Awhile back A.T. & PeterDonis helped me through several basic confusions on my way to an intuitive understanding of geodesics in curved spacetime. I looked at(adsbygoogle = window.adsbygoogle || []).push({});

http://www.relativitet.se/spacetime1.html

http://www.adamtoons.de/physics/gravitation.swf

http://www.adamtoons.de/physics/relativity.swf

But now something is bothering my feeble brain again: the inverse coefficients of dt and dr in the Schwarzschild metric.

All those diagrams seem to show space and time stretched out equally. I understand the time-coordinate lines diverging: this illustrates how proper time is shorter than coordinate time; the coefficient of dt is less than one. But the coefficient of dr is greater than one, meaning that proper length is greater than coordinate ("reduced circumference") length. So shouldn't the space-coordinate lines be getting closer together rather than further apart? (In the trumpet-shaped diagram, the trajectories being represented are straight up and down, so it is the radial distance we're talking about.)

The third page listed above shows "length contraction" that makes sense to me when gravity=0 (SR only). But with nonzero gravity, and zero initial velocity, it still shows length contraction. The coordinate time is longer than the proper time, which is okay; but the coordinate length seems to be greater than the proper length, which is backwards. No?

My general question, then, is whether there is a tradeoff between space curvature and time curvature. Perhaps the time factor overwhelms the space factor (due to conversion by c)?

Also, what's the best way to think about the fact that in SR, the space and time expansion factors are the same, whereas in GR they're inverses.

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# Inverse coefficients of dt & dr in Schwarzschild

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