“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)
Your R is actually the Ricci tensor, not the Riemann tensor. The Riemann tensor has four indices, not two. The Ricci tensor is formed by contracting the Riemann tensor on its first and third indices.
“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)
Your R is actually the Ricci tensor, not the Riemann tensor. The Riemann tensor has four indices, not two. The Ricci tensor is formed by contracting the Riemann tensor on its first and third indices.