C
Gravitational (Clock) Compass
C.1
Coordinates for Minkowskian Space-Time
We provide some supplementary material on the construction of a coordinate system
for Minkowskian space-time based on a family of space-like hypersurfaces. This
construction leads to a particularly useful form of the Minkowskian line element
which is important for the derivation of the frequency ratio in flat space-time in
Sect. 5.3. In addition basic formulas (Eqs. (C.57)–(C.61)) in the Minkowskian case
play an important role in the neighbourhood of a time-like world line in the general
curved space-time.
We begin with the Minkowskian line element in rectangular Cartesian coordinates and time X i = (X, Y, Z, T ):
ds
2
= −(dX)
2
− (dY )
2
− (dZ)
2
+ (dT )
2
= η ij dX
i dX
j .
(C.1)
Writing
X
i
= w
i (u) + r p
i (u) ,
(C.2)
with
p i p
i
= −1 ,
(C.3)
we see that p i is a unit space-like vector field defined along the world line, which
we take to be time-like with u taken to be proper-time or arc length along it. Hence
v
i (u) =
dw i
du
with v i v
i
= +1 .
(C.4)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
219