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The Local Inertial Frame orthogonal transformation from rectangular Cartesian triads to spherical polar triads.

Using the equivalence principle the domain of validity of the LIF is a 4D spacetime volume small compared to the locally variable curvature radii.

The origin of the spherical polar triads is the idealized point emission event origin of the local light cones. So this is a fiber bundle of triads.

Let @ stand for the polar angle "theta" i.e. latitude on the (anti) celestial sphere aka (future) past light cone spacelike slices. Let & stand for the longitudinal azimuthal angle.

 

For the orthonormal base triads of Cartan's mobile frames.

eR  = isin@cos& + jsin@sin& + kcos@

~ (1/2)^1/2[sin@cos& (|A+>|A'-> + |A->|A'+>) + sin@sin&(|A+>|A'-> - |A->|A'+>) + cos@(|A+>|A'+> - |A->|A'->)]


e@ = icos@cos& + jcos@sin& - ksin@

~ (1/2)^1/2[cos@cos& (|A+>|A'-> + |A->|A'+>) + cos@sin&(|A+>|A'-> - |A->|A'+>) + sin@(|A+>|A'+> - |A->|A'->)]


e& = -isin& + jcos&

~ (1/2)^1/2[-sin& (|A+>|A'-> + |A->|A'+>) + cos&(|A+>|A'-> - |A->|A'+>) These are spacelike vectors outside the light cone.

et is a timelike vector inside the light cone. ~ (1/2)^1/2(|A+>|A'+> + |A->|A'->)]


The real Wheeler-Feynman null tetrads are

lwf = (1/2)^1/2(et + eR)  the advanced destiny real null tetrad

nwf = (1/2)^1/2(et - eR)  the retarded history real null tetrad

mwf =  (1/2)^1/2(e@ - ie&) Limiting cases, along the arbitrary Cartesian z-axis, @ = 0 et is a timelike vector inside the light cone. ~ (1/2)^1/2(|A+>|A'+> + |A->|A'->) independent of @ and &

eR  ---> k

~ (1/2)^1/2(|A+>|A'+> - |A->|A'->)

lwf = (1/2)^1/2(et + eR)  the advanced destiny real null tetrad ---> |A+>|A'+>

Therefore, "+" means "advanced" or "destiny" or retro-causal back-from-the-future pointing rather than an abstract "up" spin projection state, in this new geometrodynamical context. Similarly,

nwf = (1/2)^1/2(et - eR)  the retarded history real null tetrad ---> |A->|A'->

Therefore, "-" means "retarded" or "history" or causal past-to-future pointing rather than an abstract "down" spin projection state, in this new geometrodynamical context. These geometrodynamical light cone spinors are different from, independent of the ordinary magnetic moment lepton-quark spinor degrees of freedom.


We can also use the Paul spin matrices (quaternion hyper-complex numbers) where the tetrad basis et,i,j,k set are the 2x2 matrices.

The infinitesimal quaternion (c = 1)

ds = dtet  + dxi + dyj + dzk

ds^2 = dt^2 - dx^2 - dy^2 - dz^2

taking the complex conjugate

dsds* = Euclidean metric (Wick rotation to imaginary time of temperature statistical Green's functions).

Note in spherical coordinates where the local light cone origin R = 0 is at each point emission event

ds = dtet  + dReR + Rd@e@ + Rsin@d&e&

i.e. a fiber bundle field of light cones, where curvature is the relative tipping of neighboring light cones.

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