Go to previous page Go up Go to next page

2.3 Parallel propagator

2.3.1 Tetrad on β

On the geodesic β that links x to x′ we introduce an orthonormal basis eμ(z) a that is parallel transported on the geodesic. The frame indices3 a, b, …, run from 0 to 3, and the frame vectors satisfy

μ gμν eμeν = ηab, De-a-= 0, (68 ) a b dλ
where ηab = diag (− 1,1, 1,1) is the Minkowski metric (which we shall use to raise and lower frame indices). We have the completeness relations
gμν = ηabeμaeνb, (69 )
and we define a dual tetrad a eμ(z) by
a ab ν eμ ≡ η gμν eb; (70 )
this is also parallel transported on β. In terms of the dual tetrad the completeness relations take the form
a b gμν = ηabeμeν, (71 )
and it is easy to show that the tetrad and its dual satisfy eaeμ= δa μ b b and eaeμ = δμ ν a ν. Equations (68View Equation, 69View Equation, 70View Equation, 71View Equation) hold everywhere on β. In particular, with an appropriate change of notation they hold at x ′ and x; for example, a b gαβ = ηabeαe β is the metric at x.

2.3.2 Definition and properties of the parallel propagator

Any vector field A μ(z) on β can be decomposed in the basis eμa: Aμ = Aa eμa, and the vector’s frame components are given by Aa = A μea μ. If A μ is parallel transported on the geodesic, then the coefficients a A are constants. The vector at x can then be expressed as α α′ a α A = (A eα′)ea, or

A α(x) = gα (x, x′) Aα′(x′), gα (x, x′) ≡ eα (x )ea (x′). (72 ) α′ α′ a α′
The object gαα′ = eαaeaα′ is the parallel propagator: It takes a vector at x′ and parallel-transports it to x along the unique geodesic that links these points.

Similarly, we find that

α′ ′ α′ ′ α α′ ′ α′ ′ a A (x ) = g α(x ,x) A (x ), g α(x ,x) ≡ ea (x )eα(x ), (73 )
and we see that gαα′= eαa′ eaα performs the inverse operation: It takes a vector at x and parallel-transports it back to x′. Clearly,
α α′ α α′ α α′ g α′g β = δβ, g αg β′ = δ β′, (74 )
and these relations formally express the fact that gα′ α is the inverse of gα ′ α.

The relation α α a gα′ = eaeα′ can also be expressed as α ′ a α′ gα = eαea, and this reveals that

g α′(x, x′) = gα′(x′,x), g α(x ′,x) = gα (x,x′). (75 ) α α α′ α′
The ordering of the indices, and the ordering of the arguments, are therefore arbitrary.

The action of the parallel propagator on tensors of arbitrary ranks is easy to figure out. For example, suppose that the dual vector p = p ea μ a μ is parallel transported on β. Then the frame components μ pa = pμ ea are constants, and the dual vector at x can be expressed as α′ α pα = (p α′ea )ea, or

α′ ′ ′ pα(x) = g α(x ,x)pα′(x ). (76 )
It is therefore the inverse propagator gα′ α that takes a dual vector at x ′ and parallel-transports it to x. As another example, it is easy to show that a tensor αβ A at x obtained by parallel transport from ′ x must be given by
αβ α ′ β ′ α′β′ ′ A (x) = g α′(x, x)g β′(x,x )A (x). (77 )
Here we need two occurrences of the parallel propagator, one for each tensorial index. Because the metric tensor is covariantly constant, it is automatically parallel transported on β, and a special case of Equation (77View Equation) is therefore α′ β′ gαβ = g αg β gα′β′.

Because the basis vectors are parallel transported on β, they satisfy e αa;βσβ = 0 at x and eα′′σ β′ = 0 a;β at x ′. This immediately implies that the parallel propagators must satisfy

α β α β′ α′ β α′ β′ gα′;β σ = g α′;β′σ = 0, g α;βσ = g α;β′σ = 0. (78 )
Another useful property of the parallel propagator follows from the fact that if tμ = dzμ∕dλ is tangent to the geodesic connecting x to ′ x, then α α α′ t = g α′t. Using Equations (55View Equation) and (56View Equation), this observation gives us the relations
α′ α σα = − g ασα′, σα ′ = − g α′σ α. (79 )

2.3.3 Coincidence limits

Equation (72View Equation) and the completeness relations of Equations (69View Equation) or (71View Equation) imply that

[ ] ′ gαβ′ = δαβ′. (80 )
Other coincidence limits are obtained by differentiation of Equations (78View Equation). For example, the relation gα σγ = 0 β′;γ implies gα σγ + gα σγ = 0 β′;γδ β′;γ δ, and at coincidence we have
[ α ] [ α ] g β′;γ = gβ′;γ′ = 0; (81 )
the second result was obtained by applying Synge’s rule on the first result. Further differentiation gives
g αβ′;γδεσγ + gαβ′;γδσ γε + gαβ′;γεσ γ + gαβ′;γσγ = 0, δ δε

and at coincidence we have α α [g β′;γδ] + [gβ′;δγ] = 0, or α α′ 2 [g β′;γδ] + R β′γ′δ′ = 0. The coincidence limit for gαβ′;γδ′ = gαβ′;δ′γ can then be obtained from Synge’s rule, and an additional application of the rule gives [gαβ′;γ′δ′]. Our results are

[ ] [ ] g α′ = − 1-Rα′′ ′′, gα′ ′ = 1-Rα′′ ′′, β ;γδ 2 β γδ β ;γδ 2 β γδ [ ] 1 ′ [ ] 1 ′ (82 ) gαβ′;γ′δ = − --Rαβ′γ′δ′, gαβ′;γ′δ′ = --Rαβ′γ′δ′. 2 2

  Go to previous page Go up Go to next page