# Are Geometries that contain Closed Timelike Curves Possible?

• B
• TheQuestionGuy14
There is not a consensus in the physics community about whether or not CTCs exist. Some physicists do believe that they exist, while others do not. I'm not sure what your view is, but it's not a majority view in the physics community.f

#### TheQuestionGuy14

Some exact solutions to Einstein's General Relativity show that Closed Time like Curves may theoretically exist. Could they actually exist in our universe? And if so, how would it change the understanding of physics and our universe?

No, spacetimes that have closed casual curves are not physically realistic (if following general relativity). Even spacetimes that are close to violating causality aren't physically realistic! When dealing with spacetimes in GR, there is a "stably causal" theorem. I think the best treatment I've seen of this is Chapter 8 in Wald. He even talks about what if we do violate causality, but it's what you would expect: Stuff changes.

However, if you don't have the book, there is a wikipedia article on this as well: https://en.wikipedia.org/wiki/Causality_conditions#Stably_causal

I don't see anything in that wikipedia article that supports an argument that our spacetime cannot contain closed causal loops. There is a claim

"The weaker the causality condition on a spacetime, the more unphysical the spacetime is."

but it is not supported by any argument. All it does is refer to an article on the grandfather paradox, which in fact contains the standard resolution, that a spacetime cannot contain an event in which somebody kills their own grandfather, or otherwise violates causality. This is the Novikov self-consistency principle.

This is a B thread, so I don't want to divulge into the mathematics, but you can refer to Wald Chapter 8.

I'll talk a little more on this issue, even though I thought I said enough! Locally, ALL general relativity spacetimes reduce to the causality presented in special relativity, so the only place we can have closed causal loops is somewhere globally. If you follow standard general relativity, then I will assume you believe that we have a globally hyperbolic spacetime. Globally hyperbolic spacetimes are stably causal. Thus, if our spacetime is stably causal, we can't have closed causal loops.

Chapter 8 in Wald folks, he proves this all. I might need to refresh my memory, but I believe this is the gist of it.

(Globally hyperbolic is just a fancy term for a spacetime that has a Cauchy surface).

This is a B thread, so I don't want to divulge into the mathematics, but you can refer to Wald Chapter 8.

I'll talk a little more on this issue, even though I thought I said enough! Locally, ALL general relativity spacetimes reduce to the causality presented in special relativity, so the only place we can have closed causal loops is somewhere globally. If you follow standard general relativity, then I will assume you believe that we have a globally hyperbolic spacetime. Globally hyperbolic spacetimes are stably causal. Thus, if our spacetime is stably causal, we can't have closed causal loops.

Chapter 8 in Wald folks, he proves this all. I might need to refresh my memory, but I believe this is the gist of it.

(Globally hyperbolic is just a fancy term for a spacetime that has a Cauchy surface).

The Wikipedia says this:

CTCs appear in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These include:

It says they are local and gives some examples, are any of these actually real?

are any of these actually real?

If you mean, is there any evidence for any of these solutions actually occurring, no. (Note that the Kerr vacuum only contains CTCs inside the inner horizon; that's the part that we don't have any evidence of. We do have evidence of the existence of rotating black holes, but that is only evidence of the region of Kerr spacetime outside the outer horizon, and possibly within the outer horizon but outside the inner horizon, depending on what position you want to take regarding the black hole "firewall" debate.)

If you mean, is there any evidence for any of these solutions actually occurring, no. (Note that the Kerr vacuum only contains CTCs inside the inner horizon; that's the part that we don't have any evidence of. We do have evidence of the existence of rotating black holes, but that is only evidence of the region of Kerr spacetime outside the outer horizon, and possibly within the outer horizon but outside the inner horizon, depending on what position you want to take regarding the black hole "firewall" debate.)
Do much physicists buy into Closed Timelike Curves? Also, I don't know if my view is right, but, a Closed Timelike Curve can only happen under these very specific circumstances, if they were found to be real, they couldn't happen to some person just strolling down the street, correct?

Do much physicists buy into Closed Timelike Curves?

I think most physicists do not think CTCs actually occur in our universe.

a Closed Timelike Curve can only happen under these very specific circumstances, if they were found to be real, they couldn't happen to some person just strolling down the street, correct?

I'm not sure what you mean by "happen". A CTC doesn't "happen". Either the spacetime geometry contains CTCs or it doesn't; spacetime geometry doesn't "happen", it just is.

I think most physicists do not think CTCs actually occur in our universe.

I'm not sure what you mean by "happen". A CTC doesn't "happen". Either the spacetime geometry contains CTCs or it doesn't; spacetime geometry doesn't "happen", it just is.
Are all CTCs in a certain geometry?

Are all CTCs in a certain geometry?

I'm not sure what you mean. There are certain spacetime geometries that contain CTCs, and others that don't.

Spacetime is just a structure that contains a manifold, and a metric. The geometry of spacetime depends on these. If you're asking if there is a certain class of structures that have CTCs, they're basically infinite possibilities! I can come up with a ton, but they are of no real interest!

The only way for us to classify physical realistic spacetimes is to take out those that do have the property of CTCs because we currently have no reason to expect a spacetime with that property to be physically realistic. That's the reason why people spent so much time proving that property of globally hyperbolic spacetimes. We expect all physical realistic spacetimes to be of that type.

As mentioned above, the Kerr CTCs occur within a region where we aren't too sure if the current physics even works, so the answer is maybe!

So in the end, my interpretation of my understanding of general relativity is: No, CTCs are not physical realistic. Any spacetime that has that possibility is not physically realistic. (Even though I want the answer to be yes).

Also, just another thing to add! A solution to EFE DOES NOT mean it's a physical solution, so just because you have some that satisfies ##G_{ab}=\kappa T_{ab}## does not mean it's physical. They're a lot more requirements than just satisfying EFE.

I'm not sure what you mean. There are certain spacetime geometries that contain CTCs, and others that don't.
I mean, if CTCs existed, do they require a certain geometry, or can other things make them occur? So, can a person on Earth experience one, or do they only work in specific geometries?

Spacetime is just a structure that contains a manifold, and a metric. The geometry of spacetime depends on these. If you're asking if there is a certain class of structures that have CTCs, they're basically infinite possibilities! I can come up with a ton, but they are of no real interest!

The only way for us to classify physical realistic spacetimes is to take out those that do have the property of CTCs because we currently have no reason to expect a spacetime with that property to be physically realistic. That's the reason why people spent so much time proving that property of globally hyperbolic spacetimes. We expect all physical realistic spacetimes to be of that type.

As mentioned above, the Kerr CTCs occur within a region where we aren't too sure if the current physics even works, so the answer is maybe!

So in the end, my interpretation of my understanding of general relativity is: No, CTCs are not physical realistic. Any spacetime that has that possibility is not physically realistic. (Even though I want the answer to be yes).

Also, just another thing to add! A solution to EFE DOES NOT mean it's a physical solution, so just because you have some that satisfies ##G_{ab}=\kappa T_{ab}## does not mean it's physical. They're a lot more requirements than just satisfying EFE.

Thanks. There is a spacetime called the Minser spacetime which is supposed to have CTCs, yet this type of spacetime is supposed to be everywhere in the universe, what do you think of that solution?

From the wiki; "CTCs appear in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These include:

the Misner space (which is Minkowski space orbifolded by a discrete boost) the Kerr vacuum (which models a rotating uncharged black hole)"

if CTCs existed, do they require a certain geometry, or can other things make them occur?

Remember, when we talk of a spacetime geometry, we are talking of something that already describes all of the past, present, and future, in all places, of whatever "universe" the geometry is describing. So of course CTCs "require a certain geometry" in the sense that, if there are CTCs at all in a given universe, the spacetime geometry of that universe must contain them. But this is not something different from "other things making them occur"; if CTCs are present in a given spacetime geometry, obviously that spacetime geometry--that universe--must also contain whatever is required to make CTCs occur.

Remember, when we talk of a spacetime geometry, we are talking of something that already describes all of the past, present, and future, in all places, of whatever "universe" the geometry is describing. So of course CTCs "require a certain geometry" in the sense that, if there are CTCs at all in a given universe, the spacetime geometry of that universe must contain them. But this is not something different from "other things making them occur"; if CTCs are present in a given spacetime geometry, obviously that spacetime geometry--that universe--must also contain whatever is required to make CTCs occur.
When I say spacetime, I think of a flat grid plane, but warped, as spacetime can be curved, and some irregular areas would contain CTCs, is this a correct description of a spacetime geometric?

is this a correct description of a spacetime geometric?

To the extent that a 4-dimensional geometry can be visualized at all, it's ok. But you should not expect any such visualization to be a good guide to intuition. That's why we use math for actually doing calculations and making predictions.

“Global structure of spacetimes”
by Geroch and Horowitz
http://web.physics.ucsb.edu/~phys231B/231B/Intro_files/3.globalstructure.pdf

The discussion of closed timelike curves begins on p.238.

This is a B thread, so I don't want to divulge into the mathematics, but you can refer to Wald Chapter 8.

I'll talk a little more on this issue, even though I thought I said enough! Locally, ALL general relativity spacetimes reduce to the causality presented in special relativity, so the only place we can have closed causal loops is somewhere globally. If you follow standard general relativity, then I will assume you believe that we have a globally hyperbolic spacetime. Globally hyperbolic spacetimes are stably causal. Thus, if our spacetime is stably causal, we can't have closed causal loops.

Chapter 8 in Wald folks, he proves this all. I might need to refresh my memory, but I believe this is the gist of it.

(Globally hyperbolic is just a fancy term for a spacetime that has a Cauchy surface).
FYI, your presentation does not match any argumentation in chapter 8 of Wald. The gist of that chapter is a set of definitions and proofs of relationships between different criteria for causal classification. There is no argument, nor is one possible (except by fiat) that GR only allows globally hyperbolic spacetimes. The only 'argument' in chapter 8 by Wald is the weak claim that most physicists think solutions violating these are unphysical. This is exactly as much as you can say - a belief about how our universe is.

Stronger basis to believe this, e.g. proofs they can't evolve given some plausible conditions (e.g. energy conditions), or have probability zero of evolving, have long been sought and never found. It remains true that that most GR experts suspect that something like this is true, but suspicion is not proof.

“Global structure of spacetimes”
by Geroch and Horowitz
http://web.physics.ucsb.edu/~phys231B/231B/Intro_files/3.globalstructure.pdf

The discussion of closed timelike curves begins on p.238.
And consistent with my prior point, Geroch and Horowitz conclude with:

"
Is there, for example, a good physical reason for demanding stable causality
or the existence of a Cauchy surface? Is there some theorem
expressing cosmic censorship? How, more generally, are singular
spacetimes to be dealt with?"

The answers to all of these remain currently unknown.

FYI, your presentation does not match any argumentation in chapter 8 of Wald. The gist of that chapter is a set of definitions and proofs of relationships between different criteria causal classification. There is no argument, nor is one possible (except by fiat) that GR only allows globally hyperbolic spacetimes. The only 'argument' in chapter 8 by Wald is the weak claim that most physicists think solutions violating this are unphysical. This is exactly as much as you can say - a belief about how our universe is.

Stronger basis to believe this, e.g. proofs they can't evolve given some plausible conditions (e.g. energy conditions), or have probability zero of evolving, have long been sought and never found. It remains true that that most GR experts suspect that something like this is true, but suspicion is not proof.

You claim that my presentation does not match any argument in chapter 8, so I'll open it up and make it even more clear!

Post #4 boils down to using theorem 8.3.14, and on page 202 he states "There are some good reasons for believing that all physically realistic spacetimes must be globally hyperbolic" and then gives reference to chapter 12 in Penrose 1979. If someone wants to know those reasons, I'd be happy to argue for why I take this stance. By the end, I invoke theorem 8.2.2.

He proves with 8.3.14 that a globally hyperbolic surface is stably casual. By the logic that we believe that all physically realistic spacetimes are globally hyperbolic, i can invoke 8.2.2 to state that stably causal spacetimes have no closed timelike curves. therefore, no physically realistic spacetime has CTCs. So please correct me, at what point did I misinterpret the logic of the chapter?

The only thing you can disagree with me on what is physically realistic, which is the whole point of this thread. As I said in post #11, this is how I interpret GR based off what I've learned.

I'd be happy to read a counterview if you think spacetimes with CTCs are physically realistic!

You claim that my presentation does not match any argument in chapter 8, so I'll open it up and make it even more clear!

Post #4 boils down to using theorem 8.3.14, and on page 202 he states "There are some good reasons for believing that all physically realistic spacetimes must be globally hyperbolic" and then gives reference to chapter 12 in Penrose 1979. If someone wants to know those reasons, I'd be happy to argue for why I take this stance. By the end, I invoke theorem 8.2.2.

He proves with 8.3.14 that a globally hyperbolic surface is stably casual. By the logic that we believe that all physically realistic spacetimes are globally hyperbolic, i can invoke 8.2.2 to state that stably causal spacetimes have no closed timelike curves. therefore, no physically realistic spacetime has CTCs. So please correct me, at what point did I misinterpret the logic of the chapter?

The only thing you can disagree with me on what is physically realistic, which is the whole point of this thread. As I said in post #11, this is how I interpret GR based off what I've learned.

I'd be happy to read a counterview if you think spacetimes with CTCs are physically realistic!

The part of the post I replied to that I disagreed with was:

"
If you follow standard general relativity, then I will assume you believe that we have a globally hyperbolic spacetime. Globally hyperbolic spacetimes are stably causal. Thus, if our spacetime is stably causal, we can't have closed causal loops.

Chapter 8 in Wald folks, he proves this all. I might need to refresh my memory, but I believe this is the gist of it."
The statement that standard general relativity mandates global hyperbolicity is a major overstatement.

Now, on review, I can see that I may have misread the flow of your statements. I thought you were claiming that Wald proved the necessity for global hyperbolicity, when in fact he only states most physicists think this represents physical reasonableness. I now see that you presumably only meant that Wald proved the equivalence of global hyperbolicity and stable causality, which is certaintly correct. So I concede my claim you misrepresented Wald was a misreading of what your wrote. Sorry about that.

However, it remains true that many of the physicists who expect stable causality to be true of our universe are actively seeking a better reason for this to be true than simply assuming it (or assuming something equivalent). The open questions are how to derive this from things that would not be so close to just assuming your conclusion.

Last edited:
romsofia
You claim that my presentation does not match any argument in chapter 8, so I'll open it up and make it even more clear!

Post #4 boils down to using theorem 8.3.14, and on page 202 he states "There are some good reasons for believing that all physically realistic spacetimes must be globally hyperbolic" and then gives reference to chapter 12 in Penrose 1979. If someone wants to know those reasons, I'd be happy to argue for why I take this stance. By the end, I invoke theorem 8.2.2.

He proves with 8.3.14 that a globally hyperbolic surface is stably casual. By the logic that we believe that all physically realistic spacetimes are globally hyperbolic, i can invoke 8.2.2 to state that stably causal spacetimes have no closed timelike curves. therefore, no physically realistic spacetime has CTCs. So please correct me, at what point did I misinterpret the logic of the chapter?

The only thing you can disagree with me on what is physically realistic, which is the whole point of this thread. As I said in post #11, this is how I interpret GR based off what I've learned.

I'd be happy to read a counterview if you think spacetimes with CTCs are physically realistic!
There's a thing called Minser Spacetime which apparently can hold CTCs, is this physically realistic?

There's a thing called Minser Spacetime which apparently can hold CTCs, is this physically realistic?

When I say spacetime, I think of a flat grid plane...is this a correct description of a spacetime geometric?
As long you remember that one of the dimensions on this flat plane is time not space, it's OK. If you're thinking of spacetime as a flat two-dimensional sheet, that's spacetime for one-dimensional creatures living in a one-dimensional universe in which time passes.
There's a thing called Minser Spacetime, is this physically realistic?
Misner not Minser, and no.

As long you remember that one of the dimensions on this flat plane is time not space, it's OK. If you're thinking of spacetime as a flat two-dimensional sheet, that's spacetime for one-dimensional creatures living in a one-dimensional universe in which time passes.
Misner not Minser, and no.
Since time is the fourth dimension, can we actually represent it at all on a 3D grid?

Since time is the fourth dimension, can we actually represent it at all on a 3D grid?

Sure, just leave out one of the space dimensions. In many cases this can be done without leaving out anything important; for example, in cases where there is a spatial symmetry.

Since time is the fourth dimension, can we actually represent it at all on a 3D grid?
Yes, if you leave out one or more of the spatial dimensions. For an example, look at this video by our own member @A.T.

The apple is following a path through four-dimensional spacetime, but we only care about the points in space along the one-dimensional line between where the apple is hanging in the tree and where it lands. Thus, we can use the vertical axis of the grid for space and the horizontal axis for time, and don't bother fitting the other two spatial dimensions (N/S and E/W if you're standing next to the apple tree and following the trajectory of the apple) into the diagram.

Yes, if you leave out one or more of the spatial dimensions. For an example, look at this video by our own member @A.T.

The apple is following a path through four-dimensional spacetime, but we only care about the points in space along the one-dimensional line between where the apple is hanging in the tree and where it lands. Thus, we can use the vertical axis of the grid for space and the horizontal axis for time, and don't bother fitting the other two spatial dimensions (N/S and E/W if you're standing next to the apple tree and following the trajectory of the apple) into the diagram.

Overall though, are CTCs accepted by physicists at all?

Overall though, are CTCs accepted by physicists at all?

You've asked this same question several different ways now in this thread, and the answer has already been given. The thread has run its course and is now closed.