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.
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:
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.
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.
GR predicts an innermost stable circular orbit. A stable circular orbit is possible at any radius in Newtonian physics.
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.
Newtonian physics predicts no precession of a two-body elliptical orbit. GR predicts orbital precession.
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 an expert in physics and the explanation of the others is excellent. However, I noticed a flaw in your reasoning which they did not address.
You have written:
Considering the Newton's Law of Gravity equation $F = GM/r^2$, if the radius of an object becomes super small, then it can technically have immense gravity.
Hence I deduce that you read the $r$ in the equation as the radius of the object, while in fact it is the distance between two objects. So there is no rationale based on this formula that "the smaller the radius is, the stronger the gravity must be". The proper reading would be "the closer the object is from the black hole, the stronger the gravity is", but this is valid for all the bodies, not only to black holes.
While admiring @ProfRob's answer I'll add some additional perspective/background that may serve as a helpful stepping-stone since not every Astronomy SE reader is prepared to embrace General Relativity in all its glory.
Can Newton's gravity equation explain why black holes are so strong?
- The simple equation1 $F = GMm/r^2$ doesn't explain anything but it certainly does give a result that's usually quite useful in Newtonian mechanics where space is "normal" and we can talk about gravitation as a force.
- General relativity can be said to offer an explanation of how gravitation works and it gives results that work not only where Newtonian gravity works but in extreme situations as well, where speeds are very high and/or gravity is very very strong (and other situations as well).
- Even in situations we are somewhat familliar with, like Mercury's orbit around the Sun or satellites orbiting the Earth, predictions by Newtonian mechanics is measurably off and GR gets it spot-on.
- On the flip side, if you want to approximate a trajectory of a star or bit of dust that passes far from a black hole, you certainly can go ahead and use Newton's equations. Trajectories will still be close to Keplerian orbits.
I'll make up some numbers for illustration purposes only. If a 20 solar mass star goes supernova and you add up all the ejected mass and energy and it's 12 solar masses, then you'll expect an 8 solar mass black hole left over.
If there is a companion star orbiting it at a large distance, or if you fly past it at a "safe distance" and look at what happens, it will be what you'd expect for a mass of 8 solar masses, whether it's a black hole, a neutron star, a regular star or a (magically self-supporting) ball of concrete.
It's only when you get closer that you need to use GR, and you'd have to use it whether it's a black hole or a more conventional dense object like a neutron star.
1Remember, for a force there are two masses, for acceleration there's only one $a=F/m = GM/r^2$
If Einstein's GR equations are expanded in terms of familiar coordinates (Cartesian, spherical,...), the dominant or leading terms of the expansion (for the acceleration) can be written as the single Newtonian term GM/r^2. The next terms of the expansion can be considered as GR corrections to this leading term.
Before the publication of GR, 19th-century astronomers noticed that the advance of the perihelion of Mercury was not predicted accurately by Newtonian gravitational physics. This became a major problem. At some point, astronomer/mathematicians added corrections to the Newtonian Law of Gravitation that produced a new/revised force law that accurately predicted perihelion advance. The results were accurate, but the new gravitation theories were not as rich in their "other" predictions as Einstein's GR.