Approaching a Black Hole

[RJ Emery]

TL;DR Summary

Black Holes, Event Horizon

Stephen Hawking, in his book Brief Answers to the Big Questions (2018), wrote the following (pp. 106-107):

"If you fall towards a black hole feet first, gravity will pull harder on your feet than your head, because they are nearer the black hole. The result is that you will be stretched out lengthwise, and squashed in sideways. If the black hole has a mass of a few times our Sun, you would be torn apart and made into spaghetti before you reached the horizon. However, if you fell into a much larger black hole, with a mass of more than a million times the Sun, the gravitational pull would be the same on the whole of your body and you would reach the horizon without difficulty.

“Although you wouldn’t notice anything in particular as you fell into a [supermassive] black hole, someone watching you from a distance would never see you cross the event horizon. Instead, you would appear to slow down and hover just outside. Your image would get dimmer and dimmer, and redder and redder, until you were effectively lost from sight. As far as the outside world is concerned, you would be lost for ever.”

My question is why would visible light from our hapless explorer and reaching a distant observer appear redder? I would think just the opposite, as only more energetic light (shorter wavelengths) would be escaping the clutches of gravity. I also do not grasp why time would appear to be slowing outside the event horizon. To the explorer approaching and crossing the event horizon, everything appears normal on either side of the event horizon, which he himself does not detect, so Hawking writes.

 

[Orodruin]

The energy of the light is irrelevant for whether or not it can escape the black hole. A light pulse will travel along the same worldlibe regardless of whether or not it has high or low energy. For the reddening, gravitational redshift comes into account.

 

[RJ Emery]

Ah, yes! Gravitational redshift would explain it. Thank you.

 

[Ibix]

My question is why would visible light from our hapless explorer and reaching a distant observer appear redder?

"Why" questions are a bit tricky, and there are often multiple ways to answer them. One way of answering this one is "conservation of energy". If I throw a ball upwards it loses kinetic energy to gravitational potential energy, slowing down as it does so. If I send a light pulse upwards it must also lose energy - otherwise I can exploit this to produce a free energy machine. But light can't slow down. In what sense can it be losing energy? The only way it can do so is to have its frequency lowered - i.e. become redshifted.

I would think just the opposite, as only more energetic light (shorter wavelengths) would be escaping the clutches of gravity.

If any light can escape from a given altitude, all light can. But the light becomes redshifted as it climbs. This isn't a filtering process - it's the light changing frequency. (I'm wincing slightly writing that, because there are some subtleties in that statement I'm glossing over - but the basic point is valid.)

I also do not grasp why time would appear to be slowing outside the event horizon.

This is not a helpful way of phrasing things, with all due respect to Hawking. The word "time" doesn't have a single unambiguous meaning in general relativity, and he's not (to my mind) being sufficiently careful about saying which one he's talking about at a given moment (edit: at least as quoted). It's better to think in terms of clocks and the relationships between their tick rates, because that forces you to think about what the clock is doing - free falling, or hovering near or far from the hole, or whatever.

No observer will ever notice nothing odd about their own clock. A hovering observer, however, will notice that clocks below them tick slowly (on top of a naive application of any velocity-based Doppler) and clocks above them tick fast. This is, in fact, the same phenomenon as the gravitational redshift. For example, if I, hovering above a black hole, emit one hundred cycles of 100kHz radio my clock advances 1ms. If you (hovering higher up) receive it redshifted as 50kHz radiation it'll take you 2ms to receive it. So you must see my clock tick at half the rate yours does. Thus if we accept conservation of energy we are pretty much forced into accepting gravitational time dilation.

 

[RJ Emery]

Thank you for the elaboration. It is most helpful.