Fifteen Eighty Four

The General Theory of Relativity (GTR), enunciated just over a hundred years ago by Albert Einstein, remains to this day the best available description of gravitation, the feeblest out of the four fundamental interactions and, nonetheless, the one which shapes and governs the natural world at the grandest scales.

Especially in recent decades, empirical evidence in support of it has accumulated more and more. Yet, its tests are relatively fewer when compared to those corroborating electromagnetism and the two nuclear interactions. It is so because, in most situations subject to direct experimental investigation, gravitation is far weaker than the other three rulers of Nature. In particular, GTR fully unfolds all its potential only in extreme scenarios characterized by exceptionally intense gravitational fields rapidly varying over the shortest space and temporal scales, and speeds close to that of light. Such conditions can be found uniquely in the deepest astronomical recesses.

The most spectacular confirmations of GTR recently came from the detection of gravitational waves emitted during the last stages of the cosmic dance of pairs of astrophysical compact objects (black holes and neutron stars) that would inevitably lead to their merger, and from the collection of the electromagnetic radiation at given radio frequencies emitted from matter in the close neighbours of the supermassive black holes lurking at the cores of our galaxy and of M87. It goes without saying that these great scientific and technological feats required long years of preparation and huge funds for their realization, together with the coordinated efforts of a growing number of skilled and dedicated researchers.

At less extreme scales, characterized by relatively weak fields and speeds much smaller than that of light, the footprints of GTR become more subtle and elusive, often being overwhelmed by a large number of competing classical effects of non-gravitational or Newtonian gravitational origin. This is the realm of the artificial satellites orbiting the Earth, of the interplanetary probes exploring the other planets of the solar system and their natural moons, of the exoplanets, discovered in ever-increasing numbers, that revolve almost in front of their parent stars, and, to a certain extent, also of well detached binary systems hosting at least one compact stellar corpse and of the S stars that move around the supermassive black hole at the centre of the Milky Way in Sagittarius A^∗ (Sgr A^∗). If, on the one hand, the magnitudes of the GTR effects taking place in such systems are generally rather small, being possible to calculate them often as post-Newtonian perturbations to their first order (1pN), on the other hand, such celestial laboratories are quite abundant and known with an increasingly high accuracy in view of steady progress in a number of observational techniques. Furthermore, the physics of many competing classical effects is generally rather well understood and they can be modeled with increasing detail. Last but not least, it is not always necessary to design and construct expensive and extremely sophisticated devices specifically targeted to measure the pN effects of interest, being not rarely possible to exploit a somewhat opportunistic approach which relies upon the use or the modification of techniques and spacecraft aimed to other scopes. Of course, notable exceptions exist like the ended missions Gravity Probe B (GP-B) and MICROSCOPE, and the future LISA space-based interferometry.

In this context, an important role was, is, and will be played by the orbital motions. Indeed, while the electromagnetic waves whiz by very quickly, carrying with them their information content on how their course has been influenced by relativistic gravity, planets and satellites move placidly, accumulating slowly with respect to light but constantly the changes of their otherwise Keplerian motion imposed by the Einstein’s theory. Accurately monitoring their trajectories over sufficiently long time spans should allow to finally extract the tiny relativistic part of their overall orbital evolution, provided that all the other known concurrent features of motion are modelled with good enough accuracy and subtracted. It is certainly not an easy or short task. Suffice it to think about GP-B, an extremely sophisticated and beautiful experiment specifically aimed at measuring, among other things, the small gravitomagnetic precessions of four gyroscopes made to move in the field of the spinning Earth. Its timeframe, ranging from its early conception to the release of its final results, lasted for about forty years, at a cost of about USD 750 million. Its final claimed accuracy was about 19%, after all those years it had been expected to be of the order of 1% or better. Just to stay in the field of general relativistic gravitomagnetism, attempts to measure the Lense-Thirring effect with the laser-ranged geodetic satellites of LAGEOS-type have been going on for about 30 years now.

So, if such kind of relativistic orbital features are, to the pN order, so tiny and difficult to be accurately and reliably measured, why bother so much about them and even write a book to describe them in detail? As already noted, the experimental evidence for GTR, although significantly increased over the last years, still remains considerably less numerous and generally less accurate than that supporting the other three fundamental interactions. As a crystal clear representation of this state of the art it can well be cited the fact that the Newtonian constant of gravitation G is, among all the fundamental constants of Nature, the least accurately determined. Furthermore, many of the effects discussed have their counterparts in the strong field regime in which it is more challenging to test them reliably also because the environment is much less accurately known and more difficult to model. Thus, before confidently extrapolating their validity also to such extreme domains, it is good to gain as much confidence as possible in those scenarios for which more reliable and accurate experimental and observational control can be achieved. Moreover, as the history of science has repeatedly demonstrated in the past, the possibility of unexpectedly discovering some unforeseen effect cannot be excluded. Last but not least, the steady efforts aimed at accurately modeling a huge number of competing classical features of motion can well lead, as an important byproduct, to relevant progress also in other fields.

But that’s not all. In fact, for last decades the GTR has been challenged by an increasingly broad and empirically consolidated phenomenology that points to the existence of some form of non-baryonic matter concentrated in many galaxies and galactic clusters. Furthermore, the Universe is expanding at a rate that has been accelerating since the last five billion years, driven by a still unexplained form of dark energy. To take these features into account, numerous alternative gravity models have been recently proposed. Thus, testing them independently in contexts other than the very same ones for which they were introduced is of paramount importance. In this respect, orbital motions can play an important role by putting these paradigms to the test on very different scales from each other and from those for which they were initially proposed. This is why space has also been given in the book to a lot of alternative models by calculating in detail the orbital precessions and enabling the reader to independently work out other orbital effects that are less common in the community of theorists who deal with modified models of gravity. Indeed, the methods and the approach developed for classical GTR and other Newtonian features of motion can straightforwardly be extended also to any alternative gravity model for which an acceleration or a potential can be written explicitly.

The orbiting bodies used as test probes can move along paths which greatly differ in their shapes, ranging from the nearly circular orbits of the LAGEOS-type satellites to the extremely eccentric paths followed by the stars buzzing around Sgr A^∗. Furthermore, the orientation of the spin axis of the central body, which enters both Newtonian and pN important effects like those due to its centrifugal oblateness and gravitomagnetic field, is accurately known only for the Earth, allowing to obtain simplified expressions written in reference systems whose fundamental plane can be oriented just as the Earth’s equator. For all other systems, which range from exoplanets to Sgr A^∗ via binary pulsars, this is not possible since the rotational axes of the primaries are generally poorly constrained, if not completely unknown. For all these reasons, it is necessary to work out analytical formulas whose validity is neither restricted to the case of low eccentricities nor to any particular orientation of the spin axes, precisely what has been done in this book.

Title: General Post-Newtonian Orbital Effects

Author: Lorenzo Iorio

ISBN: 9781009562874