# General Relativity, is gravity a force?

It seems to me that objects still accelerate under the space-time curvature in GR...and I suppose we could also associate an energy with respect to this. So then what is the proper definition of force?

OK, lets cut the crap. Will somebody please explain why two objects with mass (which are not moving initially) start to move toward each other? It seems that would require force, whether or not you call it gravity.

OK, lets cut the crap. Will somebody please explain why two objects with mass (which are not moving initially) start to move toward each other? It seems that would require force, whether or not you call it gravity.

The following URL helped me visualize the curvature of spacetime.

Gravity is not a force. What is a force, anyway? Newton clarified for almost the first time in Science what a force is: First I will say it, then explain it: A force is something which makes the motion of a body deviate from uniform straightline motion.

Newton pointed out that bodies have a tendency, inertia, to continue in whatever direction they are already going, with whatever velocity they have at the moment. That means uniform, rectilineal motion: steady velocity, same direction. Newton actually knew this was what would be later called a geodesic, since « a straight line is the shortest distance between two points ».

It was then Einstein (and partly Mach before him) who said this does not get to the essence of the question. For Einstein, any coordinate system had to be equally allowable, and in fact, space-time is curved (as already explained by other posters). A body or particle under the influence of gravity actually does travel in a geodesic....i.e., it does what a free particle does. I.e., it does what a particle *not under the influence of any force* does. So gravity is not a force.

Newton did not realise that space-time could be curved and that then the geodesics would not appear to our sight to be straight lines when projected into space alone. That ellipse you see in pictures of planetary orbits? It is not really there of course since the planet only reaches different points of the ellipse at different times...that ellipse is not what the planet really traverses in space-time, it is the projection of the path of the planet onto a slice of space, it is really only the shadow of the true path of the planet, and seems much more curved than the true path really is. (The curvature of space-time in the neighbourhood of the earth is really very small! The path of the earth in space-time would even appear to be nearly straight to an imaginary Euclidean observer who, in a flat five-dimensional space larger than ours, was looking down on us in our slightly curved four dimensional space-time embedded in their world.)

Since every particle under the influence of gravity alone moves in a geodesic, it does not experience any force that would make it depart from its inertia and make it depart from this geodesic. So gravity is not a force, but electric forces still do exist. They could overcome the inertia of a charged body and make it deviate from the geodesic it is headed on: change its speed and direction (when speed and direction are measured in curved space-time).

Einstein (and me too) did not want to change the definition of force in this new situation, since after all electric forces are known to exist and are still forces in GR. So the old notion of force still retains its usefulness for things *other than gravity*. To repeat: if a body is not moving in a geodesic in space-time, you go looking for a force that is overcoming its inertia....but since gravity and curvature of space-time do not make a body depart from a geodesic, neither of them is a force.

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It is possible to think of gravity in GR as a 'pseudo-force'. The equations of geodesic motion state that

$$\delta \dot{x}^\alpha = - \left( \Gamma^\alpha_{\beta \gamma} \dot{x}^{\beta} \dot{x}^{\gamma}\right) \delta \tau$$

The point being that here the Christoffel symbols act like a velocity dependent potential.

Quite right. But for some reason I was looking at Einstein's original papers the other day. Did you know he sometimes calls the Christoffel symbols « components of the gravitational field », sometimes the metric is called « gravity », and sometimes the curvature is called « the expression of gravity »? He was attacked in print for betraying his own principle of equivalence by introducing real forces of gravity, due to his calling the Christoffel symbols « gravity ». Can you imagine how he defended himself? By saying that well, callling them that was in principle « meaningless » and he only did it for the sake of continuity with our physical habits of thinking (from older theories). That is, he picked out the Christoffel symbols although they're not even a tensor, because they did describe the departure of the geodesic from an apparent straight line in space with respect to that coordinate system. And that is the definition of a pseudo-force, even in Newtonian Mechanics, whenever you use a classically disallowed non-inertial frame of reference. This puts a whole new perspective on the word « pseudo ».