained that sense in one of your older GR threads: V is time-like in the sense that a hypothetical point travelling along the V coordinate, for fixed U, would be travelling faster than light as seen by any inertial observer the point passed in the vicinity of. In GR this is determined by the metric: the metric in Kruskal coordinates will tell you that V is generally time-like everywhere, while the metric in Schwarzschild coordinates will tell you that t is time-like outside the event horizon and space-like inside it.
Agreed, I'm not making this claim because it seems there should be such a simple relationship; sitting here it seems that each party would experience redshifting because one is accelerating away from the other. I'm making this claim because this is what I recall reading. I could very well be mistaken on this point, either in my recollection or my comprehension, because I don't currently have access to my library and a quick Googling didn't help. What does standard GR theory say about how the distant observer appears to the infalling body?
OK przyk, I do have a few books on black holes and scanned through them but couldn't find a passage on the infalling body's observation of the outside world. I have a dozen or so sources that talk all about B's view of A, but cannot find anything about A's view of B. Maybe it's because it raises too many questions from the students? Please Register or Log in to view the hidden image! Frankly I'll be surprised if my recollection on this is false but it is possible. Check out the following excerpt I found from the PhysOrg archives: Now, this statement is from back in 2006, so maybe AlphaNumeric would make a different claim today, but the thread is interesting. AN is, in a way, questioning BH's himself as he makes the observation that an infalling body wouldn't cross the EH or reach the singularity before the BH had evaporated...which is sufficient logic to denounce the BH in the first place! This must have taken place before his visit to the Emperor...Please Register or Log in to view the hidden image!
I don't know. I haven't looked it up or worked it out. For all I know A may well see B blue-shifted. All I can tell just looking at the Kruskal diagram is that the rate can't be infinite. I'd imagine it was probably a misconception. Not everyone has had time to work out every last prediction GR makes, and from his very next post in that thread it's clear he wasn't exactly professing expert knowledge of GR or black holes at the time.
Your honor, I wish to submit to the court the following: Exhibit A: Exhibits B and C: Exhibit D: I assume you see where this is headed, przyk...I'd love to hear AlphaNumeric's response to this as well. An existing BH contains mass beyond the EH (by definition). Let's say that all mass in a given BH is made of a series of infalling observers labeled A1, A2, A3, etc. Run the distant observer's clock backwards and define for me when they crossed over. The answer, according to the above "exhibits", is that they never did cross, and the EH was never formed. It isn't an illusion, as you've already agreed that the distant observer could rush in and rescue any one of them, including the leading A1 infalling body. Replace the infalling observers A1, A2, A3, etc, with the molecules of a collapsing star and this extends universally.
And? That was referring to a discussion that took place in 2010. That's four years later than the old post you dug up. You'd be surprised, but that's a long time for a person's experience to change. Four years ago all the physics I knew was classical mechanics and a little about special relativity and electromagnetism.
Check again Are you suggesting he's changed his BH views in the last hour? I'm not trying to be adversarial, przyk. This shouldn't be a "debate", it should be a journey so that everyone involved understands the answer which, forgive me for saying, I'm not convinced anyone here does fully.
Well, maybe he hasn't. But then, remember what I said: I was talking about a particular series of threads that didn't include that quote of yours, and I said "generally remember" for a reason: I was speaking from memory and I didn't claim I'd scrutinized every last detail in every one of his posts. And I certainly didn't say he was infallible. Now to be honest, I'm not an expert when it comes to GR, but I think the Kruskal diagram I gave you is a pretty good reason to believe AlphaNumeric is (gasp!) wrong if he thinks A would see B speed up to an infinite rate. As you said, what A sees happen to B doesn't seem to be covered in the textbooks, I know I've never seen an "official" answer, and it's not exactly the most important detail in the context AlphaNumeric made that remark in.
Well, we don't need to beat anyone up over what was said, I would just like a specific answer as to what GR claims A would see while approaching the EH and observing B. Do you agree with me that there is a potential problem if GR does in fact claim that B's clock would appear to asymptotically accelerate? Who wants to get AN on the horn and invite him to this discussion?
Yes. Obviously. But the only direct statement of what GR claims that I know about is the one I illustrated on my old Kruskal diagram. That and, as I said, GR is constructed in such a way that the answer it gives to this sort of problem is independent of the coordinate system you happen to use. So, to me, it looks like a mathematical impossibility for GR to make any other claim about what A sees happen to B.
