Retarded coordinates

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

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 (138

) 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

which are the same as in Equation (115

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

and the parallel propagator from $x′$ to $x$ is given by

3.3.2 Definition of the retarded coordinates

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

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

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 (142

) is that

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

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 (145

) 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

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

Similarly, we can view

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

from which we also obtain $σ ′ uα′k β = 1 αβ$ . From this last result and Equation (147

) we deduce the important relation

In addition, combining the general statement $′ σα = − g αα′σα$ (cf. Equation (79

)) with Equation (144

) gives

the vector at $x$ is therefore obtained by parallel transport of $′ ′ uα + Ωaeαa$ on $β$ . From this and Equation (141

) we get the alternative expression

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

From this we infer that $kα$ satisfies the geodesic equation in affine-parameter form, $kα;βk β = 0$ , and Equation (150

) 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 (145

) and (149

), 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

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

Using the expansion for $σαα$ given by Equation (91

), we find that this becomes $′ ′ rθ = 2 − 13R α′β′σα σβ + 𝒪(r3)$ , or

after using Equation (144

). 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 (153 ) 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

This scalar field was already identified in Equation (145

): 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 (154

) 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 7

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 4

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,

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 (156 ). The identity

follows easily from Equations (160

, 161

, 162

) 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 (145 ). In these we substitute the standard expansions for $σ ′ ′ α β$ and $σ ′ α β$ , as given by Equations (88 ) and (89 ), as well as Equations (144 ) and (151 ). 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 (166

) follows immediately from Equations (152

) and (157

). From Equation (167

) and the identity $a ∂αr = Ωa ∂αˆx$ we also infer

where we have used the facts that $b Sa = SabΩ$ and $a S = SaΩ$ ; these last results were derived in Equations (161

) and (162

). It may be checked that Equation (168

) agrees with Equation (147

3.3.6 Metric near $γ$

It is straightforward (but fairly tedious) to invert the relations of Equations (164 ) and (165 ) 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 (140

). 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 (171 , 172 , 173 ) 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:

The inverse metric also is ambiguous on the world line.

To invert the curved-spacetime metric of Equations (171 , 172 , 173 ) 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 (150

) and (157

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

where we have substituted the identity of Equation (163

). Comparison with Equation (156

) 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

where the transformation matrix

satisfies the identity $a ΩaΩ A = 0$ .

We introduce the quantities

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

which satisfies the identities

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

while the transformation from $(r,θA )$ to $ˆxa$ is accomplished by

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

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 (160 , 161 , 162 ) 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 (196

, 197

, 198

) 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 (204 , 205 , 206 ) are substituted into the metric tensor of Equations (171 , 172 , 173 ) – 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 (185

). 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 (183

3.3 Retarded coordinates

3.3.1 Geometrical elements

3.3.2 Definition of the retarded coordinates

3.3.3 The scalar field and the vector field

3.3.4 Frame components of tensor fields on the world line

3.3.5 Coordinate displacements near

3.3.6 Metric near

3.3.7 Transformation to angular coordinates

3.3.8 Specialization to

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

3.3.5 Coordinate displacements near $γ$

3.3.6 Metric near $γ$

3.3.8 Specialization to $μ a = 0 = R μν$