From Osher Doctorow
Steven Weinberg's Gravitation and Cosmology, Wiley: N.Y. 1972 is in
many ways much simpler than Meisner, Wheeler, and Thorne's Gravitation
(1973) and yet is just as deep, and Weinberg clarifies the role of the
Covariant Derivative. I will use the symbol V(u; L) as the covariant
derivative of V^u with respect to L, where ; L is typically written as
a subscript and u as a superscript of V. G(u, Lk) will denote the
affine connection with superscript u and subscript Lk. We have:
1) V(u;L) = Dx^L(V^u) + G(u, Lk)V^k (Weinberg p. 103)
If c^a is a freely fallin coordinate system and x^u is any other
coordinate system, then:
2) G(L, uv) = Dc^a(x^L) Dx^u Dx^v(c^a) (Weinberg p. 71)
The two rightmost factors on the right hand side are respectively a
first partial derivative and a mixed second partial derivative with
the summation convention (that is to say, an appropriate sum of
these). The mixed second partial should be better written
Dx^u(Dxv(c^a)).
It turns out that all effects of gravitation are comprised in g_uv,
the metric tensor, and G(L, uv). G(L, uv) is a non-tensor field that
determines the gravitational force, and Weinberg proves that g_uv is
the gravitational potential in the sense that its derivatives
determine G(L, uv):
3) G(sigma, Lu) = (1/2)g^(v sigma){A + B - C)
where A = Dx^L(g_uv), B = Dx^u(g_Lv), C = Dx^v(g_uL).
From these viewpoints, the Covariant Derivative is arguably a spatial
"entanglement" of Causation with or without time.
Osher Doctorow
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