Unraveling the Mystery of Gravity and Space-Time Curvature in General Relativity

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In GR gravity is modeled well by the curving of space-time, as shown in pictures like these: http://en.wikipedia.org/wiki/Image:Spacetime_curvature.png
I've always liked GR but i don't understand why space is being bent. In the picture above, which i don't take literally as it is only a model, it looks as if the curvature of space responsible for gravity is itself caused by something like gravity, something that causes matter to bend space in always the same direction. I researched online and even thought of some models to explain this like http://www.bloo.us/branegrav.gif" but wasn't completely satisfied. I need some insight.

 

[Loren Booda]

"Mass grips spacetime, telling it how to curve, but also spacetime itself grips spacetime, transmitting curvature from near to far." -John Archibald Wheeler, A Journey into Gravity and Spacetime, p. 15.

 

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But why does it tell it to curve in a given direction instead of the other one? Why doesn't it curve in another direction, or why doesn't it curve in all kinds of directions? Why does it curve in the fourth dimension instead of our x, y, z?

 

[Loren Booda]

One usually pictures spacetime reduced to three dimensions. Where either space or time is dominant, one can approximate by keeping only the prevalent term(s). Space requires the shortest path distance between two events, and spacetime the longest!

If you were to calculate the hypotenuse of three dimensional space using the Pythagorean theorem, this is readily apparent. You can also analyze a 4-dimensional spacetime (applying the hypotenuse r as r²=x²+y²+z²-(ct)²) to show spacetime paths maximize trajectories. Much of the unique properties of spacetime arises from the negative time-squared term.

Curvature is the key quantity to ascertain the trajectory of a projectile in spacetime. (See my above reference, page 8.) In the vicinity of Earth, curvature is negligible except in rare satellite or classical experiments formulated to prove general relativity. So far I have referred primarily to special relativity's flat space (like the Minkowskian cone equation above).

General relativity approximates special relativity in spacetime local to the observer, but overall determines the curvature of such entities as the universe, black holes, and quantum gravity. One can consider the neighborhood of these objects as occupied by Minkowski cones point-by-point, denoting gradually variable curvature and the relation between time and space.

 

[JesseM]

But why does it tell it to curve in a given direction instead of the other one? Why doesn't it curve in another direction, or why doesn't it curve in all kinds of directions? Why does it curve in the fourth dimension instead of our x, y, z?

The "direction" is actually meaningless, all that is important is that objects follow geodesic paths--a geodesic along a curved 2D surface would be the path between two points on the surface with the shortest distance (like a section of a great circle on the surface of a sphere), so this path would be the same regardless of how the surface is oriented, you could equally well represent sources of gravity as bumps rather than dips. But this picture is misleading in another way--general relativity does not actually say objects take geodesic paths through curved 3D space, rather it says they take geodesic paths through curved 4D spacetime. In most situations this means the path through spacetime with the largest value of the proper time (the time ticked by a clock which follows that path), although as was pointed out to me on another thread recently, there are some cases where it can mean the path with the minimum proper time.

 

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My question regarding direction is more in the vein of why aren't there both bumps and dimps or why are there only bumps or only dimps. Assuming that there is a physical equivalent to the model of curved spacetime, why would matter curve space in a given direction, rather than another one, even if the direction itself is not relevant in producing GRs predictions.

 

[JesseM]

My question regarding direction is more in the vein of why aren't there both bumps and dimps or why are there only bumps or only dimps. Assuming that there is a physical equivalent to the model of curved spacetime, why would matter curve space in a given direction, rather than another one, even if the direction itself is not relevant in producing GRs predictions.

What I was saying is that the orientation of the surface has no physical meaning, only the curvature enters into the equations of GR. Bumps vs. dips only appears in our visualization of a curved 2D surface sitting in a higher 3D space, to a bug living on the surface who is only interested in finding the shortest path between points, the orientation of the surface in the higher space has no effect on what it experiences. And the math of GR is from a "bug's-eye" point of view, dealing only with the curvature of spacetime and the geodesic paths through it, as measured by someone within the spacetime...no higher-dimensional space appears in the equations. So it's not like you can count the number of bumps and the number of dips and compare them, the distinction is simply meaningless.