Why is the gravity a fictitious force?

[Mike_bb]

I read about equivalence principle. I tried to understand Einstein's thought experiment with elevator and I can't understand why we compare elevator in the space and elevator on the surface of the Earth and conclude that gravity is fictitious force.

Why?

Thanks!

 

[Ibix]

I can't understand why we compare elevator in the space and elevator on the surface of the Earth and conclude that gravity is fictitious force.

We don't. Newtonian gravity respects the equivalence principle, and that models gravity as a real force.

However, if gravity (in practice) did not respect the equivalence principle then metric theories of gravity, in which "the force of gravity" is a so-called fictitious force, would immediately be ruled out. The elevator thought experiment is simply a way of relating the equivalence principle to a practical situation.

To put it another way, the equivalence principle is necessary for a "fictitious force" model of gravity, but it does not rule out all other models.

 

[A.T.]

I can't understand why we compare elevator in the space and elevator on the surface of the Earth and conclude that gravity is fictitious force.

Do you understand what "fictitious forces" are in the first place?
https://en.wikipedia.org/wiki/Fictitious_force

They are all proportional to the mass of the object they act on, so combined with Newton's 2nd Law (where acceleration is anti-proportional to mass) they all accelerate everything in the same way, regardless of its mass.

Newtoninan Gravity is not a "fictitious force", but it is proportional to the mass of the object it acts on. So you can locally or for a uniform field treat it as such.

 

[Demystifier]

The geodesic equation
$$\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau} = 0$$
can be written as
$$m\frac{d^2x^{\mu}}{d\tau^2} = F^{\mu}$$
where
$$F^{\mu} \equiv - m \Gamma^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau}$$
In the second equation, the quantity $F^{\mu}$ can be interpreted as the gravitational force. However, the third equation reveals that it does not transform as a vector, because $\Gamma^{\mu}_{\alpha\beta}$ does not transform as a tensor. At any point in spacetime, one can always choose a local frame of coordinates in which it vanishes. This is the reason why we don't call it a force, but a fictitious force; its existence depends on the choice of frame.

 

[Mike_bb]

Ok.

If gravity was force then mass of object (gravity mass) in elevator on the surface of the Earth differs from inertia mass of object in elevator in the space. Is this statement true?

Could anyone provide easy explanation ? Thx.

 

[FactChecker]

No. But the current understanding explains why the exact same mass is appropriate in both equations. That is one of the strong points of General Relativity. Without that, we would have to come up with some other theory of why those two masses are exactly the same.

 

[Mike_bb]

No.

From $M_i=M_g$ we can conclude that gravity is fictitious force. Right?

 

[FactChecker]

I read about equivalence principle. I tried to understand Einstein's thought experiment with elevator and I can't understand why we compare elevator in the space and elevator on the surface of the Earth and conclude that gravity is fictitious force.

Why?

Thanks!

General Relativity hypothesizes that an object, appearing to accelerate due to gravity, is simply moving in a non-accelerated straight line in its spacetime geometry. To that object, it is not reacting to an external force, it is going in a straight, non-accelerated path. That is why gravity is called a fictitious force -- it looks like an external force to us, but it is not.
The thought experiment with the elevators is just a way of illustrating that basic part of General Relativity.

 

[FactChecker]

From $M_i=M_g$ we can conclude that gravity is fictitious force. Right?

I would call it a satisfying consequence of General Relativity. GR explains the mysterious reason behind $M_i=M_g$, not the other way around.

 

[FactChecker]

From $M_i=M_g$ we can conclude that gravity is fictitious force. Right?

There could have been some reason for $M_i=M_g$ other than GR, but nobody could come up with one. It was a mystery until Einstein explained it with GR.

 

[Mike_bb]

FactChecker,

In Russian sources I read another explanations. It confuse me.

 

[FactChecker]

FactChecker,

In Russian sources I read another explanations. It confuse me.

Give yourself time to think about it. GR is a profound change in our thinking.

 

[FactChecker]

I personally think that GR's explanation of why $M_i=M_g$ is one of the most intellectually satisfying things in physics.

 

[Ibix]

If gravity was force then mass of object (gravity mass) in elevator on the surface of the Earth differs from inertia mass of object in elevator in the space. Is this statement true?

No. It is possible to construct theories in which it may differ. For example, $F=GkMm/r^2$, where $k$ depends on the materials making up the masses would allow cannon balls of different materials to fall at different rates. This would not respect the equivalence principle. However, regular Newtonian gravity ($k=1$ for all materials) does respect it.

The equivalence principle does follow naturally from a metric theory of gravity, whereas in a force-based theory it's apparently an odd coincidence. So one might prefer a metric theory on Occam's Razor, but it is not conclusive on its own.

 

[Mike_bb]

Ibix,

Could you explain how does thought experiment with elevator lead us to the fact that gravity is geometric property (not the force)? Thx.

 

[PeterDonis]

Ibix,

Could you explain how does thought experiment with elevator lead us to the fact that gravity is geometric property (not the force)? Thx.

He can't because, as he said, it doesn't.

The logic is the other way around: if you construct a theory where gravity is a geometric property of spacetime (as it is in GR), then the equivalence principle follows as a logical consequence.

That also means that, since we observe that the equivalence principle is in fact true, adopting a theory where gravity is a geometric property of spacetime is a natural thing to do, because it makes it easy to explain why the equivalence principle is true. But it is not logically forced by the fact that the equivalence principle is true; it is logically possible to have a theory like Newtonian gravity, where gravity is an actual force, but which also obeys the equivalence principle (because in Newtonian gravity, inertial mass and gravitational mass are declared by fiat to be equal--but that means you now have to explain why they are equal, whereas in a theory like GR, the question doesn't even arise).

We treat Newtonian gravity as an approximation to GR that is valid for weak fields and slow speeds because that's what the evidence tell us, not because there's any logical prohibition on a theory like Newtonian gravity because of the equivalence principle.

 

[A.T.]

we can conclude that gravity is fictitious force

the fact that gravity is geometric property (not the force)

You seem to have a general misunderstanding about physics. It's not about what gravity "really is" according to some "fact" that we are forced to "conclude".

It's about how we can model it mathematically. We currently use both models, depending on the application. Newton's is simpler mathematically. GR is more accurate over a larger range of conditions, and has the equivalence of inertial and attractive mass naturally built in.

 

[Ibix]

Ibix,

Could you explain how does thought experiment with elevator lead us to the fact that gravity is geometric property (not the force)? Thx.

As I said (and I see Peter has also said), it does not do that, so I cannot explain it. If experiment showed that gravity did not obey the equivalence principle then it could not be modelled by curved spacetime, but the argument does not work the other way around. Just as "all poodles are dogs" does not mean that "all dogs are poodles", "curved spacetime theories respect the equivalence principle" does not mean that "all theories that respect the equivalence principle are curved spacetime theories".