How to visualize curved space-time
Intrinsic pictures of 4D metrics
How do we plot on a computer, the local frame invariant
g(x) = ds^2(x) = guv(x)dx^udx^v
r > 2rs
g(r,@) = (1-2rs/r)(cdt)^2 - (1-2rs/r)^-1dr^2 - r^2(d@^2 + sin^2@d&^2)
g(r',@') = (1-2rs/r')(cdt)^2 - (1-2rs/r')^-1dr^2 - r'^2(d@^2 + sin^2@'d&^2)
cdt, dr, rd@ & rd& << rs/r^3
These are 4 parameters to keep fixed in the computer simulation. They correspond to a kind of lattice spacing and maybe we can do something analogous to a renormalization group flow to a fixed point?
This metric is static - no t-dependence. We only have 2 effective variables, r & @ so we can plot g(r,@) with the infinitesimals as parameters.
Note that the EEP tetrad decomposition is
g(x) = (1 + B(x)^I)nIJ(1 + B(x)^J)
= (1^0 + B^0)^2 - (1^1 + B^1)^2 - (1^2 + B^2)^2 - (1^3 + B^3)^2
(1^0 + B^0)^2 = 1^0^2 + 21^0B^0 + B^0^2 = (1 + 2B^00 + B^00^2)(cdt)^2
2B^00 + B^00^2 = -2rs/r
B^00 = (hG/c^3)^1/2(1/c)dTheta/dt this must be dimensionless
Theta = Goldstone phase of the vacuum coherence.
(1^1 + B^1)^2 = 1^1^2 + 21^1B^1 + B^12 = (1 + 2B^11 + B^11^2)(dr)^2
1 + 2B^11 + B^11^2 = (1 - 2rs/r)^-1 = 1 + 2rs/r + (2rs/r)^2 + ...
B1^1 = (hG/c^3)^1/2dTheta/dr
Note that the relation between the gradients of the vacuum phase and the set of global "coordinates" is highly nonlinear especially in space.
To be continued.