Thin Spherical Matter Shell as Gravity Theory Filter

[Q-reeus]

Not easy to admit but having now done the sums I have no choice but to take everything back in respect of claiming inequivalence between SM and ISM. Just knocked up a pdf article, but am reminded files cannot be uploaded here. Anyway it turns out ∂C/∂r (proper derivative of circumference expansion rate wrt r) is less than for flat space by the frequency redshift factor √(g_tt) - for both SM and ISM. If anyone wants to see the derivations, PM me. It creates a big issue for now - that 'severe paradox' mentioned earlier. Have a hunch as to what may be going on but stick with that regardless SM is self-contradictory as per parts 1 and 2 of main article. Must go.

[Mikko]

Be careful with the definitions. Would you call a space where ds² = r² (dr² + df²) "locally Euclidean"?

Depends on what f stands for, and whether one considers a truly differential spatial expansion about some given point.

It is one of the coordinates, the other one is r.

[Q-reeus]

It is one of the coordinates, the other one is r.

My intuition of what 'coordinate isotropic' meant in terms of proper spatial gradients was wrong. Just how misleading it can be in ISM case can be gleaned from considering the limiting situation of hovering just short of a supposed BH EH. It's intuitively clear in SM case that proper ∂C/∂r -> 0 as r -> r_s (i.e. EH), an inverse of the extreme coordinate spatial anisotropy there. In ISM coordinates the radial and transverse differential lengths remain 'equal' even at EH. So the finding that it's still the case proper ∂C/∂r -> 0 as r -> r_s just shows how meaningless is 'isotropy' in ISM.

Having further done the calcs for Yilmaz metric, which is genuinely spatially isotropic (ratio of differential coordinate lengths in any two directions are exactly reflected in the proper values), it turns out proper ∂C/∂r = 2π(1-2Φ/c^2), with Φ = -GM/r the Newtonian potential. Hence the Yilmaz metric modifies the flat space differential expansion rate ∂C/∂r = 2π in opposite sense to that for SM/ISM metric - greater rather than lesser rate. At that point it clicked - even though all spatial metric components expand equally as a function of r, the mean change of r metric component between r and r+dr is just half that of the circumferential components which are evaluated only at the radial end-points r and r+dr.

So with that sorted out, it's back to pointing out that part 2 of my article is pivotal. Working consistently in SM, there is a clear mathematically valid prediction of non-zero 'redshift' for interior spatial metric. Which cannot be logically reconciled with the 'equally valid' finding in part 1 that zero interior spatial 'redshift' must apply.

Now if you think there is a logical or arithmetical error in parts 1 or 2 of current v3 of viXra article (sans any reference therein to ISM), it's long overdue for you to point to it. And that invite/challenge is open to anyone else. I maintain the only cure is to adopt a metric having Yilmaz form.

[Q-reeus]

Erratum note: For Yilmaz metric, ∂C/∂r (proper) = 2π(1-Φ/c^2), not 2π(1-2Φ/c^2) shown in last post.

[Mikko]

Which cannot be logically reconciled with the 'equally valid' finding in part 1 that zero interior spatial 'redshift' must apply.

Now if you think there is a logical or arithmetical error in parts 1 or 2 of current v3 of viXra article (sans any reference therein to ISM), it's long overdue for you to point to it.

Most of the part 1 looks good, from the beginning to formula 1-7. From that point on the discussion is affected by your confusion about coordinate systems. As you have now sorted that out, you should be able to fix it. Just be careful that your updated conclusion follows from your calculations. At that point new errors easily slip in.

[Q-reeus]

Most of the part 1 looks good, from the beginning to formula 1-7. From that point on the discussion is affected by your confusion about coordinate systems. As you have now sorted that out, you should be able to fix it. Just be careful that your updated conclusion follows from your calculations. At that point new errors easily slip in.

You are not getting it. There will be no forthcoming change in v4 to the basic methods and conclusions of current v3. Where in part 3 the provision that ISM 'isotropy' might be entirely illusory was already given (now confirmed). Since then my faulty intuition had me firm toward the other possibility given that true isotropy applied therefore SM and ISM physically differed. That mistaken notion has zero impact on all calculations and relevant conclusions from past your approved-to (1-7). Maybe you think I should just opt for the standard GR specialist line that only proper curvature invariants matter and since Weyl and Cotton curvatures are zero everywhere there is ipso facto no issue. Not true - as demonstrated in article. Global properties also matter.

If you don't appreciate what that last bit means in context of spherical shell scenario, recall that by definition the shell 'areal radius' R is defined in terms of proper area A (or equivalently circumference C) - i.e. R = √(A/(4π)) = C/(2π). So unless you have the strange view spatial components of SM are entirely meaningless (then why even have a SM?), conflicting findings as to interior spatial metric presents a 'real' paradox. In that it directly implies a physical difference in ratio of light transit time on a diameter vs circumference basis. (Given the further assumption that interior temporal redshift is uniform owing to equipotential condition.) Depending on which 'valid' interior spatial metric one opts for. As you know, I opt for that neither is believable for precisely the reason both were obtained via legitimate use of SM. And while I have not bothered to confirm it, the chances are almost certain that a more general interior metric expression derived using part 2 approach will predict non-flatness, as mentioned in article. All-in-all the sole logical conclusion is that SM itself is the problem. And yes Yilmaz metric solves all such issues in a fully self-consistent way.

You have an evident continued interest in the topic, and an apparent desire to help correct my 'errors' as you see it. I strongly suggest therefore that rather than offer vague suggestions, point to precisely why you have an evident problem beginning with (1-8) and conclusion therefrom, and so on through the v3 article. Let me repeat - there should be no paradoxes when working consistently in a given coordinate system choice. Which choice was SM in both parts 1 and 2. Unequivocally revealing a paradox in doing so (part 1 vs part 2), plus an additional physical absurdity (selective vanishing of potential dependence - part 1). Can you rescue the situation for SM?

[Note: I here correct the stab at an explanation for proper ∂C/∂r > 2π in Yilmaz case given in my second last post:

At that point it clicked - even though all spatial metric components expand equally as a function of r, the mean change of r metric component between r and r+dr is just half that of the circumferential components which are evaluated only at the radial end-points r and r+dr.

All that I now claim is, although both radial and circumferential metric components are evaluated at the end points r and r+dr, the effective spatial interval dr is owing to curvature less than in Euclidean case. Making proper ∂C/∂r greater than a naive expectation of 2π]

[Mikko]

Where in part 3 the provision that ISM 'isotropy' might be entirely illusory was already given (now confirmed).

The isotropy of isotropic coordinates is not illusory. It is a property of a coordinate system, not of space. But the space has an important property, too: some spaces allow isotropic coordinate systems, some don't. That way having an isotropic coordinate system tells something about the space, whereas having an anisotropic coordinate system does not tell the opposite.

The prose after the equation 1-7 needs correction. As written v3, it refers to spatial anisotropy as if it were a property of the space. I cannot really comment the corrected text that I havn't seen, and there is no point to comment the erroneous text now that you understand at least the most obvious error.

The weak field calculations in section 2 are easier with isotropic coordinates. In a weak field the difference between the two coordinate systems is neglible. The calculation is still easier in Cartesian coordinates. The spherical coordinates do not work that well for off-center masses. With isotropic coordinates the transition to Cartesian coordinates is simple. The comparison to Yilmaz gravity is also easier in isotropic coordinates as the expression of the line element is much simpler than in Schwarzschild coordinates. Just make sure that you have the thin shell at the right place, which is defined in terms of the Schwarzschild coordinates. The isotropic radial coordinate is different, and in Yilmaz space different from Schwarzschild space.

If the analysis of the situation in Yilmaz space were properly written, it would be the most interesting part of the paper.

[Q-reeus]

The isotropy of isotropic coordinates is not illusory.

It is in ISM case as earlier demonstrated. One has no reliable sense of how it relates to proper values, except in the trivial limit of zero gravity. Whereas SM anisotropy does at least permit an intuitive and accurate translation to the local space in Schwarzschild spacetime. And better again is genuinely isotropic Yilmaz metric.

It is a property of a coordinate system, not of space.

See above.

But the space has an important property, too: some spaces allow isotropic coordinate systems, some don't. That way having an isotropic coordinate system tells something about the space, whereas having an anisotropic coordinate system does not tell the opposite.

Exotic spacetimes not allowing (globally valid) isotropic coordinate systems are of no interest here - spherically symmetric matter distribution is the subject at hand.

The prose after the equation 1-7 needs correction. As written v3, it refers to spatial anisotropy as if it were a property of the space.

I may emphasize more often the distinction in v4, but it was already made clear numbers of times in v3 so seems to me you are just nitpicking or reading wrongly.

I cannot really comment the corrected text that I havn't seen, and there is no point to comment the erroneous text now that you understand at least the most obvious error.

Again you are imo nitpicking as I explained the context of it all in my last post. However given my thoughts below you get more than just a tidied up version of what you have now in v3. None of the present findings of self-contradictory nature of SM will change though.

The weak field calculations in section 2 are easier with isotropic coordinates. In a weak field the difference between the two coordinate systems is neglible.

Indeed but the easier use of ISM is at the cost of being intrinsically misleading aka useless - as already explained before. Might as well try portrait painting while wearing deliberately lens-distorted glasses. Simply adding (generally components of) radial and transverse components in ISM is not valid. They will have different true 'weights' in general, and to get those weights right is problematic and turns a deceptively 'simple' situation into a complex one. Ergo - stick to honest SM, to the extent one wants to know what Schwarzschild spacetime truly (multiply!) predicts.

The calculation is still easier in Cartesian coordinates. The spherical coordinates do not work that well for off-center masses. With isotropic coordinates the transition to Cartesian coordinates is simple.

I effectively worked in Cartesian coordinates in evaluation of part 2, and IF I proceed to generalize to off-center case, would stick with that approach.

The comparison to Yilmaz gravity is also easier in isotropic coordinates as the expression of the line element is much simpler than in Schwarzschild coordinates. Just make sure that you have the thin shell at the right place, which is defined in terms of the Schwarzschild coordinates. The isotropic radial coordinate is different, and in Yilmaz space different from Schwarzschild space.

If the analysis of the situation in Yilmaz space were properly written, it would be the most interesting part of the paper.

Already done for particular shell center case, in so many words in part 3 of v3. Given the genuine isotropy then applying, evaluation at center is trivial. Simple scalar addition applies - for both temporal and spatial components. The result is obvious without needing to perform integral calculus - interior with equal levels of 'redshift' for clocks and rulers, and with a smooth and natural boundary match to the external values. Something not possible with SM - as proven. Caveat follows!

For quite awhile have been slightly uneasy about two assumptions that are generally made and I have followed. That hollow interior is truly an equipotential region - probably only analytically proven for Newtonian gravity. And that exterior field truly is given by assuming an equivalent central point mass equal to the summed shell mass. Might pay to check both using part 2 method, for both SM and Yilmaz metric. That means an analysis for general r location. Won't be done anytime soon.

Another erratum note: Reference to zero Weyl curvature in my last post should have read 'spatial component of....', and best left to just saying Cotton tensor curvature everywhere zero.

[Mikko]

If you only compare Schwarzschild space with Schwarzschild coordinates to Yilmaz space with isotropic coordinates, you may get two kinds of differences: some are consequences of different coordinate systems, other of different spaces. You need to sort out which is which.

It might be easier to avoid conceptual confusions if different symbols are used for Schwarzschild's and insotropic radial coordinates. Perhaps you could use ŕ or ř or ȓ (and for the coordinate of the shell Ŕ or Ř or Ȓ) or some other decoration.

[Q-reeus]

If you only compare Schwarzschild space with Schwarzschild coordinates to Yilmaz space with isotropic coordinates, you may get two kinds of differences: some are consequences of different coordinate systems, other of different spaces. You need to sort out which is which.

In either SM or YM (Yilmaz) case, as usually formulated an underlying spherical coordinate system is modified by the metric coefficients as 'seen' by a coordinate observer. The differences are intrinsic to the different gravity theories. Ignoring mechanical strains, for SM the diameter or circumference of shell is on both a proper and coordinate basis totally unaffected by gravity. Which cannot be true on a coordinate basis for YM. Hence 'BH EH's' allowed in former but not latter. What matters is the degree of self-consistency in either theory. To the extent interior flatness actually holds, SM disagrees with itself as already shown, the converse for YM. If it turns out there is say a second-order in metric departure from flatness for either or both, it's still the case YM will likely be internally consistent, but problems for SM would remain.

It might be easier to avoid conceptual confusions if different symbols are used for Schwarzschild's and insotropic radial coordinates. Perhaps you could use ŕ or ř or ȓ (and for the coordinate of the shell Ŕ or Ř or Ȓ) or some other decoration.

Will consider such matters at the time.

[Mikko]

Ignoring mechanical strains, for SM the diameter or circumference of shell is on both a proper and coordinate basis totally unaffected by gravity.

The diameter of the shell is R because that is how the situation is specified. If you wish, you may consider that the definition of R. Anyway, that truth is unaffected by anything, for nothing can override the specification. The circumference is 2πR because space is euclidean in the interior. No choice of coordinates can alter that. Of course, in a different theory with non-zero gravity in at least a part of the interior would have a different relation between the radius and the circumference.

[Q-reeus]

[The diameter of the shell is R because that is how the situation is specified. If you wish, you may consider that the definition of R. Anyway, that truth is unaffected by anything, for nothing can override the specification.

Seem to be speaking in riddles there, but I think you are basically echoing my previous reference to 'areal radius' (6th post p5) as defined in SM:

If you don't appreciate what that last bit means in context of spherical shell scenario, recall that by definition the shell 'areal radius' R is defined in terms of proper area A (or equivalently circumference C) - i.e. R = √(A/(4π)) = C/(2π).

The circumference is 2πR because space is euclidean in the interior. No choice of coordinates can alter that.

Well for sure in SM the above reproduced 'areal radius' definition can only consistently hold if the interior is not just Euclidean but spatially non-depressed Euclidean i.e. √(g_rr) = 1 exactly for all r<R. Which automatically imposes the matching issue of part 1 in article.
On the other hand in YM the only stipulation is interior depressed Euclidean such that value of metric operator exp(Φ/c^2) has vanishingly small difference from just exterior to anywhere interior. Which gives a consistent picture of the entire shell shrinking by that factor on a coordinate basis. Contrast to SM where shell is propped up at constant R regardless of how great the negative potential becomes. As I wrote last time, it's what permits finite sized black holes/event horizons in SM. In YM, gravitational collapse (unrealistic 'dust' case) asymptotically approaches zero radius for collapsing matter without a horizon forming. No causally disconnected regions, no strange swapping of t for r, 'wormholes' to other spacetimes, 'firewalls' or some-such speculative QM effects to solve 'information paradoxes', and similar bizarre stuff. Things are boringly normal - and sensible.

Of course, in a different theory with non-zero gravity in at least a part of the interior would have a different relation between the radius and the circumference.

See above. If there is only partial 'non-zero gravity' i.e. equipotential not applying then that's an additional mess to sort out. It's already implied by part 2 of v3 article - not that equipotential failure is real, but that SM predicting it reveals it's internal inconsistency. One can just reasonably adopt the position that in weak field regime Newtonian potential Φ must be arbitrarily close to an equipotential in shell interior (declared early in part 1 btw). Thus ANY departure from flatness predicted by a direct evaluation of interior metric as per part 2 of article, spells death for such gravity theory, given the explicit dependence on Φ. Which is the case for SM. We are going round in circles and unless you have a very specific critique of a technical maths nature re my basic math results in parts 1 and 2 of v3, best to just wait for v4.

Other issues have priority right now.

[Mikko]

The circumference is 2πR because space is euclidean in the interior. No choice of coordinates can alter that.

Well for sure in SM the above reproduced 'areal radius' definition can only consistently hold if the interior is not just Euclidean but spatially non-depressed Euclidean i.e. √(g_rr) = 1 exactly for all r<R.

In Euclidean geometry the ratio of circumference to radius is always 2π. If you find anything else then your geometry is non-Euclidean.

[Q-reeus]

In Euclidean geometry the ratio of circumference to radius is always 2π. If you find anything else then your geometry is non-Euclidean.

In actuality yes. As per part 1 of article, there must be an asymptotic correspondence in the weak gravity regime between Newtonian gravity and any reasonable metric theory. So we know the interior will be equipotential thus Euclidean barring just maybe a vanishingly small correction in some higher order powers/derivatives of metric. YM demonstrably has no issue with that. But clearly part 2 direct metric evaluation for SM presents the issue shown - depressed spatial metric at center. Which contradicts the basic character of SM that transverse components have no potential dependence. The only remaining question is whether such direct method predicts interior flatness (requiring a relatively violent and unnatural transition within shell wall), or a somewhat smooth transition from center to shell wall. That has an easy match to exterior, at the obvious expense of violating the assumption of interior flatness. None of those two options is logically or physically acceptable. How many times do I have to point that out?
[PS: Are you getting email notifications re this thread? I never do!]

[FrediFizzx]

Email notifications are turned off. It was a spam problem thing for email notifications with the server we are on. I may turn them back on to see what happens.

OK, subscribing is turned back on. Send me a private message if you don't get notified.

[Mikko]

As per part 1 of article, there must be an asymptotic correspondence in the weak gravity regime between Newtonian gravity and any reasonable metric theory. So we know the interior will be equipotential thus Euclidean barring just maybe a vanishingly small correction in some higher order powers/derivatives of metric. YM demonstrably has no issue with that.

In GR the interior spatial metric is not merely asymptotically Euclidean but exactly. This corresponds exactly to the Newtonian zero gravity. But you have not made it clear why do you consider is as any anomaly, nor how would YG be any better (or otherwise different). The only difference you have shown is that you prefer to use Schwarzschild coordinates with GR but not with YG and isotropic coordinates with YG but not with GR. As both coordinate systems are quite reasonable in for both GR and YG, this hardly constitute any anomaly, even in your preferences.

[Q-reeus]

In GR the interior spatial metric is not merely asymptotically Euclidean but exactly. This corresponds exactly to the Newtonian zero gravity.

Which merely repeats precisely what was stated in part 1 of article! You think I somehow forgot that or now oppose my own writing there? No. You seem to simply never get the proper context of what I write. There is no conflict with what I wrote last post and what I wrote in part 1 of article! Despite how you may be interpreting such.

But you have not made it clear why do you consider is as any anomaly, nor how would YG be any better (or otherwise different). The only difference you have shown is that you prefer to use Schwarzschild coordinates with GR but not with YG and isotropic coordinates with YG but not with GR.

The natural metric choices - the ones appearing as originally derived, are anisotropic SM and isotropic YM respectively. Sure they can both be rejigged mathematically - as is done with SM -> ISM (but never afaik an analogous case of YM -> ?). I have previously explained the issue you get if trying to do a direct evaluation (part 2 of article) with ISM. You cannot just add (in the tensor strain field sense) elemental contributions from various locations around the shell, owing to the disguised weightings present. Far better to stick to vanilla SM for that purpose, where weightings are always unity.

As both coordinate systems are quite reasonable in for both GR and YG, this hardly constitute any anomaly, even in your preferences.

It just means you are being deliberately antagonistic out of presumably deep devotion to GR, or are unable to get it. I just hope it is merely the latter. But here's an acid test. Numbers of times over this now six pages of dialogue I have asked you to specifically find fault with in particular the math derivation (and underlying assumptions) of part 2 of v3 article. There has been none forthcoming. Strange, since you appear to know GR thus it should be child's play to do so. So finally do it! That imo will be the crunch test of your sincerity and ability. Then it is you that will have to explain the finding. Recalling my oft repeated statement - any apparent paradox found when working consistently in a given metric choice (specifically SM) spells death for such metric choice.

[Mikko]

The natural metric choices - the ones appearing as originally derived, are anisotropic SM and isotropic YM respectively.

One coordinate system is not more natural than another, they are all artificial. We often say that same choice is "natural" if it makes something simpler than any other choice, but saying so does not make it so -- we just are sloppy with the language. And anyway, simplicity depends on purpose. It your intent is to compare two spaces, it is not natural to choose different coordinate systems and let the difference obscure the comparison.

Numbers of times over this now six pages of dialogue I have asked you to specifically find fault with in particular the math derivation (and underlying assumptions) of part 2 of v3 article.

There are still problems in part 1 and in those parts of part 3 that do not depend on part 2. Discussion of part 2 does not solve them. Anyway, if you write explicitely the formula for ds² in the interior region (in the same style as you have for the exterior region), that would clarify the part 1 and probably part 2, too.

[Q-reeus]

One coordinate system is not more natural than another, they are all artificial. We often say that same choice is "natural" if it makes something simpler than any other choice, but saying so does not make it so -- we just are sloppy with the language.

In both SM and YM, proportionality between radial and transverse components permits straightforward (tensor) addition of components a la part 2. Not so for ISM. Natural, vs contrived 'isotropy'.

And anyway, simplicity depends on purpose. It your intent is to compare two spaces, it is not natural to choose different coordinate systems and let the difference obscure the comparison.

Evidently what I have been consistently saying is just going in one ear and out the other. You are making me out to be the opposite of what is so.

There are still problems in part 1 and in those parts of part 3 that do not depend on part 2.

Obscure comments as usual. Provide relevant and specific details. No - wait - just do what I asked last time - essential bit repeated below.

Discussion of part 2 does not solve them.

Solve what exactly?! Part 2 shows up the problems (with SM) more severely!

Anyway, if you write explicitely the formula for ds² in the interior region (in the same style as you have for the exterior region), that would clarify the part 1 and probably part 2, too.

What are you talking about? Have you still not grasped that math of part 2 implicitly contains such?

Bottom line: Are you or are you not prepared to both analytically critique (maths!) mine, and/or offer your own independent derivation of, my part 2? Why do you keep evading what is a quite low level maths exercise? If you are not so prepared then I must reluctantly conclude you are just TROLLING.