ldunn

The diagram on the Wikipedia page for Kruskal-Szekeres coordinates[1] does the job. There you see the trajectory of some infalling observer along with some future light cones[2] of points along that trajectory and the event horizon marked as the dashed line. The usual Schwarzschild r and t coordinates are also shown as the pale hyperbolas.

Say the trajectory that's drawn on the diagram is the trajectory of your feet. Now consider a second trajectory which begins slightly displaced "outwards" (that is, rightwards at t=0 on the diagram) from this first one - that's your head. Hopefully you agree that the head-trajectory would have to do something pretty strange to avoid crossing through the future lightcone of your feet, even behind the horizon. This doesn't require signals from your feet to travel "outward" - it's just that your head is travelling "inward".

K-S coordinates make it pretty clear that nothing drastic happens to the structure of spacetime at the event horizon - everything is perfectly regular. It's just that once you cross the horizon, the singularity (the thick hyperbola at the top of the diagram) is inevitably in your future: there is no trajectory within any future lightcone behind the horizon that doesn't run into the singularity. You're doomed to run into it in finite time, and all your future lightcones lie entirely behind the horizon.

[1] https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coord...

[2]: A useful feature of K-S coordinates is that lightcones are always at +-45 degrees

It doesn't have to move in such a direction! Look at a spacetime diagram and think about the trajectory of your head and feet! Read a book on GR! Do literally anything except have strong opinions about GR when you don't know any GR!

I agree that if you are freely falling and then you are suddenly not freely falling because you hit the surface of a planet and experienced a huge acceleration, you will notice. That doesn't have anything to do with anything I said, but it is undeniably true.

An event horizon is not like the surface of a planet - you will not be accelerated as you pass through it.

It is, once again, irrelevant that light cannot propagate outward once you're behind the horizon because, again, you are falling towards the center, and in particular you are falling through the future light cone of your feet. Please look at some spacetime diagrams if you do not believe me, preferably ones in Kruskal-Szekeres coordinates.

In GR spacetime is locally flat and for an inertial observer special relativity applies, up to tidal corrections which can be made arbitrarily small at the horizon by considering a suitably large black hole. This is a deep and important fact about GR. The idea that falling through the horizon causes you to suddenly not be able to see your feet anymore appears to obviously violate this basic principle, so if you think your assertion is true you should be able to explain why either this principle of GR is actually not true, or why your assertion does not actually violate this principle.

It is absolutely untrue that GR predicts that you would be knocked unconscious crossing the horizon. In fact one of the most fundamental aspects of GR (equivalence) predicts the exact opposite - there is no local experiment you can do as a freely falling observer to detect the horizon.

Why wouldn't you be able to see your feet? Your head is also falling through the horizon (hopefully - otherwise you are going to be very unhappy), so the light from your feet doesn't need to escape the horizon for you to see it.

Er, again, fine, but this has nothing to do with the question of "how can something so hard to detect be so abundant". The question isn't "should the non-detection of dark matter decrease my credence in it".

It's of course true that the continued non-detection restricts the available parameter space. But there is no physical principle that says that if something is as abundant as dark matter it really ought to have been detected by now - it's not as though if something is as abundant as dark matter, then it really needs to have some minimal coupling to baryonic matter that the existing experiments are now ruling out. Dark matter can just be really very hard to detect, there's no issue of how that "can be". Things that can change the rotation of galaxies are not obliged to be detectable by 2025.

If your weighting of the relative probabilities is such that you feel you should be going to bat for MOND at this point, that's your prerogative. But it's not related to the question that was asked.

This tells you that the chance of detecting dark matter is higher than it would be if it were only, say, 0.1% of the energy content of the Universe. Which is true, but so what?

Maybe you mean that this is a sense in which how hard it is to detect has something to do with how much of it there is, which is fine, but the original question is suggesting some much more constraining relationship, where the fact that dark matter is hard to detect and the fact that there is a lot of it poses some kind of apparent contradiction, or at least a puzzle. I don't know of a reason to think that any such contradiction exists.

Why do you think that how hard it is to detect should have anything to do with how much of it there is?

It bears mentioning that the situation is even more constraining than this, because you're not just looking at galactic dynamics - you're looking at galaxy _cluster_ dynamics, and gravitational lensing measurements, and the CMB, and large-scale structure formation, and whatever else.

Dark matter is not Fermi's elephant, as invoked elsewhere in the thread. It's more like the story of the blind men and the elephant - except that the blind men recognise that their individual observations, taken together, admit a coherent explanation.

Strain drops off as 1/r, not 1/r^2

I've done some very preliminary (and bumbling) work with that at https://github.com/ldunn/kern/

It's rudimentary, but it does compile and it does run, and I'm not aware of any particular obstacles to a more featureful kernel.