How does curvature of spacetime affect the motion of test particles?

[sqljunkey]

If I had a chart of Cartesian coordinates and it had four axis's, t, x, y and z, how would I induce motion of a test particle by curving those axis's ? If the test particle was standing still and only moving in the t axis, how do i get it to move in the other axis's, x, y, z.

I can understand a photon particle doing all kinds of things when I curve that spacetime, because it already is moving in space, either the z x and y. If I have an object that is standing still, and I curve that spacetime to a point, that object should keep standing still. Unless the whole coordinate system is being dragged at some rate towards the point.

Everybody says oh yeah, all these test particles are freefalling along these geodesics, and I drop a mass in there and whatever curvature happens is okay, it's just motion man.

This is the stuff that never got explained to me very well, it's like every test particle is moving everywhere, in every direction, at all times, and I drop this mass in there and all of a sudden these test particles start converging towards it. And then I ask myself what about the object that is standing still and not moving in the x, y, z coordinates.

 

[PeroK]

This seems a very confused post.

Particles move in a curved spacetime according to the principle of maximal proper time, which is the GR equivalent of Hamilton's principle of least action.

 

[Dale]

If I have an object that is standing still, and I curve that spacetime to a point, that object should keep standing still.

How do you stand still in spacetime? I can see how you can stand still in space, but not time.

 

[Nugatory]

If I had a chart of Cartesian coordinates and it had four axis's, t, x, y and z, how would I induce motion of a test particle by curving those axis's ?

You can't .
The axes are just mathematical abstractions used to assign labels/coordinates to points in spacetime, and curving the axes changes the labels but doesn't affect the distances between the points. To get the effect that you're looking for we need a curved spacetime. In such a curved spacetime it is often easiest to do the math using curved axes, but the effect is caused by the curvature and is there whether we use curved or straight axes. (There are some subtleties around the definition of "curved" and "straight" but we can fudge over these for now).

Let's look at how we do induce motion... There's a buildup here, so bear with me...

You are moving around on the curved surface of the earth, and let's say for simplicity that you start at the equator. You choose to use straight-axis x/y/z Cartesian coordinates, you choose the z axis to point straight up, the x-axis to point due north, and the y-axis to point east. This works fine as long as you don't move around too much, but when you start moving north problems appear: the x-axis starts to point up into the sky instead of due north and the z axis starts to move towards the southern horizon. Furthermore, when you try to calculate your position as a function of time the math becomes more complicated: inverse trig functions and terms like $\sqrt{x^2+y^2+z^2}$ appear, and the calculations get very hairy very quickly.
So you decide to do what every transcontinental airplane pilot does: you give up on the x/y/z Cartesian coordinates and use latitude/longitude/altitude instead. Now we are using curved axes (the latitude and longitude axes are circles, and the longitude axis isn't even a geodesic). These curved axes don't change our path across the surface of the earth, they just make the math easier.

Next, consider you and me standing at the equator, ten meters apart, and both starting to walk due north. The distance between us will shrink until it reaches zero when we meet at the north pole. It's easy to see that happens when we used the curved latitude/longitude axes: we're both following lines of constant longitude, and the axes are curved so that lines of constant longitude draw together and intersect at the north pole. It's not so easy to see if we're using the x/y/z coordinates but if we grind through the math we'll of course get the same result: distance between us shrinks until it reaches zero at the pole. That is, curved axes or straight axes don't matter, we get the same result either way; and that result comes from curvature causing our paths to intersect not anything we did with the axes.

OK, one more thing to consider: an ordinary flat-space Minkowski spacetime diagram with the time axis vertical and the x-axis horizontal. As you've said, two stationary objects will maintain constant x coordinates, their worldlines are parallel vertical lines of constant x, and the distance between them does not change. But as the discussion above shows, this is not because the t axis is not curved, it's because the spacetime is flat so initially parallel and vertical inertial worldlines remain vertical and parallel.

And with all that buildup... Suppose we draw that same Minkowski diagram on the curved surface of the earth, with the x-axis running along some line of constant latitude and the t axis running along some line of constant longitude. As we move forward in time (move north) we'll be pulled towards one another, not because some axes are curved or not curved but because our paths across the curved surafce draw together and eventually intersect.