What is 'Ricci' to 'Riemann' in Einstein eq. ???

[Widdekind]

In the Einstein equation, of GR,

 

G

_ab

comes from

Ricci

_ab

 

Ricci

_ab

comes from

Riemann

^a

_bcd

Thus, in contracting Riemann (rank 4) to Ricci (rank 2), the former is 'dumbed down' significantly, to mate up with the Stress-Energy tensor T_ab (also rank 2), in the Einstein equation. But, what about all the information that is 'lost', in 'dumbing down' Riemann to Ricci? Does that mean, that the Einstein equation, even with a 'full' specification, of all the mass-energy distributions, does not fully determine all 256 components of Riemann? To wit, mass manipulates the spacetime fabric in a 'clumsy' way, leaving many 'loose ends' indeterminate?*

 

*

What about electric charge, weak charge, and color charge? With four major forces to 'play with', each with their own constituent 'charge' densities and 'stress' tensors, is it possible, to 'augment' the Einstein equation, 'up' from rank two, to rank four, so that, with suitably sophisticated simultaneous specification of mass/charge/weak-charge/color-charge distributions, one could actually 'fully determine' all 256 components, of

Riemann

???

[Widdekind]

If, for Gravity (in 'geometric units'), w.h.t. R^k_kab = T_ab, where R is Riemann, and T is the mass-energy stress tensor... then, is there the suggestion, that the other fundamental forces, could be construed, as contractions of Riemann, over other pairs of indices?

 

R

^k

_kab

= T

_G,

_ab

 

R

^k

_akb

= T

_EM,

_ab

 

R

^k

_abk

= T

_EW,

_ab

 

R

^a

_bkk

+ R

^a

_kbk

+ R

^a

_kkb

= T

_C,

_ab

(Color Force is 'odd man out', different from other three (3) fundamental forces, increasing with distance, and having utterly dissimilar shape & form)

Such could construct the full-fledged Riemann tensor, using the 'force-generating charge densities', for the four fundamental forces (mass, charge, weak charge, color charge).