I was searching for a duplicate question when I came across this one, it's not quite the same as my question but it seems useful.
I was going to ask: "If you shot an arrow at a black hole how long would it take to hit the surface", I thought it would make for an interesting question.
The difference with this question seems to be that you're the arrow and pointing a flashlight back to the starting line.
Let me start my answer by saying that I'm a long distance and time from being qualified to answer this question.
If the event horizon of a black hole is the distance from the center from within which light cannot escape, imagine a person with a flashlight falls into the black hole.
OK, so let's not concern ourselves with:
- The length of time you expect the batteries to last, so it's ∞
- Your starting speed relative to the black hole, so it's 0
- You say: "... cannot move in an arc ...", so it's like a Light Saber (not a watering hose)
- Your strength, so it's ∞ (you can't be flattened, or drop the heavy flashlight), unless you hit your head on a rock (the surface of the black hole); that causes you to lose all your strength (you'll see how that aids the answer later)
- Motion of the black hole through space, you could match it; nor should it spin as that would add angular motion (light would "arc", to use your terminology) and increase the gravity at the equator (and reduce it at the poles) due to the fusion mass's flattening
He points his flashlight in a precisely radial direction and turns it on. Now there is a light ray moving outward at the speed of light. If it now cannot move in an arc, but rather is constrained to radial motion, it must, at some point before the horizon, switch directions and fall back into the black hole.
Your assumption is that the light travels at the speed of light, that also assumes that inside the Event Horizon is a vacuum, it seems unlikely but let's go with that. I'll address slow light below.
Imagine the event horizon like it's an enormous plastic bubble, thinner than a soap bubble and immobile at absolute zero (very strong but you can penetrate it).
If the speed of light is constant, ...
The speed of light is constant, in a vacuum. Nothing says light can't travel slower.
See Lena Hau's video: "Prof. Lene Hau: Stopping light cold" where she creates slow light conditions, causing light to travel as slow as 17 meters per second and even stopping it completely.
There are theories about faster than light travel also. See source: notational speed greater than the speed of light, otherwise inside the event horizon would be brightly lit (maybe it is, don't ask me).
... how does it suddenly change directions, ...
Imagine on Earth throwing a rubber ball directly straight at a cement wall (slightly tilted away from you); it bounces back into your hand. [Apply appropriate conventional math.] Finished, next situation.
Now instead imagine you're traveling towards a black hole head first, pointing the flashlight between your feet. You're inches away from the event horizon and there's either no accretion disk or you're in a clear spot free of debris.
The moment before you touch the event horizon you and your flashlight are traveling at virtually the speed of light (minus a tiny bit)
[Remember: We are assuming that we are in a vacuum, which is unlikely.]
If you were not strong you would stretch and your feet would travel two little bits slower than the of light. Let's assume your head, flashlight and feet all travel at the same speed.
Now instead of you being like the rubber ball and bouncing off of the event horizon you penetrate it, like a bullet through a lightbulb.
If the event horizon were infinity flat (and so were you for an instant) you'd be traveling at the of light (if you're in a vacuum), and for an instant light wouldn't be emitted from the point source bulb of your flashlight (everything, including light, is constrained to one direction; it is: 'towards the black hole or bust').
While traveling through the event horizon, and a moment later, you'd be traveling at the of light.
The moment you cross the event horizon you are traveling the same direction (physically, from your point of view, from an observer's point of view it's like their eyeball is the black hole and the event horizon a mirror) you were a moment ago, away from the starting line.
You're back to normal length (because you're so strong). The only reason to flatten you for an instant is to define a specific location where head and feet both travel at the of light. [Bad math and gross oversimplification, OK. Since we're assuming that we're in a vacuum valid calculations were thrown out already.]
Think of it as a "mandatory direction", nothing can travel in the opposite direction you are going, your shoe can not possibly fall off, you can't drop your keys (or flashlight) and light (or anything) can only travel at an angle between: slightly greater than 90° (sideways), that would require tremendous energy, and straight towards the central point.
If you had a flashlight with a wide beam pointed sideways at 90° slightly more than half of the flashlight would be black (because light couldn't push it's way out) and the remaining slightly less than half of the emitting surface would have the light sucked downwards towards the central point, like the number 7 with a short too and rounded corner (like a question mark with a flat top like a number seven).
... without either decelerating, ...
Nothing, not even light, can travel with enough force to travel downward and importantly your direction (as seen / unseeable by an outside observer) doesn't change ...
You always travel towards the center of the black hole (we're assuming it's not spinning and that if it's moving you are matching it) at an increasing speed, passing the speed of light (slightly, due to the Gravitational Constant of the black hole) at the event horizon where it is space itself that is inverted.
You are like the rubber ball bouncing off a paper thin wall except the moment you touch the wall (to bounce) you are (assuming you're paper thin) on the opposite side of the wall and bouncing from it's opposite side (space directions are inverted top to bottom but your direction of travel is the same as a moment ago).
So there's no deceleration and no energy or effort on your part to pierce the event horizon.
... or requiring an infinite amount of energy?
It is direction itself which is opposite (East and West, or North and South, flip). It's like the Earth was far bigger, hollow, and gravity reversed; you fall towards the central point. It's as though the event horizon was pushing you and shrinking at your feet; but it's the black hole and it's gravitational constant pulling you in and slowly accelerating your speed.
So light from your flashlight no longer illuminates your feet they are black (we're stretching the truth, to assume you can see and think), light can either travel towards the 'battery end' of the flashlight or it is blocked; like a laser pointer placed tip down on a mirror. Nothing has enough energy to travel towards your feet.
It's like traveling through a "two-way mirror tinfoil ball", as you continue to penetrate the layers you can only see down if the photons from the object can catch up, and objects above your head can't emit light that can escape the gravity and hit your eyes.
If the light can't possibly be emitted (like if it were an LED flashlight) then it can go nowhere, possibly heating the chip; no matter. Flip the flashlight 180° (before you crash into the black hole's liquid fusing (fusioning) surface) and the light exits the flashlight at the speed of light plus the Gravitational Constant.
You next crash into the black hole's surface at the speed of light plus terminal velocity, no bounce or splash, and because you're so tiny in comparison and your space is compressed, so (virtually) no effect.
"Infinite" energy, speed, etc. is impossible; much like all the galaxies together don't have an infinite number of atoms.
There's no acceleration or deceleration exactly at the event horizon, the direction of space flips and your inertia travels the same direction as moments before.
[That's my oversimplied no-math explanation.].