Abstract
Galileo’s first celestial observations in 1609, using the freshly invented refracting telescope, can be rightfully called the birth of electromagnetic astronomy. His observations, among which new stars, the Moon’s mountains and Jupiter’s four biggest natural satellites, were published on 13 March 1610 in a short astronomical treatise, Sidereus Nuncius.
Proper separation oscillation between two freely falling bodies could be used in principle to detect the passage of a gravitational wave, as monitored by laser interferometry. However, such detectors have so low a sensitivity that they are of little experimental interest.
C. Misner, K. Thorne and A. Wheeler,
Gravitation (1973)
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Notes
- 1.
We shall skip over the eternal debate of whether scientific facts are discovered, invented, or both, and let the reader decide for himself.
- 2.
Interestingly, Einstein mostly uses the phrasing General Theory of Relativity instead of the now common GR, thinking of the relativity theory as general, rather than relativity itself.
- 3.
Except for Mercury’s perihelion advance, which we will get back to shortly.
- 4.
Thanks to the father of his friend Marcel Grossmann [16], who we will encounter again soon.
- 5.
At the time, differential geometry was still new, following the founding works of Bernhard Riemann [23] and Felix Klein’s Erlangen program [24] in Germany, as well as the great school of Italian geometers, where Luigi Bianchi [25] and then Tullio Levi-Civita [26] played a central role. I refer to Chap. 7 of the interesting book [27] for more historical details.
- 6.
Those papers were submitted on the 4th [30], 11th [31], 18th [32] and 25th [33], each Thursday of every week !.
- 7.
This is the actual original phrasing, found on page xi of [34].
- 8.
This action is called the Einstein-Hilbert action, because of its independent discovery by mathematician David Hilbert one month prior to Einstein’s final publication [35].
- 9.
- 10.
This unicity theorem is sometimes attributed to Veirmeil, Cartan and/or Weyl, who wrote on the matter fifty years before Lovelock. For the reader’s interest, we provide in Appendix A.1 a brief summary and references for the contributions of these men.
- 11.
This comes from the assumption that the manifold representing the spacetime has no torsion. When this assumption is relaxed, one works in the Einstein-Cartan theory, a perfectly reasonable extension of GR that naturally provides some insight about the dark issues of cosmology [41].
- 12.
Except for the energy of the gravitational field itself, which, contrary to Newtonian gravity, becomes ill-defined in GR [6].
- 13.
Although at most six of them are independent, thanks to the Bianchi identity \(\nabla _a G^{ab}=0\).
- 14.
There seems to be a mistake in the original article’s title: the eclipse was on 21 September, 1922 and not 1923, as can be checked on https://eclipse.gsfc.nasa.gov/. See also this beautiful article by writer and photo editor Anika Burgess for the story behind the 1922 eclipse experiment.
- 15.
They measure how much space-curvature is produced by unit rest mass and how much “nonlinearity” there is in the superposition law for gravity, respectively [72].
- 16.
Since the polar equation of motion of Schwarzshild geodesics involves the Weierstrass elliptic function [82], exact and analytical formulae for the light deflection \(\delta \phi \) and perihelion advance \(\delta \omega \) can be given in terms of the Elliptic integrals. These beautiful formulae were derived in chap IX and X of [83], and would make for an interesting academic exam or a nice illustration of the elliptic functions/integrals duality.
- 17.
There is some irony at the core of this controversy: a strong claim in Einstein and Rosen’s paper was refuted by the referee because it simply came from a bad choice of coordinates. Let this be a lesson for any student taking a course in GR: even Einstein himself fell into the trap of his beloved diffeomorphism invariance!
- 18.
For the interested reader, this technical nomenclature encodes PSR for Pulsating Source of Radio, and 1913\(+\)16 for the equatorial coordinates \((\alpha ,\delta )\), with right-ascension \(\alpha = 19\text {hrs} 13\text {min}\) and declination \(\delta =+16^\circ \). See page 302 of [136] for additional details.
- 19.
As I am writing these lines in December 2020, parts of the Arecibo telescope have just collapsed, making additional damages to its already crumbling dish. The Arecibo telescope is responsible for a long history of discoveries, as beautifully accounted for in the review [138].
- 20.
The accordance between the GR prediction and the observed value allows for an astrophysical derivation of nontrivial series formula for the Bessel functions (see the Appendix of [139]).
- 21.
Not to be confused with Hendrik Antoon Lorentz who gave his name to the Lorentz force, Lorentz transformations and Lorentz factor [22]. Several classical textbooks [6, 123, 136, 146, 147] make this mistake. The Lorenz gauge (or coordinates that satisfy it) is also often called “harmonic” [145, 148], and more rarely De Donder [149] or Hilbert [150].
- 22.
It should be noted that in linearized theory, a rigorous derivation of (1.17) is impossible. Precisely because the equation of motion, which follows from the Lorenz gauge condition, implies that sources moves on a geodesic of Minkowski’s spacetime. This undeniably excludes any gravitationally-driven orbit. The rigorous, self-consistent derivation includes a careful inspection of the nonlinearities [136, 145].
- 23.
Theoretically, more exotic types of objects, such as dark stars [157] or boson stars [158] could also be included into this class. White dwarfs on the other hand, although compact, have too large of a radius to coalesce: they never reach the last stable orbit that would provoke a merger. Yet, they do produce GWs, which could be detected as part of a GW stochastic background or as resolved individual sources [159] (see also Sect. 1.3.3).
- 24.
As always in cosmology, we must rely on some kind of ergodic hypothesis to argue that an ensemble average (impossible as we have only access to one Universe) coincides with a time average [136, 187].
- 25.
PN theory has also been developed to handle eccentric orbits, in which case one also needs to rely on the balance of angular momentum in addition to the balance of energy.
- 26.
Although a number of recent results seem to converge on the fact that GSF theory may very-well hold for mass ratios that are not that small (for the so-called intermediate mass ratio inspirals (IMRIs), with mass ratio \(\simeq 10^{-4}-10^{-2}\)), or even with mass ratio flirting with unity [204–208].
- 27.
In practice, the light makes around 150 round trips in the vacuum cavity thanks to additional mirrors to make one Fabry-Perrot cavity in each arm, before being recombined. This effectively enhances the length of the detector’s arms.
- 28.
See [232] for a noise hunt in the detector’s output, showing how external sources such as wind, ocean waves, human activity, airplanes, thunderstorms, helicopters, etc. couple to the detector.
- 29.
I recommend taking a look at Janna Levin’s brilliant book Black Hole Blues and Other Songs from Outer Space [237]. Chapter 5 tells the story of Weber’s experiment and its outcome(s).
- 30.
The new nomenclature will also include UTC time of detection [255], in the form GWYYMMDD_HHMMSS, (Hour-Minute-Second), as several events are expected to be detected on the same day, in the near future.
- 31.
See [258] for a first principles, pedagogical derivation of these parameter estimates.
- 32.
In fact, Virgo, although fully operational, did not see anything of this event. This indicates that the source was in the detector’s blind direction (i.e. the GW produced the same displacement in both arms of the detector) which helped to find its location in the sky.
- 33.
This analogy works pretty well since it has been reported that an exceedingly large amount of gold (between 3 and 13 Earth masses !) was produced during this event [261].
- 34.
The second mass gap comes from the scarcity of observed BHs above fifty Solar masses [97].
- 35.
Coincidentally, this event is also the fifth in this selected list....
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Ramond, P. (2023). Gravitational Astronomy. In: The First Law of Mechanics in General Relativity & Isochrone Orbits in Newtonian Gravity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-031-17964-8_1
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