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My high school textbook briefly touched the topic of black holes, and this is how it defined them:

"Consider a spherical body of mass $M$ and radius $R $. Suppose,due to some reason the volume goes on decreasing while the mass remians the same. The escape velocity from such a dense material will be very high. suppose the radius is so small that $ \sqrt (2GM/R)>c$.Thus, nothing can escape from such a dense material. These are called black holes."

My question is this definition right? If this is right then why do we need general relativity to explain them? PS I am a high schooler,so it will be more helpful to me if the solution avoids complicated math like tensors or stuff.

Qmechanic
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tensorman666
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  • In short (I'm assuming someone else will give a better full answer), you should notice the 2 came out of nowhere. Newtonian gravity wouldn't predict that. The 2 comes from GR math, and its resemblence in form to Newtonian is somewhat of a coincidence. – Señor O Feb 15 '24 at 07:51
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    @SeñorO are you sure about that? Isn't that 2 always in the formula for https://en.wikipedia.org/wiki/Escape_velocity ? And according to https://en.wikipedia.org/wiki/Schwarzschild_radius, "This expression had previously been calculated, using Newtonian mechanics, as the radius of a spherically symmetric body at which the escape velocity was equal to the speed of light. It had been identified in the 18th century by John Michell and Pierre-Simon Laplace." – Nadav Har'El Feb 15 '24 at 08:03
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    @SeñorO the 2 is there in the escape velocity formula- you can see for yourself by using energy conservation and deriving it. – AVS Feb 15 '24 at 08:06
  • In Newtownian gravity, we did not talk about the speed of light itself. This arose the problem of propagation of information which seemed instantaneous in NG. This problem was solved in GR. – Proscionexium Feb 15 '24 at 08:27
  • @Proscionexium the speed of light was known to exist even before relativity (special or general) and measured experimentally, but there was no theory why gravity should effect it - given that it doesn't seem to have mass. But the point is, if you imagine light to be small particles with some tiny (but nonzero) mass, and moving at the known speed of light, then even according to Newtonian gravity they would have an escape velocity, and the correct radius limit (Schwarzschild radius) can be calculated - even without any GR formulas. – Nadav Har'El Feb 15 '24 at 09:47
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    The $c$ comes from Maxwell's equations, which are clearly not Newtonian. Einstein came up with Special Relativity (and later General Relativity) because nobody had found a way to make $c$ fit properly into Newton's theory. – m4r35n357 Feb 15 '24 at 09:48
  • Does this answer your question? Can a black hole be explained by Newtonian gravity? – John Rennie Feb 15 '24 at 10:02
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    @NadavHar'El I mean to say that perhaps Newtonian Gravity had nothing to say about speed of light being a constant and the maximum speed achievable. – Proscionexium Feb 15 '24 at 11:58
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    @tensorman666 Your textbook definition would not do, because the actual radius of any black hole is zero. Specifically, the radial distance from the horizon to the origin measured in meters is exactly zero. The formula for the so-called “Schwarzschild radius” is not actually for the radial distance, but for the length of the circumference of the horizon (divided by $2\pi$). It is not easy to visualize, but in curved spaces, a sphere with a zero radius and a zero volume can have a non-zero circumference. Once you learn the formula for the Schwarzschild metric, you’ll easily see this yourself. – safesphere Feb 16 '24 at 05:41

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Very good question. This was also how I was introduced to black holes in high school: After introducing the concept of "escape velocity", if you assume a "cosmic speed limit" - the speed of light - then you can calculate the so-called Schwarzschild radius without understanding general relativity, or even special relativity except the existence of a speed limit.

But the thing is, although you can do this specific calculation even without general relativity, you can't do more sophisticated calculations without it. What if the object is slightly less dense than a black hole, and light can escape - how would gravity near this object behave? One of the first successes of GR was correctly describing the anomalous movement of the planet Mercury that could not be reproduced by Newtonian gravity. Just knowing the speed of light would not have helped doing this calculation.

There's another example of an idea that people usually assume come from GR, but in fact can be considered even without it: In Leonard Susskind's fantastic course on Cosmology (freely available online), he begins the course by explaining how we could understand the expansion or contraction of the universe without GR. Just normal masses in normal Newtonian gravity. GR is important for understanding the details and make correct calculations, but not always necessary to understand why a certain phenomenon should exist.

Nadav Har'El
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  • Whilst everything you have said is technically correct, setting the escape speed (note than in Newtonian mechanics, the direction does not matter) to $c$ in Newtonian mechanics merely gives you a numerical answer for the Schwarzschild radius. It does not tell you the significance of the Schwarzschild radius or that nothing can escape from smaller radii or even that the escape speed depends on direction. – ProfRob Feb 16 '24 at 12:38
  • Right. Moreover, in pure Newtonian mechanics, a rocket could accelerate for a long enough time and achieve a speed over c, and escape the "black hole". As I noted above, you need to make the assumption of c somehow being the speed limit (it's impossible to reach that speed or go over it) which Newtonian mechanics can't explain. All that being said, even before relativity people knew that light has a finite speed, so one could imagine that even if a rocket could escape, light can't - and the object will be black. – Nadav Har'El Feb 16 '24 at 12:44
  • The point I am making is that you can still escape from a "Newtonian" black hole at speeds $<c$. – ProfRob Feb 16 '24 at 13:11
  • The whole point of "escape velocity" is that you start with some specific velocity, and then can't accelerate any further. If you start with speed < c, you can't escape from a "Newtonian black hole". If light was (and it isn't...) some sort of particle that moves at speed c but could slow down (like a regular particular of matter), it wouldn't be able to able to escape the "Newtonian black hole". Obviously, none of these assumptions are correct in modern relativity (even special relativity). But it still gets you the right answer :-) – Nadav Har'El Feb 19 '24 at 07:43
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Well friend to the best of my knowledge, in physics, technically there's no such absolute concept as "Right;" merely "able to adequately model & predict."

So yes, it's "Right" in the teaching sense to convey a black hole's basic conceptual ideas to one whose grasp of Newtonian mechanics is stronger than it is of more advanced physics.

"Adequate" to explain & model the basic strength of the gravity and light's challenge of escaping it at high strength.

"Not adequate" to model the full behavior of light, gravity & time near a black hole (approximations of which you may have seen in modern visual effects like in Interstellar, the optical bending of its event horizon and such).

"Not adequate" to model the shape of space, (which it didn't recognize as a thing), or see the need to couple it intrinsically with time as spacetime.

As far as I know, that needed relativity. Which itself was the means by which black holes were first predicted and theorized, that we should be on the look for.

Once found, their simplified explanation then was distilled back into Newtonian mechanics, to introduce learners to the concept of them. So sure, Newtonian mechanics is "right," but GR is "more right."

I guess as you keep going, it might serve you to think in terms of "what more completely explains the most phenomena, vs. what can only get so far," rather than "right/not."

In over-simplified terms, Einstein observed that light does not move any faster if coming from a speeding body vs. stationary one, no matter where in space and time one was. Which Newtonian mechanics did not address, so something was needed to explain that.

The result was the more comprehensive physics of spacetime; special relativity. He then generalized it to incorporate gravity, which he modeled as a curvature, not a force. That model has been able to predict and explain much more of what's since been observed, than Newton's has.

But Newtonian mechanics are still plenty right for plenty of calculations and models here on the ground, in aeronautics, and in space. They'll just only take one so far, only to certain degrees of accuracy or quantity of phenomena.

So they can't for instance, sync the clocks on GPS satellites with those in the stronger gravity in which your phone sits, (which gives it a different relative time rate), to put your car in the right place on your map. That'd hafta be Einstein, as far as I know. Good question.

jazamm
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