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Background

Here is a thought experiment I'd like to present that focuses on a photon emitted radially outwards from a black hole precisely at the event horizon. First I was thinking about the wavelength / frequency relationship there and what happens to a photon as it passes through an event horizon. I'm still thinking about the question, realizing that the root of my curiosity isn't really the photon itself but the properties of space-time in that scenario.

Granted that things get strange at an event horizon (EH), I have rarely heard much discussion about what happens to space-time at the EH. It seems to be established that nothing stops a photon from passing through an EH and it has also been established that the cosmological red-shifting of space can affect the wavelength of a photon and that the extreme curvature around a black hole effectively mimics said red-shifting. I put it this way because I'm aware that there is some discussion about how to precisely characterize cosmological red-shifting, So I've been wondering what happens in a scenario where a photon is affected by an extreme warping of space and what it tells us about space-time.

Thought Experiment

In the thought experiment I keep coming back to, we have a photon emitted radially outward from a black hole precisely at the current EH and for the sake of the experiment we will posit that no Hawking radiation will be released nor will any additional matter be added to the black hole so that the event horizon stays at a fixed distance from the singularity. Now, to an outside observer the photon would appear to have an infinite wavelength implying that it is somehow 'stuck' at the EH, which we know is misleading because the photon itself is always traveling a the speed of light, when measured locally with respect to the photon. The question becomes what can we say about this photon and the space-time in which it occupies? One theory is that time stops at the event horizon but that is only an artifact of what an observer outside the EH would be able to detect.

Questions

I did find this answer that posits that space-time itself is flowing past the photon at the speed of light which was one of my initial hypothesis but that leads to the next question which is if space-time is flowing past the photon, where is that space-time coming from and / or going to? Does that mean that the singularity is consuming space-time itself? Does the warping of space-time imply that a constant amount of space-time from outside the event horizon become stretched like elastic? If that is so then does that imply that space-time itself has a quality we would normally describe as 'density' for lack of a better term? I found another question that discusses how measuring distance inside an event horizon becomes complicated due to the distortion of space-time and wondered if this applied to the special case of the radially emitted photon at the event horizon, in that to it in its local frame of reference it still travels at c yet it never changes its distance from the singularity; if space-time isn't streaming past (see comment about the river model) then is the way to explain the conundrum to involve time dilation and if so what does that mean for the photon? Are we left with a scenario where the time-dilation becomes so strong as to become infinite even though photons don't experience time anyway?

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closed as unclear what you're asking by ACuriousMind, Jon Custer, Gert, user36790, Wolpertinger Oct 1 '16 at 14:53

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  • $\begingroup$ How can a photon be emitted "radially outwards" precisely at the event horizon? At the event horizon, the radius isn't a good coordinate at all. Much of the strangeness here comes from the fact that you posit a situation which is ill-defined. $\endgroup$ – ACuriousMind Sep 30 '16 at 23:04
  • $\begingroup$ I guess by "radially outward" you mean in a direction tangential to the event horizon. Wouldn't said photon simply move through the event horizon toward the singularity anyway? $\endgroup$ – flippiefanus Oct 1 '16 at 4:27
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    $\begingroup$ @ACuriousMind How is radius not a good coordinate? Is the concept of the distance from the center of the circle somehow changed due to the nearby singularity? Not sure I understand that. I can edit the question to say the radius is only important as a means to isolate that region of space where the curvature is such that a photon neither approaches nor recedes from the singularity, but that is want I was implying when I included the condition about the black hole neither consuming matter nor radiating (Hawking) energy so the event horizon was a fixed distance from the singularity. $\endgroup$ – Kelly S. French Oct 3 '16 at 13:43
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    $\begingroup$ The metric has a coordinate singularity at the event horizon, and the radial coordinate switches from being spacelike to being timelike. $\endgroup$ – ACuriousMind Oct 3 '16 at 15:36
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    $\begingroup$ That is fine, if we have a single photon traveling in a direction directly away from the singularity and at a distance from the singularity exactly the distance from the singularity where the event horizon is. Space isn't flowing past the photon, the photon is still traveling at speed c yet is remains a fixed distance away from the singularity so does spacelike vs timelike matter since the photon doesn't experience time anyway? I'm searching for how to describe this scenario but failing so is this just a little toy problem or can it help us understand the strangeness that is the universe? $\endgroup$ – Kelly S. French Oct 4 '16 at 2:55
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The idea of spacetime flowing inwards that you refer to is often called the river model, or more technically the Gullstrand–Painlevé coordinates. You can find a discussion of this in my answer to Why is a black hole black? But is important to understand that it is not spacetime that is flowing, it is the coordinate system that is flowing. Spacetime is not a thing and doesn't flow. That means spacetime is not piling up inside the black hole and there is no sense in which spacetime has a density that is increasing inside the black hole.

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  • $\begingroup$ Thanks for the references I plan on reading them as soon as I can. But doesn't this bring up an obvious version of my question? If a) the photon is traveling through space at the speed of light, b) the photon doesn't get any nearer or farther from the singularity, and c) spacetime isn't flowing past the photon, then how can all three of those be true at the same time? I'm not trying to be facetious I just don't know how to form the question presented by these facts, so if you can reformulate what this scenario tells us about light, energy, and spacetime, I'd be grateful. $\endgroup$ – Kelly S. French Oct 10 '16 at 13:50
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    $\begingroup$ @KellyS.French: your premise (a) is wrong. In GR the speed of light can be greater or less than $c$, and different observers will in general disagree what the speed of light is. See How can a black hole reduce the speed of light? and the links therein for more on this. $\endgroup$ – John Rennie Oct 10 '16 at 15:17
  • $\begingroup$ I read the link and understand how the speed of light can be different when not measured locally but for my premise (a) I mean the photon travels at the speed of light when measured in its local frame of reference. $\endgroup$ – Kelly S. French Oct 11 '16 at 12:53

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