5.3 Field Determination by Means of Clocks
109
Taking x 3 = r as a coordinate and labelling the remaining coordinates x i =
(x, y, r, u) we have from (5.46)
g ij
dx j
dr
= −r ,i ⇔ g i3 = −δ
3
i .
(5.47)
Hence four components of the metric tensor of the space-time are fixed. Using
straightforward algebra the remaining six components can be expressed in terms
of six functions P , α, β, a, b, c of the four coordinates x, y, r, u in such a way that
the line element of the space-time is given by
ds
2
= −(ϑ
(1) )
2
− (ϑ
(2) )
2
− (ϑ
(3) )
2
+ (ϑ
(4) )
2 ,
(5.48)
with
ϑ
(1)
= −ϑ (1)i dx
i
= r P
−1 (e
α cosh β dx + e
−α sinh β dy + a du),
(5.49)
ϑ
(2)
= −ϑ (2)i dx
i
= r P
−1 (e
α sinh β dx + e
−α cosh β dy + b du),
(5.50)
ϑ
(3)
= −ϑ (3)i dx
i
= dr,
(5.51)
ϑ
(4)
= ϑ (4)i dx
i
= c du.
(5.52)
We take r = 0 in this space-time to be a time-like world line with u as proper time
along it. Then in the neighbourhood of r = 0 (i.e. for small values of r) we expand
the functions of x, y, r, u in (5.49)–(5.52) in positive powers of r, with coefficients
functions of x, y, u, in such a way that the line element (5.48) is a perturbation of
the Minkowskian line element (C.30). Clearly this involves taking
P = P 0 + O(r) , α = O(r) , β = O(r) , a = a 0 + O(r) ,
b = b 0 + O(r) , c = 1 − h 0 r + O(r
2 ) ,
(5.53)
but we need to know the leading powers of r in the O(r)-terms here. To find these we
make use of (C.57)–(C.60). With p i given by (5.46), and using (5.49)–(5.52), and
denoting by a semicolon covariant differentiation with respect to the Riemannian
connection associated with the metric tensor g ij given by the line element (5.48) we
start by recording that
ϑ
i
(1) = r
−1 P (e
−α cosh β, −e
α sinh β, 0, 0) ,
(5.54)
ϑ
i
(2) = r
−1 P (−e
−α sinh β, e
α cosh β, 0, 0) ,
(5.55)
ϑ
i
(3) = (0, 0, 1, 0) ,
(5.56)
109
Taking x 3 = r as a coordinate and labelling the remaining coordinates x i =
(x, y, r, u) we have from (5.46)
g ij
dx j
dr
= −r ,i ⇔ g i3 = −δ
3
i .
(5.47)
Hence four components of the metric tensor of the space-time are fixed. Using
straightforward algebra the remaining six components can be expressed in terms
of six functions P , α, β, a, b, c of the four coordinates x, y, r, u in such a way that
the line element of the space-time is given by
ds
2
= −(ϑ
(1) )
2
− (ϑ
(2) )
2
− (ϑ
(3) )
2
+ (ϑ
(4) )
2 ,
(5.48)
with
ϑ
(1)
= −ϑ (1)i dx
i
= r P
−1 (e
α cosh β dx + e
−α sinh β dy + a du),
(5.49)
ϑ
(2)
= −ϑ (2)i dx
i
= r P
−1 (e
α sinh β dx + e
−α cosh β dy + b du),
(5.50)
ϑ
(3)
= −ϑ (3)i dx
i
= dr,
(5.51)
ϑ
(4)
= ϑ (4)i dx
i
= c du.
(5.52)
We take r = 0 in this space-time to be a time-like world line with u as proper time
along it. Then in the neighbourhood of r = 0 (i.e. for small values of r) we expand
the functions of x, y, r, u in (5.49)–(5.52) in positive powers of r, with coefficients
functions of x, y, u, in such a way that the line element (5.48) is a perturbation of
the Minkowskian line element (C.30). Clearly this involves taking
P = P 0 + O(r) , α = O(r) , β = O(r) , a = a 0 + O(r) ,
b = b 0 + O(r) , c = 1 − h 0 r + O(r
2 ) ,
(5.53)
but we need to know the leading powers of r in the O(r)-terms here. To find these we
make use of (C.57)–(C.60). With p i given by (5.46), and using (5.49)–(5.52), and
denoting by a semicolon covariant differentiation with respect to the Riemannian
connection associated with the metric tensor g ij given by the line element (5.48) we
start by recording that
ϑ
i
(1) = r
−1 P (e
−α cosh β, −e
α sinh β, 0, 0) ,
(5.54)
ϑ
i
(2) = r
−1 P (−e
−α sinh β, e
α cosh β, 0, 0) ,
(5.55)
ϑ
i
(3) = (0, 0, 1, 0) ,
(5.56)