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

[yuiop]

...if there is no gravitational field potential inside the hollow shell there is no time dilation...time must pass the same inside a thin hollow (mass or energy) shell as distantly outside the shell...

and from that I conclude that while it's an interesting test situation, it's probably not a horizon...

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]

dr=d\theta=d\phi=0[/tex] so that:

$$d\tau = \left(\frac{3}{2}\sqrt{1-\frac{2GM}{R}}-\frac{1}{2}\right) dt$$
...The coordinate radial speed of light in the cavity is found by setting $$d\tau=d\theta=d\phi=0$$ and solving for dr/dt and the result is:

$$\frac{dr}{dt} = \left(\frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\right)$$

Can't finger where it has entered but by inspection these don't seem right (a missing G in the second expression is just a typo). As 2GM/R ranges from 0 to 1 (which is theoretically permissible), the full expression within the parentheses starts at 1, goes to zero, and then through to -1/2. !

The radial spatial interval dS is equivalent to $$\sqrt{-d\tau^2}$$ and the radial length contraction factor is found by setting $$dt=d\theta=d\phi=0$$ giving:

$$dS = dr$$

i.e. no length contraction in the radial direction and the same is true for circumferential lengths $$d\theta$$ and $$d\phi$$.

I'm going to use someone else's work here. At https://www.physicsforums.com/showthread.php?t=404153" - entry #9 (somehow mucked up the link in entry #9 this thread), last two expressions are
[PLAIN]https://www.physicsforums.com/latex_images/27/2730817-7.png
[PLAIN]https://www.physicsforums.com/latex_images/27/2730817-8.png
which is coordinate time (take out c factor) and coordinate length respectively.
These are Schwarzschild solutions taken as valid everywhere exterior to r. Hopping from the exterior shell surface to inside what changes. As the potential is now constant, I would simply fix r = shell radius = constant, and the exterior values at r persist 'frozen' everywhere within. What do you think?

 

[yuiop]

Can't finger where it has entered but by inspection these don't seem right (a missing G in the second expression is just a typo). As 2GM/R ranges from 0 to 1 (which is theoretically permissible), the full expression within the parentheses starts at 1, goes to zero, and then through to -1/2. !

I set G=c=1 and meant to take all the G and c's out but a few got left in. Hopefully fixed now. As for the proper time of clocks in the cavity running backwards relative to external clocks once 2GM/R<9/8, that is correct. That is what the maths predicts and 2GM/R can take greater values than 1 if the the shell continues to fall inwards as GR predicts so that clocks run backwards even faster than -1/2.

I'm going to use someone else's work here. At https://www.physicsforums.com/showthread.php?t=404153" - entry #9 (somehow mucked up the link in entry #9 this thread), last two expressions are
[PLAIN]https://www.physicsforums.com/latex_images/27/2730817-7.png
[PLAIN]https://www.physicsforums.com/latex_images/27/2730817-8.png
which is coordinate time (take out c factor) and coordinate length respectively.
These are Schwarzschild solutions taken as valid everywhere exterior to r. Hopping from the exterior shell surface to inside what changes. As the potential is now constant, I would simply fix r = shell radius = constant, and the exterior values at r persist 'frozen' everywhere within. What do you think?

That does not work. You are assuming a shell with zero thickness, but since the gravitational mass is all contained in the shell that implies flat spacetime with no mass. For a shell with any non zero thickness you need to use the exterior metric, then the interior metric when transitioning the shell and then the cavity solution inside the shell. Each metric transitions smoothly from one to the other and have to match at the boundaries. You are right that once you get to the cavity, the gravitational potential and time dilation factor are "locked in" to that of the boundary. If you were to tunnel towards the centre of the Earth you would see changes in gravitational potential and time dilation as you went deeper and they are determined by the interior Schwarzschild metric.

Even if you completely ignore the interior solution and lock in the exterior solution as you pass into the cavity I think you would agree that horizontal lengths would not be length contracted (because they are not contracted outside the shell). Now if the horizontal rulers are not length contracted and the inside is Euclidean, then the vertical or radial rulers cannot be length contracted either. Agree?

 

[Q-reeus]

I set G=c=1 and meant to take all the G and c's out but a few got left in. Hopefully fixed now.

Format and my own references there weren't quite right - couldn't get cut and paste working properly and so gave up and left it as is. If there is some tutorial here at PF on editing procedures I could sure do with knowing about it! (BTW - am I the only one finding that on occasion the editing functions are frozen, requiring a browser reboot?)

As for the proper time of clocks in the cavity running backwards relative to external clocks once 2GM/R<9/8, that is correct. That is what the maths predicts and 2GM/R can take greater values than 1 if the the shell continues to fall inwards as GR predicts so that clocks run backwards even faster than -1/2.

Evidently that other thread "A New Type of BH" has been occupying much of your mind! First time I have heard of the possibility of time running backwards, but if that's what washes up, so be it! Wait and see at the moment.

That does not work. You are assuming a shell with zero thickness, but since the gravitational mass is all contained in the shell that implies flat spacetime with no mass. For a shell with any non zero thickness you need to use the exterior metric, then the interior metric when transitioning the shell and then the cavity solution inside the shell. Each metric transitions smoothly from one to the other and have to match at the boundaries.

Strictly speaking you are correct that a zero thickness shell is physically unrealistic, but as a mathematical simplification it is in this context reasonable. One example of its use is at http://cnx.org/content/m15108/latest/" under "Gravitational potential due to thin spherical shell" esp. fig.4. We both agreed early on in this thread that it is the potential and not its gradients that determine metric 'contraction' (over some infintesimal region a la 'local Lorentz invariance' condition). Essentially all the work has been done in going from infinity to the surface. Changes in potential in traversing a thin shell are smooth and minor (as opposed to the abrupt changes in gradient), and vanishes in the limit of a infinitesimally thick shell. If you like we could 'split the difference' and make r the weighted mean between outer and inner radii, but as the link above suggests, there is little point.

Even if you completely ignore the interior solution and lock in the exterior solution as you pass into the cavity I think you would agree that horizontal lengths would not be length contracted (because they are not contracted outside the shell).

As the two expressions in entry #37 show, there is at any exterior radius an equal measure of time dilation and length contraction wrt 'infinity'. This is thus so at the shell surface, and from the above reasoning, both will persist 'frozen' inside the shell. Let me put it another way. If as you maintain distance measure is uncontracted at the shell surface/interior wrt infinity, do either your eqn's or those I have 'borrowed' predict a monotonic change in length measure as a function of radius exterior to the shell? I think the answer is a clear yes. If so, how can that lead to a proper match - shell surface wrt infinity?

Now if the horizontal rulers are not length contracted and the inside is Euclidean, then the vertical or radial rulers cannot be length contracted either. Agree?

Agreed that what happens happens isotropically!

Well here in Oz it's just finished Xmas eve. Only reason I can sometimes interact with I guess about 95+% of you folks is because of having some spare time of late and not shaking off the habit of odd hours after years of night-shift work. All the best for Xmas - catch you all in a few days.[URL]http://www.christmas-corner.com/images/christmas-smileys/santa_wink.gif[/URL]

 

[yuiop]

This is thus so at the shell surface, and from the above reasoning, both will persist 'frozen' inside the shell. Let me put it another way. If as you maintain distance measure is uncontracted at the shell surface/interior wrt infinity, do either your eqn's or those I have 'borrowed' predict a monotonic change in length measure as a function of radius exterior to the shell? I think the answer is a clear yes. If so, how can that lead to a proper match - shell surface wrt infinity?
.[PLAIN]http://www.christmas-corner.com/images/christmas-smileys/santa_wink.gif[/QUOTE]

Hi Q-rees, Hope you had a good Christmas and New Year's Eve and hope you have finished partying!

I have had a closer look at how length contraction and time dilation change as you traverse a shell and rather than post all the equations I will post a graph:

[URL]http://i910.photobucket.com/albums/ac304/kev2001_photos/GravitationalShell.gif[/URL]

The yellow area is the shell that contains mass. Everywhere else is vacuum and the centre of the shell cavity is on the left.

The blue curve is the length contraction factor which starts at 1 at infinity, reaches a peak at the outer shell surface (R) and ends up back at 1 at the inner surface (r) inside the cavity for any r<R.

The red curve is the Newtonian force of gravity which is similar to the length contraction factor in that its value peaks at R and is the same at infinity and inside the cavity (zero). 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.

The light blue curve is the time dilation factor which starts at 1 at infinity and continuously drops to its lowest value inside the cavity. The last curve is the Newtonian gravitational potential which behaves in a similar way but starts at zero at infinity.

Now if the inner radius (r) is moved outward making the shell thinner but without changing the total mass the outside curves are not altered and the values inside the cavity (except for the Newtonian potential) remain unchanged. If we make r=R so the shell has zero thickness (and therefore infinite density) the result is as plotted below:
[URL]http://i910.photobucket.com/albums/ac304/kev2001_photos/Shell2.gif[/URL]
It can be seen that there is a problem here, because at a radius of r=R there are two possible results for time dilation or length contraction so there is an ambiguity or inconsistency. This is a result of having an non-physical situation of zero shell thickness and infinite density. Clearly assuming a shell of zero thickness does not work. If I allowed the mass of the shell to decrease as reduced the thickness there would be no mass at all and no gravitational field. The only thing that appears to maintain some sort of continuity across a zero thickness shell is the Newtonian potential.

So the result stands as before. Length contraction increases as you approach the outer shell and then rapidly reduces as you pass through the shell to a value of unity inside the cavity. This means that vertical and horizontal distances are not subject to gravitational length contraction inside the shell and so the geometry is Euclidean. Spatially the geometry inside the cavity is the same as in flat space far from any gravitational sources. The time dilation is different to flat space and is greater than anywhere outside the shell. This however would not be noticeable to an observer inside the shell because the speed of light is isotropically slower by the same factor inside the shell so everything appears normal and is indistinguishable from Minkowski space to a local observer.

 

[Q-reeus]

Hi Q-rees, Hope you had a good Christmas and New Year's Eve and hope you have finished partying!

Thanks yuiop. Yeah survived it all OK, and it seems you have too. All the best for 2011!

..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...

Sorry I inadvertently misrepresented your position in my last entry.

The blue curve is the length contraction factor which starts at 1 at infinity, reaches a peak at the outer shell surface (R) and ends up back at 1 at the inner surface (r) inside the cavity for any r<R...
..The light blue curve is the time dilation factor which starts at 1 at infinity and continuously drops to its lowest value inside the cavity. The last curve is the Newtonian gravitational potential which behaves in a similar way but starts at zero at infinity.

I like graphs - a picture can be worth a thousand words. There is a problem here though. Exterior to the shell your results for time dilation and length contraction are different to the solutions I 'borrowed' from https://www.physicsforums.com/showthread.php?t=404153" (#9) and used in this thread at the bottom of #37. Those results give equal values, so we had better sort out where the discrepancy lies. (Edit: Partly this is just a matter of definition. Inverting your length contraction curve will give the usual sense of a contraction, or vice versa for the time dilation curve. The relative changes for time and distance measure wrt to r will then agree for either set of eqn's. Functionally though, there is still an important difference that effects things when traversing the shell etc.) Your later comments about discontinuities in an infinitesimally thick shell will hopefully then resolve. What we agree on at the moment - there is time dilation interior to the shell, and it is everywhere 'flat' spacetime within. Still some work to do!

 

[yuiop]

Thanks yuiop. Yeah survived it all OK, and it seems you have too. All the best for 2011!

Thanks, all the best to you too!

I like graphs - a picture can be worth a thousand words. There is a problem here though. Exterior to the shell your results for time dilation and length contraction are different to the solutions I 'borrowed' from https://www.physicsforums.com/showthread.php?t=404153" (#9) and used in this thread at the bottom of #37. Those results give equal values, so we had better sort out where the discrepancy lies.

Those results do not give equal values! They are the inverse of each other. The equations I have plotted are the ratio (local measurement)/(coord measurement) so that outside the shell:

$$\frac{d\tau}{dt} = \sqrt{1-2M/x}$$

and

$$\frac{dS}{dx} = \frac{1}{\sqrt{1-2M/x}}$$

which is just a rearrangement of the equations you gave earlier except I am using x instead of r for the radial displacement, because I have reserved r to mean the inner radius of the shell (and R to mean the outer radius of the shell).

(Edit: Partly this is just a matter of definition. Inverting your length contraction curve will give the usual sense of a contraction, or vice versa for the time dilation curve. The relative changes for time and distance measure wrt to r will then agree for either set of eqn's.

You could, if you wish use the inverse ratios, but that would not substantially change the arguments and it a little inconvenient to plot because $$dt/d\tau$$ tends towards infinity as x tends towards 2M.

Functionally though, there is still an important difference that effects things when traversing the shell etc.)Your later comments about discontinuities in an infinitesimally thick shell will hopefully then resolve.

Here are the equations inside the material of the shell but outside the cavity (i.e. r<x<R) using the Schwarzschild interior solution and uniform density, where x is the radial location of an observer that makes local measurements.

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

$$\frac{dS}{dx} = \frac{1}{\sqrt{1-\frac{2M(x^3-r^3)}{x(R^3-r^3)}}}$$

It is fairly easy to see from the second equation for gravitational length contraction when traversing the shell that when x=r, the ratio always reduces to unity for any value of M or R>=r. In other words there is no gravitational length contraction at the inner radius of the shell (or in the cavity).

Here are some more graphs with similar mass and outer radius as the original graphs but I have removed the Newtonian curves for clarity. As before the yellow region represents the material of the shell, the blue curve is the length contraction ratio and the light blue curve is the time dilation ratio.

Now when inner radius is expanded without changing the total shell mass, the following graph is obtained:

The interesting aspect is that the time dilation and length contraction outside the shell and inside the cavity are completely unaffected! This thin shell can be thought of as representing the infinitesimally thin shell you speak of. It can be seen that the time dilation and length contraction curves change rapidly within the thickness of the thin shell. As long as we understand an infinitesimally thin shell is not a shell of exactly zero thickness then everything is O.K. because a shell of zero thickness and infinite density is unphysical.

What we agree on at the moment - there is time dilation interior to the shell, and it is everywhere 'flat' spacetime within. Still some work to do!

While you, me and Naty1 all seem to come to agreement that the geometry inside the cavity is Euclidean, your conjecture that the length contraction inside the cavity is the same as that immediately outside the shell cannot bring that about, because vertical lengths will be different to horizontal lengths. The equations I have given on the other hand, do agree with the conclusion that the geometry is Euclidean inside the cavity.

 

[Q-reeus]

Thanks, all the best to you too!
Those results do not give equal values! They are the inverse of each other.

Taken straight as is yes. What I meant and Edit sort of explained was one has to invert the relation dS/dx -> dx/dS to express it as a length contraction relation.

The equations I have plotted are the ratio (local measurement)/(coord measurement) so that outside the shell:

$$\frac{d\tau}{dt} = \sqrt{1-2M/x}$$

and

$$\frac{dS}{dx} = \frac{1}{\sqrt{1-2M/x}}$$

which is just a rearrangement of the equations you gave earlier except I am using x instead of r for the radial displacement, because I have reserved r to mean the inner radius of the shell (and R to mean the outer radius of the shell).
You could, if you wish use the inverse ratios, but that would not substantially change the arguments and it a little inconvenient to plot because $$dt/d\tau$$ tends towards infinity as x tends towards 2M

.
So was unaware your exterior results (sans 'inversion' issue) were as per what I had used. We therefore agree fully in that regime. Puzzled by the trans and interior results, and will need to sleep on it, like now! Thanks for putting in so much effort. Will comment more later.

 

[yuiop]

Taken straight as is yes. What I meant and Edit sort of explained was one has to invert the relation dS/dx -> dx/dS to express it as a length contraction relation.

You can invert the length contraction ratio if you wish (there is no law against it!) but it would be more consistent to invert both ratios. Even if you just invert the length contraction ratio, the curves would be superimposed outside the shell, but diverge within the material of the shell and inside the cavity everything is the same as before.

 

[Q-reeus]

You can invert the length contraction ratio if you wish (there is no law against it!) but it would be more consistent to invert both ratios.

Not really. Referring to the shell exterior situation, we have been talking all along about:
Time dilation: [PLAIN]https://www.physicsforums.com/latex_images/30/3065954-0.png. As expressed there and as per your graphics, this gives the correct sense of a reduced 'clock tick rate' wrt a distant observer. Note that the source I used has it inverse: [PLAIN]https://www.physicsforums.com/latex_images/27/2730817-7.png. This needs some interpreting. A dt/dtau >1 here means that coordinate time between two events takes longer than in proper time. Conversely though, as per your usage and graphs, the tick rate is less - I prefer that usage.

Length contraction: - the inverse of [PLAIN]https://www.physicsforums.com/latex_images/30/3065954-1.png. Note the original expression I used: [PLAIN]https://www.physicsforums.com/latex_images/27/2730817-8.png has it right. No interpretation needed here - length is less in coordinate units, period. So it properly is the inverse (ie. dx/dS) that conveys that correctly.

Even if you just invert the length contraction ratio, the curves would be superimposed outside the shell...

Yes, as per above, and that raises another issue. In #40 you said "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." Given that the two curves (inverting the length curve as graphed) match precisely everywhere outside the shell, there is no choice but to have both, or neither, to be functions of potential or potential gradient. But if functions of potential gradient (like gravitational acceleration), they would both, just as for 'gravity', disappear inside the shell! Welcome to the Twilight Zone - or not!

.., but diverge within the material of the shell and inside the cavity everything is the same as before.

This is the fundamental issue that can't just be redefined like the exterior curves can. I can find nothing amiss re your derivation in #35 of dt/dtau, dr/dS, given the usage of [PLAIN]https://www.physicsforums.com/latex_images/30/3050819-0.png. But that expression bears closer scrutiny. How was it derived, or sourced from where? In particular the abrupt gradient reversal for dS/dx (or the inverse as I prefer) shown in graphs in #40, 42 seem wholely unphysical. Think of the shell as divided into a number of concentric sub-shells, each separated by a small gap. After traversing the first sub-shell, a tiny test particle has entered a new region of slightly lower gravitational potential but otherwise as for outside the shell proper. Continuing on through successive sub-shells, the cumulative change in potential becomes progressively less, until finally entering the constant potential interior. One must expect smooth transitional curves, tangent at outer and inner radii to the respective values, both for potential and it's related functions of space and time. What physical reason can there be for the abrupt change at the outer radius? Is it possible your model is using a pathalogical black hole model a la "A new kind of BH?" Just wondering.
Was going to use an argument re traversal time for light interior to the shell to 'prove' distance must shrink equally to time dilation, but upon rediscovering the anisotropic c problem in Schwarzschild coordinates, that is on hold. Just how radial vs tangential c exterior to the shell transition to within is problematic at the moment. One or both must alter as interior there must be isotropic c. Adopting isotropic SC would 'fix' that problem, but it seems arbitrary.
In searching for someone else who has tackled or at least discussed the shell problem, could only find the following, and yes it does support my view:

http://www.bautforum.com/showthread.php/108079-Shell-Theorem-in-General-relativity?p=1796618" #2 "Ah, the interior of the shell is completely flat spacetime, Minkowski up to the boundary of the shell. The outside of the shell will be Schwarzschild from spatial infinity down to the shell. To the far, outside Schwarzschild observer, clocks inside the shell are ticking slow and radial rulers are short by just the value of the Schwarzschild factor at the shell -- this corresponds to the Newtonian potential difference. But as far as observers inside are concerned, nothing has happened, their clocks and rulers are just fine."
Only quibble here is why he specified radial rulers - being flat spacetime then by definition circumferential rulers contract by just the same. Probably just a slip up in expression.
Becoming a bit of a saga, yes?

 

[yuiop]

Your original source had something like:

$$dt = \frac{d\tau}{\sqrt{1-2M/x}}$$

and

$$dx = dS \sqrt{1-2M/x}$$

This is a consistent definition of the quantities, because the quantities on the LHS of both equations are coordinate measurements made by an observer at infinity. Your preferred method of defining the ratios is dtau/dt and dx/dS but this is inconsistent because the former is the ratio (local)/(coord) and the second is the ratio (coord)/(local). Now it is not important whether you invert the ratios or not, but if you wish to compare the ratios you should be consistent and use either (local)/(coord) for both or (coord)/(local) for both, to make a meaningful comparison.

Yes, as per above, and that raises another issue. In #40 you said "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." Given that the two curves (inverting the length curve as graphed) match precisely everywhere outside the shell, there is no choice but to have both, or neither, to be functions of potential or potential gradient. But if functions of potential gradient (like gravitational acceleration), they would both, just as for 'gravity', disappear inside the shell! Welcome to the Twilight Zone - or not!

As I said before, if you use a consistent definition of ratios, length contraction and time dilation are not the same outside the shell.

This is the fundamental issue that can't just be redefined like the exterior curves can. I can find nothing amiss re your derivation in #35 of dt/dtau, dr/dS, given the usage of [PLAIN]https://www.physicsforums.com/latex_images/30/3050819-0.png. But that expression bears closer scrutiny. How was it derived, or sourced from where?

That equation is based on the (fairly) well known and documented interior Shwarzschild solution:

$$c^2d\tau^2 = \left(\frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2M x^{2}}{R^{3}}}\right)^2 c^2 dt^2 - \left(1-\frac{2M x^2}{R^3}\right)^{-1}dx^2 -x^2(d\theta^2+\sin(\theta)^2d\phi^2)$$
where I am using x instead of r to represent the Schwarzschild radial displacement coordinate. This solution is for a solid sphere with uniform mass density and no cavity.

See Gron page 255 for example http://books.google.co.uk/books?id=IyJhCHAryuUC&pg=PA255#v=onepage&q&f=false

Even using the raw metric in its unadulterated form, the metric shows that length contraction is greatest at the outer surface (x=R) and reduces to unity at the centre (x=0), contrary to your instincts that length contraction does not reduce as you transfer from the exterior vacuum solution to the interior solution.

Now a closer look at how I derived my equations from the standard solution.

First, look at this expression that appears twice in the metric:

$$\left(1-\frac{2M x^{2}}{R^{3}}\right)$$

Multipying the top and bottom of the fraction by the density (p) and x(4/3)pi the following is obtained:

$$\left(1-\frac{2M}{x} \frac{ (4/3)\pi x^{3}p}{(4/3)\pi R^{3}p}\right)$$

Now $(4/3)\pi R^{3}p = M$ so the expression above can be written as:

$$\left(1-\frac{2}{x} (4/3)\pi x^{3}p}}\right)$$

and $(4/3)\pi x^{3}p$ is the mass enclosed ($M_x$) within a sphere of radius x and the expression can now be written as:

$$\left(1-\frac{2M_x}{x}\right)$$

Substituting this rearranged expression back into the origianl metric gives the form:

$$c^2d\tau^2 = \left(\frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2M_x }{x}}\right)^2 c^2 dt^2 - \left(1-\frac{2M_x }{x}\right)^{-1}dx^2 -x^2(d\theta^2+\sin(\theta)^2d\phi^2)$$

which is the equation you were suspicious about.

This is still the metric for a solid sphere of uniform density expressed in a different way.

To obtain the hollow shell metric we need to calculate the mass enclosed with a radius of x.

The density (p) of the shell with inner radius r and outer radius R is :

$$p = \frac{M}{(4/3)\pi (R^3-r^3)}$$

The mass enclosed within a radius of x is then:

$$M_x = M\frac{(4/3)\pi (x^3-r^3)}{(4/3)\pi (R^3-r^3)} = M\frac{(x^3-r^3)}{ (R^3-r^3) }$$

Substituting this equation into the uniform density solid sphere metric gives the uniform density shell metric as:

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

In particular the abrupt gradient reversal for dS/dx (or the inverse as I prefer) shown in graphs in #40, 42 seem wholely unphysical. Think of the shell as divided into a number of concentric sub-shells, each separated by a small gap. After traversing the first sub-shell, a tiny test particle has entered a new region of slightly lower gravitational potential but otherwise as for outside the shell proper. Continuing on through successive sub-shells, the cumulative change in potential becomes progressively less, until finally entering the constant potential interior. One must expect smooth transitional curves, tangent at outer and inner radii to the respective values, both for potential and it's related functions of space and time. What physical reason can there be for the abrupt change at the outer radius?

The physical reason for the abrupt change is that you are moving from a vacuum to a region that has a non zero mass density. This is a physical change! There is nothing unusual about this. For example the Newtonian gravitational force behaves in exactly the same way. As you get approach Earth from infinity the force of gravity increases to a peak at the surface, but if you go down a mineshaft into the interior of the Earth, the force of gravity abruptly changes and reduces as you get closer to the centre of the Earth until at the centre of the Earth the force of gravity is zero. Length contraction and the force of gravity only depend on the enclosed mass and ignore all the mass above, while time dilation and gravitational potential take all the mass above and below into account. That is what the equations are telling us.

In searching for someone else who has tackled or at least discussed the shell problem, could only find the following, and yes it does support my view:

http://www.bautforum.com/showthread.php/108079-Shell-Theorem-in-General-relativity?p=1796618" #2 "Ah, the interior of the shell is completely flat spacetime, Minkowski up to the boundary of the shell. The outside of the shell will be Schwarzschild from spatial infinity down to the shell. To the far, outside Schwarzschild observer, clocks inside the shell are ticking slow and radial rulers are short by just the value of the Schwarzschild factor at the shell -- this corresponds to the Newtonian potential difference. But as far as observers inside are concerned, nothing has happened, their clocks and rulers are just fine."
Only quibble here is why he specified radial rulers - being flat spacetime then by definition circumferential rulers contract by just the same. Probably just a slip up in expression.
Becoming a bit of a saga, yes?

Circumferential (or horizontal) rulers cannot length contract anywhere in the cavity, in the shell or outside the shell. The reason is simple. Outside the shell we are certain that circumferential rulers do not length contract. Continuity at the boundary means that circumferential rulers cannot length contract as you cross from the exterior to the material of the shell and as you pass into the cavity. For example if the exterior solution predicts that the circumference of the Earth is 40,000km it would be silly if the interior solution predicted the circumference of the Earth was 6,000,000km. Metrics have to agree at boundaries to avoid contradictions.

Now that we have determined that circumferential rulers cannot length contract inside the cavity, we must conclude that radial rulers cannot length contract inside the the cavity either, if the geometry inside the cavity is to be Euclidean. This is exactly what my model predicts, but it contradicts what publius in the other forum is saying. I would go so far as to say publius is wrong, but his statements are a bit vague. He says "radial rulers are short by just the value of the Schwarzschild factor at the shell". Does he mean the same as the inner surface of the shell or at the outer surface of the shell or some average? If he means the inner surface then he is sort of correct, but spoils it by saying rulers are "short" inside the cavity which is false. Rulers inside the cavity have length contraction ratio of unity (inverted or not) and are exactly the same length as rulers at infinity. Saying rulers are "short" inside the cavity demonstrates that publius does not understand that. To be fair, I did not understand that either earlier in this thread, until I did the actual calculations.

and ... yes

 

[Q-reeus]

A ton more info to digest, and once again my excuse is time to hit the sack. Thanks for reference to source of your 'master eq'n' in particular - seems so far to all match up but will get back later.:zzz:

 

[Q-reeus]

Your original source had something like:

$$dt = \frac{d\tau}{\sqrt{1-2M/x}}$$

and

$$dx = dS \sqrt{1-2M/x}$$

This is a consistent definition of the quantities, because the quantities on the LHS of both equations are coordinate measurements made by an observer at infinity. Your preferred method of defining the ratios is dtau/dt and dx/dS but this is inconsistent because the former is the ratio (local)/(coord) and the second is the ratio (coord)/(local). Now it is not important whether you invert the ratios or not, but if you wish to compare the ratios you should be consistent and use either (local)/(coord) for both or (coord)/(local) for both, to make a meaningful comparison...
...As I said before, if you use a consistent definition of ratios, length contraction and time dilation are not the same outside the shell...

Merely a matter of working with 'period' or 'frequency' - see below.

Even using the raw metric in its unadulterated form, the metric shows that length contraction is greatest at the outer surface (x=R) and reduces to unity at the centre (x=0), contrary to your instincts that length contraction does not reduce as you transfer from the exterior vacuum solution to the interior solution.

Yes, and it bothers me much - see below.

Now a closer look at how I derived my equations from the standard solution.

Everything checks out best I can tell. but there is still a big problem - see below.

Circumferential (or horizontal) rulers cannot length contract anywhere in the cavity, in the shell or outside the shell. The reason is simple. Outside the shell we are certain that circumferential rulers do not length contract. Continuity at the boundary means that circumferential rulers cannot length contract as you cross from the exterior to the material of the shell and as you pass into the cavity. For example if the exterior solution predicts that the circumference of the Earth is 40,000km it would be silly if the interior solution predicted the circumference of the Earth was 6,000,000km. Metrics have to agree at boundaries to avoid contradictions.

I now acknowledge that is correct when using standard Schwarzschild metric - see below.

The physical reason for the abrupt change is that you are moving from a vacuum to a region that has a non zero mass density. This is a physical change! There is nothing unusual about this. For example the Newtonian gravitational force behaves in exactly the same way. As you get approach Earth from infinity the force of gravity increases to a peak at the surface, but if you go down a mineshaft into the interior of the Earth, the force of gravity abruptly changes and reduces as you get closer to the centre of the Earth until at the centre of the Earth the force of gravity is zero.

Partly true. It is the gradient of 'g', which is the straight second derivative of Newtonian potential, that abruptly changes at the surface, but otherwise yes as far as 'gravity' is concerned.

Length contraction and the force of gravity only depend on the enclosed mass and ignore all the mass above, while time dilation and gravitational potential take all the mass above and below into account. That is what the equations are telling us.

That's what worries me. There seems to be a fundamental logical inconsistency here. I know you say we must invert or not both expressions dt/dtau and dS/dr for consistency, but that's not so. Inverse length has meaning only for crystallographers ('reciprocal lattice'), otherwise it is dr/dS not dS/dr that conveys properly coordinate 'length change'. By contrast, both clock period (dt/dtau), and clock frequency (dtau/dt) are equally valid and widely used alternate measures of time rate. As stated earlier, exterior to the shell, dr/dS and dtau/dt have exactly the same functional dependence on potential. How oh how can this identical dependence on potential magically alter once the shell is entered!? Makes no physical sense. I believe the dtau/dt term has the right general behavior, and that should be exactly matched by the dr/dS term also, but clearly isn't - according to the eqn's used. If one insists that 'inversion' must be applied to both, I should point out the dS/sr expression is already in need of inverting as per above. Either way one takes it, functional dependence on potential should in no way drastically diverge (or relatively alter in any way) merely because a shell is entered. I have a strong feeling the 'cure' may at least partly be in working with isotropic Schwarzschild metric, not the standard form which is where I believe the purely mathematical anomalies originate. The reference you gave to Gron et al "Einstein's GTR..." http://books.google.co.uk/books?id=IyJhCHAryuUC&pg=PA255#v=onepage&q&f=false" states on p216 "We will however, use the coordinates where the metric takes the form (10.4)." being the standard SC. So that is the basis for the eq'n (10.266) p255 in Gron used to give the results shown in #40, #42. Which brings me to an admission...
Embarrasing but the fact is rather than the standard SC metric:

have all this time been thinking of the isotropic SC given by:

(http://en.wikipedia.org/wiki/Schwarzschild_solution" )
where the metric operator acts equally on all lengths - radial and tangential. In standard SC radial length is contracted wrt tangential. So yes using SC 'horizontal' length is uncontracted as you have maintained. But which coordinate system accurately conveys what a distant observer sees? Not sure. Well that's about it for now.

 

[Q-reeus]

Unable to finger just where imo the maths using standard SC goes haywire, but it would sort of make sense if dr/dS inside the mass (solid sphere or shell) represented the differential change wrt the remaining mass interior to the current radius, rather than cumulative change wrt infinity. Just can't see how to demonstrate that, but - Given standard SC usage, identical functional form exterior must logically continue within and inside the shell.
A few thoughts on using the isotropic SC:

Taken from the same link in previous post, "In the terms of these coordinates, the velocity of light at any point is the same in all directions, but it varies with radial distance r1 (from the point mass at the origin of coordinates), where it has the value":

,
It is easy to find that cdt = dr so there is consistency here. Requiring the tangential components of c and r to match across an infinitesimally thick shell means, owing to isotropy, all components match from exterior to interior. So by this hand-wavy argument, using ISC gives the 'expected' result - nonzero time dilation and length contraction within the interior. The one drawback here is that now dtau/dt and dr/dS do not exactly follow the same functional form exterior (or by extension, within) to the shell. 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.

 

[Q-reeus]

Finally the penny drops - we have been arguing over eg. dr/dS, dtau/dt. Those d's mean just what we learned at school - derivatives! They are not everywhere the same as (r/S)_r, (tau/t)_r ! It is no wonder these terms seem strangely divergent within the shell. What was needed all along are the respective integrals of dr/dS, dtau/dt. So simply swapping isotropic SC for standard SC is in part barking up the wrong tree - but not entirely when it comes to final fits I would say. More later.

 

[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):

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

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):

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

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

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

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

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

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

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

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

$$dS/dx= 1$$

$$dx/dt = (3/2)(1-2M/R)^{1/2}-1/2$$

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!