Isotropic metric and circumference of sphere

[Jonathan Scott]

Schwarzschild coordinates for the Schwarzschild black hole solution become very weird near the event horizon because the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon. This is easy to see in isotropic coordinates, where the Schwarzschild radial coordinate $r$ is related to the isotropic radial coordinate $r'$ as follows:
$$r = r' \left ( 1 + \frac{Gm}{2r'c^2} \right)^2$$
By differentiating with respect to $r'$ it is easy to show that this has a minimum at $r' = Gm/2rc^2$, which is simply the event horizon in isotropic coordinates, and it then increases towards infinity as the isotropic radial coordinate decreases to zero, corresponding to the singularity.

This effectively says that radial lines become parallel in local space at the event horizon, then space gets wider as one falls past the event horizon, and is infinitely wide by the singularity. This interpretation is not valid on its own as the time coordinate also has problems at the event horizon, although another curiosity is that in isotropic coordinates the time rate seems to go negative rather than imaginary at the event horizon (although in these coordinates an object still takes an infinite time to fall to the event horizon in the first place):
$$d\tau = dt \, \frac{(1-Gm/2r'c^2)}{(1+Gm/2r'c^2)} $$
Various approximations (PPN) or alternative theories of gravity use isotropic metrics where the time and space factors are actually simply reciprocals of one another, for example $\exp(-Gm/r'c^2)$ and $\exp(+Gm/r'c^2)$. It is usually claimed that this sort of metric cannot lead to black holes, because the factors are non-singular down to $r=0$, which seems obvious enough. However, this exponential metric and other related metrics still have the property that for small enough $r$, the proper circumference of a sphere has a minimum value then starts increasing towards infinity:
$$ r = r' \exp\frac{Gm}{r'c^2}$$
$$\frac{dr}{dr'} = \exp\frac{Gm}{r'c^2} + r' \left (- \frac{Gm}{r'^2c^2} \right ) \exp\frac{Gm}{r'c^2}
= \left ( 1 - \frac{Gm}{r'c^2} \right ) \exp\frac{Gm}{r'c^2}$$
The derivative is zero at $r' = Gm/r'c^2$ and this gives the minimum value of $r$. Note that as the coordinates are isotropic, the radial coordinate behaves in the same way, in that as a spherical surface contracts, an increment in the radial coordinate corresponds to a larger and larger increment in the proper distance in the radial direction, apparently allowing a shrinking mass (regardless of the nature of the internal solution) to occupy more and more space.

A similar result occurs for any radial factor of the form $r = r' (1+Gm/nr'c^2)^n$ where $n > 1$, although there is no such problem for $n = 1$ or $n < 1$.

As far as I can see, this means that if you have an object which has starts at the radius of minimum proper circumference and has a fixed mass, it seems that it can expand into a greater amount of space by decreasing its radius, and that the more it does so, the more proper space it has to expand into, although as it does so its time rate will get slower and slower. This suggests that there would be no resistance to prevent it collapsing into a "black dot", except that it's not clear to me where the energy would go in that case!

I know this is a hypothetical metric (although a well-known one), but I'd like a better understanding of what would happen in this case, for contrast with standard "black hole" theory. Can anyone provide further clarification of what would be expected to happen in this case?

 

[PeterDonis]

the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon.

This is not correct. What is correct is that, for the region of spacetime at and outside the horizon, the proper circumference of 2-spheres has a minimum at the event horizon. But there is another region of spacetime inside the horizon that has 2-spheres with smaller proper circumferences.

This is easy to see in isotropic coordinates

That is because isotropic coordinates do not cover the region inside the horizon. It does not mean that such a region does not exist.

as the isotropic radial coordinate decreases to zero, corresponding to the singularity.

This is not correct either. The isotropic radial coordinate decreasing to zero corresponds to spatial infinity. This is obvious from the behavior of the proper circumferences of the 2-spheres in that limit, which increase without bound.

There are two possible interpretations of this behavior of isotropic coordinates. One is that these coordinates double cover the region outside the horizon. The other is that these coordinates extend into a second region outside the horizon that extends out to a second spatial infinity. (On a Kruskal diagram this region is usually marked as Region III and is the "left wedge" of the diagram, where the original region outside the horizon, where the isotropic radial coordinate goes to infinity, is Region I, the "right wedge".) I personally favor the second interpretation because it is consistent with the spacelike geodesics in a surface of constant isotropic coordinate time being smooth and infinitely extendible (whereas the first interpretation requires such geodesics to have an "end point" at the horizon where smoothness no longer holds).

in isotropic coordinates the time rate seems to go negative rather than imaginary at the event horizon

On the second interpretation above, this corresponds to the fact that in Region III in Kruskal coordinates (i.e., in the maximally extended spacetime), proper time along timelike curves "runs backwards" compared to the Kruskal time coordinate, whereas in Region I it "runs forwards". It says nothing about the behavior of proper time along timelike curves inside the horizon, since, as above, isotropic coordinates do not cover that region.

As far as I can see, this means that if you have an object which has starts at the radius of minimum proper circumference and has a fixed mass, it seems that it can expand into a greater amount of space by decreasing its radius

No, it means that it can expand into a greater amount of space in two ways: by moving towards larger values of the isotropic radial coordinate, or smaller ones. Both correspond to "increasing its radius", not decreasing it. The fact that the radial coordinate decreases in the second case does not mean the physical radius decreases; you have demonstrated this by showing that the proper circumference of 2-spheres increases as the radial coordinate decreases from the minimum point. In other words, this case is just an illustration of the maxim that coordinates alone can't tell you the physics; you have to look at the geometric invariants, which in this case are the proper circumferences.

 

[Jonathan Scott]

No, it means that it can expand into a greater amount of space in two ways: by moving towards larger values of the isotropic radial coordinate, or smaller ones. Both correspond to "increasing its radius", not decreasing it. The fact that the radial coordinate decreases in the second case does not mean the physical radius decreases; you have demonstrated this by showing that the proper circumference of 2-spheres increases as the radial coordinate decreases from the minimum point. In other words, this case is just an illustration of the maxim that coordinates alone can't tell you the physics; you have to look at the geometric invariants, which in this case are the proper circumferences.

I'm probably confused about the event horizon case and the various regions, but for this case where there is no event horizon I definitely meant the "increase" or "decrease" to refer to the location of the surface in terms of the coordinate radius; if the surface moves towards a smaller coordinate radius, the proper circumference of the surface expands, and if it moves away it also expands. I don't see any reason why the two coordinate directions should be equivalent physically as the time dilation gradient is still going the same way as the radial coordinate. I'm not sure about the relationship of the proper circumference to the proper radius as that implicitly involves an internal solution, but I assume it is likely to be in proportion.

So if that hypothetical metric could occur, what would happen if an object passed the minimum point? If I had the time and patience I might be able to come up with some calculations but I have limited time and find it quite confusing.

 

[Mihai_B]

"Where would the energy go (hypothetical) ?"

I know there are people with far better math apparatus than mine - I will let others complete me or give answers in other perspective - , but since I like very much to study Einstein-Rosen bridges I can recommend you a brilliant study article (also known as ER):
A. Einstein, N. Rosen - "The Particle Problem in the General Theory of Relativity" http://journals.aps.org/pr/pdf/10.1103/PhysRev.48.73

Eistein (and Rosen) proposed a way to get rid of the singularities by using a change of variable and for the combined field (gravity and electricity) using again a change of variable and to choose the sign of the time component (of the metric) in such a way that is always positive (one will get (1 - q²/r²) instead of Reissner-Nordström's (1 + q²/r²)).

I will detour you by bit ...

The event horizon is where r = 2mG/c² (or r = 2m if one uses G = c = 1)
and since there's no better master than the master himself ... I quote Einstein :

"
ds² = - 1/(1 - 2m/r) dr² - r² (dθ² + sin²θ dφ²) + (1-2m/r) dt²

If one introduces in place of r a new variable according to the equations

u² = r - 2m, ( u² = r - 2mG/c² )

one obtains for ds² the expresion

ds² = -4(u² + 2m)du² - (u²+2m)² (dθ² + sin²θ dφ²) + u²/(u² + 2m) dt² ."

"... a solution of the (new) field equations, which is free from singularities for all finite points.
The hypersurface u = 0 (or in the original variables, r = 2m) plays here the same role as the hypersurface x1 = 0 in the previous example. As u varies from -inf to +inf, r varies from +inf to 2m and then again from 2m to +inf. [...]
The four-dimensional space is described mathematically by two congruent parts or "sheets", corresponding to u > 0 and u < 0, which are joined by a hyperplane r = 2m or u = 0 in which g vanishes. We call such a connection between the two sheets a "bridge". We see now in the given solution, free from singularities, the mathematical representation of a an elementary particle (neutron or neutrino). Characteristic of the theory we are presenting is the description of space by means of two sheets."

So the at R = 2MG/c² we have a bridge to another "sheet", that's what the event horizon is and where it leads to .

There are further more deeper connections like "ER = EPR" but since they lead to other fields of study I won't mention them.
I end my answer with mentioning of this 2 experiments:
https://www.nasa.gov/mission_pages/sunearth/news/mag-portals.html
http://www.uab.cat/web/newsroom/new...me-1345668003610.html?noticiaid=1345689132054

 

[PeterDonis]

for this case where there is no event horizon

The case where there is no event horizon does not have the issue you describe. If there is no event horizon there must be matter present, and the matter must start at an isotropic radial coordinate that is larger than $R = M/2$ (in units where $G = c = 1$ and $R$ is the isotropic radial coordinate). Inside the matter, the metric is different, and smaller values of $R$ continue to correspond to smaller proper circumferences, down to a circumference of zero at $R = 0$.

If we are talking about the strictly vacuum case, where there is no matter present at any value of $R$, then there is no case where there is no event horizon; $R = M / 2$ is an event horizon. On the Kruskal diagram, it is the point at the very center of the diagram (which means it's a 2-sphere with radius equal to the horizon radius when you add back in the angular coordinates). This is the only case where the rate of change of the proper circumferences as a function of $R$ switches sign.

I don't see any reason why the two coordinate directions should be equivalent physically as the time dilation gradient is still going the same way as the radial coordinate.

No, it isn't. The magnitude of the metric coefficient $g_{00}$ gets larger, not smaller, with decreasing $R$ for $R < M/2$. The sign of $d/tau/dt$, which is the square root of $g_{00}$, changes (at least if you continue to take the "positive" square root, i.e., you don't start inserting an extra minus sign for $R < M/2$ that you didn't insert for $R > M/2$), but that is not the time dilation factor; as I noted in my previous post, that tells you that the direction of proper time along timelike curves is reversed relative to the time coordinate of the Kruskal diagram.

I'm not sure about the relationship of the proper circumference to the proper radius

Strictly speaking, if we restrict to the region of spacetime that is covered by isotropic coordinates, there is no "proper radius", because the 2-spheres contained in a spacelike hypersurface of constant time do not include a 2-sphere with zero circumference.

if that hypothetical metric could occur

I assume you mean, if the portion of the geometry with $R \le M/2$ could occur. The portion with $R > M/2$ does not raise any issues of the kind we are discussing.

what would happen if an object passed the minimum point?

It can't without going out of the region of spacetime covered by isotropic coordinates. If you check, you will see that there are no timelike (or even null) curves that stay entirely within the isotropic coordinate patch and also go from $R > M/2$ to $R < M/2$ (or vice versa). The only curves that traverse the $R = M/2$ boundary while staying entirely within the isotropic coordinate patch are spacelike curves.

 

[Jonathan Scott]

The case where there is no event horizon does not have the issue you describe. If there is no event horizon there must be matter present, and the matter must start at an isotropic radial coordinate that is larger than $R = M/2$ (in units where $G = c = 1$ and $R$ is the isotropic radial coordinate).

I think your response would make sense to me if it were based on the Schwarzschild solution, but I was talking about the exponential metric where the space and time factors are exact reciprocals of one another, which is not an exact solution for GR but is used as an approximate solution for PPN purposes and as a local solution for various alternative theories, where it is usually asserted that it does not lead to black holes. Although this seems to be technically true, I think that such a metric would still lead to weird effects, specifically the possibility that an object could collapse so far that the proper circumference would start to increase if it "collapsed" any further, but I'm not clear on what would theoretically happen in this case (and how for example one might distinguish that at a distance from a black hole).

 

[PeterDonis]

I was talking about the exponential metric where the space and time factors are exact reciprocals of one another

I would need a reference for this as I'm not familiar with it. My comments apply to the standard isotropic coordinates on the Schwarzschild solution in GR.

 

[PeterDonis]

for example $\exp(-Gm/r'c^2)$ and $\exp(+Gm/r'c^2)$.

Do you mean a line element as follows?

$$
ds^2 = - e^{- \frac{M}{r}} dt^2 + e^{\frac{M}{r}} \left( dr^2 + r^2 d\Omega^2 \right)
$$

 

[Jonathan Scott]

Do you mean a line element as follows?

$$
ds^2 = - e^{- \frac{M}{r}} dt^2 + e^{\frac{M}{r}} \left( dr^2 + r^2 d\Omega^2 \right)
$$

Almost, but I was giving the linear time and space factors, so the line element would involve squared factors:
$$ds^2 = - e^{- \frac{2M}{r}} dt^2 + e^{\frac{2M}{r}} \left( dr^2 + r^2 d\Omega^2 \right)$$
This matches GR closely enough to duplicate the solar system results (it is well known to have the same PPN parameters as GR) and one can more generally use $e^\phi$ as a good approximation where multiple static or slow-moving sources are present and $\phi$ is the Newtonian potential. Several theories use this metric (including Svidzinsky's dubious vector theory and the equally dubious Yilmaz theory, and I think Whitehead's theory matched it too although I can't find details right now) and each such theory is normally accompanied by a claim that it doesn't give black holes, but instead they apparently have this feature of a minimum proper circumference and I've been trying to understand the implications of that feature.

 

[PeterDonis]

I was giving the linear time and space factors, so the line element would involve squared factors

Ah, ok. I wondered about the 2 in the exponential; as you say, it needs to be there to match standard GR in the weak field slow motion limit.

I'll plug this metric into Maxima when I get a chance and see what comes out.