11 Transformation between Fermi and retarded coordinates; advanced point

A point x in the normal convex neighbourhood of a world line γ can be assigned a set of Fermi normal coordinates (as in Section 9), or it can be assigned a set of retarded coordinates (Section 10). These coordinate systems can obviously be related to one another, and our first task in this section (which will occupy us in Sections 11.111.3) will be to derive the transformation rules. We begin by refining our notation so as to eliminate any danger of ambiguity.
View Image

Figure 8: The retarded, simultaneous, and advanced points on a world line γ. The retarded point x ′ := z(u) is linked to x by a future-directed null geodesic. The simultaneous point ¯x := z(t) is linked to x by a spacelike geodesic that intersects γ orthogonally. The advanced point x ′′ := z(v) is linked to x by a past-directed null geodesic.

The Fermi normal coordinates of x refer to a point ¯x := z(t) on γ that is related to x by a spacelike geodesic that intersects γ orthogonally; see Figure 8View Image. We refer to this point as x’s simultaneous point, and to tensors at ¯x we assign indices ¯α, ¯β, etc. We let (t,sωa ) be the Fermi normal coordinates of x, with t denoting the value of γ’s proper-time parameter at ¯x, ∘ -------- s = 2σ(x, ¯x) representing the proper distance from ¯x to x along the spacelike geodesic, and a ω denoting a unit vector (a b δabω ω = 1) that determines the direction of the geodesic. The Fermi normal coordinates are defined by sωa = − ea¯ασ ¯α and σα¯u ¯α = 0. Finally, we denote by (¯eα0,¯eαa) the tetrad at x that is obtained by parallel transport of (u¯α,eαa¯) on the spacelike geodesic.

The retarded coordinates of x refer to a point ′ x := z(u) on γ that is linked to x by a future-directed null geodesic; see Figure 8View Image. We refer to this point as x’s retarded point, and to tensors at x ′ we assign indices α ′, β′, etc. We let (u,rΩa) be the retarded coordinates of x, with u denoting the value of γ’s proper-time parameter at x′, r = σ α′u α′ representing the affine-parameter distance from x′ to x along the null geodesic, and Ωa denoting a unit vector (δ Ωa Ωb = 1 ab) that determines the direction of the geodesic. The retarded coordinates are defined by a a α′ rΩ = − eα′σ and σ (x, x′) = 0. Finally, we denote by (eα0,eαa) the tetrad at x that is obtained by parallel transport of (uα′,eαa′) on the null geodesic.

The reader who does not wish to follow the details of this discussion can be informed that: (i) our results concerning the transformation from the retarded coordinates a (u,r,Ω ) to the Fermi normal coordinates a (t,s,ω ) are contained in Eqs. (11.1View Equation) – (11.3View Equation) below; (ii) our results concerning the transformation from the Fermi normal coordinates (t,s,ωa) to the retarded coordinates (u, r,Ωa) are contained in Eqs. (11.4View Equation) – (11.6View Equation); (iii) the decomposition of each member of (¯eα0,¯eαa) in the tetrad (eα0,eαa) is given in retarded coordinates by Eqs. (11.7View Equation) and (11.8View Equation); and (iv) the decomposition of each member of α α (e0,ea) in the tetrad α α (¯e0,¯ea) is given in Fermi normal coordinates by Eqs. (11.9View Equation) and (11.10View Equation).

Our final task will be to define, along with the retarded and simultaneous points, an advanced point x ′′ on the world line γ; see Figure 8View Image. This is taken on in Section 11.4.

11.1 From retarded to Fermi coordinates

Quantities at ¯x := z(t) can be related to quantities at x′ := z(u) by Taylor expansion along the world line γ. To implement this strategy we must first find an expression for Δ := t − u. (Although we use the same notation, this should not be confused with the van Vleck determinant introduced in Section 7.)

Consider the function p(τ) of the proper-time parameter τ defined by

p(τ) = σ (x,z (τ ))uμ(τ), μ

in which x is kept fixed and in which z(τ) is an arbitrary point on the world line. We have that p(u ) = r and p(t) = 0, and Δ can ultimately be obtained by expressing p(t) as p(u + Δ) and expanding in powers of Δ. Formally,

p(t) = p (u ) + p˙(u )Δ + 1-¨p(u)Δ2 + 1p(3)(u)Δ3 + O(Δ4 ), 2 6

where overdots (or a number within brackets) indicate repeated differentiation with respect to τ. We have

′′ α′ β′ ′α′ p˙(u) = σ αβ u u′ +′ σ′αa , ′ ′ ′ ¨p(u) = σ α′β′γ′uα uβ uγ + 3σα′β′uα aβ + σα′a˙α, p(3)(u) = σ ′′ ′′u α′u β′u γ′uδ′ + σ ′′ ′(5a α′u β′uγ′ + uα′uβ′aγ′) αβ γδ( ′ ′ ′ α′β) γ ′ + σα′β′ 3aαa β + 4uα ˙aβ + σα′¨aα,
where μ μ a = Du ∕d τ, μ μ a˙ = Da ∕dτ, and μ μ ¨a = D a˙ ∕dτ.

We now express all of this in retarded coordinates by invoking the expansion of Eq. (6.7View Equation) for σα′β′ (as well as additional expansions for the higher derivatives of the world function, obtained by further differentiation of this result) and the relation σα′ = − r(uα′ + Ωae α′) a first derived in Eq. (10.7View Equation). With a degree of accuracy sufficient for our purposes we obtain

[ a 1- 2 3 ] ˙p(u) = − 1 + raaΩ + 3 r S + O (r ) , ( a) 2 ¨p(u) = − r ˙a0 + ˙aaΩ + O (r ), p(3)(u) = ˙a0 + O (r ),
where a b S = Ra0b0Ω Ω was first introduced in Eq. (10.25View Equation), and where α′ ˙a0 := ˙aα′u, α′ ˙aa := ˙aα′ea are the frame components of the covariant derivative of the acceleration vector. To arrive at these results we made use of the identity aα′a α′ + ˙aα′uα′ = 0 that follows from the fact that aμ is orthogonal to uμ. Notice that there is no distinction between the two possible interpretations a˙a := daa∕d τ and ˙aa := ˙aμeμ a for the quantity ˙aa(τ); their equality follows at once from the substitution of μ μ De a∕dτ = aau (which states that the basis vectors are Fermi–Walker transported on the world line) into the identity daa ∕dτ = D (aνeνa)∕dτ.

Collecting our results we obtain

[ ] [ ] [ ] r = 1 + ra Ωa + 1r2S + O (r3) Δ + 1r a˙ + ˙a Ωa + O(r) Δ2 − 1-a˙ + O (r) Δ3 + O (Δ4 ), a 3 2 0 a 6 0

which can readily be solved for Δ := t − u expressed as an expansion in powers of r. The final result is

{ } t = u + r 1 − ra (u )Ωa + r2[a (u)Ωa ]2 − 1-r2˙a (u) − 1r2˙a (u)Ωa − 1r2R (u)Ωa Ωb + O (r3)(1,1.1 ) a a 3 0 2 a 3 a0b0
where we show explicitly that all frame components are evaluated at the retarded point z(u).

To obtain relations between the spatial coordinates we consider the functions

p (τ ) = − σ (x,z(τ))eμ(τ), a μ a

in which x is fixed and z(τ) is an arbitrary point on γ. We have that the retarded coordinates are given by rΩa = pa(u), while the Fermi coordinates are given instead by sωa = pa(t) = pa(u + Δ ). This last expression can be expanded in powers of Δ, producing

sωa = pa(u) + ˙pa(u)Δ + 1¨pa(u)Δ2 + 1pa(3)(u)Δ3 + O (Δ4 ) 2 6

with

α′ β′ ( α′)( β′) p˙a(u) = − σα′β′ea u − σα′u aβ′e a 1 2 3 = − raa − --r Sa + O(r ), 3α′ β′ γ′ ( α′ β′ α′)( γ′) α′β′ ( α′)( β′) ¨pa(u) = − σα′β′γ′ea u u − 2σα′β′u u + σα′a aγ′ea − σα′β′e a a − σα′u ˙aβ′ea ( b) 1 b 2 = 1 + rabΩ aa − r˙aa + 3rRa0b0Ω + O (r ), (3) α′ β′ γ′ δ′ ( α′ β′ γ′ α′ β′ α′ α′ β′)( δ′) pa (u) = − σα′β′γ′δ′ea u u u − 3σα′β′γ′u u u + 6σ α′β′u a + σ α′˙a + σα′u ˙aβ′u a δ′ea − σ ′ ′′eα′(2aβ′u γ′ + uβ′aγ′) − (3 σ ′′uα′uβ′ + 2σ ′aα′)( ˙a ′eγ′) − σ ′′eα′˙aβ′ (α βγ ′a)( ′) αβ α γ a αβ a − σα′uα ¨aβ′eβa = 2a˙ + O (r). a
To arrive at these results we have used the same expansions as before and re-introduced Sa = Ra0b0Ωb − Rab0cΩbΩc, as it was first defined in Eq. (10.24View Equation).

Collecting our results we obtain

[ 1 ] 1[( ) 1 ] sωa = rΩa − r aa + -rSa + O (r2) Δ + -- 1 + rabΩb aa − r˙aa + -rRa0b0Ωb + O (r2) Δ2 [ 3] 2 3 + 1- ˙aa + O (r) Δ3 + O(Δ4 ), 3
which becomes
{ } a a 1 [ b] a 1 2 a 1 2 a b 1 2 a b c 3 sω = r Ω − -r 1 − rab(u )Ω a (u) − -r a˙(u ) −--r R 0b0(u )Ω + -r R b0c(u)Ω Ω + O(r )(11.2 ) 2 6 6 3
after substituting Eq. (11.1View Equation) for Δ := t − u. From squaring Eq. (11.2View Equation) and using the identity δabωaωb = 1 we can also deduce
{ } 1- a 3-2[ a]2 1-2 1-2 a 1- 2 a b 3 s = r 1 − 2raa(u)Ω + 8r aa(u)Ω − 8r ˙a0(u) − 6r a˙a (u )Ω − 6r Ra0b0(u )Ω Ω + O (r )(11.3 )
for the spatial distance between x and z(t).

11.2 From Fermi to retarded coordinates

The techniques developed in the preceding subsection can easily be adapted to the task of relating the retarded coordinates of x to its Fermi normal coordinates. Here we use ¯x := z(t) as the reference point and express all quantities at ′ x := z(u ) as Taylor expansions about τ = t.

We begin by considering the function

( ) σ(τ ) = σ x,z(τ)

of the proper-time parameter τ on γ. We have that σ(t) = 1s2 2 and σ (u) = 0, and Δ := t − u is now obtained by expressing σ(u) as σ(t − Δ ) and expanding in powers of Δ. Using the fact that σ˙(τ ) = p(τ), we have

1 1 1 σ (u ) = σ(t) − p(t)Δ + -p˙(t)Δ2 − -¨p(t)Δ3 + ---p(3)(t)Δ4 + O (Δ5 ). 2 6 24

Expressions for the derivatives of p(τ) evaluated at τ = t can be constructed from results derived previously in Section 11.1: it suffices to replace all primed indices by barred indices and then substitute the relation σ¯α = − sωae ¯αa that follows immediately from Eq. (9.5View Equation). This gives

[ ] p˙(t) = − 1 + saa ωa + 1s2Ra0b0ωa ωb + O (s3) , 3 ¨p(t) = − s˙aaωa + O (s2), (3) p (t) = ˙a0 + O(s),
and then
2 [ a 1-2 a b 3 ] 2 1-[ a ] 3 -1-[ ] 4 5 s = 1 + saaω + 3s Ra0b0ω ω + O(s ) Δ − 3s a˙a ω + O(s) Δ − 12 ˙a0 + O(s) Δ + O(Δ )

after recalling that p(t) = 0. Solving for Δ as an expansion in powers of s returns

{ 1 3 [ ] 1 1 1 } u = t − s 1 − -saa (t)ωa + -s2 aa(t)ωa 2 + --s2a˙0(t) + -s2˙aa(t)ωa − --s2Ra0b0(t)ωa ωb + O(s3)(11,.4 ) 2 8 24 6 6
in which we emphasize that all frame components are evaluated at the simultaneous point z(t).

An expression for r = p(u) can be obtained by expanding p(t − Δ ) in powers of Δ. We have

r = − p˙(t)Δ + 1¨p(t)Δ2 − 1-p(3)(t)Δ3 + O (Δ4 ), 2 6

and substitution of our previous results gives

{ [ ] } r = s 1 + 1-saa(t)ωa − 1s2 aa(t)ωa 2 − 1s2a˙0(t) − 1s2a˙a(t)ωa + 1-s2Ra0b0(t)ωa ωb + O(s3) (11.5 ) 2 8 8 3 6
for the retarded distance between x and z (u ).

Finally, the retarded coordinates rΩa = pa(u) can be related to the Fermi coordinates by expanding pa(t − Δ ) in powers of Δ, so that

a a a 1- a 2 1- a(3) 3 4 rΩ = p (t) − ˙p (t)Δ + 2 ¨p (t)Δ − 6p (t)Δ + O(Δ ).

Results from the preceding subsection can again be imported with mild alterations, and we find

˙pa(t) = 1s2Rab0cωbωc + O (s3), 3 ( b) 1- b 2 ¨pa(t) = 1 + sabω aa + 3sRa0b0ω + O (s ), p(3)(t) = 2 ˙a (t) + O (s). a a
This, together with Eq. (11.4View Equation), gives
{ } a a 1 a 1 2 a 1 2 a b c 1 2 a b 3 rΩ = s ω + -sa (t) − --sa˙ (t) − --sR b0c(t)ω ω + --s R 0b0(t)ω + O (s ) . (11.6 ) 2 3 3 6
It may be checked that squaring this equation and using the identity δabΩaΩb = 1 returns the same result as Eq. (11.5View Equation).

11.3 Transformation of the tetrads at x

Recall that we have constructed two sets of basis vectors at x. The first set is the tetrad (¯eα0,¯eαa) that is obtained by parallel transport of (u¯α,e¯α) a on the spacelike geodesic that links x to the simultaneous point ¯x := z(t). The second set is the tetrad α α (e0 ,ea) that is obtained by parallel transport of α′ α′ (u ,ea ) on the null geodesic that links x to the retarded point ′ x := z(u). Since each tetrad forms a complete set of basis vectors, each member of (e¯α0,¯eαa) can be decomposed in the tetrad (eα0 ,eαa), and correspondingly, each member of (eα0,eαa) can be decomposed in the tetrad (¯eα0,¯eαa). These decompositions are worked out in this subsection. For this purpose we shall consider the functions

α α ( ) μ α α( ) μ p (τ) = g μ x,z(τ) u (τ), pa(τ ) = g μ x, z(τ) ea(τ),

in which x is a fixed point in a neighbourhood of γ, z(τ ) is an arbitrary point on the world line, and gα (x,z) μ is the parallel propagator on the unique geodesic that links x to z. We have ¯eα = p α(t) 0, α α ¯ea = p a(t), α α e0 = p (u), and α α ea = p a(u).

We begin with the decomposition of (¯eα0 ,¯eαa) in the tetrad (eα0 ,eαa) associated with the retarded point z(u ). This decomposition will be expressed in the retarded coordinates as an expansion in powers of r. As in Section 9.1 we express quantities at z(t) in terms of quantities at z(u) by expanding in powers of Δ := t − u. We have

1 e¯α0 = pα(u) + ˙pα(u)Δ + -¨pα(u )Δ2 + O (Δ3), 2

with

˙pα(u) = gα uα′uβ′ + gα aα′ [ α′;β′ α′ ] a 1- a b 2 α = a + 2rR 0b0Ω + O (r ) ea, ¨pα(u) = gα uα′uβ′u γ′ + gα (2a α′u β′ + uα′aβ′) + gα ˙aα′ [ α′;β′γ′ ] [ α′;β′ ] α′ = − ˙a0 + O (r) eα + ˙aa + O (r) eα, 0 a
where we have used the expansions of Eq. (6.11View Equation) as well as the decompositions of Eq. (10.4View Equation). Collecting these results and substituting Eq. (11.1View Equation) for Δ yields
[ ] [ ] ¯eα= 1 − 1r2a˙(u ) + O (r3) eα+ r(1 − a Ωb)aa(u) + 1-r2˙aa(u) + 1r2Ra (u)Ωb + O (r3) eα(.11.7 ) 0 2 0 0 b 2 2 0b0 a
Similarly, we have
α α α 1-α 2 3 e¯a = pa(u) + ˙pa(u)Δ + 2¨pa(u )Δ + O (Δ ),

with

˙pα(u) = gα eα′u β′ + (gα u α′)(a ′eβ′) a [ α′;β′ a α′ ]β a [ ] 1- b 2 α 1- b c 2 α = aa + 2 rRa0b0Ω + O (r )e 0 + − 2 rR a0cΩ + O (r ) eb, ¨pα(u) = gα eα′uβ′uγ′ + gα (2uα′uβ′a ′eγ′+ eα′aβ′) + (gα aα ′)(a ′eβ′) + (g α uα′)(˙a ′eβ′) a [ α′;β′γ′ a ] [ α′;β′ ] γ a a α′ β a α′ β a = a˙a + O (r) eα+ aaab + O(r) eα, 0 b
and all this gives
[ 1 1 ] ¯eαa = δba + -r2ab(u )aa (u ) − -r2Rba0c(u)Ωc + O (r3) eαb [ 2 2 ] + r(1 − ra Ωb )a (u) + 1r2˙a (u) + 1-r2R (u )Ωb + O (r3) eα. (11.8 ) b a 2 a 2 a0b0 0

We now turn to the decomposition of α α (e0,ea) in the tetrad α α (¯e0,¯ea) associated with the simultaneous point z(t). This decomposition will be expressed in the Fermi normal coordinates as an expansion in powers of s. Here, as in Section 9.2, we shall express quantities at z(u) in terms of quantities at z(t). We begin with

α α α 1-α 2 3 e0 = p (t) − p˙(t)Δ + 2¨p (t)Δ + O(Δ )

and we evaluate the derivatives of pα(τ) at τ = t. To accomplish this we rely on our previous results (replacing primed indices with barred indices), on the expansions of Eq. (6.11View Equation), and on the decomposition of α g ¯α(x, ¯x) in the tetrads at x and x¯. This gives

[ 1 ] p˙α(t) = aa + --sRa0b0ωb + O (s2) ¯eαa, [ 2 ] [ ] ¨pα(t) = − ˙a0 + O(s) ¯eα + a˙a + O (s) ¯eα, 0 a
and we finally obtain
α [ 1-2 3 ] α [ ( 1- b) a 1-2 a 1-2 a b 3 ] α e0 = 1 − 2s ˙a0(t) + O (s ) ¯e0 + − s 1 − 2sab ω a (t) + 2s ˙a (t) − 2s R 0b0(t)ω + O(s ) ¯e(a1.1.9 )
Similarly, we write
α α α 1-α 2 3 ea = pa (t) − ˙pa(t)Δ + 2¨pa(t)Δ + O (Δ ),

in which we substitute

α [ 1- b 2 ] α [ 1- b c 2 ] α p˙a(t) = aa + 2sRa0b0 ω + O(s ) ¯e0 + − 2sR a0cω + O (s ) ¯eb, α [ ] α [ b ] α ¨pa(t) = ˙aa + O (s) ¯e0 + aaa + O (s) ¯eb,
as well as Eq. (11.4View Equation) for Δ := t − u. Our final result is
α [ b 1 2 b 1 2 b c 3] α ea = δ a + -s a (t)aa(t) + --s R a0c(t)ω + O (s ) ¯eb [ 2( ) 2 ] + − s 1 − 1-sabωb aa(t) + 1s2˙aa(t) − 1s2Ra0b0(u)ωb + O (s3) ¯eα0. (11.10 ) 2 2 2

11.4 Advanced point

It will prove convenient to introduce on the world line, along with the retarded and simultaneous points, an advanced point associated with the field point x. The advanced point will be denoted x′′ := z(v), with v denoting the value of the proper-time parameter at ′′ x; to tensors at this point we assign indices ′′ α, ′′ β, etc. The advanced point is linked to x by a past-directed null geodesic (refer back to Figure 8View Image), and it can be located by solving σ(x,x ′′) = 0 together with the requirement that ′′ σα (x,x′′) be a future-directed null vector. The affine-parameter distance between x and x ′′ along the null geodesic is given by

α′′ radv = − σ α′′u , (11.11 )
and we shall call this the advanced distance between x and the world line. Notice that radv is a positive quantity.

We wish first to find an expression for v in terms of the retarded coordinates of x. For this purpose we define Δ ′ := v − u and re-introduce the function σ (τ ) := σ (x, z(τ)) first considered in Section 11.2. We have that σ(v) = σ(u) = 0, and ′ Δ can ultimately be obtained by expressing σ(v) as σ(u + Δ ′) and expanding in powers of Δ ′. Recalling that ˙σ(τ) = p(τ), we have

σ(v) = σ (u ) + p(u )Δ′ + 1p˙(u )Δ ′2 + 1¨p(u)Δ ′3 +-1-p(3)(u)Δ ′4 + O (Δ′5). 2 6 24

Using the expressions for the derivatives of p(τ) that were first obtained in Section 11.1, we write this as

[ ] [ ] [ ] r = 1-1 + ra Ωa + 1r2S + O(r3) Δ ′ + 1r ˙a + ˙a Ωa + O (r) Δ ′2 − -1- ˙a + O(r) Δ ′3 + O (Δ ′4). 2 a 3 6 0 a 24 0

Solving for Δ ′ as an expansion in powers of r, we obtain

{ [ ]2 1 2 1 } v = u + 2r 1 − raa(u)Ωa + r2 aa(u)Ωa − --r2˙a0(u ) − -r2˙aa(u)Ωa − -r2Ra0b0(u)Ωa Ωb + O (r3(1)1.1,2 ) 3 3 3
in which all frame components are evaluated at the retarded point z(u).

Our next task is to derive an expression for the advanced distance r adv. For this purpose we observe that ′ radv = − p(v) = − p (u + Δ ), which we can expand in powers of ′ Δ := v − u. This gives

′ 1- ′2 1-(3) ′3 ′4 radv = − p(u ) − p˙(u )Δ − 2¨p(u)Δ − 6p (u)Δ + O (Δ ),

which then becomes

[ ] [ ] [ ] r = − r + 1 + ra Ωa + 1-r2S + O(r3) Δ ′ + 1-r ˙a + ˙a Ωa + O (r) Δ′2 − 1-˙a + O(r) Δ ′3 + O (Δ ′4). adv a 3 2 0 a 6 0

After substituting Eq. (11.12View Equation) for Δ ′ and witnessing a number of cancellations, we arrive at the simple expression

[ ] r = r 1 + 2r2a˙(u )Ωa + O (r3) . (11.13 ) adv 3 a

From Eqs. (10.29View Equation), (10.30View Equation), and (11.12View Equation) we deduce that the gradient of the advanced time v is given by

[ ] [ ] a 2 0 2 a ∂αv = 1 − 2raaΩ + O(r ) eα+ Ωa − 2raa + O(r ) eα, (11.14 )
where the expansion in powers of r was truncated to a sufficient number of terms. Similarly, Eqs. (10.30View Equation), (10.31View Equation), and (11.13View Equation) imply that the gradient of the advanced distance is given by
[( b 4-2 b 1- 2 ) 2-2 1- 2 3] a ∂αradv = 1 + rabΩ + 3r ˙abΩ + 3 r S Ωa + 3r a˙a + 6 r Sa + O (r ) eα [ 1 ] + − raa Ωa −--r2S + O (r3) e0α, (11.15 ) 2
where Sa and S were first introduced in Eqs. (10.24View Equation) and (10.25View Equation), respectively. We emphasize that in Eqs. (11.14View Equation) and (11.15View Equation), all frame components are evaluated at the retarded point z(u ).


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