Gravity distortion at the center of a massive object?

[darkhorror]

I am not totally sure what I am trying to ask, but wondering about gravity at the center of a massive object such as the sun, earth, highly massive star. If you look at the time dilation at the center vs the surface or far in space time at the center time is running slowest. Now obviously for person at the center he will still see his clock ticking normally, and the other clocks running faster.

Now it seems there also has to be some sort of "length contraction" though it doesn't seem the same as length contraction in SR. More that there is more space crammed into a small space. Where the person far away in space would see a meter stick in the center of the massive object looking smaller than their meter stick. The person in the center would see an opposite effect.

Are these correct or where am I going wrong?

 

[WannabeNewton]

Are you asking if the gravitational field makes objects look bigger than they actually are to observers at infinity? If so then yes it's true. There's also the fact that proper distances get affected by the gravitational field.

Consider for simplicity the Schwarzschild geometry external to a static spherically symmetric star and an observer hovering in place outside the star at some Schwarzschild radial coordinate $r$; let $p = (t,r,\theta,\phi)$ be some event on the observer's worldline. The event $q = (t,r + dr,\theta,\phi)$ lies in the local simultaneity slice at $p$ relative to this observer; this is because the separation vector between $p$ and $q$ is given by $\xi = (0,dr,0,0)$ which is orthogonal to the 4-velocity $u = ((1-2M/r)^{-1/2},0,0,0)$ of the observer. So now imagine that this observer is using an ideal ruler to measure the spatial distance between these simultaneous events $p$ and $q$. The ruler measurement will of course just be the length of the separation vector $\xi = (0,dr,0,0)$ connecting these two events and this comes out to $ds = (g_{\mu\nu}\xi^{\mu}\xi^{\nu})^{1/2} = (g_{rr})^{1/2}dr = (1 - 2M/r)^{-1/2}dr$
(you could have arrived at this straight from the line element but I feel the above is more intuitive). In other words, the ruler measurement is affected by the curved geometry due to the gravitating central mass $M$.

As for the deflection of light making objects look bigger to an observer at infinity, consider a light ray that just grazes the surface of a star $M$ of Schwarzschild coordinate radius $R$. Let $k = (\dot{t},\dot{r},0,\dot{\phi})$ be the tangent vector to the null geodesic describing this light ray (we are restricting ourselves to trajectories in the equatorial plane). At the very instant that it grazes the surface of the star, we have $\dot{r} = 0$. Since $g(k,k) = 0$ we get that $(1 - 2M/R)\dot{t}^{2}= R^{2}\dot{\phi}^{2}$ at this instant. We know from conservation of energy and angular momentum that $\dot{t} = (1 - 2M/R)^{-1}e$ and $\dot{\phi} = \frac{l}{R^{2}}$ so $b:=\frac{l}{e}=\frac{R}{(1 - 2M/R)^{1/2}}$ where $b$ is the impact parameter for light ray trajectories. Note that $b > R$ so what this means is that an observer at infinity will see the star as having a larger diameter than it actually has because of gravity (more directly, because of the deflection of light that results from the presence of a gravitational field).

 

[A.T.]

If you look at the time dilation at the center vs the surface or far in space time at the center time is running slowest. Now obviously for person at the center he will still see his clock ticking normally, and the other clocks running faster.

Yes.

Now it seems there also has to be some sort of "length contraction" though it doesn't seem the same as length contraction in SR. More that there is more space crammed into a small space. Where the person far away in space would see a meter stick in the center of the massive object looking smaller than their meter stick. The person in the center would see an opposite effect.

This one is more subtle. Around a mass there is a spatial distortion, which can be understood in terms of real physical radial distance (ds) vs. the radial coordinate computed by dividing a real physical circumference by 2PI (dr). This is a picture for the space outside of a black hole explains how ds & dr are related:
http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/Black_Holes.htm

This includes the interior:
"commons.wikimedia.org/wiki/File:Schwarzschild_interior.jpg"

So if you put a very long ruler through the star it will appear to be shortened. This is a cumulative effect of the local distortion (ds/dr) integrated along the radial line. But you asked about spatial distortion locally at the center, and there you have ds/dr = 1. So locally at the center the spatial distortion vanishes.

And if you create a spherical cavity at the center, space will be flat throughout the entire cavity, and clocks will run at the same rate throughout the entire cavity, but still slower than at the surface. And there is no "pull" towards the center throughout the entire cavity.

 

[Bill_K]

So if you put a very long ruler through the star it will appear to be shortened.

What about the transverse direction? Does it contract in the transverse direction?

 

[pervect]

I'd have to disagree that it's a good idea to call what goes on with the metric inside a black hole as "length contraction".

As long as you use the same simultaneity convention, a 1 meter ruler / line segment has a length of 1 meter, it doesn't matter whether you are in a high gravitational potential or a low gravitational potential.

Length is observer dependent, but the observer dependence comes from simultaneity issues, if you use the simultaneity convention of a static observer, a proper length of 1 meter will always be a proper length of 1 meter. There isn't any sort of length "correction" to proper length due to gravitational potential.

There is a metric correction when you convert coordinates (which are not lengths) into lengths, but I think it's very confusing to call this length contration. Amongst the other issues, this depends on the details of the coordinates one uses.

You'll see something very lie the usual SR effects if you use the simultaneity convention appropriate for a moving observer (say, an infalling one) rather than a static one. But this is different from saying that there is a length contraction effect due to gravity, it's more akin to the "Lorentz contraction".

You can also say that the radial distance between two concentric spherical shells, one of circumference C and one of circumference C + 2*pi*delta is greater than delta due to distorted geometry of a black hole (unlike the situation in flat space-time). But I woldn't call this "length contraction" either.

 

[PAllen]

Hmm, I interpret the OP as asking a completely different question. They should come back and clarify. To simplify what I think they are asking, imagine a transparent planet with a small hollow cavity at its center. I think they want to know about clock and ruler comparisons for someone in the hollow versus at infinity.

Then, it is clear that clocks will be running slower in the hollow than at the surface (and this comparison will be mutual), and slow compared to distant clocks by amount greater than clocks at the surface. The amount of difference will, of course, depend on your assumptions about the planet, but the weak field approximation of surface to center would be straightforward to compute.

As for distance, inside the hollow you have an exact Minkowski metric for the local experience. However, to write a metric for a global coordinate patch covering all of space time, you would have to scale the flat metric to mesh with metric at the cavity boundary. Then, what you would really need to ask is the angle subtended by 1 meter of proper length in the cavity viewed orthogonally on approach to infinity, compared to Euclidean expectation. This would be a lot more work to compute, even approximately.

[edit: I see AT understood the question the same way I did.]

 

[darkhorror]

Hmm, I interpret the OP as asking a completely different question. They should come back and clarify. To simplify what I think they are asking, imagine a transparent planet with a small hollow cavity at its center. I think they want to know about clock and ruler comparisons for someone in the hollow versus at infinity.

Then, it is clear that clocks will be running slower in the hollow than at the surface (and this comparison will be mutual), and slow compared to distant clocks by amount greater than clocks at the surface. The amount of difference will, of course, depend on your assumptions about the planet, but the weak field approximation of surface to center would be straightforward to compute.

As for distance, inside the hollow you have an exact Minkowski metric for the local experience. However, to write a metric for a global coordinate patch covering all of space time, you would have to scale the flat metric to mesh with metric at the cavity boundary. Then, what you would really need to ask is the angle subtended by 1 meter of proper length in the cavity viewed orthogonally on approach to infinity, compared to Euclidean expectation. This would be a lot more work to compute, even approximately.

[edit: I see AT understood the question the same way I did.]

You are correct this is more of what I was asking, same with AT. My questions aren't so much about getting a specific numbers but understanding what explains how gravity slow's down time and how it works with the consistency of light in these situations.

 

[A.T.]

My questions aren't so much about getting a specific numbers but understanding what explains how gravity slow's down time and how it works with the consistency of light in these situations.

According to local clocks and rulers light still propagates at c. But when you do measurements over large distances based on distant clocks, you get different average speeds than c.

 

[Bill_K]

[edit: I see AT understood the question the same way I did.]

Ok, if everybody understands it, and agrees on how they understand it, maybe somebody can answer the question I raised. A.T. described the contraction in the radial direction. What about the transverse direction?

 

[yuiop]

What about the transverse direction? Does it contract in the transverse direction?

Using this interior solution for the Schwarzschild metric and setting dr=dt=0 I would say no. The result is the same as for the exterior Schwarzschild solution when dr=dt=0, so no transverse length contraction. For an extended transverse ruler, the ruler would have to curved such that all points along the ruler are at constant r from the centre.

 

[PAllen]

Ok, if everybody understands it, and agrees on how they understand it, maybe somebody can answer the question I raised. A.T. described the contraction in the radial direction. What about the transverse direction?

I don't know the answer, but I described what I would consider to be observable length contraction. For me, the indicated calculation would take far more time than I am willing to spend, and I have no intuition of how the answer would come out because I've never calculated (or followed someone else's) for an equivalent scenario.

 

[PeterDonis]

to write a metric for a global coordinate patch covering all of space time, you would have to scale the flat metric to mesh with metric at the cavity boundary.

But that scaling, counterintuitively, does *not* change the space coordinates at all!

The general metric for a static, spherically symmetric spacetime can be written in this form:

$$
ds^2 = - J(r) dt^2 +\frac{1}{1 - 2m(r) / r} dr^2 + r^2 d \Omega^2
$$

where $r$ is the standard Schwarzschild radial coordinate and $m(r)$ is the mass inside radius $r$. (This result is derived, for example, in MTW; I can look up the reference if needed.) At the outer surface of the object, $J(r) < 1$ and $m(r) = M$, where $M$ is the total mass of the object; so there is both "time dilation" due to being in the gravity well, and "space distortion" due to the presence of mass.

However, at the *inner* surface of the object (i.e., the outer edge of the hollow cavity), $J(r)$ has decreased, but $m(r) = 0$--there is no mass left inside that radius. So there is still "time dilation" compared to infinity (more time dilation than at the outer surface), but there is *no* "space distortion" any more, even in this global metric; the coefficient of $dr^2$ is $1$! To put this another way, to transform from this global metric to the local metric at any point inside the hollow shell, the *only* thing you need to do is rescale the time by the factor $\sqrt{J}$; you don't need to "rescale the space" at all.

Then, what you would really need to ask is the angle subtended by 1 meter of proper length in the cavity viewed orthogonally on approach to infinity, compared to Euclidean expectation. This would be a lot more work to compute, even approximately.

Even if you computed it and it showed that 1 meter of proper length subtended less angle (which is my guess as to what it would show), that wouldn't show that the hollow cavity was "length contracted"; see above. All it would show is that the paths of light rays are distorted when the pass through the massive object to get from the hollow cavity to the region outside.

 

[yuiop]

The general metric for a static, spherically symmetric spacetime can be written in this form:

$$
ds^2 = - J(r) dt^2 +\frac{1}{1 - 2m(r) / r} dr^2 + r^2 d \Omega^2
$$

where $r$ is the standard Schwarzschild radial coordinate and $m(r)$ is the mass inside radius $r$.

As mentioned in #10, that you might of missed, when dr=dt=0, $r d\Omega$ is a measure of transverse length and there is no effective transverse length contraction, just as in the regular Schwarzschild metric.

P.S. What is is J(r) in the equation you quote? I have seen a similar equation stated as:

$$
ds^2 = - (1-2m(r)/r) dt^2 +\frac{1}{1 - 2m(r) / r} dr^2 + r^2 d \Omega^2
$$

but I cannot recall where I saw it, so I am not sure of that equation's authenticity.

Wait ... It might have been http://preposterousuniverse.com/grnotes/grnotes-seven.pdf (See eq 7.32).

 

[PAllen]

As mentioned in #10, that you might of missed, when dr=dt=0, $r d\Omega$ is a measure of transverse length and there is no effective transverse length contraction, just as in the regular Schwarzschild metric.

P.S. What is is J(r) in the equation you quote? I have seen a similar equation stated as:

$$
ds^2 = - (1-2m(r)/r) dt^2 +\frac{1}{1 - 2m(r) / r} dr^2 + r^2 d \Omega^2
$$

but I cannot recall where I saw it, so I am not sure of that equation's authenticity.

Wait ... It might have been http://preposterousuniverse.com/grnotes/grnotes-seven.pdf (See eq 7.32).

J(r) is dependent on matter. What Peter gives covers any form of matter with possibly radially varying density and pressure as long as it exactly spherically symmetric and static.

 

[WannabeNewton]

You are correct this is more of what I was asking, same with AT. My questions aren't so much about getting a specific numbers but understanding what explains how gravity slow's down time and how it works with the consistency of light in these situations.

I don't see how this relates to your original question at all. The physical motivation behind gravitational time dilation is a completely different question and one that is covered in every GR text.

There is also no clash with the fact that $c = 1$ in local Lorentz frames. If we have an observer with 4-velocity $u^{\mu}$, a light wave with tangent vector $k^{\mu}$, and an event at which the light wave passes by the observer, then $E = -k_{\mu}u^{\mu}$ is the energy of the wave relative to this observer at that event and $p^{\mu} = g^{\mu}{}{}_{\nu}k^{\nu} + u^{\mu}u_{\nu}k^{\nu}$ its 3-momentum relative to this observer at said event.
Therefore $\left \| p \right \| = [(g^{\mu}{}{}_{\nu}k^{\nu} + u^{\mu}u_{\nu}k^{\nu})(g_{\mu\alpha}k^{\alpha} + u_{\mu}u_{\alpha}k^{\alpha})]^{1/2} = [k^{\nu}k_{\nu} + u_{\alpha}u_{\nu}k^{\nu}k^{\alpha} ]^{1/2} = E$
hence $c = 1$, as we would expect.

Gravitational time dilation is a global effect and is just a manifestation of gravitational redshift.

 

[PAllen]

Even if you computed it and it showed that 1 meter of proper length subtended less angle (which is my guess as to what it would show), that wouldn't show that the hollow cavity was "length contracted"; see above. All it would show is that the paths of light rays are distorted when the pass through the massive object to get from the hollow cavity to the region outside.

I think the only meaningful definition of length contraction/expansion is that there exists a measurement you would accept as a valid length measurement such that the locally measured length of an object (comoving if required - thus local rest length) differs from some other observer's measurement. In the case of SR, there are many ways (in principle) to realize the length measurement of a moving object so as to show the contraction.

Thus, to my mind, there are two defensible statements about length change observed at a distance:

- either it is undefined, because there is no procedure you would accept as valid
- or there is at least one procedure for remote length measurement that you would accept.

What I don't accept is that grr=1 in the fitted flat metric has any meaning at all for length contraction without an accepted way for a distant observer to measure length. If there is no procedure you would accept, then we should agree the term has no possible meaning.

 

[Naty1]

How about tidal effects, 'spaghettification' ??

 

[PAllen]

How about tidal effects, 'spaghettification' ??

The discussion concerns the interior of an ordinary massive body, not the near singular region of a BH.

 

[PAllen]

P.S. What is is J(r) in the equation you quote? I have seen a similar equation stated as:

$$
ds^2 = - (1-2m(r)/r) dt^2 +\frac{1}{1 - 2m(r) / r} dr^2 + r^2 d \Omega^2
$$

but I cannot recall where I saw it, so I am not sure of that equation's authenticity.

Wait ... It might have been http://preposterousuniverse.com/grnotes/grnotes-seven.pdf (See eq 7.32).

J(r) is dependent on matter. What Peter gives covers any form of matter with possibly radially varying density and pressure as long as it exactly spherically symmetric and static.

Actually, though your equation does come from Carroll, it looks fishy to me. Since it is normally assumed that m(r)/r->0 as r->0, it suggests clocks speed up as you go deeper into a massive body, with gravitational time dilation vanishing at the center. This seems absurd to me. This behavior is also completely different from that given by the SC interior matter solution (which, though physically implausible, would expect to get qualitative features right).

 

[Naty1]

The discussion concerns the interior of an ordinary massive body, not the near singular region of a BH.

On rereading the OP, I see while he mentioned 'surface' early in his post that was not part of his final questions. It seemed to me that everywhere except the center, some tidal effects even if minor, are at play...

 

[PAllen]

On rereading the OP, I see while he mentioned 'surface' early in his post that was not part of his final questions. It seemed to me that everywhere except the center, some tidal effects even if minor, are at play...

That's correct, though they are (or can be) minor and not really relevant to the questions.

 

[PAllen]

Actually, though your equation does come from Carroll, it looks fishy to me. Since it is normally assumed that m(r)/r->0 as r->0, it suggests clocks speed up as you go deeper into a massive body, with gravitational time dilation vanishing at the center. This seems absurd to me. This behavior is also completely different from that given by the SC interior matter solution (which, though physically implausible, would expect to get qualitative features right).

And here is another source giving a much more plausible general formula for simple stellar model:

http://tartarus.org/gareth/maths/tripos/II/General_Relativity__2013-2005.pdf

 

[PeterDonis]

I think the only meaningful definition of length contraction/expansion is that there exists a measurement you would accept as a valid length measurement such that the locally measured length of an object (comoving if required - thus local rest length) differs from some other observer's measurement.

Yes, I'll accept that definition; and by that definition, I don't think there is any valid remote measurement for length in the scenario under discussion that will show contraction, even though there is for time. In particular, I don't think using the angle subtended works because, as I said, that measurement includes the effects of distortion in the paths of light rays.

What I don't accept is that grr=1 in the fitted flat metric has any meaning at all for length contraction without an accepted way for a distant observer to measure length.

Yes, I agree; however, I think the observation that $g_{rr} = 1$ is still useful, because it shows that there is no need to "scale" the metric inside the hollow cavity spatially in order to obtain a globally valid metric; the only rescaling required is in the time coordinate.

 

[PAllen]

Yes, I'll accept that definition; and by that definition, I don't think there is any valid remote measurement for length in the scenario under discussion that will show contraction, even though there is for time. In particular, I don't think using the angle subtended works because, as I said, that measurement includes the effects of distortion in the paths of light rays.

Perhaps we just make up a fancy name for it: gravito-optical length change.

Yes, I agree; however, I think the observation that $g_{rr} = 1$ is still useful, because it shows that there is no need to "scale" the metric inside the hollow cavity spatially in order to obtain a globally valid metric; the only rescaling required is in the time coordinate.

I agree.

 

[PeterDonis]

Actually, though your equation does come from Carroll, it looks fishy to me.

It looks fishy to me too. Not only does it not match the constant-density SC interior solution (which yuiop linked to in an earlier post), it also implies, as you note, that (if we define $J(r) = - g_{tt}$ as I did in my earlier post) $dJ / dr$ becomes negative for some $r > 0$. But this appears to be inconsistent with the equation for $dJ / dr$ that I derived from the EFE in a post on my PF blog a while back:

https://www.physicsforums.com/blog.php?b=4149

The equation I obtained there (which also appears in most relativity texts, such as MTW) is (refactoring slightly):

$$
\frac{dJ}{dr} = 2J \left( \frac{m(r) + 4 \pi r^3 p(r)}{r^2 (1 - 2 m(r) / r)} \right)
$$

Since we must have $(1 - 2 m(r) / r) > 0$ and $J > 0$ for any static configuration that is not a black hole, and we must (I think) have $(m + 4 \pi r^3 p) > 0$ for any matter that doesn't violate energy conditions, it does not appear that there is any way for $dJ / dr$ to be negative.

 

[darkhorror]

Thank you, my main question has been answered and makes sense. Now I will be doing some more thinking along with looking at the extra info.

 

[Naty1]

This has been an excellent discussion for me, one of the best of the year...Initially on reading the OP question, I thought 'no relative motion, so no length change'...but then I wondered about 'length change' associated with tidal effects...

I had not consciously confronted time dilation without 'length change' [of some sort, loosely stated] that I can remember...but that survived the discussion...well done!

My own edited synopsis from posts:

With the simultaneity convention of a static observer a proper length remains unchanged regardless of gravitational potential... inside the shell is the exact Minkowski metric for the local experience...to transform from a global metric to the local metric at any point inside the hollow central shell only the time factor needs rescaling.…[you don't need to "rescale the space" at all.]

 

[yuiop]

And here is another source giving a much more plausible general formula for simple stellar model:

http://tartarus.org/gareth/maths/tripos/II/General_Relativity__2013-2005.pdf

This formula is in fact the same as the formula I linked to in post #10 although it is heavily disguised. i.e. it is the standard interior Schwarzschild metric with homogeneous density.

Actually, though your equation does come from Carroll, it looks fishy to me. Since it is normally assumed that m(r)/r->0 as r->0, it suggests clocks speed up as you go deeper into a massive body, with gravitational time dilation vanishing at the center. This seems absurd to me. This behavior is also completely different from that given by the SC interior matter solution (which, though physically implausible, would expect to get qualitative features right).

I agree that the Carroll equation contradicts the regular interior Schwarzschild metric. However, in its favour, Carroll suggests it is backed by Schutz, which is normally a reliable source. Does anyone have the Schutz text? Also, the Caroll metric clearly indicates that the spacetime inside a cavity is flat which I have seen claimed many times.

... As for distance, inside the hollow you have an exact Minkowski metric for the local experience...

As mentioned above, that seems to be the general consensus of how the metric should be inside a cavity.

However, to write a metric for a global coordinate patch covering all of space time, you would have to scale the flat metric to mesh with metric at the cavity boundary. ...

Yes, and you need to scale the metric so that interior solution meshes with the exterior Schwarzschild metric at the outer surface of the spherical massive object.

However, at the *inner* surface of the object (i.e., the outer edge of the hollow cavity), $J(r)$ has decreased, but $m(r) = 0$--there is no mass left inside that radius. So there is still "time dilation" compared to infinity (more time dilation than at the outer surface), but there is *no* "space distortion" any more, even in this global metric; the coefficient of $dr^2$ is $1$! To put this another way, to transform from this global metric to the local metric at any point inside the hollow shell, the *only* thing you need to do is rescale the time by the factor $\sqrt{J}$; you don't need to "rescale the space" at all. ...

Does your metric show that the spacetime inside a cavity is Minkowskian? Does your metric match the exterior vacuum Schwarzschild metric at the outer surface of the massive body?

... But this appears to be inconsistent with the equation for $dJ / dr$ that I derived from the EFE in a post on my PF blog a while back:

https://www.physicsforums.com/blog.php?b=4149

The equation I obtained there (which also appears in most relativity texts, such as MTW) is (refactoring slightly):

$$
\frac{dJ}{dr} = 2J \left( \frac{m(r) + 4 \pi r^3 p(r)}{r^2 (1 - 2 m(r) / r)} \right)
$$

Since we must have $(1 - 2 m(r) / r) > 0$ and $J > 0$ for any static configuration that is not a black hole, and we must (I think) have $(m + 4 \pi r^3 p) > 0$ for any matter that doesn't violate energy conditions, it does not appear that there is any way for $dJ / dr$ to be negative.

Is your metric tractabable? Can you for example state the time dilation when dr=dΩ=0 , m=1, the outer surface radius of the massive object is R=18/8 using units of G=c-1 when :

1) r=R=18/8
2) r=0 and
3) r = R/2 = 9/8?

Also, in your blog does s represent stress and p pressure, or does p represent density?

 

[WannabeNewton]

What Carroll wrote down in those notes is definitely not what Schutz has. In fact Schutz has the exact same interior solution as the one given by Padmanabhan for a constant density star. Both have $m(r) = 4\pi \rho r^3/3$,
$g_{rr} = e^{2\Lambda} = (1 - 2m(r)/r)^{-1}$,
and $e^{\Phi} = \frac{3}{2}(1 - 2M/R)^{1/2} - \frac{1}{2}(1 - 2Mr^2/R^3)^{1/2}$ where $g_{tt} = e^{2\Phi}$.

This solution of course makes perfect sense.

And further if our equation of state is such that $p = p(\rho)$ (constant entropy) then $\nabla^{\mu}T_{\mu\nu} = 0$ gives us the equation $(\rho + p) \frac{d\Phi}{dr} = -\frac{dp}{dr}$ in order to determine the pressure $p$ for completeness.

 

[nitsuj]

This has been an excellent discussion for me,

Agreed, props to darkhorror for the question Oh and implicitly to pf members who know the answer lol

For some reason from a gravitational perspective I find inside "spheres" a really interesting place conceptually even though I'm always wrong when trying to intuit an answer ahaha

 

[yuiop]

What Carroll wrote down in those notes is definitely not what Schutz has. In fact Schutz has the exact same interior solution as the one given by Padmanabhan for a constant density star. Both have $m(r) = 4\pi \rho r^3/3$,
$g_{rr} = e^{2\Lambda} = (1 - 2m(r)/r)^{-1}$,
and $e^{\Phi} = \frac{3}{2}(1 - 2M/R)^{1/2} - \frac{1}{2}(1 - 2Mr^2/R^3)^{1/2}$ where $g_{tt} = e^{2\Phi}$.

This solution of course makes perfect sense.

OK, assuming that $(1 - 2m(r)/r) = (1 - 2mr^2/R^3)$, that appears to be basically the standard interior Schwarzschild solution given by George Jones that I linked to in #10.

If that is the case then the Shutz/ standard interior Schwarzschild metric can be stated as:

$$d\tau^{2}=-\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right) ^{2}dt^{2}+\left( 1-\frac{2Mr^{2}}{R^{3}}\right) ^{-1}dr^{2}+r^{2} d\Omega ^{2}$$

or alternatively as:

$$d\tau^{2}=-\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2M(r)}{r}}\right) ^{2}dt^{2}+\left( 1-\frac{2M(r)}{r}\right) ^{-1}dr^{2}+r^{2} d\Omega ^{2}$$

Does that seem reasonable?

P.S. From the second form it is obvious that the time dilation is constant everywhere within the cavity.

 

[PAllen]

This formula is in fact the same as the formula I linked to in post #10 although it is heavily disguised. i.e. it is the standard interior Schwarzschild metric with homogeneous density.

That's true, but only at the point where A is taken to be a constant. The framework would cover A being a function of r and, and then A becomes related to density variation.

I agree that the Carroll equation contradicts the regular interior Schwarzschild metric. However, in its favour, Carroll suggests it is backed by Schutz, which is normally a reliable source. Does anyone have the Schutz text? Also, the Caroll metric clearly indicates that the spacetime inside a cavity is flat which I have seen claimed many times.

It contradicts physical plausibility. For a static solution where a potential can be introduced, gravitational time dilation is determined by potential. This metric then implies that you have to apply work to push matter down a tube through the star, and free fall would be upward. Absurd is not too strong a word for that.

Does your metric show that the spacetime inside a cavity is Minkowskian? Does your metric match the exterior vacuum Schwarzschild metric at the outer surface of the massive body?

Yes, Peter's does, as long as J(r) becomes constant inside the shell, which is the required by plausibility anyway. For the outside, J(r) has to match the exterior SC value at the outer surface.

 

[PeterDonis]

Does your metric show that the spacetime inside a cavity is Minkowskian?

Yes. Inside a cavity, $m(r) = 0$, so the only term in the line element that is not explicitly Minkowski is $g_{tt}$, and you can rescale the time coordinate to make $g_{tt} = -1$ inside the cavity.

[Edit: technically, you also have to prove that $J(r)$ is constant inside the cavity, but this is easy; see my response to PAllen.]

Does your metric match the exterior vacuum Schwarzschild metric at the outer surface of the massive body?

Yes; if you work out what $J(r)$ is at the outer surface of an isolated body surrounded by vacuum, you will obtain

$$
J(r) = 1 - \frac{2M}{r}
$$

where $M$ is the total mass of the body.

Can you for example state the time dilation when dr=dΩ=0 , m=1, the outer surface radius of the massive object is R=18/8 using units of G=c-1 when :

1) r=R=18/8
2) r=0 and
3) r = R/2 = 9/8?

Sure; this is just the case that Einstein worked out as the minimum possible radius for a body in static equilibrium. He did it using the same application of the EFE that I used in the blog post, though he didn't use the same notation that I used. (Technically, the pressure increases without bound as $r \rightarrow 0$ and $g_{tt}$ becomes zero at some point in the solution, so this case is not actually physically possible; it's a limiting case.)

Also, in your blog does s represent stress and p pressure, or does p represent density?

$p$ is radial stress, and $s$ is tangential stress. In the case of a perfect fluid, $p = s$ and the stress is normally referred to as the (isotropic) pressure.

 

[PeterDonis]

Yes, Peter's does, as long as J(r) becomes constant inside the shell

This is guaranteed by the equation for $dJ / dr$; inside the cavity, $m = 0$ and $p = 0$ so we must have $dJ / dr = 0$.