# Can special relativity contain general relativity?

I realized that considering merely an upper limit for speed per se can hint at gravity(more precisely black holes).

Very naively with Newtonian mechanics, one can derive the following formula for escape speed from an object with the mass of $$M$$:

$$v=\sqrt{2GM/R}$$

where $$R$$ is the radius of the object and $$G$$ is Newton's gravitational constant.

If one sets an upper limit for speed then the upper limit for escape speed for any object is also subject to the same limit:

$$v=\sqrt{2GM/R}=c$$

Then one can see that it surprisingly gives the Schwarzschild radius, which is somehow correct because even light can not escape black holes!

So this way one can predict black holes and their size using merely Special Relativity. Does this signal a deeper connection or it's merely a coincidence?

I know that one must use full SR to derive the correct relation for escape speed but still doesn't reject the spirit of the idea.

• Escape velocity is a very different concept from what goes on in black holes. For example, if a ball is launched from a planet with an initial speed just below escape speed then it will still reach distances arbitrarily far from the planet, but with a black hole nothing can come out of the horizon, which is a region of finite surface area (and it can be of quite modest size in practice). There are many other differences. Oct 7 '21 at 20:35
• I'm not sure if you've read my explanations. Sorry. Oct 7 '21 at 23:53
• By the way for a massive ball, it definitely can't escape the object, just like black holes. Oct 8 '21 at 1:32

Your argument is exactly the one John Mitchell gave in a letter to Henry Cavendish in 1783 about the concept of dark stars, see wikipedia.

I would classify this as an argument that works because of dimensional analysis. You have all the right basic physics, so you know your fundamental parameters are $$G$$, $$c$$, and $$M$$ and you need to end up with a length. Any logical derivation working with $$G$$, $$c$$, and $$M$$ that results in a length scale has to get the Schwarzschild radius as the answer, up to a dimensionless factor. By luck, this dimensional analysis argument gets the right dimensionless factor in this case.

There are other contexts where naive arguments about gravity without using GR get the wrong dimensionless factors. For example, the bending of light in a gravitational field.$$^\star$$ Examples like this show that it's not possible to think of GR as simply a consequence of a simpler and less complete theory of physics, like Newtonian gravity or special relativity.

That's not to say that GR can't be understood as a logical consequence of deep physical principles. There are arguments that GR is the inevitable result of special relativity and the equivalence principle (at least at low energies), and the equivalence principle follows from having a force exchanged by a spin-2 boson. But, while these arguments are beautiful and deep, they rely on extra information (or assumptions) not already implied by special relativity or Newtonian gravity.

$$^\star$$ Strictly speaking you need to assume that the photon has a small but non-zero mass in this argument, which is not consistent with special relativity. The larger point is that you need to use GR (or some better theory that reduces to GR in the regimes we've tested it) to consistently describe both relativity and gravity simultaneously.

• Thanks. Very convincing answer. The light bending of a dark star is half as much as a black hole. This smashes out the conjecture. Might it be related to the scalar gravity? Because scalar gravity predicts the same( but wrong) amount of deflection. It's also a Lorentz covariant theory. Oct 8 '21 at 17:39
• I think it's just dimensional analysis. Oct 8 '21 at 21:41

Actually, this is backwards. General relativity contains special relativity as a special case. You cannot go from the special case to the general one without additional information.

What you have found is essentially a coincidence, and it doesn’t actually work correctly since the radial coordinate in a Schwarzschild spacetime doesn’t have the same meaning as the radial coordinate in a flat spacetime.

• I guess we all already knew that GR results in SR. I'm guessing the opposite. If it's a coincidence, then there should be some contrast between such object and a black hole. Name the differences please. Oct 8 '21 at 0:00
• @BastamTajik There are more differences than similarities. The ordinary escape velocity only applies to ballistic motion, i.e. motion where you accelerate initially and then move only under the influence of gravity. If the escape speed is $c$ in Newtonian physics, you can escape at 1 m/s in a rocket if you have enough fuel. So "you can't escape from a black hole because you'd have to travel at $c$ to escape according to the Newtonian formula" doesn't work as an explanation. Oct 8 '21 at 1:03
• I think you've got the wrong attitude. If the escape speed is equal to $c$ combining this fact with SR results in no escape. This is merely a boundary condition. Calculate a physical quantity that differs in these two cases that distinguish them from each other. Otherwise, the differences are not "physical". Oct 8 '21 at 1:22
• And keep in mind that escape speed has nothing to do with the ball!!! It only depends on the features of the object, namely mass, and radius. Oct 8 '21 at 1:28
• @BastamTajik said “there should be some contrast between such object and a black hole. Name the differences please”. The biggest difference is that black holes exist whereas Newtonian objects with escape velocity = c don’t. Others include the facts that the Newtonian objects could be escaped from, have no time dilation, have less gravitational lensing, lack frame dragging, lack Hawking radiation, etc. Also, the radial coordinate has a different meaning, so saying they are the same is not meaningful.
– Dale
Oct 8 '21 at 3:37

Historically, the first people to think about the behavior of light emitted from a supermassive object thought about it in just this way: you calculate the escape velocity as a function of surface gravity and conclude that it's possible for an object to be sufficiently massive and dense for its surface gravity to yield an escape velocity equal to the speed of light- and there you have a schwarzschild black hole.

This line of reasoning is not based on, nor does it contain, special relativity. It also does not yield all the characteristics that black holes and event horizons are known to exhibit when you derive their existence using general relativity, so it's not the approach used to teach the subject today.

Special relativity is based on euclidean geometry (with nothing but perfectly straight lines to describe space). General relativity is what you get when you postulate that mass and energy can cause those straight lines to get bent. In the limit of negligible amounts of mass and energy, it reduces to special relativity.

• What do you mean historically? I need the relevant papers please. Moreover please name the features that are not in common between such object and q black hole specifically. Oct 7 '21 at 23:56
• John Mitchell and Pierre-Simon Laplace both described light being stopped by the gravity of a very heavy and dense object. Their treatment of a "black hole" did not contain a description of the event horizon nor the spacetime characteristics just outside nor just inside the black hole in terms of geodesics, hawking radiation, frame-dragging or time dilation. Oct 8 '21 at 2:51

Can special relativity contain General Relativity?

Deductively speaking, no; since, special relativity is implicit in General Relativity. However, theoretical physics is not purely a deductive discipline. No discipline is, including logic. Historically speaking, Einstein used the principle of special relativity to motivate and discover General Relativity. In a sense, it 'contained' it though it took the vision of Einstein and others to find 'it'.

Naively, with Newtonian Mechanics .... gives surprisingly, the Schwarzschild radius

The art of physics includes the the art of coming up with novel ideas and descriptions that lead to a deeper truth. A more modern incarnation of what you mention is the calculation that Bekenstein gave for black hole entropy which is correct upto a proportionality constant and then proved semi-classically by Hawking who found that constant.