# Event Horizon and Light Exit Cone Question

by Sigie
Tags: cone, event, exit, horizon, light
 P: 4 I posted a similiar question under cosmology but the question was unable to be answered. I thought I would try a reframe the question. When approaching a black holes event horizon, the exit cone for light become smaller until it is eliminated at the event horizon itself. But how can gravity bend light that is directed exactly away from the line of force? Employing the principle of equivalence, no matter how fast I accelerate in flat space, a beam of light directed in the same direction as I accelerate will never bend. Therefore, it seems to me that the event horizon breaks the principle of equivalence. For example, I am in an elevator with a laser pointer. If the elevator is accelerated upwards in flat space, from my perspective, the light bends, except if I direct the laser pointer directly upwards of downwards. If I direct the laser pointer directly upwards, no matter how fast I accelerate, the light beam will always be traveling away from me at speed c. If I use the equivalent situation of a mass below me, then, no matter how massive the object, the laser should never bend. If I am in an elevator that is crossing an event horizon (assuming a huge black hole where tidal forces won't tear me apart at the vent horizon), and I direct the laser pointer exactly away from the line of the force of gravity, at a point where the laser emiter is inside the event horizon, yet the top of the elevator is beyond it, the laser cannot by definition reach the ceiling. Yet the light cannot be slowed, and it cannot be bent if it is line with the force of gravity. Thus, it seems to be that gravity cannot completely close off light's exit cone and thus an event horizon (and a black hole) cannot truly form, no matter how massive an object. I would be very appreciative if someone could explain this paradox to me. Thanks
PF Gold
P: 1,159
 Quote by Sigie If I am in an elevator that is crossing an event horizon (assuming a huge black hole where tidal forces won't tear me apart at the vent horizon), and I direct the laser pointer exactly away from the line of the force of gravity, at a point where the laser emiter is inside the event horizon, yet the top of the elevator is beyond it, the laser cannot by definition reach the ceiling. Yet the light cannot be slowed, and it cannot be bent if it is line with the force of gravity.
It isn't actually possible to find a coordinate system where what happens inside the elevator can be mapped to the what an outside observer sees in any sensible way (for a start, it takes more than infinite time to happen, and ends up in a sort of inside-out imaginary coordinate system). However, one way of thinking about it would be that relative to the original space, an upwards ray is moving within a piece of space which is moving downwards faster than light, so overall it goes downwards, and by the time it reaches the top of the elevator, the top of the elevator has actually passed the point where the bottom of the elevator was at the time the ray was emitted.

I'd agree that this doesn't make much sense, but that doesn't actually constitute a disproof of black holes.

Note that General Relativity and Einstein's field equations do NOT actually directly predict the existence or otherwise of black holes, contrary to popular opinion. In the solutions of those equations described by the Schwarzschild solution and later extensions such as the Kerr solution for rotating central bodies, the radial coordinate involves a constant of integration which is determined by assumed boundary conditions. Karl Schwarzschild himself derived his solution using a Euclidean radial coordinate, making the assumption that a point mass was located at the origin, but then pointed out that the mathematics for describing the solution could be simplified by using a different radial coordinate, which had a value of 2GM at the physical origin. (See arXiv:physics/9905030 for a translation of Schwarzschild's original paper). When Hilbert later described Schwarzschild's solution, he did it directly in terms of the simplified radial coordinate system, but assumed that the physical origin was where Schwarzschild's modified radial coordinate was zero rather than 2GM.

It is this assumption from Hilbert which gives rise to the possibility of black holes, and for a long time it seems that no-one noticed that any assumption had been made, until Leonard S Abrams dared to point this out in 1989 (arXiv:gr-qc/0102055). Unfortunately, as there doesn't seem to be any obvious way to prove which boundary assumption is better, this has given rise to various surprisingly fierce battles based on somewhat unsatisfactory arguments which appear to boil down to "my assumptions are better than your assumptions". My personal opinion at present, on the grounds of simplicity, is that Schwarzschild's assumption makes a lot more sense than Hilbert's, in which case black holes don't happen, although it's difficult to distinguish an exceedingly dense object which hasn't quite collapsed into a black hole from one which has. Experimental evidence suggests that at least some extremely dense objects involved in quasars have strong intrinsic magnetic fields, which means they cannot be black holes because of what is known as the "no hair" theorem, and this supports Schwarzschild's original assumption.
 P: 4 Thanks for the very informative reply. Am I correct in interpreting then that the belief in black holes stems from the assumption that a singularity can occur? Does the mathematics allows for a piece of space to travel faster than light relative to the center of the black hole? What about the laser emitor what happen to the laser since the pointer is traveling faster than the speed of light attached to the floor of the elevator? It still seems like the principle of equivalence is being violated. It all seems very ineloquent. Wasn't it Einsein who said God is subtle but not malicious? I think you make a very good argument for your opinion. Thanks again!
PF Gold
P: 1,159
Event Horizon and Light Exit Cone Question

 Quote by Sigie Thanks for the very informative reply. Am I correct in interpreting then that the belief in black holes stems from the assumption that a singularity can occur? Does the mathematics allows for a piece of space to travel faster than light relative to the center of the black hole? What about the laser emitor what happen to the laser since the pointer is traveling faster than the speed of light attached to the floor of the elevator? It still seems like the principle of equivalence is being violated. It all seems very ineloquent. Wasn't it Einsein who said God is subtle but not malicious? I think you make a very good argument for your opinion. Thanks again!
The singularity isn't assumed; it is the assumed range of the radial coordinate which leads to the existence of mathematical singularities and hence to the properties known as a black hole.

The original Schwarzschild solution describes the field around a point mass assumed to be at the origin, and the simple physical assumption that the the mass cannot really be shrunk down to a point means that there are no infinities or singularities involved. Hilbert's version of the solution implicitly assumes that the radial parameter can keep going past Schwarzschild's origin into the "inside out" and imaginary space-time on the other side, and then on until the radial parameter is zero. This leads to expressions which involves dividing by zero, at r=2GM and at r=0. The first of these (at the "event horizon") can be mathematically transformed away using a different coordinate, at least from the point of a falling observer, but the second cannot.

The maths says that from the viewpoint of the falling observer, everything looks normal (apart from increasing tidal forces) until the second singularity is reached. However, there simply isn't any meaningful way to try to describe what is "actually" happening in terms of some external observer, so questions about what it would "look like" from such a viewpoint cannot be answered.

Please note that although most GR experts now agree that Schwarzschild and Hilbert made different assumptions and that the choice of radial coordinate depends on boundary conditions which have to be determined separately, the general assumption at the moment seems to be that Hilbert must have been right and anyone who questions this is labelled as a heretic or crackpot. (To some extent, they might be right: Stephen J Crothers, who has written a lot about this subject, appears to be mathematically right about many aspects, but goes over the top on many arguments and somehow manages to write like a crackpot anyway). I'm still waiting to see a rational argument on the subject.
Mentor
P: 6,246
 Quote by Jonathan Scott Note that General Relativity and Einstein's field equations do NOT actually directly predict the existence or otherwise of black holes, contrary to popular opinion. In the solutions of those equations described by the Schwarzschild solution and later extensions such as the Kerr solution for rotating central bodies, the radial coordinate involves a constant of integration which is determined by assumed boundary conditions. Karl Schwarzschild himself derived his solution using a Euclidean radial coordinate, making the assumption that a point mass was located at the origin, but then pointed out that the mathematics for describing the solution could be simplified by using a different radial coordinate, which had a value of 2GM at the physical origin. (See arXiv:physics/9905030 for a translation of Schwarzschild's original paper). When Hilbert later described Schwarzschild's solution, he did it directly in terms of the simplified radial coordinate system, but assumed that the physical origin was where Schwarzschild's modified radial coordinate was zero rather than 2GM. It is this assumption from Hilbert which gives rise to the possibility of black holes, and for a long time it seems that no-one noticed that any assumption had been made, until Leonard S Abrams dared to point this out in 1989 (arXiv:gr-qc/0102055). Unfortunately, as there doesn't seem to be any obvious way to prove which boundary assumption is better, this has given rise to various surprisingly fierce battles based on somewhat unsatisfactory arguments which appear to boil down to "my assumptions are better than your assumptions".
These ideas have been thoroughly discredited; see

http://www.physicsforums.com/showthr...ghlight=Abrams

http://arxiv.org/abs/gr-qc/0608033.
PF Gold
P: 1,159
 Quote by George Jones These ideas have been thoroughly discredited; see http://www.physicsforums.com/showthr...ghlight=Abrams http://arxiv.org/abs/gr-qc/0608033.
I'm very familiar with the first thread (which I first saw on sci.physics.research); it's certainly very dismissive of the idea, especially "tessel", whose response is one of the fiercest I've ever seen in a scientific forum. However, it does NOT answer the basic obvious question, which I feel is totally reasonable: why do people now assert that the central mass point is at R=0 when it is very clear from Schwarzschild's original paper that he assumed it to be at R=2GM on perfectly reasonable grounds?

The other paper finds fault in ideas by Antoci and others, especially in different interpretation of terminology, but again does not provide a response to that basic point. The arguments from the black hole side seem to effectively say that there is nothing wrong with assuming r=0 is the center, but I don't think that's a very strong justification for doing so.

The usual argument that seems to be used is that Antoci, Crothers and the others have failed to understand the standard theory, and are inaccurate in the use of their terminology, and the arguments always seem to revolve around peripheral points. Even if this were true it is not relevant to the main point.

From my reading of both sides, I'm convinced that Abrams, Antoci and Crothers are doing an important job in pointing out that an arbitrary assumption was initially made, calling attention to the difference between Schwarzschild's original paper and Hilbert's reinterpretation of it. I also agree with them that Schwarzschild's original assumption which does not lead to black holes seems more plausible, and I'd like to believe that simpler idea, but so far I have not found anything in their work which "proves" it, despite grand claims, although I think Crothers has some strong points and I haven't finished working through them for myself. (He tends to be so forceful and apparently arrogant that I feel he is probably likely to be making mistakes the other way, over-stating the evidence against the black hole idea, and I have to check that I understand everything myself before I can believe it). However, I've found no serious attempt from the standard theory side to address the basic point either.

It is very clear that there is an arbitrary constant included in the radial coordinate; Karl Schwarzschild explicitly chooses a value which makes sense given his initial assumption of a point mass at the origin. However, Hilbert makes a different choice, and his initial paper does not give a good reason for doing so, and even seems to assume that it doesn't make any difference. A very strong justification is needed for choosing a value which makes the physics weird. If I'm supposed to believe the extraordinary idea that it is reasonable to assume boundary conditions which cause GR to give rise to black holes, I need to see a good reason why, and so far I've only seen scientists being rude about it.

As far as I can see, with the current theoretical position there simply isn't any theoretical way to distinguish these cases based on GR alone. It either needs a breakthrough in a new level of theory, or experimental evidence to help distinguish these two cases. The experimental evidence that some super-massive "black hole candidates" appear to have significant intrinsic magnetic fields doesn't fit well with the black hole "no hair" theorem, and has already given rise to an alternative idea of Magnetospheric Eternally Collapsing objects (MECOs), which attempt to remain compatible with standard black hole theory by assuming that collapse can be resisted by sufficient radiation pressure.

I may be wrong and there may still be a convincing argument in one direction or the other in the existing literature. If anyone knows of such an argument, I'd be interested to know. If it's not suitable for the forums, please send me a PM.
P: 8,657
 Quote by Jonathan Scott However, it does NOT answer the basic obvious question, which I feel is totally reasonable: why do people now assert that the central mass point is at R=0 when it is very clear from Schwarzschild's original paper that he assumed it to be at R=2GM on perfectly reasonable grounds?
Page 43, footnote 8 of 't Hooft's notes:
http://www.phys.uu.nl/~thooft/lectures/genrel.pdf
In his original paper, Karl Schwarzschild replaced r-2M by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes ”eindeutig” (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild solution.
PF Gold
P: 1,159
 Quote by atyy Page 43, footnote 8 of 't Hooft's notes: http://www.phys.uu.nl/~thooft/lectures/genrel.pdf In his original paper, Karl Schwarzschild replaced r-2M by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes ”eindeutig” (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild solution.
Thanks for the reference. It's interesting to see another viewpoint on this area.

However, 't Hooft's assertion that "one may choose the coordinates freely" seems overstated; Schwarzschild made a choice that did not give rise to black holes but those who followed made a different specific choice which clearly has different physical consequences. It seems to me that such a choice needs a strong justification. The later conceptual model of spheres of area $4\pi r^2$ which is used to explain Schwarzschild's solution does of course suggest that r looks like a Euclidean radius, but it is clear from Schwarzschild's own original model that this is not necessarily the case.

If you've seen the original paper, the footnote is a bit backwards too. Schwarzschild started his calculations using a radial coordinate which was taken to be the coordinate distance from the origin, then after finding a solution in terms of that radial coordinate and setting the constant of integration accordingly, he points out that the mathematics can be simplified by using a different radial coordinate, which happens to have the value 2GM at the original origin, and that is the coordinate that Hilbert later assumes to be zero at the location of the mass.
P: 8,657
 Quote by Jonathan Scott Thanks for the reference. It's interesting to see another viewpoint on this area. However, 't Hooft's assertion that "one may choose the coordinates freely" seems overstated; Schwarzschild made a choice that did not give rise to black holes but those who followed made a different specific choice which clearly has different physical consequences. It seems to me that such a choice needs a strong justification.
How about 't Hooft's comment that "the singularity is not a true singularity at all"?

Edit:

Q1) Would the existence of black holes be consistent with the Einstein field equations?
Q2) Do black holes actually exist?

't Hooft and most GR texts are discussing only Q1 when talking about the Kuskal-Szekeres extension of the Schwarzschild solution.

They are not commenting on Q2. Nonetheless, in your comments on Q2, you use the no-hair theorems to tentatively exclude certain objects from being black holes. But don't the no-hair theorems themselves answer Q1 affirmatively?
PF Gold
P: 1,159
 Quote by atyy How about 't Hooft's comment that "the singularity is not a true singularity at all"?
Yes, it's well-known that there is a coordinate transform that will remove the singularity at r=2GM from the point of view of a falling observer, although there is no meaningful way to describe what happens from the point of view of an external observer. The fact that one can "get past" the "event horizon" doesn't necessarily justify adjusting the radial coordinate so that it is no longer the origin.

 Quote by atyy Reading through your posts, I think there are two distinct issues: Q1) Would the existence of black holes be consistent with the Einstein field equations? Q2) Do black holes actually exist? 't Hooft and most GR texts are discussing only Q1 when talking about the Kuskal-Szekeres extension of the Schwarzschild solution. They are not commenting on Q2. Nonetheless, in your comments on Q2, you use the no-hair theorems to tentatively exclude certain objects from being black holes. But don't the no-hair theorems themselves answer Q1 affirmatively?
On Q1 I'm happy to agree that the existence of black holes appears to be consistent with the Einstein field equations, but I'd also point out that from Karl Schwarzschild's original solution, the non-existence of black holes appears to be equally consistent with the field equations.

The field equations alone do not determine the choice of origin for the radial coordinate, but for some reason most GR texts seem to implicitly assume Hilbert's choice rather than Schwarzschild's, without justifying the choice.

If Schwarzschild was right then we would not even have any theoretical basis for hypothesising the existence of black holes. If Hilbert was right, then the equations would admit black hole solutions but there are still questions about how such solutions could be form given the complications about infinite times. If we therefore observe something which we would expect to be a black hole, but it shows evidence of being non-black (for example by having a strong intrinsic magnetic field, or having strong radio emissions apparently from the central object) this doesn't necessarily prove that Hilbert was wrong, in that there could be some other complex physical effect which isn't yet fully understood which is preventing the collapse (as in the MECO model), or some problem in the current interpretation of the black hole model. However, I feel that if all other things were equal and we had to choose between Schwarzschild's and Hilbert's interpretation, the combination of the simplicity of Schwarzschild's interpretation and the experimental evidence of intrinsic magnetic fields would provide strong support for Schwarzschild's position.

What I really want to know is how GR people justify the assumption that Hilbert's choice of origin (that is, the radial coordinate at which the central mass is located) is better than Schwarzschild's, when it gives such weird results.
P: 8,657
 Quote by Jonathan Scott On Q1 I'm happy to agree that the existence of black holes appears to be consistent with the Einstein field equations, but I'd also point out that from Karl Schwarzschild's original solution, the non-existence of black holes appears to be equally consistent with the field equations.
I would say that the existence of solutions of the field equations that do not contain true singularities show that the Einstein field equations are consistent with the non-existence of black holes. I also would say that the Kruskal-Szekeres extension of the Schwarzschild solution shows that there is at least one solution of the Einstein field equation that is consistent with the existence of black holes.

 Quote by Jonathan Scott What I really want to know is how GR people justify the assumption that Hilbert's choice of origin (that is, the radial coordinate at which the central mass is located) is better than Schwarzschild's, when it gives such weird results.
The reason for changing coordinates to "get past" the coordinate singularity is that even in Schwarzschild coordinates, the Riemann tensor at the coordinate singularity shows that there is no true singularity. Since the Riemann tensor indicates that it is the coordinates that are failing and not spacetime itself, we are obliged to change to a workable set of coordinates.

However, we can "match up" the Schwarzschild solution to other solutions, such as the interior of a star, and the Schwarzschild solution can be used to describe the solar system, which is almost certainly not a black hole. So it is not that the Schwarzschild solution must represent a black hole. Rather it is a solution that can be "matched up" in a way that what it describes is a black hole. (Obviously the matching up is done at different places in these two examples.)
PF Gold
P: 1,159
 Quote by atyy The reason for changing coordinates to "get past" the coordinate singularity is that even in Schwarzschild coordinates, the Riemann tensor at the coordinate singularity shows that there is no true singularity. Since the Riemann tensor indicates that it is the coordinates that are failing and not spacetime itself, we are obliged to change to a workable set of coordinates.
If Schwarzschild is right then the location of the coordinate singularity cannot be reached (because it is the location of the mass, which cannot be infinitely dense), so the question doesn't even arise.

 Quote by atyy However, we can "match up" the Schwarzschild solution to other solutions, such as the interior of a star, and the Schwarzschild solution can be used to describe the solar system, which is almost certainly not a black hole. So it is not that the Schwarzschild solution must represent a black hole. Rather it is a solution that can be "matched up" in a way that what it describes is a black hole. (Obviously the matching up is done at different places in these two examples.)
Mostly agreed. The specific point is that in the Hilbert interpretation of the Schwarzschild vacuum solution, the assumption is made that the central mass can physically contract so that it is within r<2GM. It is when that happens that we have a black hole. In the original Schwarzschild model, the origin of the central mass is located at r=2GM so the vacuum must end at r>2GM provided that the mass is not of infinite density.
P: 8,657
 Quote by Sigie Thus, it seems to be that gravity cannot completely close off light's exit cone and thus an event horizon (and a black hole) cannot truly form, no matter how massive an object. I would be very appreciative if someone could explain this paradox to me.
To see how things move over a large piece of spacetime, one calculates the geodesics. Light follows null geodesics, and particles follow time-like geodesics.

I'm not exactly sure of the situation you're describing, but it's not possible to use the equivalence principle over a large piece of spacetime. If spacetime is curved, the equivalence principle holds only over pieces of spacetime small enough to be locally flat. Even light bending by the sun is already a large piece of spacetime, and we cannot properly calculate that using the equivalence principle - the equivalence principle gives only half the bending predicted by General Relativity! (There are actually consistent alternative theories of gravity that incorporate the equivalence principle and therefore predict local bending of light. But in some of these alternative theories, spacetime around the sun is curved such that there is no global bending of light. We chose General Relativity over these other theories because it matched experimental observations.)

http://www.einstein-online.info/en/s...ion/index.html
P: 8,657
 Quote by Jonathan Scott Mostly agreed. The specific point is that in the Hilbert interpretation of the Schwarzschild vacuum solution, the assumption is made that the central mass can physically contract so that it is within r<2GM. It is when that happens that we have a black hole. In the original Schwarzschild model, the origin of the central mass is located at r=2GM so the vacuum must end at r>2GM provided that the mass is not of infinite density.
I believe the reason for interpreting the Schwarschild solution as we currently do is Birkhoff's theorem. It roughly states that any solution of Einstein's field equation for empty space which is spherically symmetric is locally equivalent to part of the maximally extended Schwarzschild solution. Thus if you wish to have the solution describe a spherically symmetric physical system, the centre of the physical system should be at the centre of the Schwarzschild coordinates.

If the current interpretation of the Schwarschild solution is wrong, then the textbooks are not merely wrong about black holes, they would be wrong about the perihelion precession of Mercury, solar deflection of light, and neutron stars also.
PF Gold
P: 1,159
 Quote by atyy I believe the reason for interpreting the Schwarschild solution as we currently do is Birkhoff's theorem. It roughly states that any solution of Einstein's field equation for empty space which is spherically symmetric is locally equivalent to part of the maximally extended Schwarzschild solution. Thus if you wish to have the solution describe a spherically symmetric physical system, the centre of the physical system should be at the centre of the Schwarzschild coordinates. If the current interpretation of the Schwarschild solution is wrong, then the textbooks are not merely wrong about black holes, they would be wrong about the perihelion precession of Mercury, solar deflection of light, and neutron stars also.
Sorry, but I'm sure neither of these is correct. The vacuum solution outside the event horizon is unaffected by whether the point mass location is assumed to be "at" the event horizon (as in Schwarzschild's interpretation) or "beneath" it (as in Hilbert's). The primary difference in using the original Schwarzschild interpretation is simply that the range of applicability of the vacuum solution ends outside r=2GM because of the physical constraint that the surface of the central mass lies outside that radial coordinate. The perihelion precession and deflection of light are unaffected by this difference in interpretation. The solution outside neutron stars is also unaffected, but the density profile for the star itself and the boundary conditions for the interior solution are increasingly affected for heavier neutron stars, and obviously for cases where Hilbert's version of the theory would give gravitational collapse, the original Schwarzschild version would not, but would instead give an extremely red-shifted well-defined surface.
P: 8,657
 Quote by Jonathan Scott Sorry, but I'm sure neither of these is correct. The vacuum solution outside the event horizon is unaffected by whether the point mass location is assumed to be "at" the event horizon (as in Schwarzschild's interpretation) or "beneath" it (as in Hilbert's). The primary difference in using the original Schwarzschild interpretation is simply that the range of applicability of the vacuum solution ends outside r=2GM because of the physical constraint that the surface of the central mass lies outside that radial coordinate. The perihelion precession and deflection of light are unaffected by this difference in interpretation. The solution outside neutron stars is also unaffected, but the density profile for the star itself and the boundary conditions for the interior solution are increasingly affected for heavier neutron stars, and obviously for cases where Hilbert's version of the theory would give gravitational collapse, the original Schwarzschild version would not, but would instead give an extremely red-shifted well-defined surface.
The solution stays the same. But won't the interpretation change?

Let the parameter in the Schwarzschild solution be q.
Let the radial coordinate in the Schwarzschild solution be r.
The Newtonian potential is GM/(distance from centre)

We normally interpret "distance from centre" to be r:
q/r=2GM/r
q=2GM

If we interpret "distance from centre" to be r-q:
q/r=2GM/(r-q)
q(r-q)=2GMr
q2-rq-2GMr=0
q=(r±√(r2-8GMr))/2
PF Gold
P: 1,159
 Quote by atyy The solution stays the same. But won't the interpretation change? Let the parameter in the Schwarzschild solution be q. Let the radial coordinate in the Schwarzschild solution be r. The Newtonian potential is GM/(distance from centre) We normally interpret "distance from centre" to be r: q/r=2GM/r q=2GM If we interpret "distance from centre" to be r-q: q/r=2GM/(r-q) q(r-q)=2GMr q2-rq-2GMr=0 q=(r±√(r2-8GMr))/2
Regardless of the interpretation, within the Schwarzschild solution and Schwarzschild coordinate system, the r value used in GM/r is the Schwarzschild r coordinate, so the interpretation doesn't make any difference to the vacuum solution.

Imagine a map of the system on stretchy material, with local markings showing ruler sizes and clock rates and hence describing the metric. Now make a hole at the origin and stretch it to radius 2GM, shifting the material outwards symmetrically in the radial direction (and stretching the ruler indicators as appropriate). You've distorted the original map, but what it describes is unchanged, except that you've added a hole in the middle. However, if you add the hypothesis that the hole in the map is part of the space available to the central mass, and that it can shrink down into it, which is effectively what Hilbert does, then that does make a physical difference.

(At least that's what I think, but GR experts are welcome to check it out).