### Can Newton's gravity equation explain why black holes are so strong?

• I was just wondering why black hole's gravitational forces are so powerful. I know it's usually explained by Einstein's relativity which states that when an object becomes infinitely dense (a compact mass) it can exert such a force of gravity and warp spacetime. But I also learned about Newton's Law of Gravity equation F = $$GM/r^2$$. Considering this equation, if the radius of an object becomes super small, then it can technically have immense gravity. So, can the gravitational pull of a black hole be explained by Newton's Law of Gravity or am I missing something? Thanks.

I don't know what the policy is on cross-site duplicates, but there are some good answers on the Physics Stack Exchange here: https://physics.stackexchange.com/questions/19405/can-a-black-hole-be-explained-by-newtonian-gravity

"...Einstein's relativity which states that when an object becomes infinitely dense (a compact mass) it can exert such a force of gravity and warp spacetime" - not really correct, all objects exert a force of gravity and warp spacetime. Black holes just to it to such an extreme level that not even light cannot escape.

I almost started to think the question's title started with *Ca**m** Newton's gravity equation ...*

Very simply, I am even surprised of the answers you got. They are good but made assumption on what you are really asking for. A simple answer would be that as far as you use Newton Law of gravity to describe the motion of the Earth around the Sun then nothing change - > Shall you replace the Sun with a black hole nothing would change. Again, the answer below are great but the question itself does not necessarily call for them. Unless you put your test mass near the various horizons of the black hole. In other words the subject here is unclear, the BH or the gravitating body?

one year ago

No you can't and the behaviour of bodies with mass and of light is completely different near a compact, massive object if you use Newtonian physics rather than General Relativity.

In no particular order; features that GR predicts (and which in some cases have now been observationally confirmed) but which Newtonian physics cannot:

1. An event horizon. In Newtonian physics there is a misleading numerical coincidence that the escape velocity reaches the speed of light at the Schwarzschild radius. But in Newtonian physics you could still escape by applying a constant thrust. GR predicts that no escape is possible in any circumstances.

2. Further; this numerical coincidence only applies to light travelling radially. In Newtonian physics the escape "speed" is independent of which direction you fire a body, but in GR light cannot escape from (just above) the Schwarzschild unless it if fired radially outwards. For other directions, the radius at which light can escape is larger.

3. GR predicts an innermost stable circular orbit. A stable circular orbit is possible at any radius in Newtonian physics.

4. In GR a particle with some angular momentum and lots of kinetic energy will end up falling into the black hole. In Newtonian physics it will scatter to infinity.

5. Newtonian physics predicts no precession of a two-body elliptical orbit. GR predicts orbital precession.

6. Newtonian physics predicts that light travelling close to a massive body has a trajectory that is curved by about half the amount predicted by GR. Even stranger effects are predicted close to the black hole including that light can orbit at 1.5 times the Schwarzschild radius.

The GR approach to gravity is fundamentally and philosophically different to Newtonian gravity. For Newton, gravity is a universal force. In GR, gravity is not a force at all. Freefalling bodies are said to be "inertial". They accelerate, not because a force acts upon them, but because spacetime is curved by the presence of mass (and energy).

In most cases, where Newtonian gravitational fields are weak, the consequences of this difference are small (but measureable - e.g. the orbit precession of Mercury or gravitational time dilation in GPS clocks), but near large, compact masses, like black holes and neutron stars, the differences become stark and unavoidable.

I am not familiar with the maths - is the relationship between the escape velocity and event horizon really coincidental?

@kutschkem The Schwarzchild solution to the GR equations predicts a radius at which it is impossible to escape gravity regardless of your velocity or acceleration. The algebra for computing this radius concludes with an expression that turns out to be identical to the expression obtained when setting the escape velocity equal to $c$ in Newtonian gravity. This is a "coincidence" because the Schwarzchild event horizon has no physical relation to escape velocity at all. Nevertheless, in history, Schwarzchild was aware of this algebra before he achieved the full GR solution.

@ProfRob thanks for the detailed answer!

@RossPresser I'd expect a coincidence if the escape velocity at the Schwarzschild radius were an arbitrary value. Alas, it is c. That is most emphatically *no coincidence.* (That the physics are fundamentally different notwithstanding.)

@Peter-ReinstateMonica that is a point of view; but the fact that $c$ doesn't feature in Newtonian physics at all and isn't the speed limit of anything argues the other way. I think it is an unfortunate coincidence (unfortunate because it leads to fundamental misunderstanding of the nature of the event horizon).

@Peter-ReinstateMonica a further point which I've now added is that the Newtonian escape speed argument only works for radial trajectories. For Newton it is an escape speed, independent of trajectory. In GR light sent tangentially will not escape unless $r>1.5r_s$ and the two theories disagree.

@ProfRob disregarding the fact that $c$ shows up out of nowhere seems odd to me. $c$ makes no appearance in maxwell's laws but yet $1/\sqrt{\epsilon_0\mu_0}$. For clarity, I do not disagree that Newton's gravity does not work near black holes, I just disagree that $c$ showing up in this way is a coincidence. I would suppose this just shows that the constants $G$ and $c$ are somehow connected (a connection not explained by Newton's relativity thus indeed one would still require GR to even find this connection)