Gravitational clock compass in general relativity

We show how a suitably prepared set of clocks can be used to determine all components of the gravitational field in general relativity. We call such an experimental setup a clock compass, in analogy to the usual gravitational compass. Particular attention is paid to the construction of the underlying reference frame. Conceptual differences between the clock compass and the standard gravitational compass, which is based on the measurement of the mutual accelerations between the constituents of a swarm of test bodies, are highlighted. Our results are of direct operational relevance for the setup of networks of clocks, for example in the context of relativistic geodesy.

Figure 1

Construction of the coordinate system around the reference curve $Y$. Coordinates of a point $X$ in the vicinity of $Y(s)$—with $s$ representing the proper time along $Y$—are constructed by means of a tetrad ${{\lambda }_b}^{(\alpha )}$. Here $\tau$ is the proper time along the (spacelike) geodesic connecting $Y$ and $X$. By choosing a linear ansatz for the coordinates the derivatives of the connection vanish along the geodesic connecting $Y$ and $X$.

Figure 2

Symbolical sketch of the explicit solution for linear acceleration (40). In total six suitably prepared clocks (hollow circles) are needed to determine all acceleration components. The observer is denoted by the black circle.

Figure 3

Symbolical sketch of the explicit solution for the rotational velocity (45). In total six suitably prepared clocks (hollow circles) are needed to determine all velocity components. The observer is denoted by the black circle.

Figure 4

Symbolical sketch of the explicit solution for the linear velocity (51). In total four suitably prepared clocks (hollow circles) are needed to determine all velocity components. The observer is denoted by the black circle.

Figure 5

Symbolical sketch of the explicit solution for the curvature (54)–(77). In total 21 suitably prepared clocks (hollow circles) are needed to determine all curvature components. The observer is denoted by the black circle. Note that all ${}^{(1\dots 6)}{v}^a$, but only ${}^{(1\dots 3)}{y}^a$ are needed in the solution.

Figure 6

Symbolical sketch of the explicit vacuum solution for the curvature (99)–(113). In total 11 suitably prepared clocks (hollow circles) are needed to determine all curvature components. The observer is denoted by the black circle. Note that all ${}^{(1\dots 6)}{v}^a$, but only ${}^{(1\dots 3)}{y}^a$ are needed in the solution.