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The curvature everyone talks about?... What lines are being curved? If its space being curved then when I stand still next to a gravity well, nothing would happen...
Both the space and the time are curved. A manifestation of the time curvature is that clocks at different altitudes tick at different rates. The manifestation of the spatial curvature is a little less obvious:
Suppose you build a ring in space around the Earth. You would like to know how large your ring is. So you walk all the way around it, making measurements with a meterstick, and you find that its circumference is 62831.853km. Which happens to be 2*pi*10000km. Nice. Then, looking down, you see that your friend has built another ring just below you. He does a similar measurement and reports that his ring has a circumference of 2*pi*9999km.
How far below you is his ring?
The circumference of a ring is 2 times pi times the radius R, so finding the difference in radius should be easy. Your ring is at 10,000km, and his is at 9999km, so his ring is just 1km below yours. Right?Wrong! We are safe to say that circumference is 2*pi*R if the geometry of the space is flat. But the mass of Earth curves the space around it! The spatial distance between the rings will turn out to be slightly greater than 1km because of this curvature. In fact it will be about 440 nanometers greater than 1km. Which isn't a huge difference, but it is
measurable. In Schwarzschild geometry, which describes the space-time around a spherically symmetric mass, this is actually how we define the radial coordinate. Construct in the imagination a ring or a sphere around the mass, measure its circumference, and then derive the radius as the "reduced circumference", by dividing it by 2pi. Why go to the trouble of defining radius in this weird way? Because in general, directly measuring the distance from the center of the gravitating mass is hard (the mass tends to get in the way), and the measurement is impossible for a black hole.Near more massive and compact objects, the distortion of the space becomes even stronger. An extreme example: For a ring located at a reduced circumference of r=3km around a 1 solar mass black hole, and another ring located at r=2.999km, the physical distance between those rings is not 1 meter, but 8 meters. As the shells are brought down to the event horizon, this discrepancy grows to infinity! This is one of my favorite consequences of space-time curvature in general relativity. Most are familiar with the time dilation, but space gets warped, too. There is more space near a compact massive object than meets the eye!Formulas used:
From the spatial component of the Schwarzschild metric, the infinitesimal proper distance [math]
between shells at a reduced circumference [math]
is given by
where dr is the difference in reduced circumference between them. The formula is valid for small separations dr.