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pervect is offline
Aug4-06, 09:55 PM
Sci Advisor
P: 7,443

Is the earth flat?

Using standard rulers, the surface of the Earth cannot be flat, because it has a non-zero "curvature tensor".

Similarly, space-tme cannot be flat around the Earth, either, because space-time has a non-zero "curvature tensor".

This is assuming that one defines the rather fuzzy English word "flat" with the mathematical defintion "zero Riemann curvature tensor".

The longish quote from Einstein in the thread

may give some insight. This is the quote about rulers on a heated slab, as per the title of the thread above.

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab.
For a more mathematical approach that defines what curvature means in 3dimensions, see for instance "Gaussian curvature" in the wikipedia

Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature.


An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. He runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, he would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as

K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3} [/tex]

to be successful at measuring curvature, note that it is assumed that the ants have rulers.