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Originally Posted by [url=http://www.badastronomy.com/phpBB/viewtopic.php?p=106119#106119
HankSolo[/url]]Can you straighten this out for me?
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I'll try.
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The earth does not orbit around the sun, correct? The earth and moon travel in a straight line, but the curvature of space-time makes it seem like we're going around the sun. Like drawing a line on a piece of paper and then making a cone out of the paper so that both ends of the line touch. The line is still straight based on the 2-dimensional plane on the paper, but is a circle (somewhat) to a three-dimensional observer.
But if we travel in a straight line through curved space-time, then our orbit around the sun should not be affected by velocity.
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You've taken the analogy a bit too far, some people would say, but I say you haven't taken it far enough! The Earth does follow a straight line in spacetime, basically a geodesic. However, just because a path is closed in space doesn't mean it is closed in spacetime. That may be obvious now that I mention it, but it is crucial to the understanding--part of the confusion is the difficulty in understanding why velocity would make the shape of the path any different, right? Just remember, the path is in spacetime, and it is
not a closed path.
MTW (Misner, Thorne, and Wheeler's book Gravitation) provides a very good explanation early on in the book that should clear up the whole thing. They use the two examples of the flight of a bullet and a gently tossed ball, and point out that the curvature of the two paths seem to be very different.
That's a lot like your example, isn't it? If both bullet and ball are following straight lines according to general relativity, how come the curvature of the ball's path is so much greater than the curvature of the bullet's path? That's similar to your question of why velocity would make a difference in the orbit of a planet going around the Sun.
MTW says that the curvatures are the same!
All you have to do is calculate what they are in spacetime, instead of just space. The flight of the bullet is fairly flat, but the bullet takes only a short time to travel the path. The height of its arc is a lot less than the height of the ball's path. However, the ball takes a lot longer--and we have the fourth dimension to deal with in general relativity calculations of curvature, remember. To convert time into a quantity that's compatible with space, we have to multiply by c, the speed of light. Since the flight of the ball takes so much longer than the flight of the bullet, the effect of that fourth dimension is to stretch out the path of the ball greatly.
Once, you do that, the curvatures of the ball and the bullet are the same. And that's why gravity is "constant" for both.
Hope that helps.