Future null infinity confusion

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Dale
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Summary:
Question about the definition of future null infinity
So I was trying to get a bit better handle on the definition of the difference between an event horizon and a Killing horizon. Locally they are indistinguishable, and the key difference (to my understanding) is that the event horizon is the last Killing horizon that escapes to future null infinity.

But I have found the concept of future null infinity a bit confusing. I believe that it is the set of all events in spacetime that are at the end of a null geodesic as the affine parameter goes to infinity. I have two specific concerns about this concept

1) I don't understand what is meant by the end of a geodesic as the affine parameter goes to infinity. In what way does an infinite line have an end?

2) Does this definition of future null infinity include the events on the photon sphere where the affine parameter goes to infinity as the null geodesic is a helix that loops around the circumference of the photon sphere an infinite number of times?
 

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  • #2
PeterDonis
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I was trying to get a bit better handle on the definition of the difference between an event horizon and a Killing horizon. Locally they are indistinguishable
For spacetimes in which the event horizon is a Killing horizon, they are the same globally, not just locally.

the event horizon is the last Killing horizon that escapes to future null infinity.
There aren't multiple Killing horizons to begin with; there is just one (at least if we are considering the future event horizon; in maximally extended Schwarzschild spacetime there is also a past event horizon which is a Killing horizon, but I'll ignore that case here). A Killing horizon is a null surface on which the norm of a Killing vector field vanishes. There aren't "layers" of such horizons. There is just one.

For example, in Schwarzschild spacetime, the Killing vector field of interest is the one that appears as ##\partial / \partial t## in standard Schwarzschild coordinates. The norm of this KVF is a function of ##r## only; it decreases as ##r## decreases, until it is zero at ##r = 2M##. For smaller values of ##r## the norm is nonzero again. So ##r = 2M## is the Killing horizon for this KVF. (Note that ##r## here, while it appears as a coordinate, is actually an invariant, the "areal radius"; the Killing horizon could also be specified as the null surface foliated by 2-spheres with area ##16 \pi M^2##.)

Proving that this Killing horizon is an event horizon in this spacetime is more difficult, but IIRC this is covered in detail in Hawking & Ellis (and possibly in somewhat less detail in Wald).
 
  • #3
PeterDonis
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I don't understand what is meant by the end of a geodesic as the affine parameter goes to infinity.
In the original spacetime, there isn't one. The "infinities" under discussion (which also include future timelike infinity, past timelike and null infinity, and spacelike infinity) are, strictly speaking, only contained in the conformally completed spacetime, meaning that we add conformal "boundaries" to the original spacetime according to a mathematical procedure first described by Penrose (hence the name "Penrose diagram" for the diagrams that show these things). The geodesics that have no endpoints in the original spacetime then have endpoints on the conformal boundaries in the conformally completed spacetime.

Does this definition of future null infinity include the events on the photon sphere
No. The photon sphere is within the original spacetime; it is not on any of the conformal boundaries of the conformally completed spacetime. This does illustrate that the definition of the null geodesics that do end on future null infinity has to allow for such complications. The most complete definition I know of is in Hawking & Ellis.
 
  • #4
Dale
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In the original spacetime, there isn't one. ... The geodesics that have no endpoints in the original spacetime then have endpoints on the conformal boundaries in the conformally completed spacetime.
Ah, that is a big help. I didn't realize that this concept was tied to the Penrose diagrams. I have never been terribly comfortable with those.

The photon sphere is within the original spacetime; it is not on any of the conformal boundaries of the conformally completed spacetime.
That sounds like an additional requirement then. So it is not merely that the affine parameter needs to go to infinity, but also as the affine parameter goes to infinity the geodesic needs to go to one of the conformal boundaries.

However, for a null geodesic on the photon sphere as the affine parameter goes to infinity doesn't the geodesic wind up somewhere in the boundary of the Penrose diagram?
 
  • #5
Dale
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There aren't multiple Killing horizons to begin with; there is just one (at least if we are considering the future event horizon; in maximally extended Schwarzschild spacetime there is also a past event horizon which is a Killing horizon, but I'll ignore that case here). A Killing horizon is a null surface on which the norm of a Killing vector field vanishes. There aren't "layers" of such horizons. There is just one.
I am also thinking of Rindler coordinates on flat spacetime. There every null surface is a Killing horizon. I want to understand both the Schwarzschild case and the Rindler case so that I grasp the definitions well enough to understand the salient differences.
 
  • #6
PeterDonis
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That sounds like an additional requirement then. So it is not merely that the affine parameter needs to go to infinity, but also as the affine parameter goes to infinity the geodesic needs to go to one of the conformal boundaries.
Yes. Another way to state it would be that every value of the affine parameter along the geodesic has to correspond to a distinct event. The circular photon orbits on the photon sphere don't meet this requirement, since the mapping between affine parameter values and events is periodic, not one-to-one.

I am also thinking of Rindler coordinates on flat spacetime. There every null surface is a Killing horizon.
True, but none of them are event horizons. Every event in Minkowski spacetime can send light signals to future null infinity.

Also, each null surface in Minkowski spacetime is a Killing horizon for a different Killing vector field, not the same one. All of the different KVFs "look the same" relative to their origins (the "crossing point" of the two asymptotes of the Rindler hyperbolas), but they all have different origins, so they all define different Killing vectors at each event in the spacetime. Once you pick one particular KVF, its Killing horizon is unique.
 
  • #7
Dale
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Another way to state it would be that every value of the affine parameter along the geodesic has to correspond to a distinct event. The circular photon orbits on the photon sphere don't meet this requirement, since the mapping between affine parameter values and events is periodic, not one-to-one.
No, the mapping to events on the photon sphere null geodesic is indeed one to one. It is a helix in spacetime. The mapping to points is periodic, but even that is a coordinate dependent statement.

Nonetheless it seems like the photon sphere should not go to future null infinity, I just don’t see how the rules enforce that.

Once you pick one particular KVF, its Killing horizon is unique.
Yes, understood
 
  • #8
PeterDonis
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the mapping to events on the photon sphere null geodesic is indeed one to one. It is a helix in spacetime.
Ah, yes, sorry, momentary brain failure there on my part.

it seems like the photon sphere should not go to future null infinity
Yes, it doesn't.

I just don’t see how the rules enforce that.
I don't remember the exact definition from the original sources on how the conformal completion is defined. I think there is some notion of spatial unboundedness involved (for example, in Schwarzschild spacetime the geodesics would have to extend to arbitrarily large values of ##r##), but I don't remember the details of how it is formulated. I do remember that the full definition is not as simple as just "infinitely extendible null geodesics".
 
  • #9
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I think there is some notion of spatial unboundedness involved (for example, in Schwarzschild spacetime the geodesics would have to extend to arbitrarily large values of r), but I don't remember the details of how it is formulated.
Yes, intuitively that is the sort of thing that I want, but I don’t see an obvious way to define it in a coordinate-independent manner.
 
  • #10
PeterDonis
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intuitively that is the sort of thing that I want, but I don’t see an obvious way to define it in a coordinate-independent manner.
Ok, having looked at Chapter 11 of Wald, the key piece of the puzzle that I had forgotten is that the whole conformal compactification thing only works for asymptotically flat spacetimes. The coordinate-independent definition of the conformal infinities, including future null infinity, involves the metric being Minkowski at infinity. Obviously the metric is not Minkowski on the photon sphere in Schwarzschild spacetime, so those null geodesics do not extend to future null infinity.
 
  • #11
Dale
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The coordinate-independent definition of the conformal infinities, including future null infinity, involves the metric being Minkowski at infinity. Obviously the metric is not Minkowski on the photon sphere in Schwarzschild spacetime, so those null geodesics do not extend to future null infinity.
But the Schwarzschild spacetime is asymptotically flat. So the fact that it isn’t flat on the photon sphere shouldn’t interfere with the definition, right?

Edit: or are you saying that the missing part of the definition involves the worldlines going into the asymptotically flat region
 
  • #12
PeterDonis
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are you saying that the missing part of the definition involves the worldlines going into the asymptotically flat region
Yes; a null geodesic only goes to future null infinity if the metric along the geodesic approaches Minkowski as the affine parameter increases without bound.
 
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  • #13
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Ok, thanks, that makes sense. So then the Schwarzschild event horizon does not go to null infinity, but a radial null geodesic just outside it does. Similarly a tangential null geodesic on the photon sphere does not go to null infinity, but one angled just slightly outward does.
 
  • #14
PeterDonis
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So then the Schwarzschild event horizon does not go to null infinity, but a radial null geodesic just outside it does. Similarly a tangential null geodesic on the photon sphere does not go to null infinity, but one angled just slightly outward does.
Yes.
 
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  • #17
PAllen
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Ok, having looked at Chapter 11 of Wald, the key piece of the puzzle that I had forgotten is that the whole conformal compactification thing only works for asymptotically flat spacetimes. The coordinate-independent definition of the conformal infinities, including future null infinity, involves the metric being Minkowski at infinity. Obviously the metric is not Minkowski on the photon sphere in Schwarzschild spacetime, so those null geodesics do not extend to future null infinity.
Ah, so the whole machinery of conformal infinites does not apply to any realistic cosmology, even one assumed to be infinite.
 
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  • #18
PeterDonis
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the whole machinery of conformal infinites does not apply to any realistic cosmology, even one assumed to be infinite.
Yes. It doesn't apply to any FRW spacetime, or de Sitter, or any inflationary spacetime.
 
  • #19
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Ah, so the whole machinery of conformal infinites does not apply to any realistic cosmology, even one assumed to be infinite.
Which means technically that we believe there are no event horizons in this universe.
 
  • #20
PeterDonis
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Which means technically that we believe there are no event horizons in this universe.
None due to black holes, yes. (Technically there aren't any true black holes, either--but see further comments below.) There are event horizons due to the accelerating expansion, but these are observer-dependent (each comoving worldline has its own event horizon).

As I understand it, the usual rationale for using the term "event horizon" in connection with isolated "black holes" in our actual universe is that, heuristically, in our models of such isolated objects, "the rest of the universe" plays the role of the conformal infinities. Basically, you pick some spherical surface that is far enough away from the isolated object that spacetime is flat to a good approximation, but still only contains the one isolated object (or system, such as the solar system). Then you treat that spherical surface as if it were conformal infinity for the purpose of modeling the isolated object.

Of course this is only heuristic, and a strictly accurate treatment would have to say that what we call the "horizon" of the objects we call "black holes" is only an apparent horizon (a surface where, locally, the expansion of the congruence of radially outgoing null geodesics is zero), not an event horizon.
 
  • #21
Dale
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There are event horizons due to the accelerating expansion, but these are observer-dependent (each comoving worldline has its own event horizon).
How can that be if the spacetime is not asymptotically flat and therefore there is no null infinity?
 
  • #22
PeterDonis
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How can that be if the spacetime is not asymptotically flat and therefore there is no null infinity?
The term "event horizon" is used more generally than the meaning specific to black holes, which is the one that requires the spacetime to have a future null infinity. Probably there should be separate terms for the "event horizon" in spacetimes with accelerating expansion and the event horizon in the Kerr-Newman family of black hole spacetimes, but unfortunately there isn't. If we restrict "event horizon" to refer to just the black hole concept, then your statement that there are no "event horizons" in our actual universe is correct.
 
  • #23
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How can that be if the spacetime is not asymptotically flat and therefore there is no null infinity?
The event horizon in FLRW spacetime separates regions that can send you (personally) signals from those that can't. That's a different, although similar, concept to the Schwarzschild (etc) event horizons that separate regions that can send signals to (future null) infinity from those that can't.
 
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  • #24
Dale
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The term "event horizon" is used more generally than the meaning specific to black holes, which is the one that requires the spacetime to have a future null infinity. Probably there should be separate terms for the "event horizon" in spacetimes with accelerating expansion and the event horizon in the Kerr-Newman family of black hole spacetimes, but unfortunately there isn't. If we restrict "event horizon" to refer to just the black hole concept, then your statement that there are no "event horizons" in our actual universe is correct.
Interesting. Then an argument could be made that the Rindler horizon is an event horizon too. Just not a black hole event horizon. But since we are using it in a more general sense, including an observer-dependent sense, then it would be appropriate also.

Of course, there is enough confusion with standard terminology that it is best to not deliberately introduce additional confusion. But I do feel that if the FLRW horizons merit inclusion, then so does the Rindler horizon
 
  • #25
PeterDonis
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Then an argument could be made that the Rindler horizon is an event horizon too.
Yes, but it is not quite the same as either a black hole event horizon or a de Sitter spacetime event horizon (de Sitter being the simplest example of an "accelerating expansion" spacetime). A de Sitter horizon is a horizon for one specific worldline (which is a geodesic worldline). A Rindler horizon is a horizon for a whole family of (accelerated) worldlines (all the hyperbolas that have the Rindler horizon as asymptote).

To quickly summarize my understanding of some of the similarities and differences:

Both the black hole horizon (in maximally extended Schwarzschild spacetime) and the Rindler horizon are bifurcate Killing horizons (meaning that there is a future horizon and a past horizon, which intersect in a "bifurcation surface"). The de Sitter horizon is not bifurcate (though it may be a Killing horizon--I'm not sure).

Both the de Sitter horizon and the Rindler horizon are observer-dependent. The black hole horizon is not.

Both the black hole horizon and the Rindler horizon are "shared" by a whole family of accelerated worldlines (the timelike integral curves of the Killing vector field for which they are the Killing horizon). The de Sitter horizon is not.
 

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