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3.3 Retarded coordinates

3.3.1 Geometrical elements

We introduce the same geometrical elements as in Section 3.2: We have a timelike curve γ described by relations zμ (τ ), its normalized tangent vector u μ = dzμ∕dτ, and its acceleration vector a μ = Du μ∕dτ. We also have an orthonormal triad μ ea that is transported on the world line according to

μ De-a-= aauμ + ω beμ, (138 ) dτ a b
where aa(τ) = aμeμ a are the frame components of the acceleration vector and ωab(τ) = − ωba(τ) is a prescribed rotation tensor. Here the triad is not Fermi–Walker transported: For added generality we allow the spatial vectors to rotate as they are transported on the world line. While ωab will be set to zero in most sections of this paper, the freedom to perform such a rotation can be useful and will be exploited in Section 5.4. It is easy to check that Equation (138View Equation) is compatible with the requirement that the tetrad (uμ,e μ) a be orthonormal everywhere on γ. Finally, we have a dual tetrad (e0 ,ea) μ μ, with 0 eμ = − uμ and a ab ν eμ = δ gμνeb. The tetrad and its dual give rise to the completeness relations
gμν = − uμuν + δabeμeν, g = − e0e0 + δ eaeb, (139 ) a b μν μ ν ab μ ν
which are the same as in Equation (115View Equation).

The Fermi normal coordinates of Section 3.2 were constructed on the basis of a spacelike geodesic connecting a field point x to the world line. The retarded coordinates are based instead on a null geodesic going from the world line to the field point. We thus let x be within the normal convex neighbourhood of γ, β be the unique future-directed null geodesic that goes from the world line to x, and x′ ≡ z(u) be the point at which β intersects the world line, with u denoting the value of the proper-time parameter at this point.

From the tetrad at ′ x we obtain another tetrad α α (e0,ea) at x by parallel transport on β. By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at x: e0α = − gαβeβ0 and eaα = δabgαβeβb. The metric at x can be then be expressed as

gαβ = − e0αe0β + δabeaαebβ, (140 )
and the parallel propagator from x′ to x is given by
α ′ α α a α′ ′ α′0 α′ a g α′(x,x ) = − e0 uα′ + eaeα ′, gα (x ,x) = u eα + ea eα. (141 )

3.3.2 Definition of the retarded coordinates

The quasi-Cartesian version of the retarded coordinates are defined by

ˆx0 = u, ˆxa = − eaα′(x′)σ α′(x,x ′), σ(x,x′) = 0; (142 )
the last statement indicates that ′ x and x are linked by a null geodesic. From the fact that α′ σ is a null vector we obtain
′ r ≡ (δabˆxaˆxb)1∕2 = uα′σα , (143 )
and r is a positive quantity by virtue of the fact that β is a future-directed null geodesic – this makes ′ σ α past-directed. In flat spacetime, ′ σα = − (x − x′)α, and in a Lorentz frame that is momentarily comoving with the world line, r = t − t′ > 0; with the speed of light set equal to unity, r is also the spatial distance between x′ and x as measured in this frame. In curved spacetime, the quantity ′ α′ r = u ασ can still be called the retarded distance between the point x and the world line. Another consequence of Equation (142View Equation) is that
( ) σα′ = − r u α′ + Ωae α′ , (144 ) a
where Ωa ≡ ˆxa ∕r is a spatial vector that satisfies δ ΩaΩb = 1 ab.

A straightforward calculation reveals that under a displacement of the point x, the retarded coordinates change according to

( ′ ′) ′ du = − kα dxα, dˆxa = − raa − ωabˆxb + eaα′σ αβ′uβ du − eaα′σαβ dx β, (145 )
where kα = σα∕r is a future-directed null vector at x that is tangent to the geodesic β. To obtain these results we must keep in mind that a displacement of x typically induces a simultaneous displacement of x′ because the new points x + δx and x′ + δx′ must also be linked by a null geodesic. We therefore have ′ ′ α α′ 0 = σ (x + δx, x + δx ) = σα δx + σα ′δx, and the first relation of Equation (145View Equation) follows from the fact that a displacement along the world line is described by ′ ′ δx α = u α δu.

3.3.3 The scalar field r(x) and the vector field kα (x)

If we keep ′ x linked to x by the relation ′ σ (x,x ) = 0, then the quantity

′ α′ ′ r(x) = σα′(x,x )u (x ) (146 )
can be viewed as an ordinary scalar field defined in a neighbourhood of γ. We can compute the gradient of r by finding how r changes under a displacement of x (which again induces a displacement of x ′). The result is
( ) ∂βr = − σα′a α′ + σα′β′u α′u β′ kβ + σα′βuα′. (147 )

Similarly, we can view

σα (x, x′) k α(x) = --------- (148 ) r(x)
as an ordinary vector field, which is tangent to the congruence of null geodesics that emanate from x ′. It is easy to check that this vector satisfies the identities
σ k β = k , σ ′k β = σα′, (149 ) αβ α αβ r
from which we also obtain σ ′ uα′k β = 1 αβ. From this last result and Equation (147View Equation) we deduce the important relation
kα∂ r = 1. (150 ) α
In addition, combining the general statement ′ σα = − g αα′σα (cf. Equation (79View Equation)) with Equation (144View Equation) gives
( ′ ′) kα = gαα′ u α + Ωae αa ; (151 )
the vector at x is therefore obtained by parallel transport of ′ ′ uα + Ωaeαa on β. From this and Equation (141View Equation) we get the alternative expression
kα = eα + Ωae α, (152 ) 0 a
which confirms that kα is a future-directed null vector field (recall that Ωa = ˆxa∕r is a unit vector).

The covariant derivative of k α can be computed by finding how the vector changes under a displacement of x. (It is in fact easier to first calculate how rkα changes, and then substitute our previous expression for ∂ βr.) The result is

′ ′ ( ′ ′ ′) rkα;β = σαβ − kασβγ′uγ − kβσαγ′uγ + σα′a α + σα′β′u αu β kαkβ. (153 )
From this we infer that kα satisfies the geodesic equation in affine-parameter form, kα;βk β = 0, and Equation (150View Equation) informs us that the affine parameter is in fact r. A displacement along a member of the congruence is therefore given by dxα = kα dr. Specializing to retarded coordinates, and using Equations (145View Equation) and (149View Equation), we find that this statement becomes du = 0 and a a dxˆ = (ˆx ∕r)dr, which integrate to u = const. and ˆxa = rΩa, respectively, with Ωa still denoting a constant unit vector. We have found that the congruence of null geodesics emanating from x′ is described by
a a A u = const., ˆx = r Ω (θ ) (154 )
in the retarded coordinates. Here, the two angles θA (A = 1,2) serve to parameterize the unit vector Ωa, which is independent of r.

Equation (153View Equation) also implies that the expansion of the congruence is given by

σ α − 2 θ = kα;α = ---α----. (155 ) r
Using the expansion for σαα given by Equation (91View Equation), we find that this becomes ′ ′ rθ = 2 − 13R α′β′σα σβ + 𝒪(r3), or
1- 2( a a b) 3 rθ = 2 − 3 r R00 + 2R0a Ω + Rab Ω Ω + 𝒪 (r ) (156 )
after using Equation (144View Equation). Here, α′ β′ R00 = R α′β′u u, α′ β′ R0a = R α′β′u ea, and α′ β′ Rab = R α′β′ea eb are the frame components of the Ricci tensor evaluated at ′ x. This result confirms that the congruence is singular at r = 0, because θ diverges as 2∕r in this limit; the caustic coincides with the point x ′.

Finally, we infer from Equation (153View Equation) that kα is hypersurface orthogonal. This, together with the property that α k satisfies the geodesic equation in affine-parameter form, implies that there exists a scalar field u(x ) such that

k α = − ∂ αu. (157 )
This scalar field was already identified in Equation (145View Equation): It is numerically equal to the proper-time parameter of the world line at x ′. We conclude that the geodesics to which k α is tangent are the generators of the null cone u = const. As Equation (154View Equation) indicates, a specific generator is selected by choosing a direction Ωa (which can be parameterized by two angles θA), and r is an affine parameter on each generator. The geometrical meaning of the retarded coordinates is now completely clear; it is illustrated in Figure 7View Image.
View Image

Figure 7: Retarded coordinates of a point x relative to a world line γ. The retarded time u selects a particular null cone, the unit vector a a Ω ≡ ˆx ∕r selects a particular generator of this null cone, and the retarded distance r selects a particular point on this generator. This figure is identical to Figure 4View Image.

3.3.4 Frame components of tensor fields on the world line

The metric at x in the retarded coordinates will be expressed in terms of frame components of vectors and tensors evaluated on the world line γ. For example, if α′ a is the acceleration vector at ′ x, then as we have seen,

′ aa(u) = aα′ eαa (158 )
are the frame components of the acceleration at proper time u.

Similarly,

′ ′ ′ ′ Ra0b0(u) = R α′γ′β′δ′ eαa u γeβb uδ , α′ γ′β′ δ′ Ra0bd(u) = R α′γ′β′δ′ ea u eb ed , (159 ) α′ γ′ β′ δ′ Racbd(u) = R α′γ′β′δ′ ea e c eb ed
are the frame components of the Riemann tensor evaluated on γ. From these we form the useful combinations
S (u,θA ) = R + R Ωc + R Ωc + R Ωc Ωd = S , (160 ) ab a0b0 a0bc b0ac acbd ba Sa(u,θA ) = SabΩb = Ra0b0Ωb − Rab0cΩbΩc, (161 ) S(u,θA ) = S Ωa = R Ωa Ωb, (162 ) a a0b0
in which the quantities Ωa ≡ ˆxa∕r depend on the angles θA only – they are independent of u and r.

We have previously introduced the frame components of the Ricci tensor in Equation (156View Equation). The identity

R00 + 2R0a Ωa + Rab ΩaΩb = δabSab − S (163 )
follows easily from Equations (160View Equation, 161View Equation, 162View Equation) and the definition of the Ricci tensor.

In Section 3.2 we saw that the frame components of a given tensor were also the components of this tensor (evaluated on the world line) in the Fermi normal coordinates. We should not expect this property to be true also in the case of the retarded coordinates: the frame components of a tensor are not to be identified with the components of this tensor in the retarded coordinates. The reason is that the retarded coordinates are in fact singular on the world line. As we shall see, they give rise to a metric that possesses a directional ambiguity at r = 0. (This can easily be seen in Minkowski spacetime by performing the coordinate transformation u = t − ∘x2--+-y2-+-z2.) Components of tensors are therefore not defined on the world line, although they are perfectly well defined for r ⁄= 0. Frame components, on the other hand, are well defined both off and on the world line, and working with them will eliminate any difficulty associated with the singular nature of the retarded coordinates.

3.3.5 Coordinate displacements near γ

The changes in the quasi-Cartesian retarded coordinates under a displacement of x are given by Equation (145View Equation). In these we substitute the standard expansions for σ ′ ′ α β and σ ′ α β, as given by Equations (88View Equation) and (89View Equation), as well as Equations (144View Equation) and (151View Equation). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:

( ) ( ) du = e0αdx α − Ωa ebαdx α , (164 ) [ ] ( ) dˆxa = − raa − rωabΩb + 1-r2Sa + 𝒪 (r3) e0α dxα 2 [ ( 1 ) 1 ] ( ) + δab + raa − rωacΩc + --r2Sa Ωb + -r2Sab + 𝒪 (r3) ebα dxα . (165 ) 3 6
Notice that the result for du is exact, but that dˆxa is expressed as an expansion in powers of r.

These results can also be expressed in the form of gradients of the retarded coordinates:

∂ u = e0 − Ω ea, (166 ) α α[ a α ] a a a b 1- 2 a 3 0 ∂αˆx = − ra − rω bΩ + 2r S + 𝒪 (r ) eα [ ( ) ] + δab + raa − rωacΩc + 1-r2Sa Ωb + 1r2Sab + 𝒪(r3) ebα. (167 ) 3 6
Notice that Equation (166View Equation) follows immediately from Equations (152View Equation) and (157View Equation). From Equation (167View Equation) and the identity a ∂αr = Ωa ∂αˆx we also infer
[ ] [( ) ] a 1- 2 3 0 b 1- 2 1-2 3 a ∂αr = − raa Ω + 2 r S + 𝒪 (r ) eα + 1 + rabΩ + 3r S Ωa + 6r Sa + 𝒪 (r ) eα, (168 )
where we have used the facts that b Sa = SabΩ and a S = SaΩ; these last results were derived in Equations (161View Equation) and (162View Equation). It may be checked that Equation (168View Equation) agrees with Equation (147View Equation).

3.3.6 Metric near γ

It is straightforward (but fairly tedious) to invert the relations of Equations (164View Equation) and (165View Equation) and solve for e0 dxα α and eadx α α. The results are

[ ] [ ( ) ] 0 α a 1 2 3 1 2 1 2 3 a eα dx = 1 + raaΩ + 2r S + 𝒪(r ) du + 1 + 6-r S Ωa − 6r Sa + 𝒪 (r ) dˆx , (169 ) [ ] [ ] a α ( a a b) 1- 2 a 3 a 1- 2 a 1- 2 a 3 b eα dx = r a − ω bΩ + 2r S + 𝒪 (r ) du + δb − 6 r S b + 6 r S Ωb + 𝒪 (r ) dˆx . (170 )
These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad (e0α,eaα) at x. The metric is then computed by using the completeness relations of Equation (140View Equation). We find
ds2 = guu du2 + 2guadud ˆxa + gabdˆxadxˆb,

with

( ) guu = − (1 + raaΩa)2 + r2 aa − ωabΩb (aa − ωacΩc) − r2S + 𝒪 (r3), (171 ) ( ) g = − 1 + ra Ωb + 2r2S Ω + r(a − ω Ωb ) + 2r2S + 𝒪 (r3), (172 ) ua b 3 a a ab 3 a ( 1 ) 1 1 gab = δab − 1 + -r2S ΩaΩb − -r2Sab + -r2 (SaΩb + ΩaSb ) + 𝒪 (r3). (173 ) 3 3 3
We see (as was pointed out in Section 3.3.4) that the metric possesses a directional ambiguity on the world line: The metric at r = 0 still depends on the vector a a Ω = xˆ ∕r that specifies the direction to the point x. The retarded coordinates are therefore singular on the world line, and tensor components cannot be defined on γ.

By setting Sab = Sa = S = 0 in Equations (171View Equation, 172View Equation, 173View Equation) we obtain the metric of flat spacetime in the retarded coordinates. This we express as

( ) ηuu = − (1 + raaΩa)2 + r2 aa − ωabΩb (aa − ωacΩc) , ( b) ( b) ηua = − 1 + rabΩ Ωa + r aa − ωab Ω , (174 ) ηab = δab − Ωa Ωb.
In spite of the directional ambiguity, the metric of flat spacetime has a unit determinant everywhere, and it is easily inverted:
uu ua a ab ab a a c b a( b b c) η = 0, η = − Ω , η = δ + r(a − ω cΩ )Ω + rΩ a − ω cΩ . (175 )
The inverse metric also is ambiguous on the world line.

To invert the curved-spacetime metric of Equations (171View Equation, 172View Equation, 173View Equation) we express it as gαβ = η αβ + h αβ + 𝒪(r3) and treat hαβ = 𝒪 (r2) as a perturbation. The inverse metric is then gαβ = ηαβ − ηαγηβδhγδ + 𝒪 (r3), or

guu = 0, (176 ) gua = − Ωa, (177 ) ab ab a a c b a( b b c) 1 2 ab 1 2( a b a b) 3 g = δ + r (a − ω cΩ )Ω + r Ω a − ω cΩ + -r S + -r S Ω + Ω S + 𝒪 (r ). (178 ) 3 3
The results for uu g and ua g are exact, and they follow from the general relations αβ g (∂αu)(∂βu) = 0 and gαβ(∂αu)(∂βr) = − 1 that are derived from Equations (150View Equation) and (157View Equation).

The metric determinant is computed from √ − g-= 1 + 1ηαβhαβ + 𝒪 (r3) 2, which gives

√--- 1-2( ab ) 3 1-2 ( a a b) 3 − g = 1 − 6r δ Sab − S + 𝒪 (r ) = 1 − 6r R00 + 2R0aΩ + RabΩ Ω + 𝒪 (r ), (179 )
where we have substituted the identity of Equation (163View Equation). Comparison with Equation (156View Equation) then gives us the interesting relation √ --- 1 − g = 2rθ + 𝒪 (r3), where θ is the expansion of the generators of the null cones u = const.

3.3.7 Transformation to angular coordinates

Because the vector Ωa = ˆxa∕r satisfies δabΩa Ωb = 1, it can be parameterized by two angles θA. A canonical choice for the parameterization is Ωa = (sin θcos φ,sinθ sin φ,cos θ). It is then convenient to perform a coordinate transformation from ˆxa to (r,θA), using the relations ˆxa = rΩa(θA). (Recall from Section 3.3.3 that the angles A θ are constant on the generators of the null cones u = const., and that r is an affine parameter on these generators. The relations ˆxa = rΩa therefore describe the behaviour of the generators.) The differential form of the coordinate transformation is

a a a A d ˆx = Ω dr + rΩA dθ , (180 )
where the transformation matrix
a ∂Ωa- ΩA ≡ ∂θA (181 )
satisfies the identity a ΩaΩ A = 0.

We introduce the quantities

ΩAB = δabΩaAΩbB, (182 )
which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, 2 ΩAB = diag(1,sin θ). We use the inverse of ΩAB, denoted AB Ω, to raise upper-case latin indices. We then define the new object
ΩA = δ ΩAB Ωb (183 ) a ab B
which satisfies the identities
A a A a A a a Ωa ΩB = δB, ΩAΩ b = δ b − Ω Ωb. (184 )
The second result follows from the fact that both sides are simultaneously symmetric in a and b, orthogonal to Ωa and b Ω, and have the same trace.

From the preceding results we establish that the transformation from a ˆx to A (r,θ ) is accomplished by

∂ˆxa ∂ˆxa ----= Ωa, ----= rΩaA, (185 ) ∂r ∂θA
while the transformation from (r,θA ) to ˆxa is accomplished by
∂r ∂θA 1 A --a-= Ωa, --a-= -Ω a. (186 ) ∂ˆx ∂ˆx r
With these transformation rules it is easy to show that in the angular coordinates, the metric takes the form of
ds2 = guu du2 + 2gur dudr + 2guAdud θA + gAB dθAd θB,

with

( ) guu = − (1 + raaΩa )2 + r2 aa − ωab Ωb (aa − ωacΩc ) − r2S + 𝒪 (r3), (187 ) gur = −[1, ] (188 ) ( b) 2-2 3 a guA = r r aa − ωabΩ + 3r Sa + 𝒪 (r ) Ω A, (189 ) [ ] gAB = r2 ΩAB − 1r2SabΩa Ωb + 𝒪 (r3) . (190 ) 3 A B
The results gru = − 1, grr = 0, and grA = 0 are exact, and they follow from the fact that in the retarded coordinates, α kαdx = − du and α k ∂α = ∂r.

The nonvanishing components of the inverse metric are

gur = − 1, (191 ) grr = 1 + 2raaΩa + r2S + 𝒪 (r3), (192 ) 1 [ ( ) 2 ] grA = -- r aa − ωabΩb + --r2Sa + 𝒪 (r3) ΩAa, (193 ) r [ 3 ] AB -1 AB 1- 2 ab A B 3 g = r2 Ω + 3 r S Ω aΩb + 𝒪(r ) . (194 )
The results guu = 0, gur = − 1, and guA = 0 are exact, and they follow from the same reasoning as before.

Finally, we note that in the angular coordinates, the metric determinant is given by

√ --- √ --[ 1 ( ) ] − g = r2 Ω 1 − -r2 R00 + 2R0aΩa + RabΩa Ωb + 𝒪 (r3) , (195 ) 6
where Ω is the determinant of ΩAB; in the canonical parameterization, √ -- Ω = sinθ.

3.3.8 Specialization to μ a = 0 = R μν

In this section we specialize our previous results to a situation where γ is a geodesic on which the Ricci tensor vanishes. We therefore set aμ = 0 = R μν everywhere on γ, and for simplicity we also set ωab to zero.

We have seen in Section 3.2.6 that when the Ricci tensor vanishes on γ, all frame components of the Riemann tensor can be expressed in terms of the symmetric-tracefree tensors ℰab(u) and ℬab (u ). The relations are Ra0b0 = ℰab, Ra0bc = ɛbcdℬd a, and Racbd = δabℰcd + δcdℰab − δadℰbc − δbcℰad. These can be substituted into Equations (160View Equation, 161View Equation, 162View Equation) to give

Sab(u,θA ) = 2ℰab − Ωa ℰbcΩc − ΩbℰacΩc + δabℰbcΩcΩd + ɛacdΩcℬdb + ɛbcdΩc ℬda, (196 ) A b b c d Sa(u,θ ) = ℰabΩ + ɛabcΩ ℬ dΩ , (197 ) S(u,θA ) = ℰabΩa Ωb. (198 )
In these expressions the dependence on retarded time u is contained in ℰab and ℬab, while the angular dependence is encoded in the unit vector a Ω.

It is convenient to introduce the irreducible quantities

ℰ ∗ = ℰabΩa Ωb, (199 ) ∗ ( b b) c ℰa = δa − ΩaΩ ℰbcΩ , (200 ) ℰ∗ab = 2 ℰab − 2Ωa ℰbcΩc − 2ΩbℰacΩc + (δab + ΩaΩb )ℰ ∗, (201 ) ∗ b c d ℬa = ɛabcΩ ℬ dΩ , (202 ) ℬ ∗ab = ɛacdΩcℬde (δeb − ΩeΩb ) + ɛbcdΩcℬde (δea − Ωe Ωa). (203 )
These are all orthogonal to a Ω: ∗ a ∗ a ℰaΩ = ℬaΩ = 0 and ∗ b ∗ b ℰabΩ = ℬ abΩ = 0. In terms of these Equations (196View Equation, 197View Equation, 198View Equation) become
S = ℰ ∗ + Ω ℰ∗+ ℰ∗Ω + Ω Ω ℰ ∗ + ℬ ∗ + Ω ℬ∗ + ℬ ∗Ω , (204 ) ab ab a b a b a b ab a b a b Sa = ℰ ∗a + Ωaℰ ∗ + ℬ ∗a, (205 ) S = ℰ ∗. (206 )

When Equations (204View Equation, 205View Equation, 206View Equation) are substituted into the metric tensor of Equations (171View Equation, 172View Equation, 173View Equation) – in which aa and ωab are both set equal to zero – we obtain the compact expressions

2 ∗ 3 guu = − 1 − r ℰ + 𝒪 (r ), (207 ) 2-2 ∗ ∗ 3 gua = − Ωa + 3r (ℰa + ℬ a) + 𝒪 (r ), (208 ) 1 gab = δab − ΩaΩb − -r2(ℰ ∗ab + ℬa∗b) + 𝒪 (r3). (209 ) 3
The metric becomes
g = − 1 − r2ℰ∗ + 𝒪 (r3), (210 ) uu gur = − 1, (211 ) 2 3 ∗ ∗ 4 guA = --r (ℰA + ℬ A) + 𝒪 (r ), (212 ) 3 gAB = r2ΩAB − 1-r4(ℰ∗AB + ℬ ∗AB ) + 𝒪 (r5) (213 ) 3
after transforming to angular coordinates using the rules of Equation (185View Equation). Here we have introduced the projections
ℰ∗ ≡ ℰ ∗Ωa = ℰ Ωa Ωb, (214 ) ∗A a∗ Aa b ab A a b ∗ ℰ AB ≡ ℰabΩ AΩ B = 2ℰabΩ AΩ B + ℰ ΩAB, (215 ) ℬ ∗ ≡ ℬ ∗Ωa = ɛabcΩa Ωbℬc Ωd, (216 ) ∗A a∗ Aa b A cdd a b ℬ AB ≡ ℬ abΩ AΩ B = 2ɛacdΩ ℬ bΩ (AΩ B). (217 )
It may be noted that the inverse relations are ℰ∗ = ℰ∗ ΩA a A a, ℬ ∗= ℬ∗ΩA a A a, ℰ∗ = ℰ∗ ΩA ΩB ab AB a b, and ∗ ∗ A B ℬ ab = ℬ AB ΩaΩ b, where A Ωa was introduced in Equation (183View Equation).
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