“McCabe, you’re right, it’s completely obvious, it makes you wonder why Einstein took ten years to figure it out.”
I never said it was obvious; I said that the equations were a unique solution imposed by various constraints. Proving that the equations are a unique solution is quite difficult; I can’t do it, even with a ready-made textbook in front of me. There are many examples of simple, unique-solution equations being very hard to derive- Newton’s law of gravity and Maxwell’s laws of electromagnetism come to mind.
“But selecting the tensor framework, that is of course where all the bits had to go. It is not an obvious choice at all.”
I agree that it is not at all obvious, but the search space doesn’t seem to be all that large- how many mathematical toys are there which could form a viable framework for gravity? The difficulty seems to be in understanding the math well enough to determine whether it can represent real-world phenomena. Differential geometry is not a simple Bayesian hypothesis like “the cat is blue”; to figure out whether piece of evidence Q supports a geometric theory of gravity, you have to understand what a geometric theory of gravity would look like (in Bayesian terms, which outcomes it would predict), which is quite difficult.
“Tom, is that an elaborate joke?”
No. What makes you think that?
“Sure, if we don’t mind that G and T take a full page to write out in terms of the derivatives of the metric tensor.”
The Riemann tensor is a more natural measure of curvature than the metric tensor, and even in that language it’s still pretty simple:
8piT = R (tensor) - .5gR (scalar)
where R (tensor) (subscript) ab = Riemann tensor (superscript) c (subscript) acb and R (scalar) = g (superscript) ab * R (tensor) (subscript) ab
You can make any theory seem complicated by writing it out in some nonstandard format. Take Maxwell’s equations of electromagnetism in tensor form:
dF = 0 dF = 4pi*J
Now differential form:
(divergence) E = p (divergence) B = 0 (curl) E = -dB/dt (curl) B = J + dE/dt
Now integral form:
(flux E over closed surface A) = q (flux B over closed surface A) = 0 (line integral of E over closed loop l) = - d (flux of B over surface enclosed by l)/dt (line integral of B over closed loop l) = (current I passing through surface enclosed by l) + d (flux of E over surface enclosed by l)/dt
Now in action-at-a-distance form:
E = (sum q) -q/4/pi ((r’ unit vector from q)/r’/r’ + r’ d/dt ((r’ unit vector from q)/r’/r’) + d^2/dt^2 (r’ unit vector from q)) B = (sum q) E x -(r’ unit vector from q)