|MadSci Network: Physics|
Dear Les, This is great question and leads to some fundamental ideas about space- time. What do we mean by saying that space is curved? Or, even more simply, what do we mean by saying that a surface is curved? If we were living on a curved surface, how would we know that it was curved? Well, the fact of the matter is, we are living on a curved surface, the Earth's surface, and it is certainly not flat. How could we prove this? Imagine starting a journey at the North Pole. We travel south until we reach the equator, and then turn 90 degrees to the left. We move along the equator one- fourth of the way around the Earth and then turn 90 degrees to the left again, and go north until we end up back at the North Pole. Our journey has consisted of three straight sections connected with three 90-degree turns. If we were to try and draw our trip on a flat piece of paper it would be a triangle with three ninety-degree angles! That contradicts what we know about Euclidean geometry; you can't have a flat triangle with three ninety-degree angles. That proves that the Earth is not flat. Now, how can we apply this to space? If we could take an analogous trip as described for the Earth, and if it didn't match what we would expect for a flat Euclidean three-dimensional space, then we would have to assume that space was not flat. Rather than making a big triangle, another possibility is to travel in a big circle, with the center of the circle at some massive object, like the Sun. It turns out that the distance around our circle, the circumference, is not equal to 2*PI times the radius of the circle! Again, this contradicts what we know about flat Euclidean space, so our space must be curved! A very important point about this space curvature can be seen when we compare the curvature of space with the curvature of a piece of paper. To curve a two-dimensional flat sheet, we usually think of it as bending into the third dimension. With space, we are really talking about a four- dimensional space-time, in which we combine the three spatial dimensions with the dimension of time. To curve space-time, we do not need a fifth dimension; space-time can actually curve in itself. Now, what about moving objects? You are correct in that objects follow straight lines through curved space-time. You can think of a 'straight' line in space-time as being the shortest line connecting two points. However, straight lines do not depend only on the space part of the motion, but also on the time part. A straight line that moves mostly in the time direction (that is, one with little, or slow, space-like motion) will be very different from a straight line that has equal space and time motions (as for a ray of light), for example. And so, objects moving at different speeds will travel along different paths, since the space-part of their straight lines, through curved space-time, will be different. If you would like to do some more reading about this, I think a great book that discusses the experimental verifications of Einstein's theories is "Was Einstein Right?: Putting General Relativity to the Test" by Clifford M. Will. I hope that answers your question. Please let us know if you would like any more information. Thank you for your interest. Sincerely, Jim Guinn Georgia Perimeter College
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