5 Parallel propagator

5.1 Tetrad on β

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

De μ gμνeμaeνb = ηab, ---a-= 0, (5.1 ) d λ
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, (5.2 )
and we define a dual tetrad a eμ(z) by
a ab ν eμ := η gμνeb; (5.3 )
this is also parallel transported on β. In terms of the dual tetrad the completeness relations take the form
gμν = ηabeaμebν, (5.4 )
and it is easy to show that the tetrad and its dual satisfy a μ a eμe b = δb and a μ μ eνea = δν. Equations (5.1View Equation) – (5.4View Equation) hold everywhere on β. In particular, with an appropriate change of notation they hold at x′ and x; for example, gαβ = ηabeaeb α β is the metric at x.

(You will have noticed that we use sans-serif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.)

5.2 Definition and properties of the parallel propagator

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

A α(x) = gαα′(x, x′)A α′(x′), gαα′(x,x ′) := eαa(x)eaα′(x ′). (5.5 )
The object α α a gα′ = eaeα′ 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

Aα′(x′) = gα′(x ′,x)A α(x), gα′(x ′,x) := eα′(x′)ea(x ), (5.6 ) α α a α
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,
gα ′gα′= δα , gα′gα ′ = δα′′, (5.7 ) α β β α β β
and these relations formally express the fact that ′ gαα is the inverse of gαα′.

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

α′ ′ α′ ′ α ′ α ′ gα (x, x) = g α(x ,x), g α′(x ,x) = g α′(x,x ). (5.8 )
The ordering of the indices, and the ordering of the arguments, are arbitrary.

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

p (x) = gα′(x′,x)p ′(x′). (5.9 ) α α α
It is therefore the inverse propagator ′ gαα that takes a dual vector at x ′ and parallel-transports it to x. As another example, it is easy to show that a tensor A αβ 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 ). (5.10 )
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 Eq. (5.10View Equation) is g αβ = gα′gβ′gα′β′ α β.

Because the basis vectors are parallel transported on β, they satisfy α β e a;βσ = 0 at x and ′ ′ eαa;β′σβ = 0 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. (5.11 )
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 Eqs. (3.3View Equation) and (3.4View Equation), this observation gives us the relations
σα = − gα′ασα′, σα ′ = − gαα′σ α. (5.12 )

5.3 Coincidence limits

Eq. (5.5View Equation) and the completeness relations of Eqs. (5.2View Equation) or (5.4View Equation) imply that

[ ] gαβ′ = δα′β′. (5.13 )
Other coincidence limits are obtained by differentiation of Eqs. (5.11View Equation). For example, the relation gαβ′;γσγ = 0 implies gαβ′;γδσγ + gαβ′;γσγδ = 0, and at coincidence we have
[ ] [ ] gα ′ = gα′ ′ = 0; (5.14 ) β;γ β ;γ
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 γδ + gαβ′;γσγδ𝜖 = 0,

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

[ α ] 1- α′ [ α ] 1- α′ gβ′;γδ = − 2R β′γ′δ′, gβ′;γδ′ = 2R β′γ′δ′, (5.15 ) [gα ′ ′] = − 1R α′′′ ′, [gα′ ′ ′] = 1-R α′′ ′′. β;γδ 2 βγ δ β ;γ δ 2 β γδ

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