Can Analytic Continuation Determine the Metric Over a 4-D Manifold?

[straycat]

Hello all,

Can the method of analytic continuation [1] be applied to the calculation of the metric over a 4-d manifold?

In other words, suppose that we are given the value of the metric g_ab as well as its power series at a single point p in a 4-dimensional manifold. Assume further that the metric is analytic. (Actually, I suppose we would have to assume that g_ab is the real component to some complex function f, where f is analytic over the manifold.) From what I remember from my complex analysis class of many many years ago, the metric and its power series should be enough to let us solve for g_ab over the entire 4-d manifold.

iiuc, knowledge of g_ab over all of spacetime further allows us to solve for the distribution of matter-energy throughout spacetime. The conceptual difficulty that I now have is that in principle, if an observer "observes" the point p, then she should be able to determine the matter-energy distribution throughout all of spacetime -- including regions that are outside her past light cone. This should not be allowed, I would think.

There are several possible answers I can think of:

1. I am incorrect to assume that analytic continuation can be applied to the metric over a manifold, the way I discussed above. I have only a rudimentary knowledge of differential geometry, so I cannot rule out this possibility.

2. Spacetime is not analytic. I don't really "like" this explanation -- I prefer the "niceness" of analyticity.

3. There is something that prevents us, in general, from solving for g_ab outside its radius of convergence. I don't know what this might be, though. The mathworld link below states: "under fortunate circumstances (that are very fortunately also rather common!), the function f will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition."

4. QM prevents us from knowing the precise value of the the metric and its power series at any given point. But I would like to know what GR and GR alone tells us about this issue, without having to invoke QM.

5. Spacetime topology is nontrivial. (This is the explanation I am rootin' for .) iiuc, analytic continuation assumes that the region in question is simply connected. So if we do not know the global topology of the 4-d manifold, then we cannot calculate the metric. I am imagining, for example, the "foam" picture of spacetime, as depicted here: http://universe-review.ca/I01-16-quantumfoam.jpg

Any ideas?

-- David

[Note to moderators -- I don't know if this thread would fit better into the tensor analysis and differential geometry section; feel free to move it if you see fit. ]

[1] http://mathworld.wolfram.com/AnalyticContinuation.html

 

[Hurkyl]

5) doesn't solve anything.

If we try to extend an arbitrary analytic function to a multiply-connected domain, we run the danger that the continuation around different "directions" won't match up.

But if, for some reason, we already know the function is analytic on our multiply-connected domain, then we are guaranteed that there won't be any problems when we try to recover the function from one of its restrictions.

 

[straycat]

5) doesn't solve anything.

If we try to extend an arbitrary analytic function to a multiply-connected domain, we run the danger that the continuation around different "directions" won't match up.

But if, for some reason, we already know the function is analytic on our multiply-connected domain, then we are guaranteed that there won't be any problems when we try to recover the function from one of its restrictions.

5) solves the problem if we assume that the function is analytic, but we do not know the topology. eg, imagine that the observer does not know the "number of wormholes" that populate spacetime. For every possible topology, she would calculate a distinct global solution to the spacetime metric, hence a distinct solution to the matter-energy distribution. But there would be no way to label one of the multiplicity of solutions as the "correct" one.

David

 

[Hurkyl]

(disclaimer: I'm extrapolating from what I know about complex-valued functions of a single complex variable)

When you analytically continue your fragment of the metric, you will get a unique, universal solution.

The only difference the nontrivial topologies make is whether the universe suddenly ceases to exist someplace or not -- they have absolutely no effect on what happens where the universe does happen to exist.

I think the only remaining question is philosophical -- if you get something like a periodic solution, do you leave the solution alone, or take some quotient space? Either way, the physics would remain the same.

 

[straycat]

Let me restate the issue in slightly different terms.

Suppose that the statement "Event A occurs at spacetime location p" is equivalent to the statement "the metric and its power series have such-and-such values at point p." GR tells us that event A cannot determine any event outside its future light cone. But (assuming analyticity and simply connected topology), analytic continuation allows us to calculate the metric throughout spacetime, even outside the future light cone. This seems to me to be a violation of GR.

Is it possible that the light cone defines a boundary beyond which we cannot analytically extend the function? If so, I don't know how, because one of the lessons of GR is that there is (should be) no preferred direction that corresponds to time; thus, how do we define the light cone boundary? The only way I can think of to do that would be if we assume that spacetime has local Lorentz signature (- + + +) at the point p. iow, define a 4-ball of radius r around p, and assume Lorentz signature, which allows us to divide the ball into region #1 in the future and past light cone, and region #2 outside these light cones. Perhaps (I speculate) we can analytically extend into region #1 but not region #2 ?? This would be an attractive answer to me -- but I am not enough of a mathematician to know if it is correct.

David

 

[straycat]

When you analytically continue your fragment of the metric, you will get a unique, universal solution.

But this should depend on the topology, right?

Suppose we have two 2-d manifolds, one (M1) with the topology of a sphere, the other (M2) with the topology of a donut. Pick a point p on each manifold, and specify the metric and power series (same for each manifold). Obviously the solution for M1 will be distinct from the solution for M2.

The only difference the nontrivial topologies make is whether the universe suddenly ceases to exist someplace or not -- they have absolutely no effect on what happens where the universe does happen to exist.

Well I'm assuming that we impose the requirement that the universe is sufficiently "nice," eg contains no singularities -- also no boundaries, I suppose. So if we obtain a solution that tells us the "universe ceases to exist" at some place, then we throw that solution out.

David

 

[Stingray]

The metric is not generally considered to be analytic. Hardly anything is in physics.

 

[Hurkyl]

Suppose we have two 2-d manifolds, one (M1) with the topology of a sphere, the other (M2) with the topology of a donut. Pick a point p on each manifold, and specify the metric and power series (same for each manifold). Obviously the solution for M1 will be distinct from the solution for M2.

If you're just handed a power series and a point on your manifold, it would be wrong to assume that it can be analytically extended to your entire manifold.

In fact, I think that the only power series that satisfy your example are those of the form a + 0z + 0z² + 0z³ + ... -- that is, the constant functions.

This conclusion is based on the following assumptions:
(1) The (analytic) sphere minus a point is analytically equivalent to R²
(2) R² is an analytic universal cover of the (analytic) torus.

In other words, I'm assuming that the way we usually represent these spaces can be made into an analytic map. I don't know if any two analytic spheres are analytically equivalent, and similarly for tori. (Sphere = plane plus a point, and the torus is a square with the edges glued together)

Under these representations, the analytic functions on the sphere are precisely the analytic functions on R² with a unique limit at infinity. The analytic functions on the torus are precisely the doubly periodic functions on R² with periods (0,1) and (1,0). The only functions satisfying both conditions are the constant functions.

 

[straycat]

The metric is not generally considered to be analytic. Hardly anything is in physics.

True. This would be my "explanation #2" in the first post.

Why is that, btw? I suppose it is because the assumption of analyticity is simply too restrictive. My take would be that we like our mathematics to be as "nice" as we can get away with, and analyticity is about as "nice" as it gets, but we can't always get what we want, can we?

Still, I find it an interesting exercise to assume analyticity and see where that takes us.

 

[straycat]

If you're just handed a power series and a point on your manifold, it would be wrong to assume that it can be analytically extended to your entire manifold.

Well that's my question. Can we (perhaps) extend it only through the future and past light cones?

The only functions satisfying both conditions are the constant functions.

If you're right, then I find that very interesting. Assume we are handed a non-constant metric (lots of nonzero terms in the power series). We assume analyticity, and that the manifold is not bounded (so the universe does not "cease to exist" at some boundary), but we make no assumption regarding topology. From what you are saying, we (I should say: the observer who observes the point p) can conclude, at the least, that the manifold does not have the topology of the plane (R²) (actually I should say R^4 since we're talking about 4-d manifolds), or of the (4-d) sphere. So perhaps we are able to conclude that the topology is something more complicated -- eg that it has some number N of donut holes, wormholes, generators, call them what you will. Perhaps we cannot solve for the topology exactly, but we can place some amount of restriction on it.

That would actually be pretty interesting, don't you think? The interpretation might be that each independent solution (each distinct possible topology + global metric, given the metric at point p) exists in superposition. This could be a feasible interpretation, even in the absence of injecting QM into the picture.

 

[Hurkyl]

I am going to move this to the math section, in hopes that we have a real analytic geometer who can answer questions better than I can! I'm quite worred about saying something actually false. Especially since my only real exposure to this is in the complex case, and only in one complex dimension.

From what you are saying, we (I should say: the observer who observes the point p) can conclude, at the least, that the manifold does not have the topology of the plane

No -- I said that the sphere and torus are not both candidates. And I'm going to waffle on that: there's something weird going on I don't fully understand.

I've only ever thought about the case where analytic continuation goes as far as the coordinate chart you're using. (But I didn't think about it in those terms) -- continuing beyond the coordinate patch would require a transfer to a new coordinate patch, and I'm not sure what to conclude about that.

Now, I worry if it's possible to pick a clever coordinate patch that could make an analytic function on a sphere and an analytic function on the torus look the same.

 

[Stingray]

Why is that, btw? I suppose it is because the assumption of analyticity is simply too restrictive. My take would be that we like our mathematics to be as "nice" as we can get away with, and analyticity is about as "nice" as it gets, but we can't always get what we want, can we?

Analyticity is too strong in many cases. It excludes a lot of things that are useful to talk about. Also, all of these global issues you're discussing seem to be unnecessary complications for doing physics.

I think that talking about the analyticity of physical objects is somewhat analogous to asking whether or not a physical measurement is irrational or not. We simply don't expect that our model of reality is sufficiently correct that the distinction is even meaningful.

 

[straycat]

I am going to move this to the math section, in hopes that we have a real analytic geometer who can answer questions better than I can!

I tried doing a little google search to see what is known about analytic continuation and GR -- didn't find much . Although I did run across a paper by Dieter Brill entitled "Black Hole Collisions, Analytic Continuation, and Cosmic Censorship" (arXiv:gr-qc/9501023) that has this paragraph in the introduction:

"Because interesting applications of Einstein’s equations frequently occur in manifolds of complicated topology, which cannot be covered by a single coordinate patch, the region in which a typical exact solution is first known is often incomplete. The simplest way to complete it, if possible, is by analytic extension. It is remarkable how little is known in a systematic way about this frequently encountered problem. Here we will not materially improve on this situation, but merely recall some of what is known about analytic continuation, and discuss the most common class of geometries for which a method exists."

... which makes me worry that my question may be more difficult to answer than I had hoped.

David

 

[straycat]

Analyticity is too strong in many cases. It excludes a lot of things that are useful to talk about. Also, all of these global issues you're discussing seem to be unnecessary complications for doing physics.

I think that talking about the analyticity of physical objects is somewhat analogous to asking whether or not a physical measurement is irrational or not. We simply don't expect that our model of reality is sufficiently correct that the distinction is even meaningful.

Well suppose we could draw some sort of conclusion that the only way to preserve analyticity were to assume some high degree of multiple-connectedness, along the lines of what I discussed earlier. I have no idea if that conclusion is valid -- but suppose it were. That could be one argument in support of the "quantum foam" picture of the microscopic structure of spacetime depicted in the jpg in my original post. Wouldn't you find that meaningful -- or at least, very interesting?

 

[straycat]

No -- I said that the sphere and torus are not both candidates.

Ah, OK. But we could still state that information about the metric at the point p can, in principle, confer some restrictions on the global topology of the manifold (e.g. by ruling out the sphere and the torus)... ? With the caveat that we need some more expert input here.

 

[Stingray]

Well suppose we could draw some sort of conclusion that the only way to preserve analyticity were to assume some high degree of multiple-connectedness, along the lines of what I discussed earlier. I have no idea if that conclusion is valid -- but suppose it were. That could be one argument in support of the "quantum foam" picture of the microscopic structure of spacetime depicted in the jpg in my original post. Wouldn't you find that meaningful -- or at least, very interesting?

I don't really understand what you mean.

I just can't see how it can be reasonable to ascribe any physical meaning to derivatives of arbitrarily high order. All laws of physics should be (and are) essentially "invariant" under small perturbations of any particular solution. Changing this would require a number of new concepts.

One usually says that physical fields are at least distributions. But it is always possible (I think) to generate a nice $C^{m}$ function which comes arbitrarily close to any distribution. It shouldn't matter too much which type of function is used in the end.

But I don't think this sort of thing continues to work for functions are everywhere analytic. Your space of physical possibilities gets much smaller. You could probably get away with local analyticity, but that doesn't imply anything about topology.

 

[Haelfix]

Funny you mentioned Brills paper, I had another one of his in my bookmarks

http://xxx.lanl.gov/abs/gr-qc/9507019

It will explain most of the analyticity problem in chap2.

The key thing is that in general you can't extend a smooth singlevalued function uniquely across a cauchy horizon, you will end up with a huge wad of possible continuations. Generally people impose stronger 'physical' constraints there to strengthen the result and evidently it has been done for a few small patches of interest.

 

[straycat]

Funny you mentioned Brills paper, I had another one of his in my bookmarks

http://xxx.lanl.gov/abs/gr-qc/9507019

It will explain most of the analyticity problem in chap2.

The key thing is that in general you can't extend a smooth singlevalued function uniquely across a cauchy horizon, you will end up with a huge wad of possible continuations. Generally people impose stronger 'physical' constraints there to strengthen the result and evidently it has been done for a few small patches of interest.

I think these two papers by Brill are different drafts of the same thing.

AHA! I see this on page 4:

"Because “new” information can propagate along null surfaces, such surfaces are a natural analyticity boundary."

Am I correct that a null surface is the boundary of a light cone? Let's see ... MTW box 34.1: "horizons are generated by nonterminating null geodesics" ... so if null surfaces are generated via null geodesics, and if a light cone boundary is a horizon, then: yes. Which would mean that you cannot in general analytically continue a function through the light cone boundary, which is what I speculated earlier ... am I reading this right?

Now there are lots of different types of "horizons," and I can't say that I appreciate the difference between them, so I still have some thinkin' to do

D

 

[straycat]

I don't really understand what you mean.

I just can't see how it can be reasonable to ascribe any physical meaning to derivatives of arbitrarily high order. All laws of physics should be (and are) essentially "invariant" under small perturbations of any particular solution. Changing this would require a number of new concepts.

One usually says that physical fields are at least distributions. But it is always possible (I think) to generate a nice $C^{m}$ function which comes arbitrarily close to any distribution. It shouldn't matter too much which type of function is used in the end.

But I don't think this sort of thing continues to work for functions are everywhere analytic. Your space of physical possibilities gets much smaller. You could probably get away with local analyticity, but that doesn't imply anything about topology.

Hey 'ray,

I'm not saying that we should restrict ourselves to analytic functions always and evermore. For practical application, non-analytic functions will often suffice. But suppose we want to know, eg, whether a given solution is unique? The assumption of analyticity may be useful in such a situation.

BTW here's another excerpt from the Brill paper (page 3):

"In these ambiguous cases one needs to decide on such properties as the topology of the extended manifold along with the extension of the metric. Once the whole (smooth) manifold is known, its analytic structure is essentially unique."

This seems to jive with what I was thinking earlier: if you define the topology, then there is a unique analytic extension. But if you change the topology, then you change the analytic extension.