GrayAnderson wrote:Simon: I'm going to beg you to enlighten me on that (seriously, I am...I'm no physicist and the math can make me dizzy, but I like to at least understand what's going on). I'm not sure what a given curvature would produce...or for that matter, even what curvature the surface of the Earth has (even if it's negative in a sense here).
Be warned: I may be doing something wrong here. I
think I know what I'm talking about, but don't take my word for it; get a second opinion.
Think about the Earth. At the equator, the Earth has a circumference of about 25000 miles. Therefore, if you walk 25000 miles in any direction on the Earth's surface, you will wind up back where you started, having gone around a circle.* Now, consider that there are 360 degrees in that circle. Each degree corresponds to about 25000/360, or about 70 miles on the surface.
A bit of fiddling with trig shows that if you walk one degree around the circumference of a circle, the path you take will diverge from a straight line by about 1.75%. 1.75% of 70 miles is 1.225 miles. So for every 70 miles you walk on the Earth's surface, your path is deflected away from a straight line by a distance of 1.225 miles towards the center of the Earth. That's a curvature of about 1 in 50, give or take a little.
On top of that, note that two lines that start out parallel on the Earth's surface eventually cross. You can test this by starting at the equator and heading due south, while a friend starts heading south beside you. At first, any instrument you care to use will show that you are travelling on parallel paths, and any instrument you care to use will show that both of you are travelling in a 'straight' path, not bending to the left or right. And yet somewhere between your starting point and the South Pole, you will bump into your friend.
*Assuming you can walk on the ocean bottom, or that you're Jesus and can walk on water, or that you wait for a "Snowball Earth" period in which the oceans are frozen solid and the planet looks like Hoth...
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Both these things happen on Earth because the Earth is shaped like a ball; walking around on a sphere you're inevitably going to go in circles. In Hell, it's because of "positive" spacetime curvature. In positively curved space, space bends back on itself after you walk about 25000 miles, so, just as with the math above, we can show that the curvature is about one part in 50.
But this kind of curvature is a funny animal, because it doesn't involve bending a physical surface (like the surface of the Earth). It bends
the definition of straight lines. Fire a laser through a region of curved spacetime, and it will follow a straight path... but from the point of view of an outside observer, the laser will seem to bend. In the same way, if I have a B-1 bomber with inertial guidance that tells it whether it is moving up, down, left, or right, and I fly it in a "straight" line through Hell, never changing my course... sooner or later I will come back to my starting point, even though I never changed course. And this would be true of any course I picked- as a result, parallel lines in Hell will eventually meet, just as lines of longitude on the Earth's surface meet at the poles.
Spacetime around the Earth is only
very slightly curved, to the point where it's very difficult to tell, because the only masses around that can warp spacetime with their gravity are light** (like the Earth) or distant (like the Sun). For all practical purposes, we can treat spacetime around the Earth as flat: parallel lines drawn
in space*** will not meet.
**In astronomical terms...
***Not on the surface of the Earth, which is curved, but through the air or through outer space near the planet.
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But if we have a very dense mass very close by (say, a neutron star that weighs as much as the Sun but packs it all into a space the size of Manhattan Island), then the gravity of that mass is sufficient to produce much sharper spacetime curvature. At that point you
might see situations where spacetime itself is curved to something like 1 in 100 or 1 in 50. Get close enough to a black hole and you're guaranteed to see it. But in nature, the only places where this can happen are extremely dangerous places to stand, which is why I described them as being within "danger close" range of the objects in question. Anywhere that gravity as we know it is intense enough to warp our (basically flat) spacetime into something with a curvature of 1 in 50, gravity is strong enough to be a major threat to your life. And any object capable of generating such an intense gravitational field will be doing very alarming things in the way of radiation, electromagnetic fields, and so on.
So except for the (somewhat false) analogy to the curvature of the Earth's surface, the curved space of Hell isn't like anything else we can observe in our own universe.