The problem is that this oversimplification is wrong. Curvature in general relativity cannot be expressed merely with a 2-d matrix. It's actually a hypermatrix a
I briefly looked into the section on general relativity and did not find the explanation particularly enlightening. A proper entirely geometrical description of curvature can be found in Mathematical Methods of Classical Mechanics (p. 301 ff.) by Arnold, although it is probably not accessible to a lay person. Penrose's "The Road to Reality" should be more accessible and seems to be of higher quality than the web site linked.
Somewhat on-topic, does anybody know of some decent and accurate pictures of what spacetime curvature actually looks like? No simplifications (other than suppressing dimensions), no analogies -- the real deal.
Let me describe more accurately what I'm interested in. At least one spatial dimension must be suppressed, maybe two. We're all familiar with the "trampoline" picture of general relativity, but I don't think this is accurate at all regarding the sort of curvature that actually occurs in general relativity. In particular, it's spacetime, not space, which is curved, and the trampoline picture almost invariably leaves that out.
Suppose space is one-dimensional. There's a great point-mass located somewhere on this line. In space-time, this line sweeps out a plane and the point-mass sweeps out a worldline trajectory along that plane. Now this plane must be curved according to general relativity. This curvature is something that should be fully visualizable by a human. What would it look like?
What if there were two great point-masses? Three? N, all equally spaced?
The chapters on electromagnetism and special relativity are outstanding: starting with why the magnetism we observe around a wire carrying current is a v^2/c^2 effect, and proceeding all the way to how the electric and magnetic field intensities are both part of an electromagnetic tensor. And then the field-versus-potential question, leading to a discussion of the Aharanov-Bohm effect, etc.
The discussion on the classical theories of the electromagnetic self-energy of an electron is outstanding. It's in Vol.II that we really understand how much this guy had thought through the stuff, he wasn't just drawing squiggly lines to calculate some numbers.
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[ 3.4 ms ] story [ 22.2 ms ] threadLet me describe more accurately what I'm interested in. At least one spatial dimension must be suppressed, maybe two. We're all familiar with the "trampoline" picture of general relativity, but I don't think this is accurate at all regarding the sort of curvature that actually occurs in general relativity. In particular, it's spacetime, not space, which is curved, and the trampoline picture almost invariably leaves that out.
Suppose space is one-dimensional. There's a great point-mass located somewhere on this line. In space-time, this line sweeps out a plane and the point-mass sweeps out a worldline trajectory along that plane. Now this plane must be curved according to general relativity. This curvature is something that should be fully visualizable by a human. What would it look like?
What if there were two great point-masses? Three? N, all equally spaced?
The chapters on electromagnetism and special relativity are outstanding: starting with why the magnetism we observe around a wire carrying current is a v^2/c^2 effect, and proceeding all the way to how the electric and magnetic field intensities are both part of an electromagnetic tensor. And then the field-versus-potential question, leading to a discussion of the Aharanov-Bohm effect, etc.
The discussion on the classical theories of the electromagnetic self-energy of an electron is outstanding. It's in Vol.II that we really understand how much this guy had thought through the stuff, he wasn't just drawing squiggly lines to calculate some numbers.