Notions of simultaneity in strongly curved spacetimeby PAllen Tags: curved, notions, simultaneity, spacetime, strongly 

#91
Nov2212, 09:10 AM

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PF Gold
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However, underlying all of this is just one equation, the EFE. That equation is what's really at issue here. See below. Just an "equation" isn't enough; you have to add constraintsinitial/boundary conditionsto get a particular solution. Which solution of the equation you geti.e., which spacetime geometry models the physical situation you're interested independs on the constraints. (1) The "gravitational field" is the metric. The metric (the coordinatefree geometric object, not its expression in particular coordinates) is perfectly finite and continuous at the horizon, and for reasons that both PAllen and I have explained, it can't "just stop" at the horizon without violating the EFE. (2) The "gravitational field" is the Riemann curvature tensor. Like the metric, this is perfectly finite and continuous at the horizon. (3) The "gravitational field" is the proper acceleration experienced by a "hovering" observer (an observer who stays at the same radius and does not move at all in a tangential direction). This *does* increase without bound as you get closer and closer to the horizon. However, there is *no* "hovering" observer *at* the horizon, because the horizon is a null surface: i.e., a line with constant r = 2M and constant theta, phi is not a timelike line; it's a null line (the path of a light raya radially outgoing light ray). So there is no observer who experiences infinite proper acceleration, and this definition of "gravitational field" simply doesn't apply at or inside the horizon. As far as I can see, the only possible basis you could have for claiming that "the physical reality of the gravitational field" means that the clock's worldline stops as tau>42, would be #3. However, #3 doesn't apply to infalling observers; it only applies to accelerated, "hovering" observers. Infalling observers don't feel any acceleration, so there's nothing stopping them from falling through the horizon. The "gravitational field" in the sense of #3 is simply not felt by them at all. Note that in all these cases, the physical "field" has to correspond to something invariant in the mathematical model, *not* something that only exists in a particular coordinate chart. That is something Einstein would have *agreed* with. Note also that none of the definitions of "gravitational field" I gave above used Schwarzschild coordinate time, or the fact that t>infinity as you approach the horizon. Einstein simply didn't understand that claims about t>infinity as you approach the horizon were claims about something that only exists in a particular coordinate chart. 



#92
Nov2212, 09:11 AM

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But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities. 



#93
Nov2212, 09:58 AM

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#94
Nov2212, 09:58 AM

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Peter: What is "the gravitational field"? It is not a real mathematical object Einstein: What is a "region of spacetime"? It is not a real physical object. In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless. 



#95
Nov2212, 10:01 AM

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Once more: According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation. In that different interpretation, which I still don't fully understand, the clock will pass the horizon despite Schwartzschild's t=∞. For details, see the ongoing discussion: http://www.physicsforums.com/showthread.php?t=651362 incl. an extract of OppenheimerSnyder: http://www.physicsforums.com/showpos...5&postcount=50 



#96
Nov2212, 10:06 AM

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#97
Nov2212, 10:19 AM

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PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me. 



#98
Nov2212, 12:34 PM

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What do I mean by a removable singularity? [quote] http://en.wikipedia.org/w/index.php?...ldid=507006469 IT does takes a bit of work to decide if the apparent singularity is the result of a poor coordinate choice , or is an inherent feature of the equations. It might be helpful to give a quick example of how this happens. Consider the equations for spatial geodesics on the surface of the Earth. (Why geodesics? Because that's how GR determines equation of motion. So this is an easytounderstand application of the issues involved in finding geodesics). If you let lattitude be represented by [itex]\psi[/itex] and longitude by [itex]\phi[/itex], then you can write the metrc [itex] ds^2 = R^2 (d \psi^2 + cos^2(\psi) d\phi^2)[/itex] and come up with the equations for the geodesic (which we know SHOULD be a great circle) for [itex]\psi(t)[/itex] and [itex]\phi(t)[/itex] [tex] \frac{d^2 \psi}{dt^2} + \frac{1}{2} \sin 2 \psi \left( \frac{d \phi}{dt} \right)^2 [/tex][tex] \frac{d^2 \phi}{dt^2}  2 \tan \psi \left(\frac{d\phi}{dt}\right) \left( \frac{d\psi}{dt} \right) = 0 [/tex] Now, one solution of these equations is [itex]\phi[/itex] = constant. It makes both equations zero. It's also half of a great circle. But, if we look more closely, we see that we have a term of the form 0*infinity in the second equation as we approach the north pole, because of the presence of [itex]\tan \psi [/itex] when [itex]\psi[/itex] reaches 90 degrees. THis apparent singularity is mathematical, not physical. If you're drawing a great circle around a sphere, there's no physical reason to stop at the north pole. Of course we already know what the answer is  we need to join two half circles together. In particular, we know we need to splice together two half circles, 180 degrees apart in lattitude, though as far as I know all the solution techniques (change of variable, etc) are equivalent to not using lattitude and longitude coordinates at the north pole, because the coordinates are illbehaved there. The same is in the black hole case, though to justify it you need to either do the math yourself, or read a textbook where someone else has. Note that you probably won't find this sort of thing in papers so much, it's assumed everyone knows it in the literature. Where you're more likely to find an explanation in a textbook or lecture notes. Which brings me to the next point. We don't have textooks online, but we've got several good sets of lecture notes. What does Carroll's lecture notes have to say on the topic? He defines the geodesic equation of motion  they're pretty complex looking, and I wouldn't be surprised if you didn't want to solve them yourself. But what does Caroll have to say about solving them? I'll give you a link http://preposterousuniverse.com/grno...otesseven.pdf, and a page reference (pg 182) in that link. Then I'll give you some question 1) Does Carroll support your thesis? Or does he disagree with it? 2) What do other textbooks and online lecture notes have to say? And for my own information 3) Do you think you know the difference between "absolute time" and "nonabsolute time" 4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time? 



#99
Nov2212, 01:02 PM

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Consider an ordinary boring constantvelocity special relativity problem: You are rest and you watch me passing by at some reasonable fraction of the speed of light, so you observe that my clock is ticking more slowly than yours. If the universe has a a finite age, it is certainly possible for me to observe a time on my clock that you will claim will never be reached  all that necessary is that: 1) I get to read my clock on my worldline before it terminates at the end of the universe. 2) Your worldline terminates at the end of the universe before it intersects the line of (your) simultaneity through the event of me reading my clock. But, you will say, I'm cheating by introducing this arbitrary "end of the universe" to cut off your worldline (actually, you introduced it  I'm just abusing it )before it can intersect the relevant line of simultaneity. If I didn't do that, then no matter how much of my time passes before I read my clock, you'd be able to extend your worldline to intersect the line of simultaneity. That is true enough, but then again the entire concept of "line of simultaneity" only really makes sense in flat space. The bit about a "Kruskal observer" is a red herring. The geometry around a static noncharged nonrotating mass is the Schwarzchild geometry, no matter what coordinates we use, and the only meaningful notion of time that we have is proper time along a timelike worldline. The Kruskal coordinates allow us to calculate the proper time along the infalling clock's worldline as it crosses the Schwarzchild radius, whereas the the Schwarzchild coordinates (as opposed to geometry) do not. So it's not that the "Schwarzchild observer" and the "Kruskal observer" are producing conflicting observations, it is that the Kruskal coordinates are producing a prediction for the infalling observer's worldline and the Schwarzchild coordinates are not. 



#100
Nov2212, 01:46 PM

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We have global coordinate system where we know how to get from one place to another i.e. it provides connection, but this global coordinate system does not tell anything useful about distances and angles and such. And then we have another coordinate system that tells us distances and angles but it works only locally, meaning that if we have two adjacent patches with local coordinate systems we don't know how to glue them together. So we take take global coordinate system with metric that will give us geometric values in accord with local coordinate systems. Something like that. Only I don't know how to check if this is right. 



#101
Nov2212, 07:37 PM

P: 1,162

Example: Unbounded coordinate acceleration of a system under constant proper acceleration as t >∞ Mathematically you can say this resolves to c but in this universe as we know it or believe it to be, this is not the case. What you are doing here seems to me to be equivalent to integrating proper time of such a system to the limit as v >c to derive a finite value. Thus demonstrating that such a system could reach c in finite time even if it never happens according to external clocks.. The analogy is particularly apt as by assuming the free faller reaches the horizon this is also equivalent to reaching c relative to the distant static observer yes?? What difference do you see between the two cases???? In both cases it is equivalent to directly assuming reaching c or the horizon independent of determining whether they could actually arrive there. And then determining a temporal value for your assumption. Just MHO If you are directly integrating the metric without reference to coordinate time isn't this actually integrating an infinitesimal series of static clocks between infinity and 2M??? It seems to me that either the Schwarzschild metric accurately corresponds to reality outside the hole or it doesn't. But the idea that its perfectly true up to some indeterminate pathological point "somewhere" in the vicinity of the horizon seems very shakey. Actually the idea of a horizon as a third sector of reality between inside and outside seems like a pure abstraction. Is there a surface between air and water? 



#102
Nov2212, 07:59 PM

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What if the traveling twin hangs out close to the horizon for a time before traveling back to his distant outside brother. Does his younger age indicate time slowing down at the horizon? Does it depend on an absolute time??? Is it a coordinate effect??? 



#103
Nov2212, 09:27 PM

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There's the metric that Schwarzchild discovered as a solution of the Einstein field equations. It corresponds to reality (assuming spherical symmetry, no charge, no rotation, static  the conditions under which the SW metric is solution of the EFE) inside the event horizon, outside the event horizon, and at the event horizon itself. Then there are Schwarzchild coordinates, which we often use when we want to write that metric down in a particular coordinate system. These coordinates do not work well at the event horizon. That doesn't mean that there's anything wrong there with the spacetime described by the Schwarzchild solution to the EFE; it just means that we should use some other coordinates to describe the metric there. 



#104
Nov2312, 12:58 AM

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You might well say that discussions of that type are useless too; I agree to the extent that I think quoting authorities is useless if the objective is to talk about the physics. We should be able to talk about the physics without caring what Einstein, Oppenheimer, Schwarzschild, or anyone else thought; we can talk about the mathematical model and its physical interpretation directly. You're having trouble understanding how the things PAllen and I and others have been saying about the mathematical model can all be consistent with each other; fine, I understand that. But it does no good to quote Einstein or anyone else; either you are able to construct the model yourself, or you're not. If you're not, IMO you need to learn how to do so before criticizing itor else you should be able to show your partial construction of the model and exactly where you are hitting a stumbling block. It seems to me that your current stumbling block is the fact that t>infinity as tau>42; you appear to think that this requires the infalling object to never reach tau>=42. What is your argument for this? By which I mean, what are the specific logical steps that get you from "t>infinity as tau>42" to "tau can't be >=42", and what assumptions do they depend on? I know it seems obvious to you, but it's not obvious to me, because I have a consistent mathematical model that shows how tau>=42 is possible despite the fact that t>infinity as tau>42. So one or the other of us must have a mistaken assumption somewhere. Let's see if we can find it. If it will help, I can post *my* logical argument; but that will have to wait for a separate post. 



#105
Nov2312, 01:04 AM

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The asymptotic observer may try to *interpret* this prediction as showing that the infalling observer's clock will slow down so much that it will not reach 3:00 pm before the end of this universe. But that interpretation depends on additional assumptions, such as the adoption of a particular simultaneity convention for distant events. As PAllen has pointed out repeatedly, simultaneity conventions are just that: conventions. They can't be used as the basis for making direct physical claims like those you are trying to make. Btw, all this talk about different "observers" making different predictions is mistaken as well. Predictions of physical observables are the same regardless of which coordinate chart you adopt. Also, which coordinate chart you adopt is not dictated by which worldline in spacetime you follow; there is nothing preventing the "asymptotic observer" from adopting Kruskal coordinates to do calculations. 



#106
Nov2312, 02:04 AM

P: 3,178

I let myself be held up by the continuing conversation in this thread. Consequently I will not anymore reply in this thread until that is done. http://www.physicsforums.com/showthr...=651362&page=6 PS (in contradiction to my remark above  but I won't add another post for the time being!): 



#107
Nov2312, 09:51 PM

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Consider the problem of making a map of the earth. The issue is that the Earth's surface is curved, and our paper is not. If we do a straightforwards projection, we can make a map that is "to scale" near any particular point we choose. (The further away we are from the point, the more distorted the map gets). Occasioanlly you'll see maps like this  looking up the topic for definitess, I find Goode homolosine projection : http://en.wikipedia.org/w/index.php?...ldid=508879282 So to summarize, using the example of the Earth's curved surface as a model for the similar problem of making maps of curved spacetime. Global coordinate information (lattitude and longitude in our example) does exist and does provide information on distances and angles, but the information requires decoding. We can map the surface of the Earth in a variety of ways, but while we can't make the resulting map projections appear to be in one piece and drawn to scale on a flat piece of paper. 



#108
Nov2312, 09:59 PM

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He doesn't appear to have responded to my question on the point when I asked. Perhpas he just missed it. http://preposterousuniverse.com/grno...otesseven.pdf around pg 182. Perhaps I should quote it, but I'm hoping to try and motivate people to look up references. 


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