No, that doesn't work. It doesn't address the single biggest problem with the frozen star interpretation: that it is attributing an unjustified special significance to just one of the many possible and equivalent coordinate descriptions of black holes. Kevin Brown is correct to describe general relativity as a field theory. I would personally go so far as to say it isn't even an interpretation. The metric in GR is a field, so calling GR a field theory is a simple statement of fact. There is no dichotomy between between viewing GR as a geometric or as a field theory. It is simply both. What Kevin Brown doesn't say, though, is that general relativity and modern quantum field theories are both gauge field theories. General relativity is not patterned after quantum field theory. If anything it's the other way around: the standard derivation of the fermion-photon coupling in QED for example uses an argument that is very closely analogous to GR's equivalence principle, and modern quantum field theories have a gauge symmetry and redundancy in their mathematical description that parallels GR's coordinate independence. You really need to get over this ridiculous, ignorant, and quite frankly offensive idea of the physics community being some monolithic block of people all thinking the same thoughts. The section you quote does not accomplish that at all, and the reason is given in the quote itself: You are making the mistake of attributing a literal interpretation to a coordinate description that has no required special significance. No it doesn't. It just shoots off the top of a coordinate chart that you should not be presuming represents anything physical in the first place. This is just basic GR: in general in GR there is no such thing as a globally defined inertial coordinate system. But as a practical matter we still need to work with coordinate systems to give descriptions of things, and in some cases, like with Schwarzschild coordinates, we can end up working with some really bizarre coordinate systems. (For a more everyday example of something like this, just look up a map of the entire Earth. If you look at Google Maps for instance, you'll see it makes Russia, Canada, Greenland, and Antarctica look much bigger than they really are, and it makes the North and South poles look like stretched out lines when they're really just points.) What happens with Schwarzschild coordinates is that they don't even measure everywhere what their names would imply. Specifically, inside the event horizon, the r coordinate is actually more a measure of time than of space, and the t coordinate is actually more a measure of space than of time. They switch roles on the event horizon. (I am not making this up. Like I mentioned in an earlier post, you can literally read that straight off the Schwarzschild metric, and I can walk you through it if necessary.) Whenever you see a coordinate system being referred to in GR, you should start with the presumption that it is completely arbitrary and represents nothing physical, and work up from that. You should not assume any given coordinate system represents more than you can explicitly and very carefully justify. You have never justified that a clock on the event horizon is actually "stopped" in a meaningful way. The last I recall from discussing this with you, you seemed to be presuming some well-defined global notion of simultaneity that simply doesn't exist in GR.
Yeah that guy's a complete nutzoid - I was leaning more towards stuff like Brian Greene and others had postulated.
I take the view that I do based upon what clocks measure, which is some regular local motion rather than an abstract thing called proper time. This isn't something I've made up, it comes from A World without Time: the Forgotten Legacy of Godel and Einstein. It isn't GR as at 1916, but it does come from Einstein, so it deserves some credence. If you have two identical clocks at different elevations, the lower clock goes slower than the upper one not because "time is flowing slower" but because "motion is going slower". We refer to it as gravitational time dilation. Then if we envisage a situation where gravitational time dilation goes infinite, then of necessity the rate of motion goes to zero. And that's the end of all coordinate systems, because the clock is a light clock, and when the clock is stopped the light is stopped, so you cannot measure time or space. Duly noted. Ditto. I don't think that at all. I think it consists of competing theories and shades of theories, whose advocates promote them. Think of string theory or SUSY. It is. Coordinate time is my time and your time as we observe the infalling body from a great distance. When an object "goes to future infinity (of coordinate time)" we ask ourselves "has it got there yet?" And the answer is always no. I'm not saying it's special, I'm saying it doesn't exist. But I do. A coordinate system is an abstract thing derived from measurement, usually using the motion of light through space to define your seconds and your metres. When a light clock stops, light stops. So there are no seconds and there are no metres, and whilst there's still something physical exerting its gravitational field, there's no more coordinate chart. But I can view the Earth from space and I can see it's spherical. I can see the physical territory and I can understand the limitations of the coordinate system. I'd say that you and others attach more importance to the coordinate system than to what physically exists. I know. And I know that clocks don't literally measure time flowing, they employ some regular cyclic motion in space local to the clock and give us a cumulative display called the time. So walk me through the Schwarzschild metric, and when clocks stop, we can stop walking. I know that the ethos of GR is coordinate independence, but that predates A World without Time. If gravitational time dilation goes infinite, the clock rate goes to zero. That's it. Simultaneity just doesn't matter.
This. I feel that too much emphasis is put on the model with a disregard for what is actually being modeled. I get tired of this viewpoint! The Schwarzchild coordinate description DOES have special significance: it accurately describes remote clocking behavior! If I were to lower my watch with a long cable over an event horizon for some extended period of time and then retrieve it, would I expect it have to "clocked" in accordance with the Kruskal time coordinate or the Schwarzschild time coordinate? The GR predictions of time dilation are not an illusion that we can mathematically side-step.
In this case that comes with the thread topic. If you start out this thread trying to argue that black holes don't "exist" in the context of the model that predicts them, and then go "oh well it's only a model anyway", I would view that as you shifting the goalposts for yourself. Neither. With regard to Schwarzschild coordinates, for a point at constant Schwarzschild radius outside the event horizon, proper and coordinate time are related by \(\mathrm{d}\tau \,=\, \sqrt{1 \,-\, r_{\mathrm{s}}/r} \, \mathrm{d}t\). So Schwarzschild coordinate time does not coincide with what a clock at some given Schwarzschild radius would measure. There is also no a priori reason to treat \(r = \text{constant}\) as a preferred notion of simultaneity. Schwarzschild coordinates have the nice feature that they are static in the t coordinate, so you can practically just read off certain coordinate-independent results like observed relative Doppler shifts and (if you can neglect the effects of the trips both ways) approximately how much time a clock will accumulate if you send it to a lower radius for a while and pull it back. But that's it. Like I've pointed out several times before now, the Scharzschild t coordinate is spacelike inside the event horizon. An interval of \(\Delta t\) isn't a timelike interval there. For \(r = r_{\mathrm{s}}\), the Schwarzschild t coordinate simply doesn't measure anything at all.
Actually it's my opinion that GR does not predict the existence of black holes. I'm questioning the interpretation of GR, not GR itself. No, I feel you are casually dismissing this. The t variable in Schwarzschild coordinates is directly related to the clocking of our local watch; the remote watch is linked to the clocking of our watch via the time dilation formula. Can the same be said of other time coordinates? True or false: physical limitations aside (cable strength, G forces, etc), we could use the event horizon as a form of time machine in which we could travel arbitrarily far into the future. If you agree that this is true then it is maddening to me that you consider the other effects predicted by that coordinate choice to be an illusion. The t is not "a mere choice of many"; it has a unique meaning for us as observers. I'm not sure what your point is here; the only reason the dimensional flipping is necessary is because you're making the presumption of that region's existence a priori.
Only if it comes with some good justification from Einstein, and if you had that I think you would have presented it by now rather than presenting your usual volley of quotes that appear to just jump to the conclusion. If you are arguing with laymen who have no detailed understanding of mainstream physics theories and who are completely dependent on the authority given to scientific consensus as an approximation of validity, then you might reasonably expect they should pay attention to something with Einstein's name on it. But when you are discussing a 100 year old formally defined theory with a physicist who has learned it in all the gory details, they are going to want to see the actual argument and justification, and Einstein's name in itself carries very little weight. Since science is based on reason and experimentation and not authority, that is of course exactly the way things should be. There are problems with this. The first is that gravitational time dilation is normally only given for two clocks at "rest", technically meaning at constant positions in a Schwarzschild-ish coordinate system. So gravitational time dilation might tell you that a clock sitting on the event horizon freezes, but that doesn't say anything about an infalling clock whose Schwarzschild radial position is not constant. The second is that gravitational time dilation is only really an approximately defined notion anyway. Like I've alluded to before, in order to make a comparison between what a clock in one place is doing compared with a clock in a different place, you need a definition of simultaneity. You need to say the first clock is displaying say a time \(t_{1}\) "when" (i.e. "at the same time as") the second is displaying a time \(T_{1}\), and that the first is displaying \(t_{2}\) a bit later when the second is displaying \(T_{2}\), before you can do something like calculate \(\frac{T_{2} \,-\, T_{1}}{t_{2} \,-\, t_{1}}\) and call it the relative rate of the two clocks. But GR doesn't provide that "when" for you. Simultaneity is already relative (frame-dependent) in special relativity, and it's only worse in general relativity. You've given no good justification of that. We're discussing general relativity here, and it matters what coordinates represent and how they're used according to GR. While you might think coordinates should be defined that way, it doesn't mean they're defined or used that way in general relativity. General relativity is deliberately formulated in such a way that it can be expressed in terms of completely arbitrary coordinate systems. The metric is there to keep track of how locally stretched or twisted a given coordinate system is. But you can't step outside of spacetime, so you do not have the option of settling the debate that way. You've missed the point of the analogy, as in GR the coordinate description and the metric are the only things you are given. If you are given the metric of the surface of a sphere, there are ways of telling just from the metric that you are in fact dealing with a sphere or at least something that has all the practical features of a sphere. We're using essentially the same tools in GR when we're studying some manifold that the Einstein field equation spits out in the form of a metric. I've said the exact opposite: coordinate systems are man made and do not exist. I've stressed that you should therefore not draw naive conclusions just because something looks a certain way on some coordinate chart.
Then you are discussing the model and whether something is a sensible interpretation of it. Which as I explained to Farsight above is based on treating \(r = \text{constant}\) as a preferred definition of simultaneity, and for clocks at constant Schwarzschild radius. True, for a certain trajectory, totally independently of the coordinates. (And to be clear: true in an invariant sense only in that you could, say, leave from Earth and return to a 2 million year older Earth.) I am also generally making the presumption that the region of space we live in will still exist tomorrow. You can take the GR description of say our solar system and just cut it off after some future time, and that is every bit as arbitrary as what you are doing with the black hole solutions predicted by general relativity.
No problem. But you know full well that the lower clock goes slower than the upper one, and you know full well that when you open up a clock you don't literally see proper time being measured. And you're happy enough with the gedankenexperiment where we lower the clock to r=rs and find that when we haul it back up it shows the same reading regardless of how long we left it there. I hope you're also happy that you can place clocks at various points in space round an equatorial slice of the Earth and inside the Earth to obtain a plot of gravitational potential that illustrates Riemann curvature. You know that the force of gravity at any point relates to the slope of the plot, and that this is related to the clock-rate differential. So regardless of Einstein or MTW or anything else you've been taught, you ought to be able to think for yourself and work out that when your clock can't go any slower, there is no more slope. Your rubber sheet just stops, like the clock, like the frozen-star interpretation. That's not some argument from authority. Instead the "scientific consensus" is. There are other problems with an infalling clock. It falls faster and faster, and yet the coordinate speed of light is reducing. However this isn't something for which I can refer to an interpretation that has fallen out of fashion, so I won't push it. Gravitational time dilation is real, and whilst it has some limitations, the principle of equivalence is not something we've discarded. And we're happy with the gedankenexperiment where we lower the clock to r=rs and find that when we haul it back up it shows the same reading regardless of how long we left it there. That means simultaneity just doesn't matter. The point is that according to GR, there's more than one way to skin a cat. You've been taught MTW and that there isn't. And the metric is what you measure, and when you can't measure anything any more, that's the end of the story. The metric is not space, and spacetime isn't either. As I've said before, the map is not the territory. The territory is space, and as Einstein said, it's inhomogeneous. Because of that we can plot a curvature in our measurements. But once we've lowered that clock to r=rs there isn't any more curvature, and there isn't any more slope, and there's no more measurements. Good. Now apply the same thinking to the metric and the manifold. Step outside and look up. Is that the metric up there? Is it the manifold? No. It's space, and light moves through it. Once you understand what a clock does, it's all very simple. And you don't have to discard black holes or GR, all you have to discard is the conviction that the version of GR promoted by authors who believe in time travel is the only GR there is.
But in that case I would agree that it's a reasonable presumption. Drawing the line at t=infinity does not strike me as being quite as arbitrary.
Then don't draw lines where t=infinity, you might actually end up becoming a respectable scientist one day. Black holes have jets that spew around the same time as whatever it is they are consuming is going into the black hole. If you said it falls in and its time freezes for an infinite amount of years to the outside observer it would never be able to be expelled out into a jet. There is no mathmatical principle that shows how that is infinity plus one years later. It is like the Zeno Paradox, every step you take would be more than an infinite number of smaller locations. There is no way to show that this infinite number of locations could ever be reached even if they are progressively smaller. By definition infinity is something that can never be reached, but we make leap and bounds across infinity with every step we take. You could only conclude that there is no such thing as infinite division.
I don't think the Holographic Principal could apply to any type of black hole weather systems. An object couldn't become frozen in time and then be ejected in a time frame close to when this happened. So then it means that either the principal itself is wrong, or that objects do not actually come close enough to the event horizon for this to happen. It just doesn't fit with the observational evidence, so it either doesn't take part or doesn't take place in reality. Seems to me if you wanted to track a particle pair on the event horizon is said to go beyond the horizon and the other escapes. If you tried to track these particles with a line where time goes to zero or infinity then there would be no way to know when these events actually took place to the outside observer. So observational evidence shows that there may be no events taking place on the event horizon, if the matter ejected out from jets was there it would be infinity plus one years later, not while it is falling in.
Actually...I agree with you. If a remote observer claims that matter passes the EH in infinite time but can confirm (via Hawking Radiation) that matter does indeed come out in a finite time then it is a bit of a contradiction to claim that the matter ever made it across the EH in the first place.
This is for Pryzk: From here: It's a subtle point, but by definition if a path to the singularity had a finite proper distance then spacetime would not be maximally extended. This means that it is "infinitely far away" (...or GR does not describe a maximally extended spacetime). Additionally, there is clearly not a way around the causal order issue; (unless we're located there) we cannot claim the singularity was created or exists in our past light cone. I'm not seeing it possible to consider the singularity as currently existing with any reasonable definition of the word "now".
Hawking Radiation hasn't been confirmed, we would have to risk the destruction of our planet in order to prove wrong another one of Hawkings bets. He can't lose because if he lost we would all be dead. I should take this guy to Vages!
Not in an invariantly defined sense. I don't know why you insist on constantly bringing this up. I am not saying that the relation between proper time and what clocks measure is necessarily trivial (and actually in many cases it is only an approximation), but in relativity we have the principle of relativity that implies, among other things, that if e.g. one type of clock behaves a certain way (like experiencing time dilation) under inertial conditions, then all types of clock behave that way in the same conditions. For this reason, the details of the internal functioning of clocks are generally not relevant when discussing relativity. (Incidentally this is the reason for the common derivation of time dilation based on the light clock, even though real clocks don't work that way. The relativity principle implies that what holds for a light clock must hold for any other type of clock, otherwise you could use the difference to infer your "absolute" velocity.) But that is the end result. Gravitational time dilation concerns the intermediate, and strictly coordinate-dependent, account of what's happening throughout the entire gendankenexperiment. Like I've said, any comparison between what the clock you're lowering is doing compared with your reference clock on Earth, at any time other than the beginning and end of the experiment, depends on an arbitrary simultaneity convention. In any case this is really missing the point since a clock sitting on the event horizon is actually travelling at the speed of light and really does freeze there. That fact is also depicted on the Kruskal chart for example, and is not a matter of contention. Nobody is disputing it. The real problem, and your gendandenexperiment is incapable of revealing this, is that an infalling clock crossing the event horizon proves to be a completely different animal than a clock hovering on or just outside of it. Though this doesn't have much to do with the rest of what I am saying here, there is no such object as "gravitational potential" in general relativity. That concept is a relic from Newtonian gravity and is only discussed in GR in the case of low velocities and weak gravitational fields, where GR reduces to Newtonian gravity as an approximation. Again, not in a coordinate-independent sense. Actually, Schwarzschild coordinates would have the infalling clock falling with a slower and slower coordinate velocity (it's supposed to shoot off the top of the Schwarzschild coordinate chart, remember?). If you are referring to the metric components \(g_{\mu\nu}\), then those are coordinate-dependent and so not 'measurable' in any meaningful sense of the word. I never said either. (Though a metric expression has information about the geometry of spacetime encoded in it, among other things.) Since I read this thread as a discussion about general relativity, which is a 100-year-old object of study with a formal mathematical definition, this discussion is unequivocally about the map. Whether that map is a good one or as fictional as a map of Narnia is an important but conceptually different matter than what we're discussing here. In general relativity the map is spacetime. This is made explicitly and unambiguously clear right from the beginning of any formal development of general relativity, including in Einstein's own 1916 paper. The territory is reality which we have no sixth sense about, and we can only discuss how good various maps we have of it are. Spacetime is inhomogenous in the sense that it is attributed properties (most notably, curvature), that are not required to be constant everywhere in space and time. That is true, is what Einstein says in one of your favourite passages from the Leyden address, and is not disputed by anyone. You've in the past put Einstein's 1916 paper on general relativity in front of me. I've read through it enough to convince myself it is essentially the same GR I learned in two university courses I followed as well as a number of textbooks, lecture notes, etc. I've dipped in in more or less detail. They are all describing the same theory. If there is an alternative version of GR, why isn't there a complete treatise on it in the literature, say something analogous to Einstein's 1916 paper? Where is this alternative version of GR actually formally developed and advanced? Why is it that the best you seem able to do is come up with scattered quotes mined from different places and a 2008 article in Chinese Physics Letters that is hardly contemporaneous with Einstein?
I can give you a coordinate mapping where I put tomorrow at t = infinity. I can give you an example in flat spacetime where certain events occur at t = infinity even in a coordinate system that is not defined deliberately just to make it that way. In fact, I previously have: this sort of thing happens in accelerating reference frames in SR. So a coordinate becoming infinite does not necessarily mean anything and cutting off GR black hole solutions at the event horizon is arbitrary. By the definition you are using (which I'm fairly convinced is not the definition that the source you are quoting is using), black hole solutions do not represent "maximally extended" spacetimes. That said I don't much like your definition since saying black hole solutions are not "maximally extended" carries the implication that we can actually extend them. If you know a way of extending a black hole solution past the singularity in a well-defined and non-arbitrary way, I'm all ears.
But as I said, such a coordinate mapping would not have a time parameter which is analogous to our clocking devices. In the case of accelerating reference frames I would be curious to see what you're referring to; growing without bound is different from infinity. I do. Apply the Schwarzschild coordinate time predictions around the event horizon to the center of mass at the point of creation, and take them literally.
Neither, as I have already explained, do Schwarzschild coordinates. Schwarzschild and Schwarzschild-like coordinates do have some practical use and they are arguably "natural" in some limited respects, but that mostly comes from black hole metrics being static in those coordinates, and not the time parameter being "analogous to our clocking devices", whatever that's supposed to mean. For example, suppose you are sitting at some constant Schwarzschild radius \(r_{1}\) and you are looking at an object sitting at some constant Schwarzschild radius \(r_{2}\). Then the object is remaining the same distance from you in the intuitive sense that it actually looks that way. It will also appear to tick slower or faster according to the gravitational time dilation rate you can work out from the metric, in the sense that the time dilation rate you calculate coincides with with the Doppler shift you observe. But that only holds for objects and observers remaining at constant Schwarzschild radii. You do what? Propose an extension of GR black hole metrics beyond the singularity itself? I'm not following. Are you referring to stellar collapse models? If so, from memory the interior of the star is normally defined in terms of infalling coordinates, and I think the solution has the black hole forming within finite t Schwarzschild coordinates (though it's been a while and I could be wrong). I've saved this comment for last, as the explanation is a little longer. I am referring to the fact that there is a standard and fairly "natural" way of defining accelerating coordinate systems in SR, but they don't cover the whole of spacetime and some 'finite' events in spacetime are attributed infinite coordinate values. Briefly, for an accelerating coordinate system attached to an accelerating observer, the basic idea behind the construction goes something like this: at any point along the accelerating observer's worldline, there is an inertial coordinate system in which they are instantaneously at rest. At that point, you use that instantaneous inertial rest frame's spatial axes and coordinates as the accelerating frame's spatial axes and coordinates. This defines both the spatial coordinates and the simultaneity convention. For the time coordinate, you simply use the observer's accumulated proper time. I [POST=2631153]previously[/POST] gave you a Minkowski diagram for a constantly accelerating observer, which I'll recycle here: Please Register or Log in to view the hidden image! Suppose we call the inertial coordinates depicted in the diagram \(z^{\mu} \,=\, (t,\, z)\) and work in units where \(c = 1\). In terms of such coordinates, the worldline of an accelerating observer experiencing constant proper acceleration \(a\) might be described by the trajectory \( \begin{eqnarray} t &=& \frac{1}{a} \, \sinh(a\tau) \,, \\ z &=& \frac{1}{a} \, \cosh(a\tau) \,, \end{eqnarray} \) or more compactly as \(z^{\mu}(\tau) \,=\, \bigl( \frac{1}{a} \, \sinh(a\tau) ,\, \frac{1}{a} \, \cosh(a\tau) \bigr)\), in terms of the accelerating observer's proper time \(\tau\). Such a worldline is qualitatively depicted on the Minkowski diagram above. According to the construction I described above, the accelerating observer's spatial axis at any point along the trajectory has to coincide with the spatial axis of the inertial frame the observer is instantaneously at rest in at that point. One way to identify this axis is that it has to be orthogonal to the observer's instantaneous "time" axis, which points in the same direction as their four-velocity vector. The four-velocity associated with the trajectory given above is \( u^{\mu}(\tau) \,=\, \frac{\mathrm{d}z^{\mu}}{\mathrm{d}\tau}(\tau) \,=\, \bigl( \cosh(a\tau) ,\, \sinh(a\tau) \bigr) \,. \) It is easy to find a vector that has zero Minkowski product with \(u^{\mu}(\tau)\): one is already given by the the position vector \(z^{\mu}(\tau)\) defining the worldline itself. So the accelerating observer's spatial axis, at every point along their worldline, always intersects with the coordinate origin \(z^{\mu} = (0,\, 0)\) on the Minkowski diagram above. Calling the accelerating observer's spatial coordinate \(x\), you can work out (or check, or ask me for the details), that the inertial coordinates \(t\) and \(z\) are related to the accelerating observer's coordinates \(\tau\) and \(x\) by: \( \begin{eqnarray} t &=& \bigl( \frac{1}{a} \,+\, x \bigr) \, \sinh(a\tau) \,, \\ z &=& \bigl( \frac{1}{a} \,+\, x \bigr) \, \cosh(a\tau) \,. \end{eqnarray} \) This puts the accelerating observer himself at \(x = 0\). The metric expression in the accelerating coordinates is also easy to work out: \(\mathrm{d}s^{2} \,=\, -\, (1 \,+\, ax)^{2} \, \mathrm{d}\tau^{2} \,+\, \mathrm{d}x^{2} \,.\) So the \(x\) coordinate measures distance in the same units as the inertial system, as you might reasonably expect, and you can read the gravitational time dilation factor of \(1 \,+\, ax\) straight off the metric. (Compare the latter with \(T_{d} \,=\, 1 \,+\, gh/c^{2}\) given on Wikipedia.) Now for some features of this accelerating coordinate system: 1) As pointed out above, the accelerating frame's \(x\) axis always crosses through the origin \(z^{\mu} = (0,\, 0)\). This means that the origin on the Minkowski diagram gets stretched out to an entire line in the accelerating system's coordinates. Specifically, all the coordinates \(x^{\mu} = \bigl(\tau,\, -\frac{1}{a}\bigr)\), for all \(\tau\), represent the same point \(z^{\mu} = (0,\, 0)\) in Minkowski spacetime. A clock passing through the origin would be described as permanently frozen in the accelerating frame. 2) From the coordinate transformation above, \(z^{2} \,-\, t^{2} \,=\, (1 \,+\, ax)^{2} \,\geq\, 0\), regardless of \(x\), so the accelerating coordinate system doesn't cover the whole of spacetime. It only covers the region \(|z| \,>\, |t|\), which would look like a bowtie centred on the origin on the Minkowski diagram above. Events along the lightcone \(|z| \,=\, |t|\), except for the origin \(z = t = 0\) itself, correspond to \(\tau \,=\, \pm \infty\). Events in the future and past lightcones of the origin simply aren't covered by the accelerating system at all. So most timelike trajectories you might draw on the Minkowski diagram will shoot off beyond \(\tau \,=\, +\infty\) in the accelerating reference frame, and in finite proper time. 3) You might have noticed that for events at \(x \,<\, -\,\frac{1}{a}\) or \(z \,<\, 0\) (the left hand side of the "bowtie"), the accelerating reference frame's time axis gets flipped. In this region, if two events A and B are timelike separated and \(\tau_{\mathrm{A}} \,>\, \tau_{\mathrm{B}}\), then A occurs before B. 4) Like I explained in my old post, there is an event horizon of sorts associated with a constantly-accelerating observer. From points 1) and 2), a single point on that event horizon gets streched out to the entire line \(\bigl(\tau,\, -\frac{1}{a}\bigr)\) in the accelerating frame, while most of it only appears at \(\tau \,=\, +\infty\). If some of this sounds problematic or "paradoxical", then the resolution is simple. You simply recognise that an accelerating reference frame is not an inertial reference frame and its coordinates do not have the full physical significance that inertial coordinates naturally do. It's really best to think of accelerating reference frames as an idealisation that might approximately describe the "point of view" of an accelerating observer in their immediate vicinity provided they're not accelerating too rapidly, but that become progressively less meaningful if you try to take them too seriously or push them too far. Moral: just because a coordinate system seems "natural" in a certain context does not guarantee that it won't suffer pathological features.
