Answering simple questions about black holes to someone not knowing GR

Question

In the Newtonian description of a black hole, it is a gravitating astrophysical object for which the escape velocity exceeds the speed of light in vacuum ($c$). But what does this actually mean for a photon shot radially outwards from its surface? I think, unlike a massive object shot upwards, it will not gradually decrease its speed, come to a halt and turn back. This is not how it really works. So how do I explain it to an undergraduate? Also, how will you explain with this sort of escape speed argument that any photon reaching a distance less its Schwarzschild radius is sucked into it? I am not interested in popular science analogies.

Answer 1

You could possibly explain things in terms of sonic black holes. These are systems of fluids where the bulk motion of the fluid exceeds the speed of sound in some regions.

In a 1972 lecture at the University of Oxford, a young physicist named William Unruh asked the audience to imagine a fish screaming as it plunges over a waterfall. The water falls so fast in this fictitious cascade that it exceeds the speed of sound at a certain point along the way. After the fish tumbles past this point, the water sweeps its screams downward faster than the sound waves can travel up, and the fish can no longer be heard by its friends in the river above.

The idea here is that the event horizon of a black hole works in much the same way. Beyond a certain point, the shape of spacetime means that light waves cannot make any progress towards "the outside world"; they are forced to move inexorably towards the center of the black hole.

While this might seem like a "popular-science analogy", the mathematics of these fluid flows is similar enough to that of black holes that several teams of scientists have attempted to construct these "analog black holes" in the laboratory. Their ultimate goal is to see if these "black holes" emit the sonic equivalent of Hawking radiation. Some teams have claimed success, but my impression is that the interpretation of their results is still being debated.

That said, the fluid analogy is not a perfect one. In particular, it suggests that black holes "inexorably suck things in", which is not really true.

Answer 2

You can't think of general relativity in the "space" + time picture used in Newtonian mechanics. black hole spacetimes have spacetime geometry, and can only be cleanly picked apart into seperate, global, space and time parts in special cases. The heart of this question is that the "forward in time" direction is twisted inside the horizon relative to the outside of the horizon in such a way that escape is not permitted, because the path from the inside of the horizon to the outside is spacelike, the same way that the length of a ruler is. it doesn't make sense to travel along that path.

For the very special case of a light ray pointed raidally outward at the exact event horizon, it will just evolve forward in time at a constant value of $r$. The horizon itself is a null surface.

Answer 3

As you can write the velocity of light in a spherically symmetric gravtational field in coordinate time as:

$v_{light}=c(1-\frac{2GM}{rc^2})$ in the radial direction and: $v_{light}=c(\sqrt{1-\frac{2GM}{rc^2}})$ in the pure non-radial direction you can basically show using these expression that light "stops" at the Schwarzschild radius and in that way it is "trapped"?

You are writing that "any photon reaching a distance less its Schwarzschild radius" is sucked into the black hole, but basically light stops at the event horizon and it is unclear if it can be said to be travelling inside the event horizon.

At the event horizon also time stops and the energy of a non-moving object is zero so it is also unclear if anything can actually exist in the usual sense inside the event horizon. Maybe you should avoid speaking about "escape velocity" when you talk about the GR scenario, it is perhaps not a fruitful anology.