Gravitomagnetic measurement of the angular momentum of celestial bodies

The asymmetry in the time delay for light rays propagating on opposite sides of a spinning body is analyzed. A frequency shift in the perceived signals is found. A practical procedure is proposed for evidencing the asymmetry, allowing for a measurement of the specific angular momentum of the rotating mass. Orders of magnitude are considered and discussed. A well known effect of gravity on the propagation of electromagnetic signals is the time delay: for example an electromagnetic beam emitted from a source on Earth toward another planet of the solar system, and hence reflected back, undergoes a time delay during its trip (with respect to the propagation in flat space-time), due to the influence of the gravitational field of the Sun. This effect was indeed called the ”fourth” test of General Relativity[1],[2], coming fourth after the three classical ones predicted by Einstein[3]. Though small, the time delay in the propagation of electromagnetic waves was detected by Shapiro et al. [4], timing radar echoes from Mercury and Venus, by means of the radio-telescopes of Arecibo and Haystack. Anderson et al. [5] measured the time delay of the signals transmitted by Mariner 6 and 7 orbiting around the Sun. Finally Shapiro and Reasenberg obtained more accurate results using a Viking mission that deposited a transponder on the surface of Mars: the theoretical prediction was verified within ±0.1% [6] [7]. These measurements accounted just for the presence of a massive source, described by the Schwarzschild solution of the Einstein field equations. However, there is another correction to the time delay, due to the spin of the source, which is produced by the so called ”gravitomagnetic” interaction[8]. The time delay in the gravitational field of the Sun can be studied in the weak field approximation[9]. By using Cartesian coordinates and assuming that