# How does GR handle metric transition for a spherical mass shell?

by Q-reeus
Tags: handle, mass, metric, shell, spherical, transition
PF Patron
P: 4,473
 Quote by PeterDonis The paper I mentioned some posts back, which I haven't been able to find again, basically derived this case as the limit as the shell thickness goes to zero, with the shell mass remaining at a fixed positive value.
Right, and a zillion posts ago in the black hole thread, I wrote down its metric simply from the continuity of metric requirement (using isotropic SC coords, because it was easier to state the junction that way).
P: 1,115
 Quote by PeterDonis But why would the appearance of a test sphere to someone far away make any difference, when we've already established that, to an observer right next to the test sphere, it would appear spherical, *not* distorted? To me, that local observation is a much better gauge of whether space is isotropic than the observation of someone far away, particularly when the light going to the faraway observer could be distorted by the variation in gravity in between...
There is imo an unhealthy obsession in GR circles with the tendency to look at everything from the 'local' perspective. Casting everything in terms of invariants has it's advantages, but also disadvantages in that one can lose site of the forest for the trees. Typical example - when BH sceptics bring up EH 'time freeze' and related issues, they are brow beaten with the argument that SC's are deceptive/misleading and the only 'proper' perspective is that of the invariant worldline of an infalling observer for whom 'time freeze' etc does not apply. Which to my mind brushes under the carpet serious issues evident from an external observer's perspective.
But enough of generaized criticism. To answer your specific point Peter, just consider redshift. All but the most uninformed newbe has little trouble appreciating that redshift cannot be locally observed because it is inherently a differential issue - clock-rate 'here' vs clock-rate out 'there'. We have no conceptual issue with this (well, those insisting it's exclusively a 'tired light' 'energy drain' thing might). So why should spatial measure be any different? What is the apparent fundamental divide? If effect of metric on time-rate is properly a relational, nonlocally experienced, justifiably physical thing, what allows that length measure - the spatial component of SM in particular, are *not* likewise a physically meaningful, nonlocally observed relational thing? Seems illogical to me.

Want an 'extreme' example? my 'predjudice' is that BH's are, in one important sense, propped up on the basis that, just as SC's indicate, tangent spatial components are unaffected by gravitational potential. If however *all*, spatial components at least, are subject to the J factor, we find that the physical size of a notional BH shrinks to zero before it in fact can qualify as BH - as of course referenced to a distant observer. That leaves out other matters such as whether 'gravity actually gravitates' but indicates that there are drastic consequences as to the proper, physical implications of metric on a relational, 'distant observer' basis. So it's vitally important to know just what that test sphere will, abberation free, actually be *measured* by the distant observer imo. And my assumption is SM via SC's tells us it will be an oblate spheroid with axial ratio J:1. I further think nature has a different idea - it will to first order remain spherical but shrunk. [unavoidably there will be observed second and higher order distortions simply owing to metric necessarily being a function of r in any reasonable metric theory] Hope you all get my drift here.
P: 1,115
 Quote by DaleSpam The exterior SM is a vacuum solution, so there isn't any matter contribution in the exterior SM.
Have you taken a course in "how to be an effective agent provocateur", by any chance? I consider your comments disingenuous. You surely must have read, prior to your above comments in #30 (entry timestamp: 03:30 PM), my own clear statement in #28 (entry timestamp: 12:54 PM)
"I entirely meant shell matter's contribution to the exterior, SM region, and said so explicitly in #25."
You are capable of discerning the fundamental significance of replacing 'in' with 'to' in your above distortion of what I was on about in #25, right?
 That transition requires non-zero spatial components of the stress-energy tensor, i.e. stress and pressure. The large contribution of the energy density is in the wrong place to matter for the spatial components of the curvature.
Will have more to say on that claim in responding to your #35
P: 1,115
 Quote by DaleSpam Originally Posted by Q-reeus: "Agree with essentially all the rest above, but not the probable implication that stress in the shell can account for anything remotely significant re transition through shell wall." Why not? Your argument regarding the relative size of the pressure and energy density is not relevant. So what logical reason could you have to disagree?
Because it's entirely relevant.
 Originally Posted by Q-reeus: "There is a severe logical chasm imo." Agreed. The reference that I posted derived the pressure terms directly from the Einstein field equations. So there is no logical gap there. But you, without touching the math, claim to know that the effect is too small. A logical chasm indeed.
Yes, and it's you, and the formal system you trust is right, that has the logical chasm problem. You keep implying I'm some kind of ill-informed dumbo, but do so on the basis of distorting what I have actually argued. OK , gloves are off. Should you not duck the challenge, think I'm about to make you look stupid, and happy to do so. Sure, I'm not up with the tensor math and you are. Good for you. But I will claim a certain insight on this issue that either you entirely lack, or are unwilling to acknowledge for whatever reason. So stress in the shell wall solves it all? OK, here's the task for you - who knows the math. In #8 values were given for a monster, 8 ton 'toy globe' subject to ~ atmospheric pressure. Go ahead and work out the self-gravity value instead (floating in space, fully evacuated interior). My 'lazy' estimate - ca 10-30 times the mass contribution to T00 as source of g00, which is all here that determines exterior SM, and the depressed interior MM values (give or take half a dozen orders of magnitude, as if it matters really). The incredibly tiny shell stresses, virtually pure biaxial compressive, somehow can effect a reduction in the tangent metric components going from rb to ra? Convince me please.
Oh, here's a possible fly in the ointment. Add the tiniest puff of fresh, pure mountain air inside the shell. Just a touch. Just enough to reverse the sign of shell hoop stresses and blow the amplitude up by, say, a mere factor of one million. And this is still looking anything but Alice-in -Wonderland nonsense?! Good luck, genius!
Mentor
P: 15,610
 Quote by Q-reeus To answer your specific point Peter, just consider redshift. All but the most uninformed newbe has little trouble appreciating that redshift cannot be locally observed because it is inherently a differential issue - clock-rate 'here' vs clock-rate out 'there'. We have no conceptual issue with this (well, those insisting it's exclusively a 'tired light' 'energy drain' thing might). So why should spatial measure be any different? What is the apparent fundamental divide?
The difference is simply that there is no agreed upon standard for comparing distant lengths as there is for comparing distant times. Because there is no standard you simply have to clearly define the experiment you want to perform in order to determine the length as measured by a distant observer. Depending on the complexity of the experiment it may be difficult to calculate, but in principle you should be able to determine the result of that experiment which you would call the length as measured by the distant observer.
Mentor
P: 15,610
 Quote by Q-reeus But I will claim a certain insight on this issue that either you entirely lack, or are unwilling to acknowledge for whatever reason.
Your insight is simply wrong. It is not based on logic, but comes from ignorance and a prejudice against GR. You dismiss it as illogical and non-self-consistent without bothering with the effort of learning the math which ensures its self-consistency.

 Quote by Q-reeus So stress in the shell wall solves it all? OK, here's the task for you - who knows the math. In #8 values were given for a monster, 8 ton 'toy globe' subject to ~ atmospheric pressure. Go ahead and work out the self-gravity value instead (floating in space, fully evacuated interior).
I will do that. It will take a few days.

 Quote by Q-reeus My 'lazy' estimate - ca 10-30 times the mass contribution to T00 as source of g00, which is all here that determines exterior SM, and the depressed interior MM values (give or take half a dozen orders of magnitude, as if it matters really).
Sure, that is not in doubt. The issue is the relative contribution of the stresses and the mass to g11, g22, and g33. The relative contribution is infinite since the mass does not contribute at all.

 Quote by Q-reeus The incredibly tiny shell stresses, virtually pure biaxial compressive, somehow can effect a reduction in the tangent metric components going from rb to ra? Convince me please.
Yes, but I doubt that you will be convinced.
P: 1,115
 Quote by DaleSpam The difference is simply that there is no agreed upon standard for comparing distant lengths as there is for comparing distant times. Because there is no standard you simply have to clearly define the experiment you want to perform in order to determine the length as measured by a distant observer. Depending on the complexity of the experiment it may be difficult to calculate, but in principle you should be able to determine the result of that experiment which you would call the length as measured by the distant observer.
My angle on this - either SC's have readily testable, remotely determinable, physical consequences for *all* components, or not. If not, we are giving heed to a fantasy.
P: 1,115
 Quote by DaleSpam Your insight is simply wrong. It is not based on logic, but comes from ignorance and a prejudice against GR. You dismiss it as illogical and non-self-consistent without bothering with the effort of learning the math which ensures its self-consistency.
I will accord that this tirade is genuinely motivated. I'll even commend you for doggedly pursuing the topic. But we shall see.
 I will do that. It will take a few days.
My 'challenge' was partly rhetorical. You did stop and think a bit about the 'fly in the oinment'? If you do labour on for a few days and somehow manage to stitch a credible answer (boundary fitting magic), I will immediately point you to the 'puff of air' dilemma. This is the difference between knowing the math of received wisdom, and having some insight that looks outside the square. Please, concede on this one. The task of getting a consistent physics here is impossible.
PF Patron
P: 4,773
 Quote by Q-reeus To answer your specific point Peter, just consider redshift. All but the most uninformed newbe has little trouble appreciating that redshift cannot be locally observed because it is inherently a differential issue - clock-rate 'here' vs clock-rate out 'there'. We have no conceptual issue with this (well, those insisting it's exclusively a 'tired light' 'energy drain' thing might). So why should spatial measure be any different? What is the apparent fundamental divide? If effect of metric on time-rate is properly a relational, nonlocally experienced, justifiably physical thing, what allows that length measure - the spatial component of SM in particular, are *not* likewise a physically meaningful, nonlocally observed relational thing?
First of all, as DaleSpam noted, there is an agreed standard for comparing times at distant locations; in fact, the redshift basically *is* that standard. There is no agreed standard for comparing distant lengths.

Second, the redshift is coordinate independent. The "spatial measure" as you are using the term is not, which is why I was so careful in previous posts to describe everything in terms of areas and the "non-Euclideanness" of space, without saying anything definite about "distance measure". See below.

 Quote by Q-reeus If however *all*, spatial components at least, are subject to the J factor
I think you mean "K factor" since that's the one I defined relative to the spatial components; the J factor affects the time component. The point is that which spatial components are affected by the K factor is coordinate dependent. So the "spatial measure" as you are using the term is not a good way to judge the actual physics.

It's worth walking through this in some more detail. Go back to the picture I gave in terms of 2-spheres with gradually decreasing areas. The areas of these 2-spheres, and the volume in between neighboring 2-spheres, are physical observables; we can measure them by covering the area or packing the volume with little identical objects and counting them. So the factor K, that I defined, is a coordinate-independent quantity and represents actual physics.

However, there are different ways in which this actual physics can be represented in a coordinate system. The different radial coordinate definitions that I described are different ways of *labeling* the 2-spheres, and different labelings lead to different conclusions about which "spatial components" are affected by the K factor. If we use the Schwarzschild r coordinate, we label each 2-sphere with a coordinate r equal to the square root of its physical area divided by 4 pi. With this labeling, only the radial component of the metric is affected by the K factor; the tangential components are not. However, if we use the isotropic R coordinate, meaning that a 2-sphere gets labeled with a "radius" R that does *not* equal the square root of its physical area divided by 4 pi (in the case we're discussing, R will be smaller than that), then all three spatial components *are* affected by the K factor.

 Quote by Q-reeus we find that the physical size of a notional BH shrinks to zero before it in fact can qualify as BH
It's worth discussing this in a little more detail too. First of all, as I showed above, whether or not all spatial components are affected by the K factor is coordinate dependent, so your logic here is not correct as it stands since coordinate dependent quantities can't describe the actual physics. However, there is a legitimate physical question to be asked: what *is* the K factor, as I defined it physically (the "non-Euclideanness" of space, in terms of the volume between two adjacent 2-spheres compared to their areas), at the EH of a black hole?

The answer is that the question is not valid, because the physical definition of the K factor requires that the 2-spheres in question are spacelike surfaces. But the 2-sphere at the EH, where r = 2M in Schwarzschild coordinates, is not spacelike; it's null. So it is physically impossible to perform the comparison I described using a 2-sphere at the EH, and there is therefore no way to physically define the K factor (or the J factor, for that matter) at the EH.
Mentor
P: 15,610
 Quote by Q-reeus Please, concede on this one. The task of getting a consistent physics here is impossible.
Prove it.
PF Patron
P: 4,773
 Quote by PeterDonis the 2-sphere at the EH, where r = 2M in Schwarzschild coordinates, is not spacelike; it's null. So it is physically impossible to perform the comparison I described using a 2-sphere at the EH, and there is therefore no way to physically define the K factor (or the J factor, for that matter) at the EH.
On re-reading, I should re-state this. The 2-sphere at the EH can still be said to have a physical "area", which is 4M^2 in geometric units. So it may not be technically correct to say the 2-sphere itself is null. (When I compute the norms of the tangential unit vectors, I don't get zero at r = 2M; the norms are still positive, so the unit vectors are still spacelike, assuming I'm doing the computation right).

However, there can't be a static 2-sphere "hovering" at the EH, because the EH, as a surface in spacetime, is a null surface, and therefore there can't be a surface of "constant time" that is orthogonal to the EH, in which the 2-sphere could be said to lie, and in which the area of the 2-sphere could be compared with the volume between it and a neighboring 2-sphere. So the main point in what I said above still holds: it's impossible to do the physical measurement at the EH that I was using to define the K factor.

I should also note that the above does not entirely apply to the J factor; since there are still timelike worldlines passing through the EH, it is still possible to define a "gravitational redshift" factor there, for an infalling observer. However, this factor cannot apply to a static, "hovering" observer at the EH, since as we've seen there can't be one. So what I said does apply to the J factor for static observers.
P: 1,115
 Quote by DaleSpam Originally Posted by Q-reeus: "Please, concede on this one. The task of getting a consistent physics here is impossible." Prove it.
Was meant as good advice, based on what I wrote in #40
"Oh, here's a possible fly in the ointment. Add the tiniest puff of fresh, pure mountain air inside the shell. Just a touch. Just enough to reverse the sign of shell hoop stresses and blow the amplitude up by, say, a mere factor of one million."
If you choose to reject the basic logic of that bit, then recall - you have committed to proving me wrong by calculations I consider doomed to failure - but go ahead and show that I'm the mistaken one.
P: 1,115
 Quote by PeterDonis I think you mean "K factor" since that's the one I defined relative to the spatial components; the J factor affects the time component.
Took them to be identical in vacuum SM region, but upon looking back in your #9 I see that the defined relation is J = K-1 there. Had assumed the redshift factor J was in terms of frequency, since it declines with descent into lower potential. That is my expectation of how 'coordinate' measure of lengths will go, hence did mean J, but formally should have used K-1.
 Originally Posted by Q-reeus: "we find that the physical size of a notional BH shrinks to zero before it in fact can qualify as BH"
Allright, probably should have just said the area of a collapsing 'almost, approaching notional 'BH'' body by this reckoning shrinks indefinitely since tangent metric components would actually shrink by the factor J = K-1, that relation holding everywhere, not just in vacuum. Hence area follows as K-2. By contrast with standard BH there is infinite redshift at EH but finite area. [I believe actual collapsing body would never shrink to a point, but stabilize by matter/radiation ejection and spin at a perfectly finite size where J > 0. Gravity as source term in T would imo have to figure in that.]
So what follows in your comments here are I think perfectly OK only if one already accepts spatial metric permitting finite EH area. Catch 22, seems to me. thanks for your clarification in #47, but same deal.

As in my first passage in #19, still see this thing of defining distance in terms of area as another Catch 22. How do you define area divorced from linear length measure? Those 'packing objects' are LxLxL entities, and one must have a clear definition of L, and same goes with the area thing - area A is an LxL object! If A is the primitive, how do you determine it apart from L measure! Could this be a conundrum forced by need to accomodate difficulties with the BH EH issue? Maybe not, but it's my hypothesis.
Much later.
PF Patron
P: 4,773
 Quote by Q-reeus Allright, probably should have just said the area of a collapsing 'almost, approaching notional 'BH'' body by this reckoning shrinks indefinitely since tangent metric components would actually shrink by the factor J = K-1, that relation holding everywhere, not just in vacuum.
I'm not sure I understand you here. Two comments:

(1) Did you read that previous post where I defined J and K carefully? You'll note that I specified there that the relation J = K-1 does *not* hold in the non-vacuum region. I was talking about the "shell" scenario there, but the same would apply for the interior of a collapsing body such as a star.

(2) The factor J does not apply to the tangential metric components; it applies to the time component, since it's the "redshift factor". So I'm not sure how you're concluding that the tangential metric components would "shrink" by the factor J.

 Quote by Q-reeus By contrast with standard BH there is infinite redshift at EH but finite area.
This is not a contrast with a standard BH. A standard BH does have infinite redshift but finite area at the EH. (More precisely, it has "infinite redshift" for "static" observers at the EH--more precisely still, the limit of the redshift for static observers as r goes to 2M is infinity; there are no static observers exactly at the EH so there is no "redshift" for them at that exact point).

 Quote by Q-reeus [I believe actual collapsing body would never shrink to a point, but stabilize by matter/radiation ejection and spin at a perfectly finite size where J > 0. Gravity as source term in T would imo have to figure in that.]
Perfect spherical non-rotating collapse is certainly an idealization. But there have been many numerical calculations done of non-idealized collapses, and they still show an EH forming and the body collapsing inside it.

 Quote by Q-reeus As in my first passage in #19, still see this thing of defining distance in terms of area as another Catch 22. How do you define area divorced from linear length measure? Those 'packing objects' are LxLxL entities, and one must have a clear definition of L, and same goes with the area thing - area A is an LxL object! If A is the primitive, how do you determine it apart from L measure! Could this be a conundrum forced by need to accomodate difficulties with the BH EH issue?
The little identical objects used for packing have to have some linear dimension, yes. But that doesn't commit you to very much since it's for very small objects, so the effects of spacetime curvature can be ignored. As soon as you start trying to deal with size measures over a significant distance, where spacetime curvature comes into play, you have to be a lot more careful. The reason for using those little objects to define area first is that the spacetime has spherical symmetry, so the areas of 2-spheres centered on the origin can be defined without worrying about the curvature of the spacetime. That is not true for radial distance measures, as I have shown.
P: 1,115
 Quote by PeterDonis Originally Posted by Q-reeus: "Allright, probably should have just said the area of a collapsing 'almost, approaching notional 'BH'' body by this reckoning shrinks indefinitely since tangent metric components would actually shrink by the factor J = K-1, that relation holding everywhere, not just in vacuum." I'm not sure I understand you here. Two comments: (1) Did you read that previous post where I defined J and K carefully? You'll note that I specified there that the relation J = K-1 does *not* hold in the non-vacuum region. I was talking about the "shell" scenario there, but the same would apply for the interior of a collapsing body such as a star. (2) The factor J does not apply to the tangential metric components; it applies to the time component, since it's the "redshift factor". So I'm not sure how you're concluding that the tangential metric components would "shrink" by the factor J.
I understand your comments, but they are referencing to the standard GR view of things. I was addressing things assuming my notion of isometric metric applies, for which the product JK is invariant - same in exterior vacuum as shell wall matter region. Which gets back to finding a self-consistent answer to the shell metric transition problem. With no hope of a resolution via shell stresses, there is what to fall back on?
 Originally Posted by Q-reeus: "By contrast with standard BH there is infinite redshift at EH but finite area." This is not a contrast with a standard BH. A standard BH does have infinite redshift but finite area at the EH. (More precisely, it has "infinite redshift" for "static" observers at the EH--more precisely still, the limit of the redshift for static observers as r goes to 2M is infinity; there are no static observers exactly at the EH so there is no "redshift" for them at that exact point).
Ha ha. Blame myself for not having added a comma after the word 'contrast'. Hope you get the unambiguous meaning now. So as per last passage, you are referring to standard picture, I was contrasting my idea of 'how it ought to be' with the standard, BH's are real, picture.
 Originally Posted by Q-reeus: "As in my first passage in #19, still see this thing of defining distance in terms of area as another Catch 22. How do you define area divorced from linear length measure? Those 'packing objects' are LxLxL entities, and one must have a clear definition of L, and same goes with the area thing - area A is an LxL object! If A is the primitive, how do you determine it apart from L measure! ..." The little identical objects used for packing have to have some linear dimension, yes. But that doesn't commit you to very much since it's for very small objects, so the effects of spacetime curvature can be ignored. As soon as you start trying to deal with size measures over a significant distance, where spacetime curvature comes into play, you have to be a lot more careful. The reason for using those little objects to define area first is that the spacetime has spherical symmetry, so the areas of 2-spheres centered on the origin can be defined without worrying about the curvature of the spacetime. That is not true for radial distance measures, as I have shown.
Your efforts to educate me haven't been entirely wasted. Finally appreciate, I think, that this definition allows the only unambiguous locally observable measure of curvature effects - as you say, packing ratios vary with 'radius'. But the persistent opinion one cannot decently relate length measure 'down there' to 'out here' is not true if what I have just realized makes sense. Simply apply the well known radial vs tangent c values cr, ct, which in terms of J factor, are cr = J, ct = J1/2 (e.g. http://www.mathpages.com/rr/s6-01/6-01.htm Last page or so). These are naturally the coordinate values. Now as locally to first order in metric everything is observed isotropic, we must have that spatial metric components scale identically to their cr, ct counterparts according to cdt = dx. Settles the matter for me. So with I think a proper handle on how SM predicts metric scale in coordinate measure, will try to test their consistency.
Interestingly, while SC's were telling me tangent component was invariant, from this directionally dependent c perspective, there is in fact tangent shrinkage of a collapsing objects perimeter by factor J1/2. Still not isotropic, but not as 'bad' as I thought before.
PF Patron
P: 4,773
 Quote by Q-reeus I understand your comments, but they are referencing to the standard GR view of things. I was addressing things assuming my notion of isometric metric applies, for which the product JK is invariant - same in exterior vacuum as shell wall matter region.
Have you checked to see that this notion of yours can even be satisfied at all consistent with the Einstein Field Equation? All the things DaleSpam and I have been saying about how the J and K factors change in the non-vacuum shell region are based on the EFE, relating the stress-energy tensor to the curvature. If you're going to just throw out notions without caring if they're consistent with the EFE, then there's no point in discussion, since you're not going to convince anyone else here that the EFE might not be valid under these conditions.

 Quote by Q-reeus Which gets back to finding a self-consistent answer to the shell metric transition problem. With no hope of a resolution via shell stresses, there is what to fall back on?
DaleSpam and I have both given self-consistent answers that resolve it via shell stresses. The fact that you don't accept them doesn't make them wrong.

 Quote by Q-reeus But the persistent opinion one cannot decently relate length measure 'down there' to 'out here' is not true if what I have just realized makes sense.
We are not saying there is *no* way to relate local length measures to distant length measures. We are saying there is not a *unique* way to do it, so you have to specify how; and you can't just hand-wave it, you have to actually do some calculating to see how it works out. For example, if you want to do it based on what the distant observer actually sees, you have to work out the paths of light rays.

 Quote by Q-reeus Simply apply the well known radial vs tangent c values cr, ct, which in terms of J factor, are cr = J, ct = J1/2 (e.g. http://www.mathpages.com/rr/s6-01/6-01.htm Last page or so). These are naturally the coordinate values.
Only in Schwarzschild coordinates, as the page you link to makes clear. In other coordinates the c values work out differently. You can't make any valid claims about the actual physics using things that are only true in a specific coordinate system.
P: 1,115
 Quote by PeterDonis Have you checked to see that this notion of yours can even be satisfied at all consistent with the Einstein Field Equation?
No, because the shell problem was thrown up to indicate, imo, that EFE's, or at least the SM, has problems, so it seems kind of circular to then use EFE's as the yardstick. I threw the problem to the pros for consideration, and have appreciated some useful feedback, but see nothing to this point satisfactorally answering it.
 DaleSpam and I have both given self-consistent answers that resolve it via shell stresses. The fact that you don't accept them doesn't make them wrong.
You've thrown me there completely. I recall you suggesting there is room for one via stresses, but can't recollect any actual detailed argument. Could you point to where I may have missed it? As you know, DaleSpam says he will have such an answer, but that's still future. You are aware of my reasons for total scepticism on that. Perhaps you wouldn't mind telling me why the 'puff of air' bit I brought up with DaleSpam in #40 is wide of the mark. To me it seems devastating, but sure I may not understand something basic here.
 We are not saying there is *no* way to relate local length measures to distant length measures. We are saying there is not a *unique* way to do it, so you have to specify how; and you can't just hand-wave it, you have to actually do some calculating to see how it works out. For example, if you want to do it based on what the distant observer actually sees, you have to work out the paths of light rays.
Have been looking at some kind of thought experiments along those lines, but re below that is in limbo for the moment.
 Only in Schwarzschild coordinates, as the page you link to makes clear. In other coordinates the c values work out differently. You can't make any valid claims about the actual physics using things that are only true in a specific coordinate system.
Thinking about that again, I was too hasty and will probably have to withdraw my claim - there is possible ambiguity about splitting the spatial and temporal contributions to cr, ct not fully thought through. I detect stormy weather here.
PF Patron