10 Retarded coordinates

10.1 Geometrical elements

We introduce the same geometrical elements as in Section 9: 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 eμ a that is FW transported on the world line according to

μ De-a-= aauμ, (10.1 ) dτ
where aa(τ) = aμeμ a are the frame components of the acceleration vector. Finally, we have a dual tetrad (e0,ea) μ μ, with e0 = − u μ μ and ea = δabg eν μ μν b. The tetrad and its dual give rise to the completeness relations
gμν = − uμuν + δabeμaeνb, gμν = − e0μe0ν + δabeaμebν, (10.2 )
which are the same as in Eq. (9.4View Equation).

The Fermi normal coordinates of Section 9 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

g = − e0e0+ δ eaeb , (10.3 ) αβ α β ab α β
and the parallel propagator from x′ to x is given by
gαα′(x,x′) = − eα0 uα′ + eαaeaα ′, gαα′(x ′,x) = u α′e0α + eαa′eaα. (10.4 )

10.2 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; (10.5 )
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
a b1∕2 ′ α′ r := (δabˆx ˆx ) = uα σ , (10.6 )
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 ′ r = u α′σ α can still be called the retarded distance between the point x and the world line. Another consequence of Eq. (10.5View Equation) is that
α′ ( α′ a α′) σ = − r u + Ω ea , (10.7 )
where a a Ω := ˆx ∕r is a unit spatial vector that satisfies a b δabΩ Ω = 1.

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

α a ( a a b a α′ β′) a α′ β du = − kαdx , dˆx = − ra − ω bˆx + eα′σ β′u du − eα′σ βdx , (10.8 )
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 ′ 0 = σ (x + δx,x′ + δx ′) = σ αδxα + σα′δxα, and the first relation of Eq. (10.8View Equation) follows from the fact that a displacement along the world line is described by δx α′ = u α′δu.

10.3 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 ) (10.9 )
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
∂ r = − ( σ ′a α′ + σ ′ ′u α′u β′)k + σ ′uα′. (10.10 ) β α α β β αβ

Similarly, we can view

α ′ k α(x) = σ--(x,-x-) (10.11 ) 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′, (10.12 ) αβ α αβ r
from which we also obtain σα′βu α′k β = 1. From this last result and Eq. (10.10View Equation) we deduce the important relation
kα∂αr = 1. (10.13 )
In addition, combining the general statement α α α′ σ = − g α′σ – cf. Eq. (5.12View Equation) – with Eq. (10.7View Equation) gives
α α ( α′ a α′) k = g α′u + Ω ea ; (10.14 )
the vector at x is therefore obtained by parallel transport of uα′ + Ωaeαa′ on β. From this and Eq. (10.4View Equation) we get the alternative expression
kα = eα0 + Ωae αa, (10.15 )
which confirms that α k is a future-directed null vector field (recall that a a Ω = ˆx ∕r is a unit vector).

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

γ′ γ′ ( α′ α′ β′) rk α;β = σαβ − k ασβγ′u − kβσ αγ′u + σα′a + σ α′β′u u kαkβ. (10.16 )
From this we infer that kα satisfies the geodesic equation in affine-parameter form, kα kβ = 0 ;β, and Eq. (10.13View Equation) informs us that the affine parameter is in fact r. A displacement along a member of the congruence is therefore given by α α dx = k dr. Specializing to retarded coordinates, and using Eqs. (10.8View Equation) and (10.12View Equation), we find that this statement becomes du = 0 and dˆxa = (ˆxa∕r)dr, which integrate to u = constant 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 = constant, ˆx = rΩ (𝜃 ) (10.17 )
in the retarded coordinates. Here, the two angles 𝜃A (A = 1,2) serve to parameterize the unit vector Ωa, which is independent of r.

Eq. (10.16View Equation) also implies that the expansion of the congruence is given by

α σ αα − 2 𝜃 = k ;α = ---r----. (10.18 )
Using Eq. (6.10View Equation), we find that this becomes r𝜃 = 2 − 1R α′β′σ α′σ β′ + O (r3) 3, or
1-2( a a b) 3 r𝜃 = 2 − 3r R00 + 2R0a Ω + RabΩ Ω + O (r ) (10.19 )
after using Eq. (10.7View Equation). Here, R = R ′ ′uα′uβ′ 00 α β, R = R ′′uα′eβ′ 0a αβ a, and R = R ′ ′eα′eβ′ ab α β a b 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 Eq. (10.16View 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. (10.20 )
This scalar field was already identified in Eq. (10.8View Equation): it is numerically equal to the proper-time parameter of the world line at ′ x. We conclude that the geodesics to which α k is tangent are the generators of the null cone u = constant. As Eq. (10.17View 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 Ωa := ˆxa∕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.

10.4 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 ′ a α is the acceleration vector at x′, then as we have seen,

a (u) = a ′eα′ (10.21 ) a α a
are the frame components of the acceleration at proper time u.

Similarly,

α′ γ′ β′δ′ Ra0b0(u ) = R α′γ′β′δ′ea u eb u′ , Ra0bd(u ) = R α′γ′β′δ′eαa′uγ′eβb eδd′, α′ γ′β′ δ′ Racbd(u ) = R α′γ′β′δ′ea ec eb ed (10.22 )
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 , (10.23 ) ab a0b0 a0bc b0ac acbd ba Sa(u,𝜃A ) = SabΩb = Ra0b0Ωb − Rab0cΩbΩc, (10.24 ) S(u,𝜃A ) = S Ωa = R Ωa Ωb, (10.25 ) 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 Eq. (10.19View Equation). The identity

R00 + 2R0a Ωa + Rab ΩaΩb = δabSab − S (10.26 )
follows easily from Eqs. (10.23View Equation) – (10.25View Equation) and the definition of the Ricci tensor.

In Section 9 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.

10.5 Coordinate displacements near γ

The changes in the quasi-Cartesian retarded coordinates under a displacement of x are given by Eq. (10.8View Equation). In these we substitute the standard expansions for σα′β′ and σα′β, as given by Eqs. (6.7View Equation) and (6.8View Equation), as well as Eqs. (10.7View Equation) and (10.14View Equation). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:

( ) ( ) du = e0αdx α − Ωa ebαdxα , (10.27 ) [ 1 ]( ) dˆxa = − raa + --r2Sa + O (r3) e0αdx α [ 2( ) ] + δa + raa + 1r2Sa Ω + 1-r2Sa + O (r3) (ebdx α). (10.28 ) b 3 b 6 b α
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:

0 a ∂αu = eα[ − Ωaeα, ] (10.29 ) ∂ xˆa = − raa + 1r2Sa + O (r3) e0 α 2 α [ a ( a 1 2 a) 1 2 a 3] b + δ b + ra + 3-r S Ωb + 6r S b + O (r ) eα. (10.30 )
Notice that Eq. (10.29View Equation) follows immediately from Eqs. (10.15View Equation) and (10.20View Equation). From Eq. (10.30View Equation) and the identity ∂ r = Ω ∂ ˆxa α a α we also infer
[ a 1 2 3 ] 0 [( b 1 2 ) 1 2 3 ] a ∂αr = − raa Ω + --rS + O (r ) eα + 1 + rabΩ + -r S Ωa + -r Sa + O (r ) eα, (10.31 ) 2 3 6
where we have used the facts that b Sa = SabΩ and a S = SaΩ; these last results were derived in Eqs. (10.24View Equation) and (10.25View Equation). It may be checked that Eq. (10.31View Equation) agrees with Eq. (10.10View Equation).

10.6 Metric near γ

It is straightforward (but fairly tedious) to invert the relations of Eqs. (10.27View Equation) and (10.28View Equation) and solve for e0dx α α and ea dxα α. The results are

0 α [ a 1 2 3] [( 1 2 ) 1 2 3 ] a eαdx = 1 + raaΩ + 2-r S + O (r )du + 1 + 6-r S Ωa − 6r Sa + O (r ) dˆx , (10.32 ) [ 1 ] [ 1 1 ] eaαdxα = raa + -r2Sa + O (r3) du + δab − -r2Sab + -r2Sa Ωb + O (r3) dˆxb. (10.33 ) 2 6 6
These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad (e0,ea) α α at x. The metric is then computed by using the completeness relations of Eq. (10.3View Equation). We find
( a)2 2 2 2 3 guu = − 1 + raaΩ + r a − r S + O (r ), (10.34 ) ( b 2- 2 ) 2- 2 3 gua = − 1 + rabΩ + 3 r S Ωa + raa + 3 rSa + O(r ), (10.35 ) ( 1 ) 1 1 ( ) gab = δab − 1 + -r2S Ωa Ωb − -r2Sab + --r2 SaΩb + ΩaSb + O (r3), (10.36 ) 3 3 3
where a2 := δabaaab. We see (as was pointed out in Section 10.4) that the metric possesses a directional ambiguity on the world line: the metric at r = 0 still depends on the vector Ωa = ˆxa∕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 Eqs. (10.34View Equation) – (10.36View Equation) we obtain the metric of flat spacetime in the retarded coordinates. This we express as

( ) ηuu = − 1 + raaΩa 2 + r2a2, ( b) ηua = − 1 + rabΩ Ωa + raa, (10.37 ) ηab = δab − ΩaΩb.
In spite of the directional ambiguity, the metric of flat spacetime has a unit determinant everywhere, and it is easily inverted:
ηuu = 0, ηua = − Ωa, ηab = δab + r(aaΩb + Ωaab). (10.38 )
The inverse metric also is ambiguous on the world line.

To invert the curved-spacetime metric of Eqs. (10.34View Equation) – (10.36View Equation) we express it as g = η + h + O (r3) α β αβ αβ and treat 2 hαβ = O (r ) as a perturbation. The inverse metric is then αβ αβ αγ βδ 3 g = η − η η hγδ + O (r ), or

guu = 0, (10.39 ) ua a g = − Ω , (10.40 ) ab ab ( a b a b) 1-2 ab 1- 2( a b a b) 3 g = δ + r a Ω + Ω a + 3r S + 3 r S Ω + Ω S + O (r ). (10.41 )
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 Eqs. (10.13View Equation) and (10.20View Equation).

The metric determinant is computed from √ --- − g = 1 + 12ηαβhαβ + O (r3), which gives

√ --- 1 ( ) 1 ( ) − g = 1 − -r2 δabSab − S + O (r3) = 1 − -r2 R00 + 2R0a Ωa + Rab ΩaΩb + O(r3), (10.42 ) 6 6
where we have substituted the identity of Eq. (10.26View Equation). Comparison with Eq. (10.19View Equation) gives us the interesting relation √ --- − g = 12r𝜃 + O(r3), where 𝜃 is the expansion of the generators of the null cones u = constant.

10.7 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 a ˆx to A (r,𝜃 ), using the relations a a A ˆx = rΩ (𝜃 ). (Recall from Section 10.3 that the angles A 𝜃 are constant on the generators of the null cones u = constant, 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

dˆxa = Ωadr + rΩaAd 𝜃A, (10.43 )
where the transformation matrix
a ΩaA := ∂-Ω- (10.44 ) ∂ 𝜃A
satisfies the identity ΩaΩaA = 0.

We introduce the quantities

ΩAB := δabΩa Ωb , (10.45 ) A B
which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, ΩAB = diag(1,sin2𝜃 ). We use the inverse of ΩAB, denoted ΩAB, to raise upper-case Latin indices. We then define the new object
ΩA := δ ΩAB Ωb (10.46 ) a ab B
which satisfies the identities
ΩA Ωa = δA, ΩaΩA = δa − ΩaΩb. (10.47 ) a B B A b b
The second result follows from the fact that both sides are simultaneously symmetric in a and b, orthogonal to Ωa and Ωb, and have the same trace.

From the preceding results we establish that the transformation from ˆxa to (r,𝜃A) is accomplished by

∂ˆxa ∂ˆxa ----= Ωa, --A-= rΩaA, (10.48 ) ∂r ∂𝜃
while the transformation from (r,𝜃A ) to ˆxa is accomplished by
A ∂r--= Ω , ∂𝜃--= 1ΩA . (10.49 ) ∂ˆxa a ∂ˆxa r a
With these transformation rules it is easy to show that in the angular coordinates, the metric is given by
( a)2 2 2 2 3 guu = − 1 + raa Ω + r a − r S + O (r), (10.50 ) gur = − 1, (10.51 ) [ ] guA = r raa + 2r2Sa + O (r3) ΩaA, (10.52 ) [ 3 ] 2 1-2 a b 3 gAB = r ΩAB − 3r SabΩ AΩ B + O (r ) . (10.53 )
The results gru = − 1, grr = 0, and grA = 0 are exact, and they follow from the fact that in the retarded coordinates, kαdxα = − du and kα ∂α = ∂r.

The nonvanishing components of the inverse metric are

ur g = − 1, (10.54 ) grr = 1 + 2raa Ωa + r2S + O (r3), (10.55 ) [ ] grA = 1-raa + 2-r2Sa + O (r3) ΩAa, (10.56 ) r [ 3 ] AB 1- AB 1- 2 ab A B 3 g = r2 Ω + 3 r S Ωa Ωb + O (r ) . (10.57 )
The results uu g = 0, ur g = − 1, and uA g = 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

[ ] √ − g-= r2√ Ω- 1 − 1-r2(R + 2R Ωa + R Ωa Ωb) + O (r3) , (10.58 ) 6 00 0a ab
where Ω is the determinant of Ω AB; in the canonical parameterization, √ Ω-= sin𝜃.

10.8 Specialization to aμ = 0 = R μν

In this subsection we specialize our previous results to a situation where γ is a geodesic on which the Ricci tensor vanishes. We therefore set μ a = 0 = R μν everywhere on γ.

We have seen in Section 9.6 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 ℰ (u) ab and ℬ (u ) ab. The relations are Ra0b0 = ℰab, d Ra0bc = 𝜀bcdℬ a, and Racbd = δabℰcd + δcdℰab − δadℰbc − δbcℰad. These can be substituted into Eqs. (10.23View Equation) – (10.25View Equation) to give

A c c c d c d c d Sab(u,𝜃 ) = 2ℰab − Ωa ℰbcΩ − ΩbℰacΩ + δabℰbcΩ Ω + 𝜀acdΩ ℬ b + 𝜀bcdΩ ℬ a, (10.59 ) S (u,𝜃A ) = ℰ Ωb + 𝜀 Ωbℬc Ωd, (10.60 ) a A ab a b abc d S(u,𝜃 ) = ℰabΩ Ω . (10.61 )
In these expressions the dependence on retarded time u is contained in ℰ ab and ℬ ab, while the angular dependence is encoded in the unit vector a Ω.

It is convenient to introduce the irreducible quantities

∗ a b ℰ := ℰabΩ Ω , (10.62 ) ℰ∗ := (δ b− Ωa Ωb)ℰbcΩc, (10.63 ) a∗ a c c ∗ ℰ ab := 2ℰab − 2ΩaℰbcΩ − 2Ωb ℰacΩ + (δab + Ωa Ωb)ℰ , (10.64 ) ℬ∗a := 𝜀abcΩb ℬcdΩd, (10.65 ) ∗ c d ( e e ) c d ( e e ) ℬ ab := 𝜀acdΩ ℬ e δ b − Ω Ωb + 𝜀bcdΩ ℬ e δ a − Ω Ωa . (10.66 )
These are all orthogonal to Ωa: ℰ∗Ωa = ℬ∗Ωa = 0 a a and ℰ ∗Ωb = ℬ ∗ Ωb = 0 ab ab. In terms of these Eqs. (10.59View Equation) – (10.61View Equation) become
∗ ∗ ∗ ∗ ∗ ∗ ∗ Sab = ℰ ab + Ωa ℰb + ℰaΩb + Ωa Ωbℰ + ℬ ab + Ωa ℬb + ℬaΩb, (10.67 ) Sa = ℰ ∗a + Ωaℰ ∗ + ℬ ∗a, (10.68 ) ∗ S = ℰ . (10.69 )

When Eqs. (10.67View Equation) – (10.69View Equation) are substituted into the metric tensor of Eqs. (10.34View Equation) – (10.36View Equation) – in which aa is set equal to zero – we obtain the compact expressions

2 ∗ 3 guu = − 1 − r ℰ + O (r ), (10.70 ) 2-2( ∗ ∗) 3 gua = − Ωa + 3r ℰa + ℬ a + O (r ), (10.71 ) 1 ( ) gab = δab − ΩaΩb − -r2 ℰa∗b + ℬ∗ab + O (r3). (10.72 ) 3
The metric becomes
g = − 1 − r2ℰ∗ + O (r3), (10.73 ) uu gur = − 1, (10.74 ) 2 3( ∗ ∗) 4 guA = --r ℰA + ℬ A + O(r ), (10.75 ) 3 ( ) gAB = r2ΩAB − 1-r4 ℰ∗AB + ℬ ∗AB + O (r5) (10.76 ) 3
after transforming to angular coordinates using the rules of Eq. (10.48View Equation). Here we have introduced the projections
ℰ ∗ := ℰ∗Ωa = ℰ Ωa Ωb, (10.77 ) ∗ A a∗ Aa b ab A a b ∗ ℰAB := ℰabΩ AΩ B = 2ℰabΩ AΩ B + ℰ ΩAB, (10.78 ) ℬ ∗ := ℬ∗Ωa = 𝜀abcΩa Ωb ℬc Ωd, (10.79 ) ∗ A a∗ Aa b A cdd a b ℬAB := ℬabΩA ΩB = 2𝜀acdΩ ℬ bΩ(AΩ B). (10.80 )
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 Eq. (10.46View Equation).


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