A Light from the singularity of a charged black hole

[Elnur Hajiyev]

According to Cosmic Sensorship Conjecture, naked singularities are prohibited in General Relativity. To my knowledge, naked singularity means light from the singularity can escape to infinity.

In Reissner-Nordström metric, references say naked singularity appears only if $GM^2<P^2+Q^2$. However in $GM^2>P^2+Q^2$ case, according to the Penrose diagram below, light from the singularity which is a timelike surface can take a path which leads to the New Universe.

Does it mean, an observer can see the singularity while he/she is in the wormhole, white hole and in the new universe? Isn't it considered as a naked singularity?

 

[PeterDonis]

To my knowledge, naked singularity means light from the singularity can escape to infinity.

That's part of the definition, but not all of it. See below.

light from the singularity which is a timelike surface can take a path which leads to the New Universe.

This is correct. But from the standpoint of the new universe, the light from the singularity is coming out of the white hole. That means nobody in the new universe can reach the singularity, even with a light signal, because they can't go into the white hole. And that means the singularity isn't a naked singularity, because for a naked singularity, observers in the same universe have to be able to both see light from the singularity and send light signals to the singularity.

 

[Elnur Hajiyev]

That's part of the definition, but not all of it. See below.
This is correct. But from the standpoint of the new universe, the light from the singularity is coming out of the white hole. That means nobody in the new universe can reach the singularity, even with a light signal, because they can't go into the white hole. And that means the singularity isn't a naked singularity, because for a naked singularity, observers in the same universe have to be able to both see light from the singularity and send light signals to the singularity.

But an observer in the wormhole part of the spacetime can both send and receive light from the singularity, right? If I get it correctly, it will not be considered as a naked singularity, will it be, because wormhole is a limited zone of the diagram, enclosed with horizons.

 

[Elnur Hajiyev]

That's part of the definition, but not all of it. See below.
for a naked singularity, observers in the same universe have to be able to both see light from the singularity and send light signals to the singularity.

Why is it descripted with the incomplete form in Wikipedia, GR books(S. Carroll) then? Actually it is the first time I have seen the second part of the definition. Can you please mention any reference, in which I can find a description of naked singularity in this form?

 

[PeterDonis]

an observer in the wormhole part of the spacetime can both send and receive light from the singularity, right?

Yes, but that part of the spacetime does not include infinity.

If I get it correctly, it will not be considered as a naked singularity, will it be, because wormhole is a limited zone of the diagram, enclosed with horizons.

Yes.

Why is it descripted with the incomplete form in Wikipedia, GR books(S. Carroll) then?

Wikipedia is not a good source to begin with. In GR textbooks the term "naked singularity" is not defined explicitly at all, as far as I can see; for example, in Carroll's online lecture notes on GR, in Chapter 7, the term is used without ever being defined. So the only definition we have is implicit, by looking at the context, at which cases are described as having a naked singularity and which are not. Since the geometry with $GM^2 > P^2 + Q^2$ is always described as not having a naked singularity, there must obviously be a condition of the sort I have described in the (implicit) definition that is being used.

 

[Elnur Hajiyev]

@PeterDonis , I study from S. Carroll's book and it is from there, that's why I am confused.

 

[PeterDonis]

I study from S. Carroll's book and it is from there

Ok, this might be a somewhat different definition than the one I was using. Note that here he says that the conjecture refers to the formation of naked singularities, not to their existence. In the quote he explicitly says that the Schwarzschild white hole singularity is a naked singularity. He also says that the super-extremal charged black hole (i.e., the case $GM^2 < P^2 + Q^2$) singularity is. He does not explicitly say that the singularity in the sub-extremal charged black hole (where $GM^2 > P^2 + Q^2$) is a naked singularity, but the analogy with the Schwarzschild white hole would suggest that it is. So it might be that the textbooks are not entirely clear or consistent about the definition.

There is also another important point made in this quote: note that it says the cosmic censorship conjecture is about the formation of naked singularities, not their existence. In other words, singularities like the Schwarzschild white hole or the super-extremal charged black hole singularities exist in mathematical solutions, but they are not physically realistic because there is no way for them to form in the first place. Similar remarks would apply to the singularity in the sub-extremal charged black hole.

 

[Elnur Hajiyev]

Ok, this might be a somewhat different definition than the one I was using. Note that here he says that the conjecture refers to the formation of naked singularities, not to their existence. In the quote he explicitly says that the Schwarzschild white hole singularity is a naked singularity. He also says that the super-extremal charged black hole (i.e., the case $GM^2 < P^2 + Q^2$) singularity is. He does not explicitly say that the singularity in the sub-extremal charged black hole (where $GM^2 > P^2 + Q^2$) is a naked singularity, but the analogy with the Schwarzschild white hole would suggest that it is. So it might be that the textbooks are not entirely clear or consistent about the definition.

There is also another important point made in this quote: note that it says the cosmic censorship conjecture is about the formation of naked singularities, not their existence. In other words, singularities like the Schwarzschild white hole or the super-extremal charged black hole singularities exist in mathematical solutions, but they are not physically realistic because there is no way for them to form in the first place. Similar remarks would apply to the singularity in the sub-extremal charged black hole.

@PeterDonis , after reading your comment and doing some research, I think I have finally got it.
I have came to this conclusion that naked singularity is only defined by the fact whether it is seen from infinity or not. So, we say, RN black hole has a naked singularity for only super-extremal case, because in sub-extremal black hole case, the singularity is not seen from black hole, but from the white hole which opens to another spacetime. The singularity is considered as naked for an observer in the other spacetime, and because it is not created within that spacetime, CSC is not violated.

 

[PeterDonis]

the white hole which opens to another spacetime

This is not correct. All of the different "universes" are part of the same spacetime.

because it is not created within that spacetime, CSC is not violated

It is true that the singularities in this spacetime do not violate CSC, but not for the reason you give. The reason is that these singularities are not "created" from "generic intial conditions", so the precondition for the CSC to apply at all is not met.

 

[Elnur Hajiyev]

This is not correct. All of the different "universes" are part of the same spacetime.

I think it is not important in this situation, whether the second patch is of a different or the same one since they are different locations in space and/or time. However as far as I know, the metric does not imply anything about the connection of the patches, that is why I used the word "spacetime" instead of "universe".

It is true that the singularities in this spacetime do not violate CSC, but not for the reason you give. The reason is that these singularities are not "created" from "generic intial conditions", so the precondition for the CSC to apply at all is not met.

But sub-extremal RN black hole can be formed from generic initial conditions and there will be singularity behind the inner horizon. This singularity still cannot be seen from the vicinity of the black hole, that is why it is not considered as a naked singularity from the perspective of spacetime, where the black hole is formed. On the other hand, the singularity will be seen from the different patch, so it will be considered as a naked singularity in the vicinity of the white hole.

 

[PeterDonis]

sub-extremal RN black hole can be formed from generic initial conditions

No, it can't. The interior of this solution (everything inside the outer horizon) is unstable against small perturbations. (In fact this is true of all the idealized black hole solutions, including Schwarzschild). "Generic initial conditions" requires stability against small perturbations.

I think it is not important in this situation, whether the second patch is of a different or the same one

It doesn't matter. They are connected through the "wormhole" regions, so they're part of the same spacetime.

the metric does not imply anything about the connection of the patches

Yes, it does; they are connected through the "wormhole" interior. See above.

 

[Elnur Hajiyev]

No, it can't. The interior of this solution (everything inside the outer horizon) is unstable against small perturbations. (In fact this is true of all the idealized black hole solutions, including Schwarzschild). "Generic initial conditions" requires stability against small perturbations.

@PeterDonis , I would really like to see the references from which you come to these conclusions. But as I mentioned before, I read from Carroll and everything he says seems to me logical, so is it too. Other sources that I read are also consistent with it. You can see it here:

I know in the realistic conditions neither Schwarzschild black hole nor RN black hole can be exist, but we are discussing ideal black holes here.

Yes, it does; they are connected through the "wormhole" interior. See above.

Can you please show which part of the metric implies mathematically that the infinite spacetime patches should belong to the same universe? I agree that, they are likely belong to the same universe, existence of wormhole does not definitely mean that it must be true. If it is, please clarify it with reasonable arguments or references.

 

[PeterDonis]

I would really like to see the references from which you come to these conclusions.

The fact that the interiors of the idealized black holes are unstable against small perturbations is common knowledge at the graduate level in GR; since this thread is an "A" level thread, that background knowledge is assumed. IIRC it is discussed in MTW and Wald.

Can you please show which part of the metric implies mathematically that the infinite spacetime patches should belong to the same universe?

I didn't say they "belong to the same universe" (by which I assume you mean the infinities in each of them are the same infinity). I said they are all part of the same spacetime, because they are connected through the "wormhole" interior regions. That is obvious from the diagram in the OP of this thread, and it's true regardless of the relationship of the infinities in each exterior region to each other.

 

[Elnur Hajiyev]

The fact that the interiors of the idealized black holes are unstable against small perturbations is common knowledge at the graduate level in GR; since this thread is an "A" level thread, that background knowledge is assumed. IIRC it is discussed in MTW and Wald.

No. You said sub-extremal black hole cannot be formed from generic initial conditions. I showed you a reference from S. Carroll's book in which it says it can and requested from you to explain your contradictory statement and giving some noble references if you think the book is wrong.

I didn't say they "belong to the same universe" (by which I assume you mean the infinities in each of them are the same infinity). I said they are all part of the same spacetime, because they are connected through the "wormhole" interior regions. That is obvious from the diagram in the OP of this thread, and it's true regardless of the relationship of the infinities in each exterior region to each other.

By saying different spacetime, I meant different spacetime patches with different infinities. I assumed that you meant the same.

 

[PeterDonis]

You said sub-extremal black hole cannot be formed from generic initial conditions. I showed you a reference from S. Carroll's book in which it says it can

I don't see such a statement in any of the references you have given. If you are referring to the quote you gave in post #12, it does not say that. It says that a "realistic gravitational collapse" should obey the sub-extremal condition $GM^2 > P^2 + Q^2$, but it does not say such a collapse will form a sub-extremal black hole with the singularities inside the "wormhole" region from generic initial conditions.

By saying different spacetime, I meant different spacetime patches with different infinities. I assumed that you meant the same.

You assumed incorrectly. "Spacetime" is standardly used to refer to an entire connected manifold. Everything shown in the diagram in your OP, including the multiple exterior patches with different infinities, is all one spacetime according to the standard usage of that term. Again, this is an "A" level thread so that kind of background knowledge is assumed. The word "patch" is used to refer to a portion of a spacetime; it is not a synonym for "spacetime".

 

[Elnur Hajiyev]

I don't see such a statement in any of the references you have given. If you are referring to the quote you gave in post #12, it does not say that. It says that a "realistic gravitational collapse" should obey the sub-extremal condition $GM^2 > P^2 + Q^2$, but it does not say such a collapse will form a sub-extremal black hole with the singularities inside the "wormhole" region from generic initial conditions.

@PeterDonis , I don't understand, why are you making the discussion more complicated than it is. You see, I wrote that naked sigularity is naked from white hole infinity and is hidden from black hole infinity. Black hole infinity does have horizon, from which "light cannot escape", so I am convinced CSC is not violated. You say it is not for that reason. Can I ask, do you mean, "no, singularity is considered as naked even from black hole infinity"? If no, then you did not get reply correctly, and it means we were arguing on a pointless basis. But if you meant it, then this time you contradict your previous answer trying to define naked singularity.
You are trying to say, explanation of why CSC is not violated is even subextremal black hole is not formed by the way of which CSC says, but according to the book it is. (Again, if this is not what you are trying to say, then the whole disuccion after #8 is a bit of pointless, I had figured out the answer thanks to your previous replies and other sources.)

You assumed incorrectly. "Spacetime" is standardly used to refer to an entire connected manifold. Everything shown in the diagram in your OP, including the multiple exterior patches with different infinities, is all one spacetime according to the standard usage of that term. Again, this is an "A" level thread so that kind of background knowledge is assumed. The word "patch" is used to refer to a portion of a spacetime; it is not a synonym for "spacetime".

Yes, I missed the "patch" word, I think it caused a controversy. I did not marked the thread because I am graduate level student, but because the topic is graduate level, and I expect that this question can be answered by graduate level users more correctly. If I misunderstood the use of this functionality, sorry for that.

 

[PeterDonis]

I wrote that naked sigularity is naked from white hole infinity and is hidden from black hole infinity

Yes, and after consideration, I agreed with you; the way the term "naked singularity" appears to be used by Carroll, it implies that a white hole singularity is naked and a black hole singularity is not.

Black hole infinity does have horizon, from which "light cannot escape", so I am convinced CSC is not violated. You say it is not for that reason.

No, I said that the white hole singularity being naked does not violate CSC. That is because the white hole singularity cannot form from generic initial conditions, so the precondition for CSC to apply at all is not met. It has nothing to do with the exterior region that the white hole emits light signals into "not being part of the same spacetime", which is what you claimed the reason was.

You are trying to say, explanation of why CSC is not violated is even subextremal black hole is not formed by the way of which CSC says, but according to the book it is.

I said that the interior region (inside the outer horizon) of a sub-extremal black hole cannot form from generic initial conditions, yes. (That also implies that a white hole singularity cannot form from generic initial conditions, as I said above.) But this has nothing to do with CSC either, because a black hole singularity is not "naked" to begin with, so whether or not it can form from generic initial conditions is irrelevant to CSC. That is not my opinion; it's explicitly stated in one of the quotes you gave from Carroll that CSC is about the formation of naked singularities.

if this is not what you are trying to say, then the whole disuccion after #8 is a bit of pointless, I had figured out the answer thanks to your previous replies and other sources

IMO it's never pointless to clarify and correct misstatements. That's what I have been doing. You had "figured out the answer" to how Carroll was using the term "naked singularity", yes (and I agree with your answer to that, as I said above). But you said other things as well, which were incorrect, so I responded to correct them.

I did not marked the thread because I am graduate level student, but because the topic is graduate level, and I expect that this question can be answered by graduate level users more correctly.

That is probably true, but you also have to have the requisite background to properly understand the answers and put them in context.

 

[Elnur Hajiyev]

No, I said that the white hole singularity being naked does not violate CSC. That is because the white hole singularity cannot form from generic initial conditions, so the precondition for CSC to apply at all is not met. It has nothing to do with the exterior region that the white hole emits light signals into "not being part of the same spacetime", which is what you claimed the reason was.

That is exactly what I was trying to express in post #8. Maybe you did not fully comprehend my reply. So, I again have been sure that the conclusion which I had came to in #8 was right after this reply.

I see in almost all of the threads in this category you try to help users to find an answer, thank you for your effort and contribution, but I have also noticed that you try to catch "formalism errors" in the posts after which discussions go a bit off-topic.
I know, you say these are important definitions to know with the exact forms if you study physics. Yes. But I try to understand/explain physical concepts with logical reasoning, rather than focusing to learn definitions by heart, that is why I enjoy to study physics not law :). Of course there is still many things that I do not know, and to learn I try to do more readings, post questions, try to discuss some topics which confuses me etc. But for example after all the conversation we do here 3 days, seeing that there was nothing wrong in my understandings of these concepts that we argue on after #8, again I can definitely say that it was pointless and a waste of time to argue on those concepts in terms of learning new thing or changing my fundamental understandings. On the other hand if you see in my any post that I understood any physical/mathematical concept totally wrong, please let me know of it, I would very appreciate it, because it is possible to occur to me.

I hope I expressed myself correctly. Thanks. Take care.

 

[PeterDonis]

I don't have anything to add to my previous posts as far as clarification of anything is concerned.