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I think you've got a typo here: points should be 0-dimensional.

by pflats 8y ago

curlypaul924 8y ago
Are you sure it's a typo? https://physics.stackexchange.com/a/194947 https://physics.stackexchange.com/a/194947 (I'm not a physicist.)
- checkyoursudo 8y ago
  I believe the previous poster is referring to a typo following from this: >A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point. In Euclidean geometry, A cube is 3 dimensions. A plane is 2 dimensions. A line is 1 dimension. A point is ...
- codethief 8y ago
  Not sure why you're getting downvoted, this is a legitimate question. However, in this particular case, I think the final sentence of the accepted answer on Physics.SE, namely that > In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime. is wrong or at least very misleading – the answer does (correctly) say that asking for the "dimensionality of a singularity […] is a meaningless question because the spacetime geometry is undefined at a singularity".
cuspycode 8y ago
A point is 0-dimensional, but in a space-time diagram the time dimension is added, which makes it 1-dimensional. Penrose diagrams for black holes assume spherical symmetry, so all of space is represented by a single radial coordinate, which makes it possible to display such diagrams in 1+1=2 dimensions.
- codethief 8y ago
  A singularity doesn't have a dimension. It is a portion of spacetime that is missing, not a point or set of points. We can't define its dimensionality, either.(×) What we can say is that the singularity in Schwarzschild black holes is spacelike. ×) Counterexample: Consider the manifold M := R³\B, where B is the closed unit ball, equipped with the standard Euclidean metric. This manifold is certainly not Cauchy-complete and we can reach the singularity at r=1 in finite time. Now, if we had to define the dimension of the singularity, what dimension n should it have? n=2 (a sphere)? Maybe. At least we could extend M by the unit sphere to make it complete. But could the singularity also be a point (i.e. n=1)? Yes, certainly. By diffeomorphism invariance, we could simply find new coordinates and map R³\B to R³\{0}, so the singularity would suddenly become a point. So, as you can see, interpreting the singularity as a point or set of points that have a topological dimension doesn't work.

I think you've got a typo here: points should be 0-dimensional.

by pflats 8y ago

I think you've got a typo here: points should be 0-dimensional.

Are you sure it's a typo? https://physics.stackexchange.com/a/194947 https://physics.stackexchange.com/a/194947 (I'm not a physicist.)

checkyoursudo 8y ago
I believe the previous poster is referring to a typo following from this: >A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point. In Euclidean geometry, A cube is 3 dimensions. A plane is 2 dimensions. A line is 1 dimension. A point is ...
codethief 8y ago
Not sure why you're getting downvoted, this is a legitimate question. However, in this particular case, I think the final sentence of the accepted answer on Physics.SE, namely that > In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime. is wrong or at least very misleading – the answer does (correctly) say that asking for the "dimensionality of a singularity […] is a meaningless question because the spacetime geometry is undefined at a singularity".

cuspycode 8y ago

A point is 0-dimensional, but in a space-time diagram the time dimension is added, which makes it 1-dimensional. Penrose diagrams for black holes assume spherical symmetry, so all of space is represented by a single radial coordinate, which makes it possible to display such diagrams in 1+1=2 dimensions.

codethief 8y ago
A singularity doesn't have a dimension. It is a portion of spacetime that is missing, not a point or set of points. We can't define its dimensionality, either.(×) What we can say is that the singularity in Schwarzschild black holes is spacelike. ×) Counterexample: Consider the manifold M := R³\B, where B is the closed unit ball, equipped with the standard Euclidean metric. This manifold is certainly not Cauchy-complete and we can reach the singularity at r=1 in finite time. Now, if we had to define the dimension of the singularity, what dimension n should it have? n=2 (a sphere)? Maybe. At least we could extend M by the unit sphere to make it complete. But could the singularity also be a point (i.e. n=1)? Yes, certainly. By diffeomorphism invariance, we could simply find new coordinates and map R³\B to R³\{0}, so the singularity would suddenly become a point. So, as you can see, interpreting the singularity as a point or set of points that have a topological dimension doesn't work.