Notions of simultaneity in strongly curved spacetime


by PAllen
Tags: curved, notions, simultaneity, spacetime, strongly
PAllen
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Nov19-12, 12:04 PM
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Quote Quote by harrylin View Post
However, that brings us immediately to the real sticking point that has all the time been lurking over the discussions of the last weeks:

If you use a valid coordinate system, then there is nothing against it. The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.

We know that this can happen with earth maps; however that is without consequence, as it's easily verified (I can give a simple example). It appears that the same problem occurred in GR, but without the possibility for a direct "reality check".

On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?
For the case of earth maps, do you claim there is case of conflicting prediction for maps as used in differential geometry:

- associated with each map is a metric expression, such that each map expressed the same geometry
- if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere.

Who decides what is a valid coordinate system? Differential geometry has a well defined, precise, answer to this (see any definition of topological manifold, refined further to become a pseudo-riemannian manifold).

If you think this is wrong, then what is your precise criteria for a valid coordinate system? If it is different from above, you have a new theory, not GR as understood by everyone else. And in this new theory, general covariance is rejected, because that requires that any coordinates allowed by the criteria in the prior paragraph or good.

One possible analogy for our disagreement is:

- Imagine a 2-sphere in polar coordinates. Bob doesn't like what happens at or near the poles. So Bob decides to analyze only different object: a sphere missing a little disk around each pole. This is a valid, different geometric object. It is easy to demonstrate that you have holes using only polar coordinates with metric.

Now in the case of O-S collapse, the hole you are proposing 'must' be accepted as the correct prediction of GR is rather strange. A clock in the middle of collapsing dust ball stops for no reason. It stops in a strange sense - locally everything proceeds at a normal rate until it is declared to stop.

Note that for Krauss, et. all, assuming their quantum simulation is correct, they have good physical justification for this - this central clock is not acatually stopping; it evaporates in finite local time. Then it makes sense to talk about chopping a classical model at similar point.

If, instead, you accept the the interior clock proceeds normally, there is no escaping (using any coordinates), that the clock is proceeding for some time after an event horizon has formed around it. Any signals it sends will not escape, but it can readily receive signals from an external observer.
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Quote Quote by zonde View Post
With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.
No, this is not correct. The correct statement is: SC type time slicing cannot *cover* the EH and interior region.

Quote Quote by zonde View Post
In order to have EH and interior region with SC type time slicing you have to have eternal BH.
This is not correct either. SC type time slicing cannot cover the EH and interior region for *any* black hole spacetime; it only covers the exterior. But in both cases you cite (collapsing mass and eternal BH), the EH and interior region are part of the spacetime; they are just not covered by the SC type time slicing.
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Nov19-12, 12:17 PM
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Quote Quote by zonde View Post
So basically you are saying that simultaneity is a convention, right?
But it does not mean that you can use different conventions at the same time.
Who says? It is no different than saying: I have two problems in analytic plane geometry. One is easier to compute in cartesian coordinates, one in polar coordinates. So I do one calculation one way, the other a different way.
Quote Quote by zonde View Post

With SC type time slicing there is no EH and no interior region for collapsing mass. All you can get is "frozen star". EH appears at infinite future i.e. never.

In order to have EH and interior region with SC type time slicing you have to have eternal BH.
This is just false. For an O-S collapse, there is always an interior because it is a collapsing ball of dust. If you restrict yourself to what a distant observer sees, what they see is a ball that freezes throughout, at a radius just larger than the SC radius.

Outside the frozen ball, you can apply SC coordinates and metric (or others). Inside (where matter is), you must do something else.

Now, SC time slicing is a specific choice of simultaneity (of which there are infinite choices, including simple, physically defined ones: http://physicsforums.com/showpost.ph...0&postcount=23 ). If I pick a different choice, and use it throughout, then the external observer assigns well defined time coordinates to the dust ball collapsing inside the horizon, and even a well defined time to the formation of a singularity.
PeterDonis
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Nov19-12, 12:20 PM
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Quote Quote by harrylin View Post
The issue is about what kind of coordinate systems are valid in GR, and if perhaps contradictory mapping models can be made that match the mathematics of GR (but perhaps not all equally well matching the foundations), thus resulting in contradictory predictions.
I don't think this can happen, because any valid coordinate system in GR has to preserve geometric invariants, and all of the physical predictions of GR depend only on geometric invariants. So any valid coordinate system in GR must lead to the same physical predictions as any other valid coordinate system.

Quote Quote by harrylin View Post
We know that this can happen with earth maps; however that is without consequence, as it's easily verified (I can give a simple example).
Please do; I don't understand what you're referring to here. See my comments above about what a "valid" coordinate system is. Any valid map of the Earth would also have to preserve geometric invariants; that is, you would have to be able to calculate, say, the correct great-circle distance between New York and Sydney using any valid map (though the calculation might be easier in some maps than in others). Note that this is *not* the same as how the distance "looks" on the map: the NY-Sydney great circle looks very different on a Mercator projection than it does on a stereographic projection, but both allow you to calculate that the physical distance is the same; it's just represented differently in terms of the coordinates.

Quote Quote by harrylin View Post
On a side note there is a somewhat similar case in SR, with tachyons. Are tachyons really SR? Must they exist if one can "fix" the math to contain their mathematical possibility?
I'm not aware of any requirement that everything mathematically possible according to any theory must exist. The maximally extended Schwarzschild spacetime, including a white hole and a second exterior region, is mathematically possible in GR, but nobody, AFAIK, thinks it's physically possible.
zonde
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Nov19-12, 12:42 PM
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Quote Quote by PAllen View Post
Who says? It is no different than saying: I have two problems in analytic plane geometry. One is easier to compute in cartesian coordinates, one in polar coordinates. So I do one calculation one way, the other a different way.
Hmm, I meant it differently. You can't use different conventions within single calculation/reasoning. Because there can be convention dependent statements.

Quote Quote by PAllen View Post
This is just false. For an O-S collapse, there is always an interior because it is a collapsing ball of dust. If you restrict yourself to what a distant observer sees, what they see is a ball that freezes throughout, at a radius just larger than the SC radius.

Outside the frozen ball, you can apply SC coordinates and metric (or others). Inside (where matter is), you must do something else.
With "inside" I mean "inside EH" not "inside gravitating mass".
And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.
PAllen
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Nov19-12, 01:18 PM
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Quote Quote by zonde View Post

With "inside" I mean "inside EH" not "inside gravitating mass".
And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.
OK, if you assume a transparent pressure less dust, and do this radially from infinity, you get this SC time slicing outside and something similar inside. The whole point of this thread is that neither SR nor GR say this is the only allowed way to slice spacetime. And it is provable that you have hole in spacetime (world lines of particles that just end for no reason, at finite proper time along them), or a region not covered by these coordinates.

The point of this thread, is that you can choose many other intuitive simultaneity conventions for a distant observe (and simultaneity convention identically equals time sicing), which don't have a hole, and assign finite time coordinates to events inside the EH, which itself labeled with a finite time coordinate.
PeterDonis
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Nov19-12, 02:18 PM
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Quote Quote by zonde View Post
And you can apply SC type time slicing inside collapsing mass. With SC type time slicing I mean equal time for forward and backward trip of light signal (after factoring out dynamics of collapse) just like it is for outside coordinates.
You can do this, as PAllen said, but it's important to note that if you do, this time slicing still won't cover the horizon or the region of spacetime inside it. That includes the portion of the collapsing matter that is inside the horizon.
pervect
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Nov19-12, 03:52 PM
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I've got several comments:

On the map issue. There ARE coordinates that are mildly "special" at any given point. These coordinates are the ones where your map is drawn to scale. The mathematical feature of such maps is that the metric is diag(-c, 1, 1, 1).

Because these maps are to scale, one can freely interchange coordinate distances (i.e. changes in coordinate) with physical distances. They're easy to work with. Discussions don't go off in as many strange tangents when you use maps that are to scale, and people don't get "lost" so much, I find.

Some recent discussion reads to me something like this:

"Your'e reading the map wrong - here , use this one, It's almost to scale - well, not really to scale, actually, but the distortions are at least finiite."

"No - I like the one that's infinitely distorted better, because - nevermind why, I just like it better."

People for some reason seem to have a really hard time dealing with distances in GR but if done correctly it's not that complex - just make sure your map is to scale (and if its not, learn enough math to covert your maps so they are to scale, this can reasonably be motivated with as little math as algebra).

The only remaining issue with maps is the SR issue. This is understanding that time is not absolute, that simultaneity is relative, that as you change your notion of simultaneity your notion of distance also changes correspondingly because space-time is a continuum, and space and time are fundamentally linked.

Mathematically: the maps of SR and GR preserve the Lorentz interval.

This is where I keep feeling the communication is lacking - but many people who say they "get" this point obviously don't :-(.

The notion of "now" is relative. The local notion of now is determined by backwards compatibility with Newton's laws, but said notion of now (necessary for this backwards compatibility) is purely local, and not universal. This relativity means that this, or any other , notion of "now" is not used to determine cause and effect, but rather one uses light cones. To put it succinctly, "now" is relative, light cones are absolute.

The last issue. I think there are some people who believe in "white holes", but not in the context of classical GR. See the thread on Nikodem Poplawski I started
harrylin
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Nov19-12, 04:33 PM
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Quote Quote by PAllen View Post
For the case of earth maps, do you claim there is case of conflicting prediction for maps as used in differential geometry:

- associated with each map is a metric expression, such that each map expressed the same geometry
- if one is talking about the same sphere using different maps, and one map doesn't cover all of the sphere, you use other maps to cover the rest, such that you are always describing the same complete sphere.[..]
No, I meant nothing like that. Instead my illustration is about different mapping systems, which are related by conformal transformations. It's a very simple example that I thought of when I awoke one morning at the time that we were discussing Hamilton's model, which as we all agreed, in the way he literally pictures it isn't exactly GR. My example is a bit silly, please don't laugh - but you may smile.

There is a flatland country near the equator, and sailors are setting out on voyages to distant destinations. Now comes a cartographer, who decides to make a map that can serve as a travel guide. Based on intuition (inspiration?) he then develops a map of the world with the strange property that it has medians that converge when going away from the equator towards the North. As a result everything ends in a singularity, which he gives the name North Pole. Thus it appears that one cannot get further away than the North pole, which is intellectually unsatisfactory.

Then comes along a different cartographer who thinks up a conformal transformation that results in a very similar map, but now with the medians running parallel. The funny thing of that map is that the singularity is gone; on that map people can continue beyond the North pole. Of course, the two maps can be transformed from one into the other without problems up to that singularity, but predictions definitely differ from that point onward.

Now they have a problem; at best one of the two will match reality. Either Flatlander Earth physics law can tell them which one to choose as the most correct one, or Earth science is incomplete, so that the flatlanders don't know yet which mapping would best, even just in theory.
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Quote Quote by PeterDonis View Post
I don't think this can happen, because any valid coordinate system in GR has to preserve geometric invariants, and all of the physical predictions of GR depend only on geometric invariants. So any valid coordinate system in GR must lead to the same physical predictions as any other valid coordinate system.
I agree with you about valid coordinate systems. However, it appears that the disagreements in the physics community that I noticed relate to the issue of what is valid. If so, then this is a "hot potato".

Note: regretfully I can merely choose like a voter about a political issue: Yes, I do have an opinion, and No, I probably don't have enough GR expertise to make a case. So, please regard me as a science reporter who meddles in expert discussions and asks annoying questions.
[..] I'm not aware of any requirement that everything mathematically possible according to any theory must exist. The maximally extended Schwarzschild spacetime, including a white hole and a second exterior region, is mathematically possible in GR, but nobody, AFAIK, thinks it's physically possible.
Yes, that is what I think too. I got the impression that PAllen was heading for the contrary opinion, although within certain limits.
PAllen
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Nov19-12, 04:56 PM
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Quote Quote by harrylin View Post
Yes, that is what I think too. I got the impression that PAllen was heading for the contrary opinion, although within certain limits.
Not at all. I believe (physically intuit?) that tachyons are not likely to exist in our universe; nor white holes; nor closed time like curves; nor super-extremal kerr black holes; nor alcubierre drive; nor actual singularities. I also admit that all of these have mathematically consistent treatment.

But in each of these cases, I see a clear reason to 'draw a line'. Tachyons have to be added to SR in violation of the causal interpretation of SR. White holes have no process by which they can form. Similarly, for most of the others, there is no known process they can form out of plausible initial conditions. Singularities I interpret as a clear sign that GR has broken in this domain.

With black holes, treated classically, we have, instead, the singularity theorems: With almost any reasonable starting point, once collapse has gotten close to a critical radius, it must proceed all the way to a singularity. Further, all the approaches, classically, to try to avoid an event horizon formation amount to my example chopping the poles off a sphere because I don't like what they do to my coordinate preference.
PeterDonis
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Quote Quote by harrylin View Post
I agree with you about valid coordinate systems. However, it appears that the disagreements in the physics community that I noticed relate to the issue of what is valid.
I honestly haven't seen this in what I've seen of the reputable physics community (which is not a lot as I am not an academic). There is certainly a lot of disagreement, but it doesn't look to me like disagreement about which coordinate systems are valid. It looks to me like disagreement on which physical principles should be retained, and which discarded, when they conflict.

For example, the black hole information loss problem basically comes down to: which do you want to keep when push comes to shove, general relativity or quantum mechanics? GR predicts that gravitational collapse leads to the formation of regions of spacetime from which information can't get back out; that means a quantum state that falls in gets lost, so unitarity, a central principle of QM, is violated. Hawking's position (at least until about 2004-ish; he seems to have switched sides in what Susskind calls the "Black Hole War" about then) was, so much the worse for QM. Susskind's and t'Hooft's position (there were others, but they seem to have been the primary ones who held to this position throughout) was, so much the worse for GR. They can't both be right. But nobody, as far as I can see, was arguing one way or the other based on which coordinates were valid and which weren't.

Quote Quote by harrylin View Post
Note: regretfully I can merely choose like a voter about a political issue: Yes, I do have an opinion, and No, I probably don't have enough GR expertise to make a case. So, please regard me as a science reporter who meddles in expert discussions and asks annoying questions.
Fair enough. You ask better questions than a lot of science reporters do.
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Nov19-12, 11:29 PM
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Quote Quote by PAllen View Post
OK, if you assume a transparent pressure less dust, and do this radially from infinity, you get this SC time slicing outside and something similar inside. The whole point of this thread is that neither SR nor GR say this is the only allowed way to slice spacetime. And it is provable that you have hole in spacetime (world lines of particles that just end for no reason, at finite proper time along them), or a region not covered by these coordinates.
There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.

And wordlines of particles do not end. They just extend toward infinite future.
You would not claim that there is some problem with this expression, right?
[tex]\lim_{t\to\infty}f(t)=a[/tex]
Let's say that t is coordinate time and f(t) is proper time of infalling particle.

Quote Quote by PAllen View Post
The point of this thread, is that you can choose many other intuitive simultaneity conventions for a distant observe (and simultaneity convention identically equals time sicing), which don't have a hole, and assign finite time coordinates to events inside the EH, which itself labeled with a finite time coordinate.
You can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.
PAllen
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Nov19-12, 11:36 PM
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I thought it would be useful to do something Peter Donis suggested in another thread. That is, to look classically at the complete geometry of a collapsing shell of matter, which is like the Krauss case. I found a couple of references, and have done some order of magnitude calculations based on the arxiv paper (adjusted for a matter shell rather than a null dust shell).

The follwoing gives a Kruskal diagram (and other coordinates) for a collapsing shell. This one is pressure-less, and only the final stages (where the matter is almost light like) is shown (the r=0 line and the shell line need to slanted closer to vertical to consider a collapse from rest not too far from the SC radius):

http://casa.colorado.edu/~ajsh/collapse.html

This paper discusses collapse of a null dust shell, including Kruskal coordinates (section 4):

http://arxiv.org/abs/gr-qc/0502040

While this is not quite the case of interest, the techniques shown for matching interior and exterior are general.

I think the interesting case to consider is a clock sitting inertially at the center of a collapsing shell, under the further assumption that the shell is transparent. There are a number of interesting features:

- The spacetime inside the shell is pure Minkowski flat spacetime; there are no tidal forces at all. There is no matter density at all. In the whole region inside the shell, SR physics applies. The analog of this is obviously true for Newtonian physics; it is also well known that this is exactly true for GR as well (that inside a spherical shell, physics is indistinguishable from SR with no gravity).

- The clock inside operates according to SR physics, time locally flowing normally, until shell collapse reaches singularity. The clock at the center cannot detect the shell passing the event horizon. Nothing about physics within the shell changes on approach to or passing the event horizion - the interior is pure SR until the singularity.

- The clock at the center sees the external 'universe' proceeding normally except highly (but finitely) blue shifted until the moment of singularity. Here, the clock world line ends for a very physical reason: the shell has collapsed to a singularity, bringing the clock with it. One of the things I wanted to get right here was the blueshift relation. Unlike an O-S collapse, where an infaller (with right trajectory) can have very mild or even no blue shift relative to the outside universe, the boundary matching conditions for a shell collapse require that the whole interior (for observers stationary with respect to shell center) experiences (in the limit of zero thickness shell) the same blueshift as a clock riding exactly on the shell.

- A distant observer, of course, never sees the shell reach the event horizon. Similarly, they see the inside clock stop at a time before the clock hits the singularity.

- Later free fall clocks will be seen, from a distance, to stop on reaching the event horizon. Such a clock, itself, will experience no such thing, and its time will progress further until it hits the singularity (the shell is 'long gone' into the singularity).

Classically, one may ask by what possible rationale should one declare that a clock operating according to pure SR physics, be declared to stop for no local physical reason? What does what a distant observer sees have anything to do with their local, pure SR physics?

All of the above is indisputable for classical GR. The Krauss paper, by choice, only covered what the outside observer sees, for the classical case, because of what they discovered about the quantum case. The quantum analysis (if true) showed that, a lot of the above does not happen in our universe. It shows, instead, that (no matter what coordinates are used), the interior clock actually evaporates into not quite thermal radiation well before its classical end. The Kruskal diagram changes in significant ways as a result. Only with this justification, does it make any sense to 'chop' the classical analysis based on Schwarzschild time coordinate.

Note, that in the classical analysis without the quantum chop, perfectly reasonable, physically based simultaneity conventions, establish simultaneity between the interior clock and exterior clocks well after the shell has crossed the event horizon. One simple example is:

http://physicsforums.com/showpost.ph...0&postcount=23
PeterDonis
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Quote Quote by zonde View Post
There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.
You think that this is a logical proof, but it isn't. What you have actually proven is only that there is no hole in the region of spacetime covered by the SC type time slicing. You have *not* proven that that region of spacetime is the entire spacetime, and in fact it's easy to prove that it's not. See below.

Quote Quote by zonde View Post
And wordlines of particles do not end. They just extend toward infinite future.
No, they don't, at least not the way you mean. When you have extended an infalling worldline all the way to t = infinity by your clock, that worldline still only has a *finite* length. "Length" for worldlines means proper time, and the proper time for an object to fall to the horizon is finite.

Quote Quote by zonde View Post
You would not claim that there is some problem with this expression, right?
[tex]\lim_{t\to\infty}f(t)=a[/tex]
Let's say that t is coordinate time and f(t) is proper time of infalling particle.
This isn't quite the right expression. The right expression is:

[tex]\lim_{t\to\infty} \int_{t_0}^t dt' f(t')[/tex]

where [itex]t_0[/itex] is the time by the distant observer's clock that the infalling object starts falling. Substitute the correct function f(t'), for the proper time of the infalling object, do the integral, and take the limit. You will find that it gives a finite answer. This shows that "extending the worldline to infinity" by the distant observer's clock only extends it by a finite length. The SC time coordinate is so distorted at the horizon that it makes finite lengths look like infinite lengths.

This also shows why the region of spacetime covered by the SC time slicing can't be the entire spacetime: what happens to the worldlines once they reach the horizon? They have only covered a finite length, and spacetime is perfectly smooth and well-behaved at the horizon: the curvature is finite, there is nothing there that would stop the objects from going further. The only physically reasonable conclusion is that they *do* go further; that is, that there is a region of spacetime on the other side of the horizon, where the infalling objects go, but which can't be covered by the SC time slicing.

Quote Quote by zonde View Post
You can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.
This is a good idea; do you have any suggestions? I would suggest comparing the description of flat spacetime in Rindler coordinates to its description in Minkowski coordinates.
PAllen
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Quote Quote by zonde View Post
There is no hole in spacetime with SC type time slicing for collapsing mass. EH forms at infinite future i.e. never. So there is no hole in spacetime.

And wordlines of particles do not end. They just extend toward infinite future.
You would not claim that there is some problem with this expression, right?
[tex]\lim_{t\to\infty}f(t)=a[/tex]
Let's say that t is coordinate time and f(t) is proper time of infalling particle.
This is false. You detect readily in SC coorinates that there is a hole in space time. You integrate proper time along an infall trajectory and find that proper time stops at a finite value (unlike for various other world lines). You ask, what stops the clock? There is no local physics to stop the clock - tidal gravity may be very small; curvature tensor components are finite. The infinite coordinate time is not a physical quantity in GR. Einstein spoke of rulers and clocks, as Harrylin likes to point out. This clock stops for no conceivable local reason. If you add SC interior coordinates, and use limiting calculations, you smoothly extend this world line to the real singularity (with infinite curvature). All of this is exactly as if you chopped a disk around the pole from a sphere - you would find geodesics ending for no reason.
Quote Quote by zonde View Post

You can choose different simultaneity conventions but then we need good understanding of the things that depend on simultaneity convention as they would change along with it.

And I think that in order to improve that understanding it would be good idea to start with some examples from flat spacetime.
There is no physical observable, anywhere in SR or GR, that depends on simultaneity convention at all. This is part of what Pervect was saying above. Belief that simultaneity convention has physical consequence reflects complete, total, misunderstanding of SR and GR.

As for flat spacetime, the Rindler example Dr. Greg has posted beautiful pictures of, is relevant. The belief that there is no hole in SC exterior coordinates is 100% equivalent to the belief that most of the universe doesn't exist because a uniformly accelerating rocket can't see it.
zonde
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Nov19-12, 11:53 PM
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Quote Quote by pervect View Post
Mathematically: the maps of SR and GR preserve the Lorentz interval.

This is where I keep feeling the communication is lacking - but many people who say they "get" this point obviously don't :-(.
I would like get this point better and as I understand you are confident about your understanding of that point.

Transformations between inertial SR coordinates preserve Lorentz interval. And they don't change metric just as well.
But in GR transformations between coordinates don't have to preserve metric intact. That's how Lorentz interval is left the same, right?
PeterDonis
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Quote Quote by zonde View Post
But in GR transformations between coordinates don't have to preserve metric intact. That's how Lorentz interval is left the same, right?
No. GR coordinate transformations leave the metric intact; at least, they do in the normal meaning of that term, that geometric invariants are preserved. The metric may *look* different, as a formula, in a different coordinate chart; for example, the metric of the exterior vacuum region of Schwarzschild spacetime looks different in Painleve coordinates than it does in Schwarzschild coordinates. But if you calculate any geometric invariant, such as the length of a curve, in different coordinate charts, you will get the same answer.


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