Comment on Open Thread: November 2009

Tue, 03 Nov 2009 20:52:08 GMT

What I meant is that the notion of gravity as "something that pulls on matter" goes away.

There're a couple of concepts that're needed to see this. First is "locality is really important"

For instance, you're in an elevator that's freefalling toward the earth... or it's just floating in space. Either way, the overall average net force you feel inside is zero. How do you tell the difference? "look outside and see if there's a planet nearby that you're moving toward"? Excuse me? what's this business about talking about objects far away from you?

Alternately, you're either on the surface of the earth, or in space accelerating at 9.8m/s^2

Which one? "look outside" is a disallowed operation. It appeals to nonlocal entities.

Once again, return to you being in the box and freefalling toward the ground. What can you say locally?

Well... I'm going to appeal to Newtonian gravity briefly just to illustrate a concept, but then we'll sort of get rid of it:

Place two test particles in the elevator, one above the other. What do you see? You'd see them accelerating away from each other, right? ie, if one's closer to the earth than the other, then you get tidal force pulling them away from each other.

Similarly, placing them side by side, well, the lines connecting each of them to the center of the earth make a bit of an angle to each other. So you'll see them accelerate toward each other. Again, tidal force.

From the perspective of locality, tidal force is the fundamental thing, it's the thing that's "right here", rather than far away, and regular gravity is just sort of the sum (well, integral) of tidal force.

Now, let's do a bit of a perspective jump to geometry. I'll get back to the above in a moment. To help illustrate this, I'll just summarize the "Parable of the Apple" from Gravitation:

Imagine you see ants crawling on the side of the apple. You see them initially seem to move parallel, then as they crawl up, you see them moving toward each other.

"hrm... they attract each other perhaps?" you suppose.So you get out your knife and you cut a bit of the apple, a thin cut a millimeter to either side of the path of one of the ants. when you peel off the bit of the apple's skin, you find that the path is... straight!

"huh! Well, maybe it's the other one that's curving its path..." you think to yourself. So you go and cut out the other one's path in the same way... and lo and behold, that one's straight too.

"augh! WTF? what sort of witchery is this?"

The answer? You look again and you see that it's the shape of the apple that brings those paths together, even if they're individually straight (ie, geodesics).

Also, you might note that as the ants crawl toward the stem, as they move on the indentation near the stem, their paths seem to change a bit more... Is the stem exerting some sort of mysterious force on them?

Having learned your lesson once, you look closer, and you see that it's simply the different curvature of the apple over there.

The shape of the apple there affects what sorts of curvature there can be nearby, etc etc.

Hrm... the behavior of the ants sounds similar to tidal force. Perhaps then the apparent gravitational "forces" are really just geometric properties. Everything in freefall moves in straight lines, it's just that curvature changes how geodesics relate to each other.

There is one other catch: in basic GR, it's not space that's curved, but spacetime that's curved. Objects in freefall have geodesic paths through spacetime.

It's not that the earth is pulling down on you, it's that the earth is pushing up on you, and in a local inertial reference frame, you would be accelerating downwards. If you zoom in on any small part of spacetime, it's locally flat. (I'm ignoring stuff like zooming in so close that stuff like quantum foam may or may not show up. I'm just talking classical GR here.) Curvature could be viewed as controlling how all those locally flat bits are "stitched together"

Does that make sense? So the idea of gravity "pulling" on you goes away completely. The "force" of gravity amounts to nothing more than the geometry of spacetime.

What's left is the Einstein field equation which tells how matter shapes spacetime. (It doesn't control the metric or the curvature directly, but rather it controls a certain "average" or sum of certain properties of the curvature.)

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