Gravitomagnetism and the significance of the curvature scalar invariants

L. Filipe O. Costa, Carlos A. R. Herdeiro, Lode Wylleman

The curvature invariants have been subject of recent interest due to the ongoing debate concerning the questions of whether the gravitomagnetic interaction has already been detected with Lunar Laser Raging [e.g. PRL 98 (2007) 071102; PRD 78 (2008) 024033], and in the observations of the binary pulsar PSR 1913 +16 [IJTP 27 (1988) 1395], and whether there is a fundamental difference between that form of gravitomagnetism, which arises from the translational motion of the sources, and the gravitomagnetic field produced by the rotation of the Earth, detected in the LAGEOS Satellites data [Nature 431 (2004) 958] (whose detection was also the primary goal of the Gravity Probe-B mission). These phenomena have been dubbed, respectively, ``extrinsic'' and ``intrinsic gravitomagnetism'', such distinction being based on the second order scalar invariants of the Riemann tensor [New Astronomy 15 (2010) 332].
Starting from previous knowledge on the classification of purely electric and magnetic spacetimes [e.g. PRD. 74 (2006) 084001], and using the gravito-electromagnetic analogy based on tidal tensors [PRD 78 (2008) 024021] as a physical guiding principle, we clarify the physical meaning of the curvature invariants. Exploring both the similarities and the differences with the analogous electromagnetic apparatus, we interpret the invariant structure of relevant gravitational fields and discuss the implications on the motion of test particles, dissecting its relationship with the gravitomagnetic effects.