# Thread: Black hole inside a black hole

1. Hello everone,
I had a question regarding the possibility of a black hole existing inside another black hole.
Theoretically, an objet that goes through the horizon of a black hole sees nothing special ( except for the firewall theory but let's keep things simple for now ). That should also be true for a black hole in particular. Lets take the example of a small black hole, let's say of stellar mass, falling in a very large black hole like a galactic black hole. At the horizon of the galactic black hole the gravitational field is not very strong and space is flat at its horizon. When the small BH enters the large one, its gravitationnal field adds to that of the large one, so obviously the large one has a stronger gravitionnal field and its horizon expands accordingly. The 2 black holes can be said to "merge". However the small black hole should remain intact inside the large one. In other words a photon trapped inside the small black hole continues to be trapped inside that small region of space-time after the small black hole has fallen into the large one.
As for what happens to the small black hole once inside the large one, I don't know but I guess several possibilities could be imagined, such as:
- when the small BH gets close to the center the gravitationnal field becomes so strong that the small BH is torn apart and destroyed.
- the small BH evaporates before it gets to the center
- the small black hole swallows all the matter located in the center and grows until all the content of the large BH in the center has been swallowed and thus it becomes the large BH.
I guess that scientists have already thought about that kind of things, so I was wondering if you knew what they thought was happening.
Thank you,
Regards,
Nic321.

2.

3. Originally Posted by Nic321
However the small black hole should remain intact inside the large one.
A black hole is not a "thing" like that which can remain intact. It is simply the radius of an event horizon which is solely defined by the mass within it. (OK, and the charge and angular momentum.)

So if you add mass, in any form, to a black hole the radius of the event horizon increases. That is all you need to know. There is no mathematical meaning (as far as I know) to an event horizon within a black hole.

4. Hi Strange,
I agree with you, there is nothing concrete at the horizon of a BH.
A BH is just a region of space-time from which light can't escape and from outside it is only defined by its mass, angular momentum and charge.
However, you could be inside a large BH and have a small BH next to you. You would only know its mass, angular momentum and charge. Light would be trapped in this small very dense lump of matter. That lump of matter would have a gravitationnal field which would add to the gravitationnal field of the large BH, so obviously the horizon of the large BH gets larger.
Nic.

5. Originally Posted by Strange
Originally Posted by Nic321
However the small black hole should remain intact inside the large one.
A black hole is not a "thing" like that which can remain intact. It is simply the radius of an event horizon which is solely defined by the mass within it. (OK, and the charge and angular momentum.)

So if you add mass, in any form, to a black hole the radius of the event horizon increases. That is all you need to know. There is no mathematical meaning (as far as I know) to an event horizon within a black hole.
But one can have an event horizon inside an event horizon. This occurs in the case of a charged and/or rotating blackhole. In the case of two merged blackholes, the two separate central singularities may represent an additional parameter of the total blackhole that might give rise to an additional event horizon. However, the outer event horizon would be governed by the total mass of the two blackholes.

6. Originally Posted by Nic321
Lets take the example of a small black hole, let's say of stellar mass, falling in a very large black hole like a galactic black hole.
I like this example. If an observer who is measuring the properties of the small blackhole falls into the large blackhole along with the small blackhole, then he could continue to measure the properties of the small blackhole without noticing that anything untoward had happened as he crossed the event horizon of the large blackhole.

7. Originally Posted by KJW
I like this example. If an observer who is measuring the properties of the small blackhole falls into the large blackhole along with the small blackhole, then he could continue to measure the properties of the small blackhole without noticing that anything untoward had happened as he crossed the event horizon of the large blackhole.
Is that really the case? (I am expressing surprise, rather than disagreeing. ) I have seen simulations of black hole mergers and they show that the two black holes are "distorted", with their event horizons being stretched towards one another until the merge is complete (followed by a brief oscillation or "ring down").

I can't find the nice pictures I have seen before that show that, but this might be of interest: Steve Drasco's black hole movies

8. Originally Posted by Strange
Originally Posted by KJW
I like this example. If an observer who is measuring the properties of the small blackhole falls into the large blackhole along with the small blackhole, then he could continue to measure the properties of the small blackhole without noticing that anything untoward had happened as he crossed the event horizon of the large blackhole.
Is that really the case? (I am expressing surprise, rather than disagreeing. ) I have seen simulations of black hole mergers and they show that the two black holes are "distorted", with their event horizons being stretched towards one another until the merge is complete (followed by a brief oscillation or "ring down").

I can't find the nice pictures I have seen before that show that, but this might be of interest: Steve Drasco's black hole movies
To be honest, I had never considered this example until I saw this thread and realised that the principle that one doesn't notice anything special when crossing the event horizon must also apply to a blackhole as well. I'm guessing the simulations to which you refer are only dealing with the outermost event horizon.

9. Originally Posted by KJW
To be honest, I had never considered this example until I saw this thread and realised that the principle that one doesn't notice anything special when crossing the event horizon must also apply to a blackhole as well. I'm guessing the simulations to which you refer are only dealing with the outermost event horizon.
It occurred to me after I posted, that an infalling observer might not notice the distortion of the event horizons because it is their space time that is being affected and so all would look the same. But I do know that modelling this stuff is bloody hard and needs complicated numerical simulations.

10. Originally Posted by Strange
Originally Posted by KJW
To be honest, I had never considered this example until I saw this thread and realised that the principle that one doesn't notice anything special when crossing the event horizon must also apply to a blackhole as well. I'm guessing the simulations to which you refer are only dealing with the outermost event horizon.
It occurred to me after I posted, that an infalling observer might not notice the distortion of the event horizons because it is their space time that is being affected and so all would look the same. But I do know that modelling this stuff is bloody hard and needs complicated numerical simulations.
I think the outer event horizon would be the global threshold of light escaping the total system, whereas the inner event horizon would be the more local threshold of light escaping the small blackhole. Bear in mind that the small blackhole will eventually hit the large blackhole singularity, so that the threshold is essentially whether or not light reaches the large blackhole singularity from outside or inside the small blackhole.

Event horizons interchange time and one space dimension, so the inside of the small blackhole, being two event horizons from the outside world, would have the same space/time sense as the outside world.

11. Hello KJW,
Originally Posted by KJW
Originally Posted by Strange
Originally Posted by KJW
To be honest, I had never considered this example until I saw this thread and realised that the principle that one doesn't notice anything special when crossing the event horizon must also apply to a blackhole as well. I'm guessing the simulations to which you refer are only dealing with the outermost event horizon.
It occurred to me after I posted, that an infalling observer might not notice the distortion of the event horizons because it is their space time that is being affected and so all would look the same. But I do know that modelling this stuff is bloody hard and needs complicated numerical simulations.
I think the outer event horizon would be the global threshold of light escaping the total system, whereas the inner event horizon would be the more local threshold of light escaping the small blackhole. Bear in mind that the small blackhole will eventually hit the large blackhole singularity, so that the threshold is essentially whether or not light reaches the large blackhole singularity from outside or inside the small blackhole.
It is very speculative but I was wondering, could there be at the center of the large black hole some sort of foam of small black holes. BHs would form, change shape, desintegrate because of the very strong gravitational field, evaporate, etc...
Event horizons interchange time and one space dimension, so the inside of the small blackhole, being two event horizons from the outside world, would have the same space/time sense as the outside world.
I guess in theory there could even be any number of imbrication of black holes. An observer's space/time would be the same as those at 2n levels of imbrication, and reversed at levels 2n+1.
Nic.

12. Originally Posted by Nic321
It is very speculative but I was wondering, could there be at the center of the large black hole some sort of foam of small black holes. BHs would form, change shape, desintegrate because of the very strong gravitational field, evaporate, etc...
I wouldn't like to speculate too much on the spacetime near the singularity. My guess is that the inner blackhole would shrink as it approached the singularity of the large blackhole, vanishing entirely at the singularity.

13. Why is there an assumption the black hole masses would necessarily meld instead of becoming a rotating two body system inside the expanded event horizon?

14. Originally Posted by dan hunter
Why is there an assumption the black hole masses would necessarily meld instead of becoming a rotating two body system inside the expanded even horizon?
There are no stable orbits within 1.5 times the Schwarzschild radius.

15. Originally Posted by Strange
Originally Posted by dan hunter
Why is there an assumption the black hole masses would necessarily meld instead of becoming a rotating two body system inside the expanded even horizon?
There are no stable orbits within 1.5 times the Schwarzschild radius.
Also, the singularity isn't a location in space. It is in the future. Recall the interchange between time and the radial direction at the event horizon.

16. There is a property of the interior of a Schwarzschild blackhole worth noting: Because it is symmetric about the two equivalent angular directions and the spatial direction, the three dimensional space inside a Schwarzschild blackhole is homogenous. That is, it is the same everywhere.

17. Originally Posted by KJW
There is a property of the interior of a Schwarzschild blackhole worth noting: Because it is symmetric about the two equivalent angular directions and the spatial direction, the three dimensional space inside a Schwarzschild blackhole is homogenous. That is, it is the same everywhere.
Just to clarify this, which one of the following statement is true inside a Schwarzschild blackhole:
- space-time is curved and space is flat ( option I would guess )
- space-time is curved and space is curved
- space-time is flat and space is curved ( false I guess because of the gravitationnal field)
Gee I am always confused about which one is curved...
And does it change anything if the BH is spinning?

18. - space-time is flat and space is curved
Space-time includes space and time and so when space-time is curved both space and time are curved.

And does it change anything if the BH is spinning?
Yes. You then have a Kerr black hole which is more complex but almost certainly more realistic.
Kerr black hole

19. Originally Posted by Strange
- space-time is flat and space is curved
Space-time includes space and time and so when space-time is curved both space and time are curved.
What about the expansion of the universe? For instance the universe was flat at the emission of the CMB, but space-time should not be flat given that there is a gravitationnal field present. No?

And does it change anything if the BH is spinning?
Yes. You then have a Kerr black hole which is more complex but almost certainly more realistic.
Kerr black hole
Thank you for the link, I'm gonna check it out.

20. Originally Posted by Nic321
What about the expansion of the universe? For instance the universe was flat at the emission of the CMB, but space-time should not be flat given that there is a gravitationnal field present. No?
We are probably getting beyond my level of competence here. My understanding is that there are two different things here. One is the overall geometry (and topology) of the universe. This appears to be (spatially) flat overall, as far as we can measure. What this means is that if you created a humongous triangle then the angles would add up to 180. This limits the possible topologies of the universe.

Locally, the presence of mass causes space-time curvature (within that overall flat space) which we perceive as the force of gravity, etc.

21. I am not sure but I believe that at the time of the CMB emission, space-time was curved by the mass of the entire universe. Indeed if you add up the 3 angles you get 180 degrees. But if you do that at very small scale you wouldn't get 180 degrees because of the inhomogeneities of matter.
I don't know what it really means mathematically to say that "space is curved" or "space-time is curved". Maybe it has to do with the values in the curvature tensor. "space-time is curved" would mean that the numerical values in the time components of the tensor are non zero. "Space is flat" maybe means that the spatial components are null. I don't know how it works.
(The Kerr BH is really cool with the 2 horizons and the naked singularity, Kerr must have been one hell of a mathematician...)

22. Originally Posted by Nic321
I am not sure but I believe that at the time of the CMB emission, space-time was curved by the mass of the entire universe.
I'm not sure either, but I assumed that if the universe is flat overall now, then it must have always been so. But maybe someone who knows what they are talking about will put us right...

23. Originally Posted by Strange
Originally Posted by Nic321
I am not sure but I believe that at the time of the CMB emission, space-time was curved by the mass of the entire universe.
I'm not sure either, but I assumed that if the universe is flat overall now, then it must have always been so. But maybe someone who knows what they are talking about will put us right...
I know that scientists have measured that the universe is flat with the CMB probes, but I don't know why they would conclude that the universe is still flat 13.8 billion years after.
If the universe started to rotate or do something weird ( maybe like the acceleration of the universe by dark energy ), maybe it would cause it to change from flat to curved.
All this is not clear to me. mmh...

24. Originally Posted by Nic321
I don't know what it really means mathematically to say that "space is curved" or "space-time is curved". Maybe it has to do with the values in the curvature tensor. "space-time is curved" would mean that the numerical values in the time components of the tensor are non zero. "Space is flat" maybe means that the spatial components are null. I don't know how it works.
You can't really separate space from "space-time" in the context of GR. Curvature along the time direction is what is responsible for what we experience as gravity, whereas curvature in the three spatial directions give you tidal forces. These two go hand-in-hand, though at vastly different orders of magnitude; it isn't possible to have "curved time" without "curved space", and vice versa, in physically realistic solutions to the field equations.

A space-time is flat if the Riemann curvature tensor vanishes at all points in it.

25. Originally Posted by Nic321
Originally Posted by KJW
There is a property of the interior of a Schwarzschild blackhole worth noting: Because it is symmetric about the two equivalent angular directions and the spatial direction, the three dimensional space inside a Schwarzschild blackhole is homogenous. That is, it is the same everywhere.
Just to clarify this, which one of the following statement is true inside a Schwarzschild blackhole:
- space-time is curved and space is flat ( option I would guess )
- space-time is curved and space is curved
- space-time is flat and space is curved ( false I guess because of the gravitationnal field)
Gee I am always confused about which one is curved...
And does it change anything if the BH is spinning?
The spacetime is curved. I don't know about the space defined by constant coordinate. My guess is that it would be curved. The point to be taken from this is that the flatness or curvature of one doesn't imply the flatness or curvature of the other. Also, one needs to define the space that is embedded in the higher-dimensional space. For example, the space defined by the set of points equidistant from a single point in flat spacetime is hyperbolically curved (analogous to a sphere in Euclidean three-dimensional space). Conversely, a flat space whose points are equidistant from a single point in spacetime does imply that the spacetime is curved.

26. Hi Markus,
Originally Posted by Markus Hanke
Originally Posted by Nic321
I don't know what it really means mathematically to say that "space is curved" or "space-time is curved". Maybe it has to do with the values in the curvature tensor. "space-time is curved" would mean that the numerical values in the time components of the tensor are non zero. "Space is flat" maybe means that the spatial components are null. I don't know how it works.
You can't really separate space from "space-time" in the context of GR. Curvature along the time direction is what is responsible for what we experience as gravity, whereas curvature in the three spatial directions give you tidal forces. These two go hand-in-hand, though at vastly different orders of magnitude; it isn't possible to have "curved time" without "curved space", and vice versa, in physically realistic solutions to the field equations.
A space-time is flat if the Riemann curvature tensor vanishes at all points in it.
I would agree with you but what about the expansion of the universe where space is flat and space-time curved ( if I am not mistaken )?
I am under the impression that positive energy causes a positive space curvature and negative energy a negative space curvature.
I have heard several time scientists ( like Lawrence Krauss and Alan Guth ) say that the total energy of the universe is zero because the gravitationnal energy of the universe is negative and exactly compensate for the positive energy of matter. They also say that space is flat because of that.
But I don't know how it is supposed to work in the Einstein equation. Are the 2 components, positive and negative, put in the energy momentum tensor? Then how can the equation give a not flat space-time?
For normal situations with no negative energy like a mass ( including a BH ), I am under the impression that the positive energy in the E-M tensor creates a positive curvature of space in the curvature tensor.
Nic.

27. Originally Posted by KJW
The spacetime is curved. I don't know about the space defined by constant coordinate. My guess is that it would be curved. The point to be taken from this is that the flatness or curvature of one doesn't imply the flatness or curvature of the other. Also, one needs to define the space that is embedded in the higher-dimensional space. For example, the space defined by the set of points equidistant from a single point in flat spacetime is hyperbolically curved (analogous to a sphere in Euclidean three-dimensional space).
It hard to visualize.
Conversely, a flat space whose points are equidistant from a single point in spacetime does imply that the spacetime is curved.
I am not sure, but does that correspond to the situation of the expansion of the universe: space is flat and all the points of the flat space are equidistant in spacetime from the big bang?

28. Originally Posted by Nic321
Originally Posted by KJW
The spacetime is curved. I don't know about the space defined by constant coordinate. My guess is that it would be curved. The point to be taken from this is that the flatness or curvature of one doesn't imply the flatness or curvature of the other. Also, one needs to define the space that is embedded in the higher-dimensional space. For example, the space defined by the set of points equidistant from a single point in flat spacetime is hyperbolically curved (analogous to a sphere in Euclidean three-dimensional space).
It hard to visualize.
The difference is due to the indefinite nature of the Minkowskian metric compared to the definite nature of the Euclidean metric (the signs of the coefficients in the Pythagorean formula).

Note that describes a circle, whereas describes a hyperbola.

Originally Posted by Nic321
Originally Posted by KJW
Conversely, a flat space whose points are equidistant from a single point in spacetime does imply that the spacetime is curved.
I am not sure, but does that correspond to the situation of the expansion of the universe: space is flat and all the points of the flat space are equidistant in spacetime from the big bang?
Yes it does. One can treat the big bang as a single point in the above. From that, it is simply a coordinate transformation to the Friedmann–Lemaître–Robertson–Walker (FLRW) metric of cosmology for which the big bang is a null three-dimensional space.

29. Originally Posted by Nic321
I would agree with you
Actually, I was wrong in what I said. Consider for example a spherical cavity inside a thin shell of matter - the curvature tensor vanishes everywhere inside the cavity, so space-time there is flat. However, if you place a clock there and compare it to a clock very far away ( at infinity ), you will find that the cavity-clock is gravitationally time dilated. It therefore is possible to get gravitational effects even though there are no tidal forces present.

30. The difference is due to the indefinite nature of the Minkowskian metric compared to the definite nature of the Euclidean metric (the signs of the coefficients in the Pythagorean formula).
Note that describes a circle, whereas describes a hyperbola.
Ok, it is an hyperbola in spacetime.
Yes it does. One can treat the big bang as a single point in the above. From that, it is simply a coordinate transformation to the Friedmann–Lemaître–Robertson–Walker (FLRW) metric of cosmology for which the big bang is a null three-dimensional space.
I see.
I have a question however. Gravity is supposed to travel at the speed of light, so do the solutions to Einstein's equation take that into account, the FLRW metric for instance.
What I mean is that each mass in the universe is attracted gravitationnaly by all the other masses, but it seems to me you'd have to use the past light cone going back to each source to determine its gravitationnal effect. A source that is close would have emitted its gravitation not long ago to get to "here and now", whereas a source located very far in the universe would have emitted its gravitationnal signal a very long time ago. The calculus of the overall gravitationnal effect of all the masses and the evolution of the scale factor seems very complicated indeed.

31. Actually, I was wrong in what I said. Consider for example a spherical cavity inside a thin shell of matter - the curvature tensor vanishes everywhere inside the cavity, so space-time there is flat. However, if you place a clock there and compare it to a clock very far away ( at infinity ), you will find that the cavity-clock is gravitationally time dilated. It therefore is possible to get gravitational effects even though there are no tidal forces present.
Wow, that's weird. So the cavity has the same time dilation as the shell?
Apart from this, has Newton's theorem ( or an equivalent ) been proven mathematically for general relativity? That sounds more complicated to demonstrate than in newtonian gravity, because of the non-linear nature on General Relativity. Does the finite transmission speed of gravity ( c ) play a role here?

32. Originally Posted by Nic321
Wow, that's weird. So the cavity has the same time dilation as the shell?
Yes, as compared to a reference clock far away.

Apart from this, has Newton's theorem ( or an equivalent ) been proven mathematically for general relativity?
Yes, it's called Birkhoff's theorem.

33. Originally Posted by Nic321
Ok, it is an hyperbola in spacetime.
It's a hyperbola in two-dimensional spacetime. In four-dimensional spacetime, it's a three-dimensional hyperboloid.

Originally Posted by Nic321
Gravity is supposed to travel at the speed of light, so do the solutions to Einstein's equation take that into account
The Einstein equation will correctly describe the propagation of gravitational effects whether they travel at the speed of light or not.

Originally Posted by Nic321
... the FLRW metric for instance. What I mean is that each mass in the universe is attracted gravitationnaly by all the other masses, but it seems to me you'd have to use the past light cone going back to each source to determine its gravitationnal effect. A source that is close would have emitted its gravitation not long ago to get to "here and now", whereas a source located very far in the universe would have emitted its gravitationnal signal a very long time ago. The calculus of the overall gravitationnal effect of all the masses and the evolution of the scale factor seems very complicated indeed.
The FLRW metric is homogenous and isotropic, and doesn't contain propagating gravitation.

34. The FLRW metric is a three-dimensional space of constant negative, positive, or zero curvature that is expanding or contracting over time in an arbitrary manner. From this metric, the curvature tensors are readily obtained (this is much easier than solving an Einstein equation). The Einstein tensor, which is equivalent to the energy-momentum tensor (with or without the cosmological constant), can then be checked against the equation of state to determine if it corresponds to ordinary matter, radiation, or something else (such as dark energy).

35. Originally Posted by Markus Hanke
Originally Posted by Nic321
Wow, that's weird. So the cavity has the same time dilation as the shell?
Yes, as compared to a reference clock far away.
Apart from this, has Newton's theorem ( or an equivalent ) been proven mathematically for general relativity?
Yes, it's called Birkhoff's theorem.
Ok, thanks for the explanations Markus.
Nic.

36. Originally Posted by KJW
Originally Posted by Nic321
Ok, it is an hyperbola in spacetime.
It's a hyperbola in two-dimensional spacetime. In four-dimensional spacetime, it's a three-dimensional hyperboloid.

Originally Posted by Nic321
Gravity is supposed to travel at the speed of light, so do the solutions to Einstein's equation take that into account
The Einstein equation will correctly describe the propagation of gravitational effects whether they travel at the speed of light or not.

Originally Posted by Nic321
... the FLRW metric for instance. What I mean is that each mass in the universe is attracted gravitationnaly by all the other masses, but it seems to me you'd have to use the past light cone going back to each source to determine its gravitationnal effect. A source that is close would have emitted its gravitation not long ago to get to "here and now", whereas a source located very far in the universe would have emitted its gravitationnal signal a very long time ago. The calculus of the overall gravitationnal effect of all the masses and the evolution of the scale factor seems very complicated indeed.
The FLRW metric is homogenous and isotropic, and doesn't contain propagating gravitation.
Ok thanks, but do some metrics take into account propagating gravitation? Or does the fact that gravitation propagates at the speed of light never change anything?
The FLRW metric is a three-dimensional space of constant negative, positive, or zero curvature that is expanding or contracting over time in an arbitrary manner. From this metric, the curvature tensors are readily obtained (this is much easier than solving an Einstein equation). The Einstein tensor, which is equivalent to the energy-momentum tensor (with or without the cosmological constant), can then be checked against the equation of state to determine if it corresponds to ordinary matter, radiation, or something else (such as dark energy).
But so basically you do it backwards. You define the FLRW metric, then you show that it's a solution to Einstein's equation, instead of solving the equation directly.
I guess scientists try to use all the tricks they can to avoid having to solve the equations, since they are so complicated.

37. Originally Posted by Nic321
Ok thanks, but do some metrics take into account propagating gravitation?
Yes, any metric describing gravitational waves does that, for example the Bondi-Pirani plane wave metric.

But so basically you do it backwards. You define the FLRW metric, then you show that it's a solution to Einstein's equation, instead of solving the equation directly.
You need an initial ansatz to be able to solve the equations. The Einstein field equations are a system of non-linear coupled partial differential equations, and there is no general solution or general procedure to find a solution. Hence you have to start with some form of a metric that reflects the symmetries of your problem, and then solve the field equations for the metric coefficients. See here for a simple (??) example :

Solving the Einstein Field Equations

38. Yes, any metric describing gravitational waves does that, for example the Bondi-Pirani plane wave metric.
Ok, the metric predicts the existence of waves propagating at c, but from what I understand, that's not the propagation of gravitation itself. The propagation of
gravitation would be the propagation of gravitons (if they exist) at c.
So does the metric itself change as gravitation propagates from a source at the speed of light?
I am not sure if my question is really clear.
You need an initial ansatz to be able to solve the equations. The Einstein field equations are a system of non-linear coupled partial differential equations, and there is no general solution or general procedure to find a solution. Hence you have to start with some form of a metric that reflects the symmetries of your problem, and then solve the field equations for the metric coefficients. See here for a simple (??) example :
Solving the Einstein Field Equations
That's a great link, I understand a little bit how it works. I am going to check it out again in more detail.
Are there cases where scientists would like to solve the equations but they don't know how?

39. Originally Posted by Nic321
but from what I understand, that's not the propagation of gravitation itself
Gravitational waves are periodic oscillations in the curvature of space-time; what this means is that the components of the metric become dependent on time. So yes, it is the metric that changes with time.

Are there cases where scientists would like to solve the equations but they don't know how?
You can only obtain analytic solutions for very simplified cases. More complicated scenarios will either require numerical solutions, or perhaps approximated treatments using linearised gravity ( which has a general solution in the form of retarded potentials ).

40. You guys have gotten a long way from the original question. I think an essential point to keep in mind for The Rest of Us is that the geometry of spacetime inside a black hole's event horizon is so distorted that the only difference someone already inside a smaller black hole might notice would be a change in how far in their future the singularity was. We don't have a good enough theory of quantum gravity to answer how the singularities of the two holes might combine, though Kip Thorne has speculated on it in one of his popular science books. A lot of people think there would be a "ringing" coalescence inside the larger hole as the two singularities merged. They might orbit one another tighter and tighter first and spiral together, or the little one might just head directly for the big one, or they might make some sort of region between them of extremely unusual character, but totally inaccessible since it's inside the event horizon of a black hole. What we do know is that once you cross the event horizon the singularity is now in your future, from the POV of someone outside the hole. We don't know what that means to an observer inside the horizon; we need a quantum gravity theory for that. Also compliments to the IP; this was a great question to ask. I'm looking for if I can give you a reputation point.

41. Gravitational waves are periodic oscillations in the curvature of space-time; what this means is that the components of the metric become dependent on time. So yes, it is the metric that changes with time.
Ok. But let's take the case for instance of 2 masses passing each other at close to the speed of light. Each mass is attracted in the direction from where the other mass was in the past, right? So does the curvature tensor, which depends on time, take that delay into account?
You can only obtain analytic solutions for very simplified cases. More complicated scenarios will either require numerical solutions, or perhaps approximated treatments using linearised gravity ( which has a general solution in the form of retarded potentials ).
What about the problem of the rotation speed of galaxies ( the problem of dark matter ). You talked previously about Birkhoff's theorem, so does that theorem enable us to know for sure that there are no weird phenomona of general relativity that could account for the speed of the stars? Is a galaxy with hundreds of billions of gravitationnal sources ( stars ) a case where we are sure of what happens with regard to general relativity? It seems to me that solving Einstein's equations with hundreds of billions of sources would be nearly impossible, except with approximations, including Birkhoff's theorem, and we would only obtain a mere approximation.

42. One of the really important things about physics is that it's almost all subject to the Superposition Theorem, which means you can add a gravity/electric field from two separate objects to the field of free space, and get a resultant single field. If you move very much relative to the two objects, then you can perform a non-local experiment that identifies the difference between the real field and the single fictitious field that is their sum; but no local experiment, that does not move far enough to explore more of the field, can determine this. We EEs facilely equivocate the two, but real physicists know there are actually two fields and we don't have the deep understanding of math that lets us delve under GR to find this more basic theory of gravity that would allow us to define the whole field instead of the field at this point.

43. Hi Schneibster,
You guys have gotten a long way from the original question. I think an essential point to keep in mind for The Rest of Us is that the geometry of spacetime inside a black hole's event horizon is so distorted that the only difference someone already inside a smaller black hole might notice would be a change in how far in their future the singularity was. We don't have a good enough theory of quantum gravity to answer how the singularities of the two holes might combine, though Kip Thorne has speculated on it in one of his popular science books. A lot of people think there would be a "ringing" coalescence inside the larger hole as the two singularities merged. They might orbit one another tighter and tighter first and spiral together, or the little one might just head directly for the little one, or they might make some sort of region between them of extremely unusual character, but totally inaccessible since it's inside the event horizon of a black hole. What we do know is that once you cross the event horizon the singularity is now in your future, from the POV of someone outside the hole. We don't know what that means to an observer inside the horizon; we need a quantum gravity theory for that. Also compliments to the IP; this was a great question to ask. I'm looking for if I can give you a reputation point.
Thank you for the compliment.
Does a mass that falls into the black hole even reach the center or does it keep on falling for ever, from his POV?
Or could the mass be falling, falling, and be caught up by the shrinking horizon from behind as the black hole evaporates before it reaches the center?
I read once about hypothetical black membranes inside the black hole in string theory, or something to the effect that there was one less dimension inside, I don't remember exactly. It was rather nebulous for me.
Nic.

44. Originally Posted by Schneibster
One of the really important things about physics is that it's almost all subject to the Superposition Theorem, which means you can add a gravity/electric field from two separate objects to the field of free space, and get a resultant single field. If you move very much relative to the two objects, then you can perform a non-local experiment that identifies the difference between the real field and the single fictitious field that is their sum; but no local experiment, that does not move far enough to explore more of the field, can determine this. We EEs facilely equivocate the two, but real physicists know there are actually two fields and we don't have the deep understanding of math that lets us delve under GR to find this more basic theory of gravity that would allow us to define the whole field instead of the field at this point.
How do 2 curvatures of space (and space-time) by 2 sources add up is rather unclear to me. I believe that if there are 2 sources of gravitation, it is impossible to just add the 2 metrics and the 2 curvature tensors, there are non linear effects. I once heard that if you increase the kinetic energy of an object it had the same effect as increasing its mass, so it should curve spacetime more I guess.
Mmh I am not sure.

45. Originally Posted by Nic321
Hi Schneibster,
You guys have gotten a long way from the original question. I think an essential point to keep in mind for The Rest of Us is that the geometry of spacetime inside a black hole's event horizon is so distorted that the only difference someone already inside a smaller black hole might notice would be a change in how far in their future the singularity was. We don't have a good enough theory of quantum gravity to answer how the singularities of the two holes might combine, though Kip Thorne has speculated on it in one of his popular science books. A lot of people think there would be a "ringing" coalescence inside the larger hole as the two singularities merged. They might orbit one another tighter and tighter first and spiral together, or the little one might just head directly for the little one, or they might make some sort of region between them of extremely unusual character, but totally inaccessible since it's inside the event horizon of a black hole. What we do know is that once you cross the event horizon the singularity is now in your future, from the POV of someone outside the hole. We don't know what that means to an observer inside the horizon; we need a quantum gravity theory for that. Also compliments to the IP; this was a great question to ask. I'm looking for if I can give you a reputation point.
Thank you for the compliment.
Does a mass that falls into the black hole even reach the center or does it keep on falling for ever, from his POV?
Or could the mass be falling, falling, and be caught up by the shrinking horizon from behind as the black hole evaporates before it reaches the center?
I read once about hypothetical black membranes inside the black hole in string theory, or something to the effect that there was one less dimension inside, I don't remember exactly. It was rather nebulous for me.
Nic.
Actually you should check out the Shell Theorem. And remember that reaching the event horizon is something that only happens theoretically.

There is a possibility that black holes are shells of matter frozen in relativistic time on its way to the event horizon. This would prevent the necessity for the infinities implicit in making the future the direction you see as forward in spacetime.

46. Originally Posted by Nic321
Originally Posted by Schneibster
One of the really important things about physics is that it's almost all subject to the Superposition Theorem, which means you can add a gravity/electric field from two separate objects to the field of free space, and get a resultant single field. If you move very much relative to the two objects, then you can perform a non-local experiment that identifies the difference between the real field and the single fictitious field that is their sum; but no local experiment, that does not move far enough to explore more of the field, can determine this. We EEs facilely equivocate the two, but real physicists know there are actually two fields and we don't have the deep understanding of math that lets us delve under GR to find this more basic theory of gravity that would allow us to define the whole field instead of the field at this point.
How do 2 curvatures of space (and space-time) by 2 sources add up is rather unclear to me. I believe that if there are 2 sources of gravitation, it is impossible to just add the 2 metrics and the 2 curvature tensors, there are non linear effects. I once heard that if you increase the kinetic energy of an object it had the same effect as increasing its mass, so it should curve spacetime more I guess.
Mmh I am not sure.
Here's a gedankenexperiment:

Let's ignore Roche's limit, and what would be required to hold it up, and propose that we can place a mass overhead of a spot on the Earth's surface and nullify gravity at that spot.

The problem with this as a "anti-gravity" solution is that it only nullifies gravity at that spot; that's because gravity is a field that spreads spherically. The two spheres will contact at one spot, but at all others there will be a complex interaction.

Real gravity fields are spherical which considerably complicates their interactions. It's merely a matter of geometry, but the closest we've come to the real geometry involved is plane gravity fields, not spherical ones. Even GR fails at this; that's why we can't make quantum gravity from it without getting unrenormalizable infinities.

47. Originally Posted by Schneibster
I think an essential point to keep in mind for The Rest of Us is that the geometry of spacetime inside a black hole's event horizon is so distorted that the only difference someone already inside a smaller black hole might notice would be a change in how far in their future the singularity was.
In the case of a small black hole he would also experience very strong tidal forces, even long before reaching the event horizon.

One of the really important things about physics is that it's almost all subject to the Superposition Theorem, which means you can add a gravity/electric field from two separate objects to the field of free space, and get a resultant single field.
I should note though that in the case of scenarios where GR becomes dominant, the superposition will not necessarily be linear, due to self-coupling effects of space-time curvature.

Does a mass that falls into the black hole even reach the center or does it keep on falling for ever, from his POV?
The mass reaches the center in a well defined, finite amount of proper time as measured by a clock falling in with it.

And remember that reaching the event horizon is something that only happens theoretically.
It's rather the other way around. The impossibility of anything reaching the horizon as seen from a far-away observer's point of view is merely a coordinate effect. Locally, an infalling clock will just continue to tick as normal, and will record a finite time until the horizon is reached.

Even GR fails at this
I am not certain why you would think that. Even the most basic of solutions to the GR field equations ( the Schwarzschild metric ) is a spherically symmetric metric; it can be thought of as a family of nested spheres. Note however that there is no requirement for spherical symmetry; give the central body angular momentum or charge, and the symmetry disappears, as in the Kerr-Newman metric.

48. Actually you should check out the Shell Theorem. And remember that reaching the event horizon is something that only happens theoretically.
There is a possibility that black holes are shells of matter frozen in relativistic time on its way to the event horizon. This would prevent the necessity for the infinities implicit in making the future the direction you see as forward in spacetime.
I once heard Leonard Susskind say that before the 1950's scientists, including Einstein, thought black holes could not form. I don't know exactly what made them change their mind, but there seems to be a bit of controversy right now about their formation since the problem of firewall has been put forward.
In order to solve the firewall problem, Hawking has pubished a paper a couple months ago that made sensastion where he said that black holes don't exist. He said that event horizons don't exist and that light is only "trapped" temporarily inside what he calls an "apparent horizon". This is his paper:
http://arxiv.org/pdf/1401.5761v1.pdf
I read it but unfortunately I don't understand much of what it says.
Here is an article about this whole debate about Hawking's paper:
Stephen Hawking: 'There are no black holes' : Nature News & Comment
It's a very controversial for sure and from what I have read most scientists still believe that black holes form.
Apart from this, I read there is also the theory of gravastar, which is an alternative to black holes:
Gravastar - Wikipedia, the free encyclopedia
It is quite hard to understand how black holes could form, because indeed it would take an infinite amount of time to reach the horizon. From the infalling observer's POV it takes a finite time, but it is only for him because time passes so slowly for him. The universe could eventually end before he reaches the horizon.
In any case I think scientists have reasonably good reasons to believe that they form, and the debate about the firewall is very interesting, although hard to understand.

49. The mass reaches the center in a well defined, finite amount of proper time as measured by a clock falling in with it.
Ok, but what about from the POV of an observer at infinity, admitting that the mass passes through the horizon? From looking at the Schwarzschild metric, I believe it also takes an infinte amount of time ( for the observer at infinity ).

50. Originally Posted by Nic321
It is quite hard to understand how black holes could form, because indeed it would take an infinite amount of time to reach the horizon.
You must remember that "time" is a local phenomenon. The only meaningful measurement of in-fall time is the one taken by a clock that falls with that object. And that is finite and well defined. You can send an object onto a highly elliptical free-fall trajectory which will take it very close to the event horizon, and then back out to a far-away reference point. At all points of that trajectory the clock will tick as normal, and the total travel time is quite finite, and well defined - no one will see the end of the universe.

What an observer at infinity sees on his clock is meaningless for what an in falling object actually does. The fact that he never sees anything hit the event horizon is solely due to the fact that the light emitted by the in falling object travels back to the observer along null-geodesics, the geometry of which is not the same as in flat space-time. The curved space-time itself is everywhere smooth and regular outside the singularity, so there is really nothing special about the event horizon, in terms of local structure. This is easily seen by choosing a different set of coordinates.

From the infalling observer's POV it takes a finite time, but it is only for him because time passes so slowly for him.
No - everyone agrees on the proper time of an in-falling observer, because that is what a clock falling with him physically measures - it is quite simply the length of his world line through space-time. What an observer at infinity measures, on the other hand, is valid only in his own, local frame of reference - concluding that a BH cannot form because an observer at infinity never sees anything reach the event horizon is a logical fallacy. Besides, the coordinate time of such an observer is dependent on the coordinate system used - replace Schwarzschild coordinates with, say, Kruskal-Szekeres coordinates, and you suddenly get a finite coordinate time also. This quantity - coordinate time of far-away observer - has no physical meaning for the in-falling object, or an occurring gravitational collapse.

The universe could eventually end before he reaches the horizon.
No, that is clearly not the case - the easiest way to see that is to plot everything onto a Kruskal-Szekeres chart. See here : general relativity - Does someone falling into a black hole see the end of the universe? - Physics Stack Exchange

51. Originally Posted by Strange
Originally Posted by Nic321
What about the expansion of the universe? For instance the universe was flat at the emission of the CMB, but space-time should not be flat given that there is a gravitationnal field present. No?
We are probably getting beyond my level of competence here. My understanding is that there are two different things here. One is the overall geometry (and topology) of the universe. This appears to be (spatially) flat overall, as far as we can measure. What this means is that if you created a humongous triangle then the angles would add up to 180. This limits the possible topologies of the universe.

Locally, the presence of mass causes space-time curvature (within that overall flat space) which we perceive as the force of gravity, etc.
Nice. But , I doubt your first sentence. I know nothing about math except basic arithmetic. But that is satisfyingly expressed.

52. Originally Posted by Nic321
Ok, but what about from the POV of an observer at infinity, admitting that the mass passes through the horizon? From looking at the Schwarzschild metric, I believe it also takes an infinte amount of time ( for the observer at infinity ).
He will never see anything reach or cross the horizon, because of the geometry of the null geodesics along which the light emitted by such an object travels, so the question is moot. What I am trying to point out is that what a far-away observer sees is completely irrelevant to what the in-falling object does in its own local frame, because his judgment of "infinite coordinate time" is merely an artefact of the way he chooses to label events in space-time ( i.e. his coordinate system ). On the other hand, all observers agree on the length of the in-falling observer's world line, which is an invariant, and which is what a clock falling along with it physically measures. And that is finite and well defined.

Even the far-away observer can eliminate that infinity by choosing different coordinates.

Consider again the example I gave previously - you can send an object onto an elliptical orbit around a black hole which brings it very close to the event horizon, and then back out. Will such an object see the end of the universe ? Will the far away observer have to wait for billions and billions of years for it to come "back out" ? No - in fact, what you will find is that even for a very close approach to the event horizon, you will get differences in proper time readings of only a few 10s percentage points. See here for an explicit calculation of such an orbit : http://arxiv.org/pdf/1201.5611.pdf

53. Hi Markus,
I see what you mean. What you have explained makes things clearer for me now, although not quite completely yet.
Thank you for your link, I am going to check it out.
I had watched Leonard Susskind's lecture on the Schwarzchild metric some time ago, I am going to check it again.
Nic.

54. Originally Posted by Markus Hanke
Originally Posted by Schneibster
I think an essential point to keep in mind for The Rest of Us is that the geometry of spacetime inside a black hole's event horizon is so distorted that the only difference someone already inside a smaller black hole might notice would be a change in how far in their future the singularity was.
In the case of a small black hole he would also experience very strong tidal forces, even long before reaching the event horizon.
The observer is inside the smaller hole, IIRC. We don't know if there are tidal forces inside a black hole, not even from an intersection with another hole. We have no gravity theory that reaches past event horizons, and a lot of pretty good reasons to believe it's not as simple as "it's just like out here." OTOH your point is fair when we consider how said observer got in there in the first place.

Originally Posted by Markus Hanke
One of the really important things about physics is that it's almost all subject to the Superposition Theorem, which means you can add a gravity/electric field from two separate objects to the field of free space, and get a resultant single field.
I should note though that in the case of scenarios where GR becomes dominant, the superposition will not necessarily be linear, due to self-coupling effects of space-time curvature.
That's correct and becomes of ever-increasing importance near the event horizon. And in fact the reason GR is intractable to a quantum gravity theory is partly because of this non-linearity. Such situations are notoriously difficult and intractable mathematically. Feynman, Tomonaga, and Schwinger were able to renormalize them away in the case of QED. This doesn't work in attempted quantum gravity theories, at least not any way we've tried it.

Originally Posted by Markus Hanke
Does a mass that falls into the black hole even reach the center or does it keep on falling for ever, from his POV?
The mass reaches the center in a well defined, finite amount of proper time as measured by a clock falling in with it.
We don't know that.

Originally Posted by Markus Hanke
And remember that reaching the event horizon is something that only happens theoretically.
It's rather the other way around. The impossibility of anything reaching the horizon as seen from a far-away observer's point of view is merely a coordinate effect. Locally, an infalling clock will just continue to tick as normal, and will record a finite time until the horizon is reached.
Assuming it's not spaghettified by the tides, yes. However, in your previous statement, you claimed it would do that inside the event horizon. That's a completely different proposition. We have no idea about that because we have no useful theory of quantum gravity.

Originally Posted by Markus Hanke
Even GR fails at this
I am not certain why you would think that. Even the most basic of solutions to the GR field equations ( the Schwarzschild metric ) is a spherically symmetric metric; it can be thought of as a family of nested spheres. Note however that there is no requirement for spherical symmetry; give the central body angular momentum or charge, and the symmetry disappears, as in the Kerr-Newman metric.
I think it because it's inside the event horizon and we don't know what happens there.

Reviewing my statement I can see that was not clear and I should have stated it explicitly.

55. Originally Posted by Nic321
Hi Markus,
I see what you mean. What you have explained makes things clearer for me now, although not quite completely yet.
Thank you for your link, I am going to check it out.
I had watched Leonard Susskind's lecture on the Schwarzchild metric some time ago, I am going to check it again.
Nic.
Perhaps I could also recommend Edwin Taylor's "Exploring Black Holes" - a good book which can be followed given just standard algebra and calculus knowledge. You might consider this.

56. Originally Posted by Schneibster
We don't know if there are tidal forces inside a black hole
We don't know that.
We have no gravity theory that reaches past event horizons
General Relativity describes the geometry of space-time both inside and outside the event horizon, until a point is reached where quantum effects can no longer be neglected. You are correct that we do not currently have a consistent theory of gravity which incorporates quantum effects; however, a large part of the volume enclosed by the event horizon can be well approximated by classical GR, especially for the case of larger black holes. Within the domain of applicability of GR, it is fairly trivial to calculate in-fall times and tidal forces.

And in fact the reason GR is intractable to a quantum gravity theory is partly because of this non-linearity.
Yes, I agree.

This doesn't work in attempted quantum gravity theories, at least not any way we've tried it.
I think a strong case can be made for the assertion that standard renormalisation techniques of QFT will never work in the case of gravity; in fact, I would go so far as to say that there is no reason to assume that gravity can ever be formulated as a QFT in the first place.

Assuming it's not spaghettified by the tides, yes. However, in your previous statement, you claimed it would do that inside the event horizon.
Tidal forces are inversely proportional to distance; hence, if the black hole is large enough, such forces will be negligible even at the event horizon. On the other hand, for small black holes you might get spaghettification long before the infalling object ever gets anywhere close to the event horizon. In the simple case of Schwarzschild geometry the relevant components of the Riemann tensor become ( I'm lazy so I'm using geometrised units ) :

The upshot is that it all depends on the total mass of the black hole.

I think it because it's inside the event horizon and we don't know what happens there.
I think we might be arguing from different points of view. My original statements were based on purely classical GR ( space-time smooth and regular everywhere outside a point singularity ), whereas you are considering quantum gravitational effects as well.

I agree with you of course that a lot of what classical GR predicts will no longer be valid once quantum effects are fully accounted for, hence there is no way to tell for certain just what would happen to an in-falling object; we quite simply don't know what will happen during the final stages of a gravitational collapse, and what the final state will then be. However, I should think that even with quantum effects accounted for, a substantial portion of the 4-volume enclosed by the horizon can still be treated classically ( at least for macroscopic black holes ), in the same way that most cases of weak gravity can be treated via Newtonian gravity.

Ultimately, the jury is out as to what actually happens to the infalling object in the real universe; however, if one sets aside quantum effects and looks at the scenario within classical GR, then the prediction is clear and straightforward. At the moment, in the absence of a consistent QG model, that is all we can really do.

57. Hi Markus,
Perhaps I could also recommend Edwin Taylor's "Exploring Black Holes" - a good book which can be followed given just standard algebra and calculus knowledge. You might consider this.
Thanks for the reference. That book sounds interesting, but I hope it's not too complicated. My level in physics and General Relativity is quite primitive.
I am going to check the issue of the coordinate singularity at Rs tonight, I didn't have much time yesterday.
But may I ask just a question? If, by using Schwarzschild's metric I calculate that an infalling observer would reach the horizon in say 10 seconds of his proper time, how much does that represent for me at infinity? I can't know that using Schwarzschild's metric?

58. I think you are a relativist, and a professional one. It's great to talk to a real expert.
Originally Posted by Markus Hanke
Originally Posted by Schneibster
We don't know if there are tidal forces inside a black hole
We don't know that.
We have no gravity theory that reaches past event horizons
General Relativity describes the geometry of space-time both inside and outside the event horizon, until a point is reached where quantum effects can no longer be neglected. You are correct that we do not currently have a consistent theory of gravity which incorporates quantum effects; however, a large part of the volume enclosed by the event horizon can be well approximated by classical GR, especially for the case of larger black holes. Within the domain of applicability of GR, it is fairly trivial to calculate in-fall times and tidal forces.
As you note below, we're talking from different viewpoints. We can extrapolate what classical GR would predict beyond the event horizon; and we have. But we don't know. Most probably you are correct and from a proper time perspective the falling observer will not see a difference when crossing the event horizon. But until someone does, and then figures out how to return or send a message to an observer outside, we will not know. OTOH, we have found out what's outside the universe, so I hesitate to call that return or message "impossible," especially when regions exist around a spinning black hole that are neither wholly within nor wholly outside the horizon; this is the ergosphere, which is only present around spinning holes. This region may hold the key to some of the mysteries we're currently working out.

I've been busy and haven't had time to look at Hawking's latest, but hopefully things have quieted down and I'll be able to go over it with the folks on this forum.

Originally Posted by Markus Hanke
And in fact the reason GR is intractable to a quantum gravity theory is partly because of this non-linearity.
Yes, I agree.
I was pretty sure.

Originally Posted by Markus Hanke
This doesn't work in attempted quantum gravity theories, at least not any way we've tried it.
I think a strong case can be made for the assertion that standard renormalisation techniques of QFT will never work in the case of gravity; in fact, I would go so far as to say that there is no reason to assume that gravity can ever be formulated as a QFT in the first place.
OTOH, the field formulation of ED-- Maxwell's Equations-- is as available as the quantum formulation, QED, and in field terms it is only a simple matter of adding one dimension to spacetime and one can derive the field formulation directly as GR was derived. Kaluza showed this to Einstein in the 1940s, but because he didn't have the key insight- a small dimension- Einstein found irreparable flaws in it that would cause violation of mass-energy conservation. Do you believe the fact that all the other forces work in small dimensions, and gravity works in the large ones, that causes the intractability to a QFT of gravity?

Originally Posted by Markus Hanke
Assuming it's not spaghettified by the tides, yes. However, in your previous statement, you claimed it would do that inside the event horizon.
Tidal forces are inversely proportional to distance; hence, if the black hole is large enough, such forces will be negligible even at the event horizon. On the other hand, for small black holes you might get spaghettification long before the infalling object ever gets anywhere close to the event horizon. In the simple case of Schwarzschild geometry the relevant components of the Riemann tensor become ( I'm lazy so I'm using geometrised units ) :

The upshot is that it all depends on the total mass of the black hole.
Yes. But we don't know if that holds true beyond the event horizon.

Originally Posted by Markus Hanke
I think it because it's inside the event horizon and we don't know what happens there.
I think we might be arguing from different points of view. My original statements were based on purely classical GR ( space-time smooth and regular everywhere outside a point singularity ), whereas you are considering quantum gravitational effects as well.

I agree with you of course that a lot of what classical GR predicts will no longer be valid once quantum effects are fully accounted for, hence there is no way to tell for certain just what would happen to an in-falling object; we quite simply don't know what will happen during the final stages of a gravitational collapse, and what the final state will then be. However, I should think that even with quantum effects accounted for, a substantial portion of the 4-volume enclosed by the horizon can still be treated classically ( at least for macroscopic black holes ), in the same way that most cases of weak gravity can be treated via Newtonian gravity.

Ultimately, the jury is out as to what actually happens to the infalling object in the real universe; however, if one sets aside quantum effects and looks at the scenario within classical GR, then the prediction is clear and straightforward. At the moment, in the absence of a consistent QG model, that is all we can really do.
We agree completely. Now that that is understood I won't keep reminding you about the horizon.

59. Originally Posted by Nic321
Hi Markus,
Perhaps I could also recommend Edwin Taylor's "Exploring Black Holes" - a good book which can be followed given just standard algebra and calculus knowledge. You might consider this.
Thanks for the reference. That book sounds interesting, but I hope it's not too complicated. My level in physics and General Relativity is quite primitive.
I am going to check the issue of the coordinate singularity at Rs tonight, I didn't have much time yesterday.
But may I ask just a question? If, by using Schwarzschild's metric I calculate that an infalling observer would reach the horizon in say 10 seconds of his proper time, how much does that represent for me at infinity? I can't know that using Schwarzschild's metric?
A more popular science title in the same line is Kip Thorne's Black Holes and Time Warps: Einstein's Outrageous Legacy, 1995. If calculus is scary, then this is a less challenging choice. OTOH, you may enjoy challenging yourself or not be afraid of scary integrals (I am!).

60. A more popular science title in the same line is Kip Thorne's Black Holes and Time Warps: Einstein's Outrageous Legacy, 1995. If calculus is scary, then this is a less challenging choice. OTOH, you may enjoy challenging yourself or not be afraid of scary integrals (I am!).
Thanks for the reference.
I was at the library this evening and I saw a book about GR, it sounded interesting too. I need something relatively basic.
That's why I like Susskind's lectures on youtube, he is a good teacher and it doesn't get too complex so I can follow.

61. Actually Albert himself wrote a popular science book named Relativity and many folks ignore it. I don't think you can understand what he had in mind until you read it, so I highly recommend it. He is very gentle about the math; he was a genius and had analogies that he used that remain some of the best visualizations I have ever come across of these moderately difficult concepts. It's also cheap, because it's out of copyright, and surprisingly short.

62. They have the book at my Library. Almost every time I go them I check it out. It's funny to read Einstein himself.

63. He is after all The Man.

64. Right.

65. Originally Posted by Nic321
If, by using Schwarzschild's metric I calculate that an infalling observer would reach the horizon in say 10 seconds of his proper time, how much does that represent for me at infinity? I can't know that using Schwarzschild's metric?
The far-away observer will always calculate an infinite amount of Schwarzschild coordinate time from his own frame. The important and crucial thing to realise here is that time is a local notion in GR - so, if the far-away observer determines that it takes an infinite amount of his own time to reach the event horizon, then that is true and meaningful only in his own frame of reference. His concept of time is not the same as the one of an in-falling object - therefore, the fact that far-away coordinate time becomes infinite does not mean that nothing can fall through the event horizon, as determined from the frame of the object itself. There is no "right" and "wrong" - both observers are right, but only in their own frames of reference. There is no globally valid point of view here that can give the one, true answer; all we can fall back on is what clocks physically measure. And a clock that falls into a black hole will measure a finite amount of time, even if an outside observer never sees it reach the horizon.

Do you know what I am trying to say ?

66. Originally Posted by Schneibster
We can extrapolate what classical GR would predict beyond the event horizon; and we have. But we don't know.
Yes, I agree with this.

Do you believe the fact that all the other forces work in small dimensions, and gravity works in the large ones, that causes the intractability to a QFT of gravity?
I don't know the scientific answer to this, nor do I think anyone knows for sure, as things stand. My personal opinion is that it might pay to revisit the status of gravity as a "fundamental force", because I am not sure that it actually is one. My own personal pet theory (!!!!) is that space-time itself ( along with its properties, and hence gravity ) might be better regarded as an emergent property of a more fundamental, underlying system of degrees of freedom that are not in themselves spatio-temporal in nature. A good analogy might be ice, a substance which emerges from water through a phase transition; likewise I regard space-time to emerge from some other system through some form of phase transition. At some point beyond the event horizon of a black hole, for example, space-time would quite simply cease to exist, and transition back into its underlying degrees of freedom.

I must stress again that this merely a personal opinion.

Yes. But we don't know if that holds true beyond the event horizon.
It must hold true in classical GR so long as quantum effects are negligible, since otherwise the theory is false. I am quite sure it won't hold true in quantum gravity except as an approximation.

67. Originally Posted by Markus Hanke
My personal opinion is that it might pay to revisit the status of gravity as a "fundamental force", because I am not sure that it actually is one.
I certainly don't think it is one. I think we've been too greatly influenced by Newton. Gravitation actually can't be a force, even in quantum mechanics (especially in quantum mechanics!). The notion of gravity as a force contradicts itself.

68. The far-away observer will always calculate an infinite amount of Schwarzschild coordinate time from his own frame. The important and crucial thing to realise here is that time is a local notion in GR - so, if the far-away observer determines that it takes an infinite amount of his own time to reach the event horizon, then that is true and meaningful only in his own frame of reference. His concept of time is not the same as the one of an in-falling object - therefore, the fact that far-away coordinate time becomes infinite does not mean that nothing can fall through the event horizon, as determined from the frame of the object itself. There is no "right" and "wrong" - both observers are right, but only in their own frames of reference. There is no globally valid point of view here that can give the one, true answer; all we can fall back on is what clocks physically measure. And a clock that falls into a black hole will measure a finite amount of time, even if an outside observer never sees it reach the horizon.
Do you know what I am trying to say ?
Ok, I get it now.
However, if the falling observer measures 10 seconds of proper time before he crosses the horizon, can I say, if I am at infinity, that the falling observer has crossed the horizon after 10 seconds in my reference frame, knowing that it will take an infinite time for me to observe?
Let's say I observe a star collapse. I know by calculation that it will take say 1 min to form a black hole to form but if I observe it, I will never see the horizon form. And the matter will measure 1 min of proper time before it crosses the horizon.

69. Originally Posted by Nic321
However, if the falling observer measures 10 seconds of proper time before he crosses the horizon, can I say, if I am at infinity, that the falling observer has crossed the horizon after 10 seconds in my reference frame, knowing that it will take an infinite time for me to observe?
No. This suggests the existence of a universal time. I should also remark that the notion of the "observer at infinity" is perhaps misleading. The observer doesn't have to be at infinity. The observer only has to remain outside the event horizon. Then that observer will never observe an object crossing the event horizon, or even the formation of the event horizon.

70. Then how come we can say definitely that black holes do form. For instance if astronomers see that a star is about to collapse, they would say that it would take say 1 min for the collapse to form a black hole. The astronomers look at their clock and after one minute they say 'ok that's it the black hole has formed'. (even though they can't observe its formation before an infinite time ). Where does that 1 min duration come from in the calculations?

71. Originally Posted by Nic321
Where does that 1 min duration come from in the calculations?
From a calculation of radial velocity, which can be obtained from the metric. For coordinate time ( far-away observer ) you get a radial velocity of

If you integrate this up from some point far away down to the event horizon to find out how long it would take, the result is infinite. On the other hand, if you do the same with proper time ( in-falling clock ), you get

Integrating this proper radial velocity will yield a finite and well defined amount of proper time.

72. Thank you for the detail. I see what you are doing with your calculation, and I agree with you on the fact that the in-falling observer will measure a finite time to get to the horizon. Let's say this proper time in 1 min. But why is an astronomer far away in a position to say that on his clock, the in-falling observer will have taken in 1 min to reach the horizon? ( he will never be able to observe it however ). The in-falling observer and the astronomer don't have the same clock and don't have the same gravitational potential, so shouldn't there be a time dilation effect or something like that which would do in sort that for the astronomer it does not represent 1 min?
And thank you for your patience. I am sorry if I am kind of slow.

73. Originally Posted by Nic321
Then how come we can say definitely that black holes do form.
I don't know that we can say definitely that blackholes do form. Not long ago I posted a thread challenging that notion, so you could say that I'm somewhat of a blackhole sceptic.

Originally Posted by Nic321
For instance if astronomers see that a star is about to collapse, they would say that it would take say 1 min for the collapse to form a black hole. The astronomers look at their clock and after one minute they say 'ok that's it the black hole has formed'. (even though they can't observe its formation before an infinite time ). Where does that 1 min duration come from in the calculations?
I don't think astronomers ever actually say that.

The belief in the reality of blackholes comes from topological considerations. The spacetime that extends toward the event horizon simply has to continue somewhere. In my challenge I mentioned above, I considered the possibility that the blackhole is a wormhole, the spacetime continuing to another "sheet" of external geometry. However, my real point was that blackholes are not as straightforward as they seem to be.

74. KJW, would you have the link to the thread you are refering to? Thank you.

75. Markus, I am watching sports today. I expect to be paying close enough attention to respond properly to your post the day after tomorrow. You will find this is common with me on the weekend. Your last response to me requires careful thought; thank you very much! But today I am not thinking carefully. I prefer to take a bit off than reply without proper attention.

76. Originally Posted by Nic321
KJW, would you have the link to the thread you are refering to? Thank you.
A hypothesis regarding blackholes and dark energy

77. Thanks KJW.

78. Thanks for this info: I will find 'Relativity'. As a non-mathematician, I find his analogies quite helpful. This thread is fascinating.

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