5
Gravitational (Clock) Compass
Abstract
A central question in general relativity is how the components of the Riemann
curvature tensor (the gravitational field) can be determined in an operational way.
Here we review two different methods which allow for a complete determination
of the Riemann curvature tensor. The first method relies on measuring the
accelerations of a suitably prepared set of test bodies relative to an observer.
The second method utilizes a set of suitably prepared clocks.
5.1
Determination of the Gravitational Field by Means of Test
Bodies
Historically, Felix Pirani [1] was the first to point out that one could determine the
full Riemann tensor with the help of a (sufficiently large) number of test bodies in
the vicinity of an observer’s world line. Pirani’s suggestion to measure the curvature
was based on the equation which describes the dynamics of a vector connecting two
adjacent geodesics in spacetime. In the literature this equation is known as the Jacobi
equation or the geodesic deviation equation. Its early derivations in a Riemannian
context can be found in [2–4]. See Sect. 1.3 for a derivation and extension based on
Synge’s world function.
In the following we explicitly show how a suitably prepared set of test bodies can
be used to determine all components of the curvature of spacetime (and thereby to
measure the gravitational field) with the help of an exact solution for the components
of the Riemann tensor in terms of the mutual accelerations between the constituents
of a cloud of test bodies and the observer. The analysis follows the presentation
given in [5].
This can be viewed as an explicit realization of Szekeres’ “gravitational compass” [6], or Synge’s “curvature detector” [7]. The operational procedure, see
Fig. 5.1, is to monitor the accelerations of a set of test bodies w.r.t. to an observer
© 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_5
99
Gravitational (Clock) Compass
Abstract
A central question in general relativity is how the components of the Riemann
curvature tensor (the gravitational field) can be determined in an operational way.
Here we review two different methods which allow for a complete determination
of the Riemann curvature tensor. The first method relies on measuring the
accelerations of a suitably prepared set of test bodies relative to an observer.
The second method utilizes a set of suitably prepared clocks.
5.1
Determination of the Gravitational Field by Means of Test
Bodies
Historically, Felix Pirani [1] was the first to point out that one could determine the
full Riemann tensor with the help of a (sufficiently large) number of test bodies in
the vicinity of an observer’s world line. Pirani’s suggestion to measure the curvature
was based on the equation which describes the dynamics of a vector connecting two
adjacent geodesics in spacetime. In the literature this equation is known as the Jacobi
equation or the geodesic deviation equation. Its early derivations in a Riemannian
context can be found in [2–4]. See Sect. 1.3 for a derivation and extension based on
Synge’s world function.
In the following we explicitly show how a suitably prepared set of test bodies can
be used to determine all components of the curvature of spacetime (and thereby to
measure the gravitational field) with the help of an exact solution for the components
of the Riemann tensor in terms of the mutual accelerations between the constituents
of a cloud of test bodies and the observer. The analysis follows the presentation
given in [5].
This can be viewed as an explicit realization of Szekeres’ “gravitational compass” [6], or Synge’s “curvature detector” [7]. The operational procedure, see
Fig. 5.1, is to monitor the accelerations of a set of test bodies w.r.t. to an observer
© 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_5
99