Curvature vs matter by KST


My general thinking is to begin with the 4th rank
Riemann Christoffel (RC) Curvature tensor I'll denote
R_abcd. If the RC tensor is zero then the field can
be transformed to a state where the metric "g_uv"
are constants, indicative of no perturbing fields,
either EM or gravitational, thus defining empty space,
given by,

R_abcd = 0 == empty space. (1)

(I invite comments here).

Presuming (1) is true, then

R_abcd =/=0 == a field effect, (2)

causal by the presence of matter, (non empty).
Should we experiment with using (2) to provide
us with a description of matter and fields?

Let me write the RHS of (2) as,

R_abcd = T_abcd (3).

Of course one can find an invariant in a non-empty
space by,

R_abcd - T_abcd = 0 (4),

proving a "non-empty" space inclusive of matter
and fields can be described by a 4th rank tensor,
(invariant) defined by (4).

To recap, the components in (3) are not all zero,
hence the "g_uv" are not constant,
(for background review Weinberg's G&C Eq.(6.10.1),
sometimes fella's call that a tidal effect, also see
Eq.(6.9.1) that involves the relations of geodesic
curves and special note to Jay, "-R/2" just below that.)

There are (IMO) equally tangible interpetations,
from (3) in view of the relations between geodesics,

G_ab (W_cd) = T_ab (W_cd) (5)

rendering AE's law,

G_ab = T_ab (6)

where the W_cd may be equated to the metric or to
power, the dimensions are the same, as are the
effects on measurements, (I can explain that if asked).

Now from (6) we can define energy conservation
by using a covariant derivative wrt "w",

(G_ab = T_ab); w = 0,

where G_ab and T_ab are symmetric in AE's law.

Now lets apply that same procedure to (5), and find

(W_cd) ;w =0, (W_cd) , w =/=0, (7)

enabling the partials " , "

W_cd,w + W_dw,c + W_wc,d =0 , (8)

because W_cd can have asymmetrical components.
To verify that please see Weinbergs G&C pg 141,
Eqs (6.6.3-6.6.5), and find both Antisymmetry and
Cyclicity embodied within R_abcd.

That math seems ok, my *physical* understanding
of (8) would describe the propagation relation
between geodesics "a" and "b" as defined by
Weinberg G&C pg 148, as a wave equation and
quantizable as Jay suggests.
Best Regards
Ken S. Tucker

.

Relevant Pages

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