What makes matter travel along geodesics?

The relativistic explanation of gravity is geometric, the motion of a body in a field of space-time distortion can be described as being at rest and travelling along a geodesic of that field, but why does it travel along that geodesic? What causes it to move?

• Possible duplicates: physics.stackexchange.com/q/24359/2451 and links therein. – Qmechanic Jan 3 '14 at 18:07
• You shouldn't think of the world line of an object as its trajectory. It is the objects history. – MBN Jan 6 '14 at 22:05

You ask what motivates it to move so I wonder if you're puzzled why matter is in some way forced to move along a geodesic instead of just staying still. If so, it's because a geodesic is a trajectory in spacetime i.e. in time as well as space. Since nothing can avoid moving in time that means everything is moving in spacetime even when it appears to be stationary in space. We measure the rate of movement by the four velocity, and in fact the magnitude of the four velocity is always equal to $c$, the speed of light.

So given that everything is moving, the only question is what controls the trajectory in spacetime that the object follows. The geodesic is simply the path followed if no external forces act on the object. It's the general relativistic equivalent of moving in a straight line.

• would that be, because the time coordinate is always changing the position has also to change? (excuse my parlance) – user36093 Jan 5 '14 at 16:16
• @user36093: I'm not sure what you're asking. Have a look at my answer to physics.stackexchange.com/questions/90592/… and the comments to it and see if that is related. – John Rennie Jan 6 '14 at 8:48

One way of looking at the situation is that a particle samples all possible (though unphysical) paths through spacetime, and the amplitude of it ending up at its destination is the sum of the amplitude of each path, weighted by, say, $$e^{i S} = e^{i\frac{1}{2}\int\!d\tau\,\left[ \left(\frac{dx^\mu}{d\tau}\right)^2+m^2\right]}$$ Now, most of these paths/phases interfere destructively (think of a random walk in the 2-dimensional complex plane), but close to the paths that minimize the action $S$, this phase factor varies more slowly, and because a lot of those paths have approximately the same phase, there is constructive interference.

So the total probability amplitude (sum of phases) depends mostly on the paths that minimize the action, and the equation that describes paths that minimize the action is just the geodesic equation, $\nabla_\mu \left( \frac{dx^\mu}{d\tau}\right) = 0$.

You may ask why the point particle is governed by this principle, and the answer to that lies in the correlation functions of quantum field theory, but after you follow that rabbit hole, you'll be back where you started, because quantum fields also behave this way. Nobody really knows why.

Actually "to move" along a geodesic does not seem very appropriate, in my view. "To evolve" along a geodesic sounds more appropriate, since motion is a notion given in space, while we are dealing with spacetime. An object evolves along its story even if it remains at rest in the reference space of some observer: no motion but evolution.

However, I think that your question is epistemologically inappropriate at least relying upon GR. Physics mostly deals with how phenomena are and not why they are.

Actually sometimes physics even considers the problem of "why", but only after having stated the basic assumptions of a theory. The fact that bodies subjected to the gravitational interaction evolve along geodesic is a basic assumption of GR. Conversely, the observed perihelion precession of Mercury has a reason in GR and one is allowed to ask why it happens: it happens because of the nature of geodesics of the spacetime surrounding Sun's story.

If one wishes to answer your question he/she should look for that answer within a theory more fundamental than GR.