JHEP06(2017)042
where lowercase Latin letters denote (for now arbitrary) integration functions that do not
depend on r, with
i
m
jk
= −
j
m
ik
and
i
m
j1
= −
j
m
i1
. (To express L
11
, we used the equation
(n − 2)Φ = −D(L
−1
ji
Φ
ij
), which follows from (B.5, [11]).)
Taking the affine parameter r as one of the coordinates, we can write the basis vectors as
` = ∂
r
, n = U∂
r
+ X
A
∂
A
, m
(i)
= ω
i
∂
r
+ ξ
A
i
∂
A
, (2.5)
where U, X
A
, ω
i
and ξ
A
i
are spacetime functions (to be determined), ∂
A
= ∂/∂x
A
, and
the x
A
represent any set of (n − 1) scalar functions such that (r, x
A
) is a well-behaved
coordinate system. The r-dependence of the functions in (2.5) can be determined using
the commutators (B.12) and (B.13), and reads [38]
ω
i
= −L
1i
r + L
ji
ω
0
j
, ξ
A
i
= L
ji
ξ
A0
j
, (2.6)
X
A
= X
A0
, U = −l
11
r −
1
n − 2
Z
L
−1
ji
Φ
ij
dr + U
0
, (2.7)
where a superscript
0
denotes r-independent quantities.
Clearly, the r-dependence of (2.3), (2.4) and the second of (2.7) is, at this stage, de-
termined only implicitly, since some Weyl components appear there — their r-dependence
will be given below (at least at the leading order in 1/r). Let us also observe that the above
result holds also without the assumption (1.6), and can be easily extended to include a
cosmological constant [27, 38].
2.2 Weyl components of boost weight zero
Using now also assumption (1.6), the r-dependence of the Weyl b.w. 0 components has
been already studied in [25, 27].
4
At the leading order in 1/r it reads
Φ
S
ij
=
Φ
0
n−2
δ
ij
r
n−1
+
n−1
2(n−3)
b
kl
b
lk
δ
ij
+2b
ik
b
kj
r
n+1
+O
r
−n−3
, Φ
A
ij
=
(n−1)Φ
0
(n−2)(n−3)
b
[ij]
r
n
+O
r
−n−2
,
(2.8)
C
ijkm
=
4Φ
0
(n − 2)(n − 3)
δ
i[m
δ
k]j
r
n−1
+ O
r
−n−1
, (2.9)
where Φ
0
6= 0 is an integration function independent of r (for Φ
0
= 0 all the b.w. 0 com-
ponents vanish [25, 38] and the spacetime is flat). For our purposes, terms of higher order
are not needed. It is just important to observe that these can be determined recursively to
any desired order once the leading terms in (2.8), (2.9) are known [38], and do not involve
any integration functions other that Φ
0
and b
ij
. This implies (cf. (2.5) with (1.5), (2.6)
and (2.7)) that the full spacetime metric is uniquely determined by knowing b
ij
, l
1i
, l
11
, ω
0
i
,
ξ
A0
i
, X
A0
, U
0
and Φ
0
.
By a suitable (r-independent) rescaling of the affine parameter r and a corresponding
rescaling (boost) of `, such that ` = ∂
r
and (2.1) are preserved, one can always set (cf.,
e.g., [5, 18])
Φ
0
= const 6= 0. (2.10)
This choice simplifies considerably several expressions to be obtained in the following.
4
In [25] it was assumed that positive powers of r vanish, however this assumption is not necessary [27].
– 6 –
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