Parallel propagator

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

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

and we define a dual tetrad $a eμ(z)$ by

this is also parallel transported on $β$ . In terms of the dual tetrad the completeness relations take the form

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 (68

, 69

, 70

, 71

) 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

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

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,

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

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

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

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 (77

) 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

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 (55

) and (56

), this observation gives us the relations

2.3.3 Coincidence limits

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

Other coincidence limits are obtained by differentiation of Equations (78

). 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

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