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.
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.
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?