go to next pagego upgo to previous page

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 g described by relations zm(t), its normalized tangent vector um = dzm/dt, and its acceleration vector m m a = Du /dt. We also have an orthonormal triad m ea that is transported on the world line according to

m De-a-= aaum + w bem, (138) dt a b
where aa(t) = amem a are the frame components of the acceleration vector and wab(t) = - wba(t) 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 wab 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 (um,em ) a be orthonormal everywhere on g. Finally, we have a dual tetrad (e0,ea) m m, with 0 em = - um and a ab n em = d gmneb. The tetrad and its dual give rise to the completeness relations
gmn = - umun + dabemen, g = - e0e0 + d eaeb, (139) a b mn m n ab m n
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 g, b be the unique future-directed null geodesic that goes from the world line to x, and x' =_ z(u) be the point at which b 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 a a (e0,ea) at x by parallel transport on b. By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at x: b e0a = -gabe 0 and eaa = dabgabeb b. The metric at x can be then be expressed as

0 0 a b gab = - eaeb + dabeaeb, (140)
and the parallel propagator from x' to x is given by
a ' a a a a' ' a'0 a' a g a'(x,x ) = -e0 ua'+ eaea', ga(x ,x) = u ea + ea ea. (141)

3.3.2 Definition of the retarded coordinates

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

^x0 = u, ^xa = - eaa'(x')sa'(x,x'), s(x,x') = 0; (142)
the last statement indicates that ' x and x are linked by a null geodesic. From the fact that a' s is a null vector we obtain
' r =_ (dab^xa^xb)1/2 = ua'sa , (143)
and r is a positive quantity by virtue of the fact that b is a future-directed null geodesic - this makes ' sa past-directed. In flat spacetime, ' sa = - (x - x')a, 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 ' a' r = ua s can still be called the retarded distance between the point x and the world line. Another consequence of Equation (142View Equation) is that
( ) sa'= - r ua'+ _O_aea' , (144) a
where _O_a =_ ^xa/r is a spatial vector that satisfies d _O_a_O_b = 1 ab.

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

( ' ') ' du = -ka dxa, d^xa = - raa - wab^xb + eaa'sab'ub du - eaa'sab dxb, (145)
where ka = sa/r is a future-directed null vector at x that is tangent to the geodesic b. 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 + dx and x'+ dx' must also be linked by a null geodesic. We therefore have ' ' a a' 0 = s(x + dx,x + dx ) = sa dx + sa'dx, and the first relation of Equation (145View Equation) follows from the fact that a displacement along the world line is described by ' ' dxa = ua du.

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

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

' a' ' r(x) = sa'(x,x )u (x ) (146)
can be viewed as an ordinary scalar field defined in a neighbourhood of g. 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
( ) @br = - sa'aa'+ sa'b'ua'ub' kb + sa'bua'. (147)

Similarly, we can view

sa(x, x') ka(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
s kb = k , s 'kb = sa', (149) ab a ab r
from which we also obtain s ' ua'kb = 1 a b. From this last result and Equation (147View Equation) we deduce the important relation
ka@ r = 1. (150) a
In addition, combining the general statement ' sa = -gaa'sa (cf. Equation (79View Equation)) with Equation (144View Equation) gives
( ' ') ka = gaa' ua + _O_aeaa ; (151)
the vector at x is therefore obtained by parallel transport of ' ' ua + _O_aeaa on b. From this and Equation (141View Equation) we get the alternative expression
ka = ea + _O_aea , (152) 0 a
which confirms that ka is a future-directed null vector field (recall that _O_a = ^xa/r is a unit vector).

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

' ' ( ' ' ') rka;b = sab - kasbg'ug - kbsag'ug + sa'aa + sa'b'ua ub kakb. (153)
From this we infer that ka satisfies the geodesic equation in affine-parameter form, ka;bkb = 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 dxa = ka dr. Specializing to retarded coordinates, and using Equations (145View Equation) and (149View Equation), we find that this statement becomes du = 0 and a a d^x = (x^ /r)dr, which integrate to u = const. and ^xa = r_O_a, respectively, with _O_a 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., ^x = r_O_ (h ) (154)
in the retarded coordinates. Here, the two angles hA (A = 1,2) serve to parameterize the unit vector _O_a, which is independent of r.

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

sa - 2 h = ka;a = ---a----. (155) r
Using the expansion for saa given by Equation (91View Equation), we find that this becomes ' ' rh = 2- 13Ra'b'sa sb + O(r3), or
1- 2( a a b) 3 rh = 2 - 3 r R00 + 2R0a_O_ + Rab_O_ _O_ + O(r ) (156)
after using Equation (144View Equation). Here, a' b' R00 = Ra'b'u u, a'b' R0a = Ra'b'u ea, and a'b' Rab = Ra'b'e a eb are the frame components of the Ricci tensor evaluated at ' x. This result confirms that the congruence is singular at r = 0, because h diverges as 2/r in this limit; the caustic coincides with the point x'.

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

ka = -@au. (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 ka 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 _O_a (which can be parameterized by two angles hA), 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 g. The retarded time u selects a particular null cone, the unit vector a a _O_ =_ x^ /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 g. For example, if a' a is the acceleration vector at ' x, then as we have seen,

' aa(u) = aa'eaa (158)
are the frame components of the acceleration at proper time u.

Similarly,

' ' ' ' Ra0b0(u) = Ra'g'b'd'eaa ug ebb ud , a' g'b' d' Ra0bd(u) = Ra'g'b'd'ea u eb ed , (159) a' g' b' d' Racbd(u) = Ra'g'b'd'ea e c eb ed
are the frame components of the Riemann tensor evaluated on g. From these we form the useful combinations
S (u,hA) = R + R _O_c + R _O_c + R _O_c_O_d = S , (160) ab a0b0 a0bc b0ac acbd ba Sa(u,hA) = Sab_O_b = Ra0b0_O_b - Rab0c_O_b_O_c, (161) S(u,hA) = S _O_a = R _O_a_O_b, (162) a a0b0
in which the quantities _O_a =_ ^xa/r depend on the angles hA 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_O_a + Rab_O_a_O_b = dabSab - 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 - V~ 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 g

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 s ' ' a b and s ' a b, 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 = e0adxa - _O_a ebadxa , (164) [ ]( ) d^xa = - raa - rwab_O_b + 1-r2Sa + O(r3) e0a dxa 2 [ ( 1 ) 1 ]( ) + dab + raa - rwac_O_c + --r2Sa _O_b + -r2Sab + O(r3) eba dxa . (165) 3 6
Notice that the result for du is exact, but that d^xa 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 - _O_ ea, (166) a a[ a a ] a a a b 1- 2 a 3 0 @a^x = - ra - rw b_O_ + 2r S + O(r ) ea [ ( ) ] + dab + raa - rwac_O_c + 1-r2Sa _O_b + 1r2Sab + O(r3) eba. (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 @ar = _O_a@a ^x we also infer
[ 1 ] [( 1 ) 1 ] @ar = - raa_O_a + --r2S + O(r3) e0a + 1 + rab_O_b + -r2S _O_a + -r2Sa + O(r3) eaa, (168) 2 3 6
where we have used the facts that S = S _O_b a ab and S = S _O_a a; 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 g

It is straightforward (but fairly tedious) to invert the relations of Equations (164View Equation) and (165View Equation) and solve for e0a dxa and eaa dxa. The results are

[ ] [( ) ] 0 a a 1 2 3 1 2 1 2 3 a ea dx = 1 + raa_O_ + 2r S + O(r ) du + 1 + 6-r S _O_a - 6r Sa + O(r ) d^x , (169) [ ] [ ] a a ( a a b) 1- 2 a 3 a 1- 2 a 1- 2 a 3 b ea dx = r a - w b_O_ + 2r S + O(r ) du + db - 6 r S b + 6 r S _O_b + O(r ) d^x . (170)
These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad (e0a,eaa) at x. The metric is then computed by using the completeness relations of Equation (140View Equation). We find
ds2 = guu du2 + 2guadud^xa + gabd^xad^xb,

with

( ) guu = - (1 + raa_O_a)2 + r2 aa - wab_O_b (aa - wac_O_c) - r2S + O(r3), (171) ( ) g = - 1 + ra _O_b + 2r2S _O_ + r(a - w _O_b)+ 2r2S + O(r3), (172) ua b 3 a a ab 3 a ( 1 ) 1 1 gab = dab - 1 + -r2S _O_a_O_b - -r2Sab + -r2 (Sa_O_b + _O_aSb) + 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 _O_ = ^x /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 g.

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

( ) juu = - (1 + raa_O_a)2 + r2 aa - wab_O_b (aa - wac_O_c) , ( b) ( b) jua = - 1 + rab_O_ _O_a + r aa- wab_O_ , (174) jab = dab- _O_a_O_b.
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) j = 0, j = - _O_ , j = d + r(a - w c_O_ )_O_ + r_O_ a - w c_O_ . (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 gab = jab + hab + O(r3) and treat hab = O(r2) as a perturbation. The inverse metric is then gab = jab- jagjbdhgd + O(r3), or

guu = 0, (176) gua = - _O_a, (177) ab ab a a c b a( b b c) 1 2 ab 1 2( a b a b) 3 g = d + r (a - w c_O_ )_O_ + r_O_ a - w c_O_ + -r S + -r S _O_ + _O_ S + O(r ). (178) 3 3
The results for uu g and ua g are exact, and they follow from the general relations ab g (@au)(@bu) = 0 and gab(@au)(@br) = - 1 that are derived from Equations (150View Equation) and (157View Equation).

The metric determinant is computed from V~ -g-= 1 + 1jabhab + O(r3) 2, which gives

V~ -- 1-2( ab ) 3 1-2 ( a a b) 3 - g = 1 - 6r d Sab - S + O(r ) = 1- 6r R00 + 2R0a_O_ + Rab_O_ _O_ + O(r ), (179)
where we have substituted the identity of Equation (163View Equation). Comparison with Equation (156View Equation) then gives us the interesting relation V~ --- 1 - g = 2rh + O(r3), where h is the expansion of the generators of the null cones u = const.

3.3.7 Transformation to angular coordinates

Because the vector _O_a = ^xa/r satisfies dab_O_a_O_b = 1, it can be parameterized by two angles hA. A canonical choice for the parameterization is _O_a = (sin hcos f,sinh sin f,cos h). It is then convenient to perform a coordinate transformation from ^xa to (r,hA), using the relations x^a = r_O_a(hA). (Recall from Section 3.3.3 that the angles A h are constant on the generators of the null cones u = const., and that r is an affine parameter on these generators. The relations x^a = r_O_a therefore describe the behaviour of the generators.) The differential form of the coordinate transformation is

a a a A d^x = _O_ dr + r_O_A dh , (180)
where the transformation matrix
a @_O_a- _O_A =_ @hA (181)
satisfies the identity a _O_a_O_ A = 0.

We introduce the quantities

_O_AB = dab_O_aA_O_bB, (182)
which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, 2 _O_AB = diag(1,sin h). We use the inverse of _O_AB, denoted AB _O_, to raise upper-case latin indices. We then define the new object
_O_A = d _O_AB_O_b (183) a ab B
which satisfies the identities
A a A a A a a _O_a _O_B = dB, _O_A_O_ b = d b- _O_ _O_b. (184)
The second result follows from the fact that both sides are simultaneously symmetric in a and b, orthogonal to _O_a and b _O_, and have the same trace.

From the preceding results we establish that the transformation from a ^x to A (r,h ) is accomplished by

@^xa @^xa ----= _O_a, ----= r_O_aA, (185) @r @hA
while the transformation from (r,hA) to ^xa is accomplished by
@r @hA 1 A --a-= _O_a, --a-= -_O_ a. (186) @^x @^x 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 + 2guAdudhA + gAB dhAdhB,

with

( ) guu = - (1 + raa_O_a)2 + r2 aa- wab_O_b (aa - wac_O_c) - r2S + O(r3), (187) gur = -[1, ] (188) ( b) 2-2 3 a guA = r r aa - wab_O_ + 3r Sa + O(r ) _O_ A, (189) [ ] gAB = r2 _O_AB - 1r2Sab_O_a _O_b + 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, a kadx = - du and a k @a = @r.

The nonvanishing components of the inverse metric are

gur = - 1, (191) grr = 1 + 2raa_O_a + r2S + O(r3), (192) 1 [ ( ) 2 ] grA = -- r aa - wab_O_b + --r2Sa + O(r3) _O_Aa, (193) r [ 3 ] AB -1 AB 1- 2 ab A B 3 g = r2 _O_ + 3 r S _O_ a_O_b + 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

V~ --- V~ --[ 1 ( ) ] - g = r2 _O_ 1- -r2 R00 + 2R0a_O_a + Rab_O_a_O_b + O(r3) , (195) 6
where _O_ is the determinant of _O_AB; in the canonical parameterization, V~ -- _O_ = sin h.

3.3.8 Specialization to m a = 0 = Rmn

In this section we specialize our previous results to a situation where g is a geodesic on which the Ricci tensor vanishes. We therefore set am = 0 = Rmn everywhere on g, and for simplicity we also set wab to zero.

We have seen in Section 3.2.6 that when the Ricci tensor vanishes on g, 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 = ebcdBd a, and Racbd = dabEcd + dcdEab- dadEbc - dbcEad. These can be substituted into Equations (160View Equation, 161View Equation, 162View Equation) to give

Sab(u,hA) = 2Eab- _O_aEbc_O_c - _O_bEac_O_c + dabEbc_O_c_O_d + eacd_O_cBdb + ebcd_O_cBda, (196) A b b c d Sa(u,h ) = Eab_O_ + eabc_O_ B d_O_ , (197) S(u,hA) = Eab_O_a_O_b. (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 _O_.

It is convenient to introduce the irreducible quantities

E * = Eab_O_a_O_b, (199) * ( b b) c Ea = da - _O_a_O_ Ebc_O_ , (200) E*ab = 2Eab - 2_O_aEbc_O_c- 2_O_bEac_O_c + (dab + _O_a_O_b)E *, (201) * b c d Ba = eabc_O_ B d_O_ , (202) B*ab = eacd_O_cBde (deb - _O_e_O_b) + ebcd_O_cBde (dea - _O_e_O_a). (203)
These are all orthogonal to a _O_: * a * a Ea_O_ = Ba_O_ = 0 and * b * b E ab_O_ = Bab_O_ = 0. In terms of these Equations (196View Equation, 197View Equation, 198View Equation) become
S = E * + _O_ E*+ E*_O_ + _O_ _O_ E *+ B* + _O_ B* + B*_O_ , (204) ab ab a b a b a b ab a b a b Sa = E *a + _O_aE *+ 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 wab are both set equal to zero - we obtain the compact expressions

2 * 3 guu = - 1 - r E + O(r ), (207) 2-2 * * 3 gua = - _O_a + 3r (Ea + B a) + O(r ), (208) 1 gab = dab - _O_a_O_b - -r2(E *ab + B*ab) + 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 = r2_O_AB - 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 *_O_a = E _O_a _O_b, (214) *A a* Aa b ab A a b * E AB =_ Eab_O_ A_O_ B = 2Eab_O_ A_O_ B + E _O_AB, (215) B* =_ B* _O_a = eabc_O_a _O_bBc _O_d, (216) *A a* Aa b A cdd a b B AB =_ B ab_O_ A_O_ B = 2eacd_O_ B b_O_ (A_O_ B). (217)
It may be noted that the inverse relations are E* = E *_O_A a A a, B* = B* _O_A a A a, E* = E * _O_A _O_B ab AB a b, and * * A B B ab = B AB_O_a_O_ b, where A _O_a was introduced in Equation (183View Equation).

go to next pagego upgo to previous page