How to visualize curved space-time
bcc

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

SSS example

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)

The infinitesimals

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

Therefore

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.

Similarly,

(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.