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mkline55
2014-Nov-21, 03:26 PM
This is another of those random Friday questions.

Does general relativity define a limit to the curvature of spacetime? In other words, inside the Schwarzchild radius, does curvature continue to increase?

For descriptive purposes, spacetime is often represented with a flat 2-dimensional grid, and a mass is shown as something that creates a depression in the grid. I assume a similar representation for a black hole would show that at the Schwarzchild radius, the grid is stretched into a cylinder perpendicular to the flat grid. The cylinder would have a radius equal to that of the Schwarzchild radius of the mass.

Although this is only an illustration, it still leads to the question of what shape, if any, the grid has on the interior of that cylinder. Apparently, the grid does not exist there, so may be undefined.

But again, this is just an illustration. What does the theory say when applied to three spatial dimensions plus time? Does the theory even apply inside the Schwarzchild radius? Do the Eddington–Finkelstein coordinates provide an answer?

ShinAce
2014-Nov-21, 08:12 PM
Check out the first paragraph of the relevant wiki article:
http://en.wikipedia.org/wiki/Gravitational_singularity

There's no limit to curvature in GR. But that happens at the center of the black hole, not the event horizon. The event horizon can be solved with the right metric.

mkline55
2014-Nov-21, 08:20 PM
Check out the first paragraph of the relevant wiki article:
http://en.wikipedia.org/wiki/Gravitational_singularity

There's no limit to curvature in GR. But that happens at the center of the black hole, not the event horizon. The event horizon can be solved with the right metric.
I'm not sure I read that right. This first paragraph, or some other first paragraph?

A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter.

grapes
2014-Nov-21, 09:17 PM
It looks to me that that first paragraph (which you've quoted) says some quantities become infinite, and one of the quantities is curvature.

Jeff Root
2014-Nov-23, 03:24 PM
The way I see it...

In the grid diagram, which is essentially a graph, space far
from any mass-energy is represented as a flat, level surface.
There is no rotation of space into the time (vertical) direction.
The presence of mass-energy makes it curve downward (on
the graph!). Let's assume that when spacetime is so curved
that time is as "important" as space, the slope of the graph
is 45 degrees. That is the situation at the event horizon of
a black hole. The 45-degree angle is the angle on the graph
of the path of a light ray through spacetime. So the angle is
exactly the right steepness at the event horizon that light
heading outward makes no progress. It falls at the same
rate that it rises, and so is stuck there.

The steepness of the curve increases toward the center of
the black hole. A vertical line on the graph would represent
infinite curvature of spacetime, which is what theoretically
would result at the very center of a black hole an infinite
time after it formed. The mass-energy which collapses to
form the black hole would keep on collapsing forever, since
there is nothing to stop it. That means its density keeps
increasing, forever approaching infinite density and infinite
spacetime curvature at that point.

-- Jeff, in Minneapolis

Cougar
2014-Nov-23, 03:56 PM
...some quantities become infinite, and one of the quantities is curvature.

Isn't that the same as saying infinite gravity? But demonstrably, neither black holes nor the big bang have infinite gravity. So apparently that's another reason why it is said that when your calculations reach a 'singularity', it means the theory you're working with is not working anymore.


"We mentioned that the FLRW cosmology begins with a singularity. This is a much more serious breakdown than a flat tire or a cracked engine block. It is, in fact, a physical impossibility -- a region where the laws of physics break down altogether and even spacetime comes to an end." - Rothman