How is Gravity, assuming only General Relativity, *not* like Centrifugal Force?

Question

It is common to state that "Gravity is not a force" due to its interpretation as a curvature effect in general relativity. By this, is it right to say that gravity is a fictitious force due to a change in reference frame, in the exact same way as centrifugal force, or is there some way in which this is not quite correct, or an oversimplication? Is gravity in GR not a "real" force in some other way different from the fictitious forces arising from a rotating/accelerating reference frame? If this is the case, why is gravity not more often grouped in with other fictitious forces, on par with centrifugal, Corriolis, etc., when discussing relativity.

Simply put, why do we ever state that "Gravity is not a force", as if it is unique in that regard, instead of simply referring to it as a pseudoforce or inertial force. Is it simply for the sake of ease of explanation / brevity, or is there really some sense in which gravity is a different sort?

Answer 1

I like to think of gravity as being analogous to the centrifugal force. The reason we discuss them as separate things is that in non-relativistic mechanics it is not at all obvious that gravity is an inertial force: within non-relativistic mechanics, the standard point of view is that gravity is a force, modeled by Newton's universal law of gravitation. To view gravity as an inertial force, you would need to get to general relativity.

Within general relativity, inertial forces can all be thought, in a certain sense, as a gravitational effect. I don't think there is any particular reason to not mention them, but there is also no particular reason to mention them. The geodesic equation, for example, will trivially predict the Coriolis force, the centrifugal force, and etc. Perhaps most references don't pay much attention to inertial forces because they simply do not find them that interesting when compared to other gravitational effects that cannot be described at the level of non-relativistic mechanics. I'd say this is mostly a sociological question.

Answer 2

In Newton's framework, making the one philosophical concession of postulating gravity as a force described by $GMm/r^2$, without further explanation, allows us to model nearly everything else in mechanics and astronomy based on inertial frames.

If we insist on explaining gravity from deeper principles, which General Relativity does, we must uproot the Newtonian framework and redefine inertial frames, and how space and time are structured. But the Newtonian framework still works for almost everything, and is much easier to deal with practically.

It's a bit like using the assumption that the Earth is flat and the weight of anything is $mg$, with constant $g$. This is a great framework to use for teaching students about the world and for engineering and building bridges, buildings, or nearly anything. Yes, we could start high school physics from the beginning using only spherical coordinates, writing all object trajectories in terms of $(r,\theta,\phi)$ rather than $(x,y,z)$, and engineers could compute the length of a bridge as the arc length of a curve with radius $R_{Earth}$ and the weight of each segment as $GmM_{Earth}/R_{Earth}^2$ rather than $mg$. This would perhaps be "more conceptually correct" in a sense. But the end results would not be any different, and the learning and calculations would be much more difficult unnecessarily.

In the same way, using Newtonian mechanics and gravitational forces to describe and calculate most things, rather than the Riemannian geometry of GR, is the more practical approach, while keeping in mind the limitations and cases where GR is needed.

To more directly address your question, I would say in GR gravity is the "same type" of inertial or fictitious force as an acceleration or centrifugal force. This is due to the Equivalence Principle which states a uniform gravitational field is indistinguishable from a constant acceleration. Also, you can always choose a reference frame in which gravitational forces vanish (i.e. the local inertial frame of the observer), just as you can with inertial and rotational forces.

Answer 3

The "fictitious" contribution to the acceleration of a particle arises from choice of coordinates. For e.g. , flat spacetime described using spherical coordinates have non-trivial Christoffel symbols even though there is no curvature. These Christoffel symbols encodes the centrifugal contribution to the acceleration. Same is true when moving from an inertial frame of reference to, say, an uniformly accelerated frame of reference (see Rindler coordinates). But if you look at the equation of motion of a free falling particle on a curved spacetime, then the gravitational contribution to the acceleration also comes from these Christoffel symbols. So the question is, are these two forces the same? A distinction can be made between forces of centrifugal type vs gravitational type if we compare trajectories of nearby particles. The worldlines of two free falling particles can distort in presence of intrinsic curvature which is described using geodesic deviation equation (aka the tidal force) , while this won't be the case for just centrifugal force alone. Likewise, there can be relative frame dragging effects due to the magnetic component of the curvature. These non-trivial effects which arise from curvature alone are what set apart the gravitational force from the centrifugal ones.

A different way to think about this is that if gravity was indeed fictitious in the same sense as centrifugal force, then there should exist a global inertial coordinate system... which would imply zero intrinsic curvature for the spacetime as a whole, which isn't the case for any metric theory of gravity like in Einstein's field equations.

Answer 4

The thing that makes fictitious forces fictitious is not, as is sometimes implied in textbooks, that they arise due to the acceleration of the reference frame (though they do), but rather that a real force always involves the interaction of two bodies. There must be an agent exerting the force and an object experiencing it. That's not true, e.g., for a centrifugal force - there is no agent pulling the object outward from the axis of rotation of the frame of reference. It is true, however, for gravity. That may be why it's not mentioned along with fictitious forces - it's not in the same category. (One might split hairs and make a distinction between "fictitious" and "inertial" forces, and call gravity an "inertial" force because it is indistinguishable from an acceleration of the reference frame, but not "fictitious," because it involves an agent, but I have never seen that distinction made in practice.) More likely, though, it's just because people like to jump to the statement that gravity is not a force at all - at that point, what value would there be in likening it to any force, real or fictitious?

Actually, I will be controversial and say that I have never heard a convincing argument that gravity is not a force, GR and the equivalence principle notwithstanding. The arguments usually go, as OP says, along the lines of, "Gravity is not a force. Rather, objects move in a spacetime that is curved by mass, and all objects in the same curved spacetime experience the same acceleration." But I don't see why that disqualifies it as a force, one that is mediated by spacetime curvature. Maybe it comes down to a philosophical question of what you consider to be a force.

Answer 5

Gravity works exactly like a low pressure system. That could be a low pressure system in the weather, where things are sucked in towards the centre of the low. Or it could be a low pressure system in water, where things move to where the water pressure is lower, i.e higher up in the water.

So you would not say that a low pressure is a force. Things move towards the low pressure because that is where they give up potential energy. You would not say that if you hold a stick under water, and release it, it rises to the top because of a force on the stick. Rather you would simply say that it rises because the water pressure above the stick is lower than the water pressure below the stick. (And before anyone says that the stick rises because is is more buoyant, I'll ask you to look at a photo of an air bubble stationary inside a water ball up on the space station. It remains in the centre of the water because the pressure of the water is equal all around the air bubble.)

In the case of gravity, the pressure system is in time. Nobel prize winner Kip Thorne said that things like to move where they age more slowly. I would then say that things move to where the pressure of time is the least, i.e. where time runs slower. In moving in this direction things give up potential energy. And this is the exact reason why all objects fall at the same speed and acceleration in a vacuum. Because the falling rate of an object has nothing to do with the object, but only with the pressure in time above and below the object.