Does a Spherical Hollow Shell Represent a New Type of Gravitational Horizon?

[Q-reeus]

Last post wasn't quite right. Integrating the differential expressions dr/dS, dtau/dt will not give correct results either. Need to start from scratch - define the dependence on potential of coordinate distance and time wrt to their proper values. Dusting off some old photocopies of "Gravitation and Relativity" (M.G.Bowler), we have in ch.6 'The Distortion of Reference Frames', for light speed c, frequency f, and isotropic distance r (see p75 re isotropy of r),:
c_φ = c(1+2φ), f_φ = f(1+φ), r_φ = r(1+φ), where φ = -GM/(c²r) is the Newtonian potential, and φ sub-scripted terms are equivalent to coordinate values. These are the fundamental dependencies on φ, valid for weak gravity, and quite unlike the φ dependence of gravitational acceleration g = -dφ/dr. Note that clock rate (f_φ) and distance r_φ have exactly the same φ dependence - everywhere. Also that c_φ has the right value to ensure flat spacetime physics applies within the shell cavity - given that time dilates to the same extent space isotropically contracts. This match up is only correctly portrayed using isotropic SC, as discussed in entry #49.
Interestingly, unlike standard SC, isotropic SC implies divergence of radial metric lines akin to lines of E in a region containing a nonzero volume charge density. Properly reflecting that 'gravity gravitates' perhaps, or at least that nonlinearity should be reflected in all directions of metric? This is about as far as I can and want to go here, but feel free to comment.

 

[Naty1]

I lost track of this thread...now I believe post #36 is correct and answers my original post question:

There is a uniform potential inside a hollow shell and it is lower than outside the shell and it is potential that causes time dilation. It is more accurate to say there is no gravitational field potential gradient inside the hollow (so that is why there no gravitational force) but that is irrelevant to the time dilation at a given point.

 

[Q-reeus]

Not sure where yuiop stands on this one now but as I have been at pains to point out, time dilation is one half the story - there is equally length contraction, both exterior and interior to the shell. Not specifically mentioned in #51 is the fact that for a static, non-rotating mass (our shell for instance) time dilation and length contraction factors are scalar functions of potential, guaranteeing isotropy in both cases. In hindsight, even isotropic SC does not really fix things consistently. In #49 I wrote: "dtau/dt varies by an extra factor 1-(GM/(c²r₁))², wrt dr/dS, which is second order and only becomes important in strong gravity situations." Problem is it becomes very important in strong gravity, pointing to a fundamental inconsistency - a proper coordinate system (for an observer not in free fall) should accurately reflect that clock rate and length scale have equal and isotropic dependence on potential everywhere. The one cure as I see it is to adopt what is known as an exponential metric, where the standard SC

is modified such that all of the spatial components are equally acted on by the term (1-r_s/r)^-1. And guess what happens to a black hole when that is done - 'event horizon' shrinks to a point. This situation arises 'naturally' when a la H Yilmaz gravity, a negative energy density and gravitational mass is assigned to spacetime curvature itself - ie it becomes part of the energy-momentum stress tensor. This is considered heretical but then again Clifford Will amongst other 'heavyweights' believes 'gravity is a source of further gravity' (though he would be against Yilmaz). Suffice to say as of a few days ago I no longer believe in BH's and a fair swag of GR. But that's purely my own pov.

 

[yuiop]

Not sure where yuiop stands on this one now but as I have been at pains to point out, time dilation is one half the story - there is equally length contraction, both exterior and interior to the shell

I have been considering length contraction in my posts and I have already stated my position in various posts which I will now summarise in a single post.

First I will define 3 regions for the spacetime of a uniform density non-rotating shell with outer radius R and inner radius r. One is the exterior region outside the shell which is a vacuum and covered by the Schwarzschild metric. The second is the region in the material of the shell itself covered by the shell metric and the third is the cavity inside the shell which is also a vacuum and covered by the cavity metric. In each case the radial position where the local measurements are made is denoted by x, the location of the coordinate observer is at infinity and units are such that c=1.Exterior vacuum metric (x >= R):

<br /> d\tau^2 = \left(1-2M/x}\right) dt^2 - \left(1-2M/x\right)^{-1}dx^2 -x^2(d\theta^2+\sin(\theta)^2d\phi^2)<br />

Here time dilation, length contraction and the coordinate speed of light are given by:

dt/d\tau = ({1-2M/x})^{1/2}

dS/dx = ({1-2M/x})^{-1/2}

dx/dt = (1-2M/x)Uniform density shell metric (r <= x <= R):

<br /> d\tau^2 = \left(\frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2M}{x}\frac{(x^3-r^3)}{ (R^3-r^3) }}\right)^2 dt^2 - \left(1-\frac{2M}{x}\frac{(x^3-r^3)}{ (R^3-r^3) }\right)^{-1}dx^2 -x^2(d\theta^2+\sin(\theta)^2d\phi^2)<br />

Here time dilation, length contraction and the coordinate speed of light are given by:

d\tau/dt = \frac{3}{2}\sqrt{1-\frac{2M}{R}} - \frac{1}{2}\sqrt{1-\frac{2M(x^3-r^3)}{x(R^3-r^3)}}

dS/dx = \left(1-\frac{2M(x^3-r^3)}{x(R^3-r^3)}\right)^{-1/2}

<br /> dx/dt = \frac{3}{2}\sqrt{\left(1-\frac{2M}{R}\right)\left(1-\frac{2M}{x}\frac{(x^3-r^3)}{ (R^3-r^3) }\right)} -\frac{1}{2}\left(1-\frac{2M}{x}\frac{(x^3-r^3)}{ (R^3-r^3) }\right) <br />Vacuum cavity metric (0 <= x <= r):

<br /> d\tau^2 = \left(\frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\right)^2 dt^2 - dx^2 -x^2(d\theta^2+\sin(\theta)^2d\phi^2)<br />

The time dilation factor, length contraction factor and coordinate speed of light within the cavity are:

<br /> d\tau/dt = (3/2)(1-2M/R)^{1/2}-1/2<br />

<br /> dS/dx= 1<br />

<br /> dx/dt = (3/2)(1-2M/R)^{1/2}-1/2<br />

If I have done the calculations correctly, then at the boundary of any two metrics above, you can use the metric of the region either side of the boundary and get the same answers. The above metrics also give the result that length contraction, time dilation and the coordinate speed of light is isotropic and constant everywhere inside the cavity as we would expect, so I am not sure why you are not satisfied with the results.

 

[Q-reeus]

I had expected this thread to sink quietly out of sight till Naty1 revived it - #47->#51 just petered out as a monologue!
Once again, as I wrote in #45, your maths seems OK - given you are working with formulas based around standard SC. However...
To put it simply, my conclusion is working with standard SC in particular (and ISC is only a half-way house) completely butchers the correct perspective expected of coordinate quantities; dtau/dt, dx/dS, that are inherently equal scalar functions of potential (as per #51). This distortion reaches extreme proportions in the vicinity of a notional BH EH, where local Lorentzian physics is maintained only because one extreme distortion (radial vs tangential length) is canceled by another (radial vs tangential c). No great surprise that multiplying zero by infinity can give rise to some strange creatures like BH's with finite EH's for instance. It's also probably the main or sole reason for those divergent diagrams in #40, #42. It sort of explains why on the one hand you can state in #36:
"There is a uniform potential inside a hollow shell and it is lower than outside the shell and it is potential that causes time dilation.", (and as per #48, whether you choose dt/dtau or it's inverse dtau/dt is "Merely a matter of working with 'period' or 'frequency'")while in #40:
"This hints at what I said earlier that length contraction is a function of potential gradient rather than the value of the potential itself like gravitational force."
It's the graphs that make one think that, forgetting that exterior to the shell, clock rate and length have precisely equal scalar dependency on potential. It is completely illogical to imagine that can radically alter within the shell, as though some kind of bizarre spacetime transformation selectively alters functional form for one and not the other. If equal scalar functions here, then equal scalar functions everywhere! Can't overemphasize that point - it's my fundamental position. Moral - recognize that relevant quantities are scalar functions of potential, hence inherently isotropic functions. Next, choose a metric that fully respects that fact - and exponential metric is imho the only proper candidate here. Finally, compute the potential, and then determine the relevant time and distance parameters directly from that. Expected result - isotropic and equal contraction of clock rate and length. I don't possesses the maths software to attempt that myself - sorry. Can we make a pact to have this all sorted by or before it gets to #60 - what do you say?

 

[yuiop]

If equal scalar functions here, then equal scalar functions everywhere! Can't overemphasize that point - it's my fundamental position.

This is how I see it. Length contration is a function of (Mass_below) and time dilation is a function of (Mass_below & Mass_above). Outside the shell there is no Mass_above so both effects behave as if they are purely functions of Mass_below. Once you pass into the shell or cavity, there is Mass_above and the differences manifest themselves. This difference is masked outside the shell. Since length contraction is purely a function of Mass_below (consistently everywhere inside and outside the shell) there is no length contraction inside the cavity as there is no Mass_below there, while time does dilate inside the cavity because it is a function of Mass_above.

Can we make a pact to have this all sorted by or before it gets to #60 - what do you say?

There you go.. all done and dusted by post #56

 

[Q-reeus]

This is how I see it...
There you go.. all done and dusted by post #56

Guess you figured I couldn't let it end just yet. First a few corrections re #55:
"..dtau/dt, dx/dS, that are inherently equal scalar functions of potential (as per #51)..."
"..clock rate and length have precisely equal scalar dependency on potential..."
"If equal scalar functions here, then equal scalar functions everywhere!"
Actually that depends in each case on which coordinate system one adopts. Clearly in standard Schwarzschild metric length is a tensor quantity, while in isometric Schwarzschild metric and exponential metric it is purely scalar. So I ought to have prefixed 'in my opinion...'. Nevertheless only one of these can truly be reflecting the actual physics wrt a distant fixed observer. My choice is the latter - owing to what I see as anomalies when adopting either of the first two. As is now evident, part and parcel of adopting exponential metric is to accept this entails some version of 'gravity gravitates' whether Yilmaz or a similar competing theory. That is non-kosher. It has become increasingly obvious this forum is not about debating various theories but largely about correcting persons with initially non-kosher ideas. So for the record, I consider myself 'kosher' re SR but as of a short while ago 'non-kosher' re GTR. In summary, this thread has been a real personal educational experience. Came in as a novice who just accepted GTR as 'gospel', but have finished up as a non-believer. I accept where you are coming from, so best thing is I think to just agree to disagree, shake hands, and move on.

 

[yuiop]

In summary, this thread has been a real personal educational experience. Came in as a novice who just accepted GTR as 'gospel', but have finished up as a non-believer. I accept where you are coming from, so best thing is I think to just agree to disagree, shake hands, and move on.

Hi Q-reeus. I am a little bit saddened that I seem to have somehow made you lose faith in GR. That was never my intention and I obviously failed spectacularly.

Before you quit, could you answer this? You obviously believe that whatever happens outside the shell should continue to happen within the shell. If we follow your premise, then both radial length contraction and time dilation reach a maximum within the shell. Right? Outside the shell circumferential length contraction does not occur, so being consistent with your premise, circumferential length contraction does not occur within the cavity either. Now if we do NOT have circumferential length contraction within the cavity and we DO have maximum radial length contraction within the cavity, then the space inside the cavity cannot be Euclidean. Is that your position? Do you not accept that the space within the cavity should be Euclidean? Do you disagree with my conclusion that the coordinate speed of light is isotropic inside the cavity?

 

[Q-reeus]

Hi Q-reeus. I am a little bit saddened that I seem to have somehow made you lose faith in GR. That was never my intention and I obviously failed spectacularly.

Not at all yuiop. You have actually performed a real service in that your hard work with detailed shell calculations, boundary matching etc, and charts have highlighted the critical role choice of metric makes, something that had never crossed my mind previously. More on that below.

Before you quit,..

Please don't misunderstand. When I said "..move on." that simply meant 'let's bury this thread here and now', not that I was exiting PF. However I have been thinking that anyway, but for reasons other than belief or not in GTR. But that's another matter.

could you answer this? You obviously believe that whatever happens outside the shell should continue to happen within the shell. If we follow your premise, then both radial length contraction and time dilation reach a maximum within the shell. Right?

It's what I believe is the actual physics, yes.

Outside the shell circumferential length contraction does not occur, so being consistent with your premise, circumferential length contraction does not occur within the cavity either.

If you adopt SSM (standard Schwarzschild metric), as you have done (commented on that in #48 - Gron et al used SSM), that position necessarily follows, agreed. But it is not consistent with my premise because I believe SSM is not reflecting the actual physics. More below.

Now if we do NOT have circumferential length contraction within the cavity and we DO have maximum radial length contraction within the cavity, then the space inside the cavity cannot be Euclidean. Is that your position?

I agree, but it's a monster that could only arise by mixing different coordinate schemes. SSM, ISM (isotropic Schwarzschild metric), and EM (exponential metric) all predict Euclidean interior owing to an equipotential situation. They disagree however on other things.

Do you not accept that the space within the cavity should be Euclidean?

It has to be, as per my last comment. In fact in #27 I first came to that firm conclusion, quoting Birkhoff's theorem in support.

Do you disagree with my conclusion that the coordinate speed of light is isotropic inside the cavity?

No, but that's not the real issue. Please re-read #49, where I point out that adopting ISM inevitably requires an interior contracted length. And while EM (which in weak gravity yields essentially the same result) is considered by some to be 'fringe physics', ISM is a perfectly 'respectable' metric choice. So here's the rub: use SSM and as you have shown, tangent matching forces an uncontracted interior spatial metric - hence a severe jump in radial component. Using ISM by contrast forces a contracted interior spatial metric (with no sudden jumps anywhere), but one that does not exactly match the temporal (frequency) component. EM goes one further and has a smooth transition for all components and an equal match of spatial and temporal contractions. As per #51, reading Bowler's simplified explanation of how energy, mass, length, time, light speed all have scalar dependence on gravitational potential was something I could follow and it impressed me. So I asked "how to modify SSM to make it reflect fully scalar dependence, and equally so for spatial and temporal components", and it became obvious one simply applies the metric operator (1-r_s/r)^-1 to all the spatial components. It then struck me the circumference of a notional BH EH then goes to zero - ie is non-existent. So next did a web search for "gravity theory without BH's" and eventually hit upon Yilmaz gravity, where was immediately struck by the fact the metric used, EM (derived on the basis gravitational energy density should be part of T), had the form I surmised based on a different requirement, of scalar = isotropic response as 'true physics'. Certainly nothing rigorous on my part, but it's my opinion such an outlook is more consistent than otherwise with SSM in particular.

In summary, for our static spherically symmetric gravitational case, the shell, there can be only one 'true physics' seen by a stationary observer. Trouble is, SSM, ISM, and EM are all contenders that predict different and incompatible versions of what that true physics is. So a correct choice needs to be made. And one has to base that on some clear principle. Mine is that scalar dependence applies everywhere, equally for temporal and spacial, and hence EM - which is 'fringe' but that doesn't make it wrong imho. There is afaik no current observational test that distinguishes between the three. So there you have it!