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Gravitatio mundi, a brief historical prelude
The gravitational two-body problem, in its broadest form, has always occupied a role apart in the historical development of physics, astronomy, mathematics, and even phi-losophy: How does one massive object move around another, and why that particular mo-tion? From labyrinthine epicycles, to Keplerian orbits, to the notion of a universal grav-itational “force” and beyond, the centuries-old struggle to tackle this question directly precipitated—more so, arguably, than any other single physical problem—the emergence of modern scienti c thought around the turn of the 18th century. Up to the present day, with vast opportunities currently presented by the revolutionary expansion of observa-tional astronomy into the domain of gravitational waves, understanding and solving this problem has remained as galvanizing an incentive as ever for both technical as well as conceptual advances.
From our contemporary point of view, the two parts of the problem as formulated above—on the one hand, the empirical question of how motion occurs in a gravitational two-body system, and on the other hand, the theoretical question of why it is that (rather than any other conceivable) motion—are indisputably regarded as having reached their rst true synthesis in the work of Newton, above all in the Principia [Newton 1687]1. Certainly, hardly any of Newton’s preeminent predecessors, from the ancient Greeks to the astronomers of the Renaissance, fell short of taking an avid interest in not only how the Moon and the planets moved, but why they moved so—or, perhaps o ering a better sense of the epochal mindset, “what” moved them so. Still, pre-Newtonian “explanations” of heavenly mechanics generally appear to us today to rest rather closer to the realm of myth than to that of scienti c theory.
The gure which stood at the point of highest in ection in the evolution of the intellec-tual mentality towards answering this latter, theoretical type of question was at the same time one of the greatest empiricists and mystics—Johannes Kepler (1571-1630). A restlessly contradictory character throughout his life, we can glean a brief sense of the dramatic psy-chological uxes that marked it—and therethrough, ultimately, his entire era—by simply recalling Kepler’s two most famous theoretical models for Solar System motion [Koestler 1959]. When he was in his mid-20s, he developed in a book called Mysterium Cosmo-graphicum [Kepler 1596] a model in which the orbits of the planets around the Sun are determined by a particular embedding of Pythagorean solids2 centered thereon, see Fig. 1.1. Then, a little over a decade later, in Astronomia Nova [Kepler 1609], he put forth an For an English translation with excellent accompanying commentary by Chandrasekhar for to-day’s “common reader”, see [Chandrasekhar 2003].
2 Also known as Platonic solids, or perfect solids, these are the set of three-dimensional solids with identical faces (regular, convex polyhedra). It was shown by Euclid that only ve such solids exist. They are [Koestler 1959]:
Gravitatio mundi, a brief historical prelude
empirical model of elliptical orbits, based on the observations of Tycho Brahe, establish-ing what we nowadays refer to as Kepler’s laws of planetary motion3. See Fig 1.2. What may be called the (neo-) Platonic basis of “explanation” underlying the former stands, to the modern reader, in radically sharp contrast with the manifestly quasi-mechanistic one at the basis of the latter. This reasoning is brought by Kepler to its logical end in a let-ter to Herwart, which he wrote as Astronomia Nova was nearing completion (taken from [Koestler 1959]):
My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork […] insofar as nearly all the manifold motions are caused by a most simple […] and material force, just as all motions of the clock are caused by a simple weight. And I also show how these physical causes are to be given numerical and geometrical expression.
One discerns in these lines an approach towards the sort of thinking that ultimately led to the paradigmatic Newtonian explanation of the elliptical shapes of the planetary orbits.
Arthur Koestler, in his authoritative history of pre-Newtonian cosmology The Sleep-walkers [Koestler 1959], to which we have referred so far a few times, traces out in great detail the work of Kepler and especially his “giving of the laws” of planetary motion. He summarizes their signi cance:
Some of the greatest discoveries […] consist mainly in the clearing away of psy-chological road-blocks which obstruct the approach to reality; which is why, post factum, they appear so obvious. In a letter to Longomontanus33 Kepler quali ed his own achievement as the “cleansing of the Augean stables”.
the tetrahedron (pyramid) bounded by four equilateral triangles; (2) the cube; (3) the octahedron (eight equilateral triangles); (4) the dodecahedron (twelve pentagons) and (5) the icosahedron (twenty equilateral triangles).
The Pythagoreans were fascinated with these, and associated four of them (1,2,3, and 5, in the above numbering) with the “elements” ( re, earth, air, and water, respectively) and the remaining one (4, the dodecahedron) with quintessence, the substance of heavenly bodies. The latter was considered dangerous, and so “[o]rdinary people were to be kept ignorant of the dodecahedron” [Sagan 1980].
In fact, only the rst two of what we today refer to as the three Keplerian laws of planetary motion were proposed in this work (the third he found a bit later): (1) the orbits of planets are ellipses with the Sun at a focus; (2) the planets move such that equal areas in the orbital plane are “swept out”, by a straight line with the Sun, in equal time. It is interesting to remark that these were actually discovered in reverse order. For a detailed historical account, see Part Four, Chapter 6 of [Koestler 1959].
Figure 1.1. Detail of Kepler’s model of Solar System motion based on Pythagorean solids, taken from [Koestler 1959] (adapted from Mysterium Cosmographicum [Kepler 1596]). A property of all Pythegorean solids, of which ve exist, is that they can be exactly inscribed into—as well as circumscribed around—spheres. As only six planets were then known (from Mercury to Jupiter), this seemed to leave room for placing exactly these ve perfect solids between their orbits (determined as an appro-priate cross-section through the inscribing/circumscribing spheres). This gure shows the orbits of the planets up to Mars inclusive.
Figure 1.2. A gure of an ellipse (dotted oval) circumscribed by a circle from Astronomia Nova [Kepler 1609].
Gravitatio mundi, a brief historical prelude
But Kepler not only destroyed the antique edi ce; he erected a new one in its place. His Laws are not of the type which appear self-evident, even in retrospect (as, say, the Law of Inertia appears to us); the elliptic orbits and the equations governing planetary velocities strike us as “constructions” rather than “discoveries”. In fact, they make sense only in the light of Newtonian Me-chanics. From Kepler’s point of view, they did not make much sense; he saw no logical reason why the orbit should be an ellipse instead of an egg. Accord-ingly, he was more proud of his ve perfect solids than of his Laws; and his contemporaries, including Galileo, were equally incapable of recognizing their signi cance. The Keplerian discoveries were not of the kind which are “in the air” of a period, and which are usually made by several people independently; they were quite exceptional one-man achievements. That is why the way he arrived at them is particularly interesting.
Nonetheless, the basic new concepts involved in articulating this new, clockwork-type worldview presented great conceptual di culties. In the Astronomia Nova, for example, Kepler wrestled profusely with the concept of the “force” causing the motions in his imag-ined clockwork universe [Kepler 1609] (taken from [Koestler 1959]):
This kind of force […] cannot be regarded as something which expands into the space between its source and the movable body, but as something which the movable body receives out of the space which it occupies… It is propagated through the universe … but it is nowhere received except where there is a movable body, such as a planet. The answer to this is: although the moving force has no substance, it is aimed at substance, i.e., at the planet-body to be moved…
Koestler remarks, interestingly, that Kepler’s description above actually seems to be “closer to the modern notion of the gravitational or electro-magnetic eld than to the classic Newtonian concept of force” [Koestler 1959].
With Newton’s arrival on the scene, the vision of a mechanistic clockwork uni-verse took de nitive shape in the form of three laws of motion and the inverse-square law of universal gravitation—with Kepler’s three laws recovered from these as particu-lar consequences [Newton 1687]. What was particularly crucial here was the veritable introduction—or, at the very least, the unprecedented clari cation—of a new sort of rea-soning, one rooted in the idea that any useful description of fundamental physical phe-nomena must assume a universal and mathematical4 character—a sort of reasoning then called natural philosphy, and which later came to be referred to more commonly as science.
The speci c mathematical form of such descriptions—invented by Newton himself and, since then, amply developed but still lying at the basis of all physical laws formulated to this day—is that of the di erential equation.
Koestler once again does better than we can to contextualize the relevance of this moment [Koestler 1959]:
It is only by bringing into the open the inherent contradictions, and the meta-physical implications of Newtonian gravity, that one is able to realize the enor-mous courage – or sleepwalker’s assurance – that was needed to use it as the basic concept of cosmology. In one of the most reckless and sweeping general-izations in the history of thought, Newton lled the entire space of the universe with interlocking forces of attraction, issuing from all particles of matter and acting on all particles of matter, across the boundless abysses of darkness.
But in itself this replacement of the anima mundi by a gravitatio mundi would have remained a crank idea or a poet’s cosmic dream; the crucial achievement was to express it in precise mathematical terms, and to demon-strate that the theory tted the observed behaviour of the cosmic machinery – the moon’s motion round the earth and the planets’ motions round the sun.
Newton, of course, was famously aware of the “inherent contradictions” to which Koestler is referring. While comments to this e ect appear in the Principia itself [Newton 1687], in a letter to Bentley just a few years later, he could not have been clearer vis-à-vis what he thought about his proposed theory—and in particular, the physical conception of gravita-tion o ered by it (taken from [Koestler 1959]):
It is inconceivable, that inanimate brute matter should, without the mediation of something else which is not material, operate upon and a ect other matter without mutual contact, as it must be, if gravitation in the sense of Epicurus, be essential and inherent in it. And this is one reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left open to the consideration of my readers.
No less di cult for the consideration of Newton’s readers at that time was the new mathematics describing this metaphysically mysterious “agent”. In fact, Newton noto-riously avoided publishing his work on calculus—which he referred to as the “method of uxions”—for decades, leading to the infamous controversy with Leibnitz over its dis-covery [Gleick 2004]. Meanwhile, the Principia [Newton 1687], though clearly bearing the basic elements of the in nitesimal analysis at the basis of calculus, was written essentially, one might say in “brute-force” style, in the technical language then commonly understood: Euclindean geometry. Newton presented his solution of the two-body problem—the proof of elliptical planetary motion as a consequence of his laws—in the Principia, Book I, Section XI, Propositions LVII-LXIII [Newton 1687]. See Fig. 1.3.
The advent of relativity
While there certainly existed some known empirical discrepancies with Newton’s the-ory by the end of the 19th century—among the most notable being, especially in view of the two-body problem, the perihelion precession of Mercury known since 1859—what pri- marily led to its overthrow had, at least in the vision of its chief perpetrator, much more to do with its eminently long-standing “inherent contradictions”. Einstein, indeed, often re-garded his development of relativity5 as merely the proverbial cleansing of the Newtonian stables [Einstein 1954]:
Let no one suppose […] that the mighty work of Newton can really be super-seded by [general relativity] or any other theory. His great and lucid ideas will retain their unique signi cance for all time as the foundation of our whole modern conceptual structure in the sphere of natural philosophy.
There is a good deal of di erence between the circumstances surrounding the emer-gence of general relativity compared with that of the Newtonian theory. While the latter went hand in hand with strong empirical contingencies—primary among these being, as we have seen, solving the two-body problem—the former was driven much more by basic conceptual and logical questions. Cornelius Lanczos, a mathematician contemporary with Einstein, comments [Lanczos 1949]:
Einstein’s Theory of General Relativity […] was obtained by mathematical and philosophical speculation of the highest order. Here was a discovery made by a kind of reasoning that a positivist cannot fail to call “metaphysical,” and yet it provided an insight into the heart of things that mere experimentation and sober registration of facts could never have revealed.
Viewed from such a standpoint, the local e ects of special relativity—time dilation, length contraction and all the rest—as well as the globally curved (non- at) geometry of the space-time we inhabit can be regarded as following, essentially, as logical consequences from:
on the one hand, demanding consistency between the physical laws then known (in particular, as concerns the Maxwellian theory of electromagnetism), and (ii) on the other hand, dispensing with what appeared to be the most unnecessary assumptions causing the “inherent contradictions” of the Newtonian theory: in particular, the notion of abso-lute space, and connected with this, the formulation of physical laws in a privileged—the so-called inertial—class of coordinate reference frames. It is quite remarkable how what may look from this point of view as a sort of exercise in logic has ultimately produced such wonderfully diverse physical insights into the nature of gravity, and even—though this generally took longer to understand—the sorts of basic objects that can exist in our Universe, such as black holes and gravitational waves.
An issue that attracted much of Einstein’s attention throughout his development of general relaitivity was that of the motion of an idealized “test” mass, that is, one provoking In fact, Einstein wished to call it the “theory of invariance” (to highlight the invariance of the speed of light and that of physical laws in di erent reference frames), but the term “theory of relativity” coined by Max Planck and Max Abraham in 1906 quickly became, to Einstein’s dissatisfaction, the more popular nomenclature, and the one which has persisted to this day [Galison et al. 2001].
The advent of relativity
no backreaction in the eld equations of the theory [Renn 2007; Lehmkuhl 2014]. Already in 1912, in a note added in proof to [Einstein 1912], he stated for the rst time that the geodesic equation, that is, the extremization of curve length, ds = 0 ; (1.2.1) is the equation of motion of point particles “not subject to external forces”. In this case, ds is an in nitesimal distance element in any curved four-dimensional spacetime. By this point, Einstein understood that the basic mathematical methods for studying spacetime curvature, logically identi ed as gravity, were those of di erential geometry pioneered during the previous century by Gauss, the Bolyais (Farkas and his son János), Lobachevsky, Riemann, Ricci and Levi-Civita, to name a few of the main players6. Thus the basic object, in a theory fundamentally concerned with length measurements (in the broadest sense), is the metric tensor, denoted g in Einstein’s original notation. This object de nes the notion of in nitesimal distance ds, and hence also that of motion [Einstein 1913] (taken from [Lehmkuhl 2014]):
A free mass point moves in a straight and uniform line according to [our Eq.
[. . . ] In general, every gravitational eld is going to be de ned by ten compo-nents g , which are functions of [local coordinates] x1, x2, x3, x4.
In 1914, Einstein actually used the word “geodesic” for the rst time to refer to length-extremizing curves [Lehmkuhl 2014]. Then, in 1915, the theory was completed with the promulgation of the nal form of the gravitational eld equations governing his g [Ein-stein 1915] (English translation in [Einstein 1996a]). In a paper consolidating the theory the following year, Einstein summarizes the main ideas [Einstein 1916] (taken from [Ein-stein 1996b]):
We make a distinction hereafter between “gravitational eld” and “matter” in this way, that we denote everything but the gravitational eld as “matter.” Our use of the word therefore includes not only matter in the ordinary sense, but the electromagnetic eld as well.
Much of the development of di erential geometry had to do with attempts to prove Euclid’s famous fth postulate. Ever since the appearance of the Elements, which is based on ve postulates, there had been skepticism regarding the necessity of the last of these. In its original form it is much more complicated to state than the rst four, but it is equivalent to the statement that the sum of the three angles of a triangle is always equal to two right angles. The advent of di erential (“non-Euclidean”) geometry is essentially related to the relaxation of this condition, permitting the description of globally curved surfaces. See [Aczel 2000] for a brief history.
Table of contents :
Part I. Fundamentals of General Relativity: Introduction, Canonical Formulation and Perturbation Theory
Chapter 1. Introduction
1.1. Gravitatio mundi, a brief historical prelude
1.2. The advent of relativity
1.3. Geometry, gravity and motion
1.4. Gravitational waves and extreme-mass-ratio inspirals
1.5. The self-force problem
Chapter 2. Canonical General Relativity
2.1. Introduction
2.2. Lagrangian formulation
2.3. Canonical formulation of general eld theories
2.4. Canonical formulation of general relativity
2.5. Applications
Chapter 3. General Relativistic Perturbation Theory
3.1. Introduction
3.2. General formulation of perturbation theory
3.3. Perturbations of the Schwarzschild-Droste spacetime
3.4. Perturbations of the Kerr spacetime
Part II. Novel Contributions: Entropy, Motion and Self-Force in General Relativity
Chapter 4. Entropy Theorems and the Two-Body Problem
4.1. Introduction
4.2. Entropy theorems in classical mechanics
4.3. Entropy theorems in general relativity
4.4. Entropy in the gravitational two-body problem
4.5. Conclusions
Chapter 5. The Motion of Localized Sources in General Relativity: Gravitational Self-Force from Quasilocal Conservation Laws
5.1. Introduction: the self-force problem via conservation laws
5.2. Setup: quasilocal conservation laws
5.3. General derivation of the gravitational self-force from quasilocal conservation laws
5.4. Application to the Gralla-Wald approach to the gravitational self-force
5.5. Discussion and conclusions
5.6. Appendix: conformal Killing vectors and the two-sphere
Chapter 6. A Frequency-Domain Implementation of the Particle-without-Particle Approach to EMRIs
6.1. Introduction
6.2. The scalar self-force
6.3. The Particle-without-Particle method
6.4. Frequency domain analysis
6.5. Numerical implementation and results
Chapter 7. Conclusions
