I'm trying to understand gravity in General Relativity and I'm having some questions. I can understand that an object in orbit around another more massive object is free falling and simply following a geodesic. What I can't understand is, if those objects are standing still relative to each other, why would they ever "start moving" along a geodesic until they collide? My feeling is that the problem is somewhat related to my definition of "standing still", which in this case it would be that the objects pop up spontaneously out of nowhere without any force acting on them and they happen to be close.
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3 Answers
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my definition of "standing still"
Exactly - you are not standing still in four space. Just look at the hands of a clock!
I always liked this one:
the reason you're sitting in your chair is that the shortest distance between today and tomorrow is through the center of the Earth.
answered Oct 17, 2018 at 16:02
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1$\begingroup$ Everything "stands still" in 4D spacetime. The spacetime representation of a particle moving through 3D Space is a static, 4D world line. World-lines in a spacetime diagram don't move around. $\endgroup$ Commented Oct 17, 2018 at 16:06
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$\begingroup$ I would like to continue with your example of the chair but first I have a question. In order to wrap around my ahead on the complexity of visualizing time, I'm simplifying the 4D spacetime continuum as a 3D space: 2D (x,y) for spatial coordinates and the third dimension (z) as time. In other words, I'm swapping the time for a spatial dimension. This means that moving through time would actually be moving up in my model. Can I continue with this simplification or is the time dimension special in some way that voids my model? $\endgroup$– PaulCommented Oct 17, 2018 at 20:29
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$\begingroup$ That is how I think about it Paul, whether that's "actually correct" is another matter. But there is a key difference: you are always moving along that one axis. But this is key to the whole idea, we're all going along our merry way expanding outward from the center of the universe (in time) and keep bumping into each other along the way due to our local effects. $\endgroup$ Commented Oct 18, 2018 at 19:29
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You must remember that objects are never "standing still" in general relativity, as the four-velocity in spacetime is always nonzero for any object. The value of this four vector is what defines the initial condition for geodesics, not the three-velocity in any one reference frame.
answered Oct 17, 2018 at 16:09
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"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Wheeler
Anywhere there is mass/energy, space time is distorted. However, a small mass distorts it less than a big mass. Geodesics explain how small masses move in a background geometry. However, they do not account for how masses affect the geometry itself.
In your scenario, you do not want to neglect the mass of either object. Therefore, in order to understand "why" the masses attract, you would need some understanding of the full Einstein field equations. This is more complicated than small masses moving along geodesics.
In other words, you need to not only understand that "spacetime tells matter how to move." You also need to understand how "matter tells spacetime how to curve."
answered Oct 17, 2018 at 16:16
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$\begingroup$ If it makes any difference I can neglect the mass of one of the objects. For example, the objects that spontaneously pop up can be a planet and a golf ball. $\endgroup$– PaulCommented Oct 17, 2018 at 20:04
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$\begingroup$ You would still need to know how the earth warps space time. Once you knew that, the golf ball would follow a geodesic. $\endgroup$ Commented Oct 17, 2018 at 23:03
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$\begingroup$ I don't see any reason for the downvote or anything wrong with this answer +1 $\endgroup$ Commented Aug 30, 2019 at 22:42