Einstein

The general theory of relativity, Einstein’s premier contribution to physics, displaced Newton’s law of gravity, for two hundred years the acme of classical mechanics and epitome of an exact law of nature. General relativity remains the cornerstone of the contemporary understanding of gravity, even though it is a classical, not a quantum, field theory. Today, most cosmological models of a homogeneous and isotropic expanding universe are based on an exact solution of Einstein’s field equations of gravitation, the Friedmann-Lemaître-Robinson-Walker metric. But general relativity was largely ignored by physicists (though not by astronomers) after the advent of quantum mechanics in 1925–1926; on account of the paucity of empirical data supporting it, general relativity was taught for decades mainly in mathematics departments, as having more mathematical than physical significance. A renaissance occurred in the 1960s with the discovery of the cosmic microwave background (CMB) signature of the “Big Bang” (1964), followed by the theorems of Roger Penrose, Stephen Hawking, and Robert Geroch on the existence of singularities in a wide class of solutions to Einstein’s field equations, and John A. Wheeler’s coining of the term “black hole” in 1967 (see Chapter 11). Gravitational radiation, a prediction of the theory in 1916, was not observed for nearly one hundred years (in 2015). Yet to those physicists who knew it well, the theory always retained its luster; to Paul Dirac, it was “probably the greatest scientific discovery ever made” while Russian Nobel physicist Lev Landau pronounced it “probably the most beautiful of all existing physical theories”.²

Born in the throes of World War I, general relativity was communicated to the wider world in London on November 6, 1919 following the theory’s confirmation by data gathered by English scientists from a solar eclipse the previous May. The announcement was made at a rare joint meeting of the Royal Society of London and the Royal Astronomical Society, in an atmosphere of intense expectation compared by the philosopher Whitehead to that of a Greek drama. Standing under a portrait of Newton, J.J. Thomson, Nobel laureate and President of the Royal Society, pronounced Einstein’s gravitational theory “one of the highest achievements of human thought”. Coming just a year after the Armistice of November 11, 1918 ended an unimaginable slaughter in the trenches of Flanders and France, the public found irresistible solace, even romance, in the fact that the scientific establishment of Great Britain showered its highest accolades on a hitherto obscure Berlin physicist whose theory had surpassed the defining achievement of England’s greatest scientist. To a world weary of war, revolution, and a raging influenza pandemic, Einstein and his incomprehensible theory quickly became a spectacle possessing seemingly inexhaustible potential to sell journalistic copy. In short order, Einstein acquired worldwide renown and the mystique of iconoclastic genius, accolades he did not seek but was never to lose, even in death.

A tortured path

Although possibly contravening details are lost in the mists of history, Einstein appears to have formulated special relativity in a flash of insight, the theory emerging fully grown suddenly in the first half of 1905, Minerva springing from the brow of Jove. Such was not the case with general relativity; its author, already widely known, left an extended and considerable trail of evidence. Eight years in gestation, the theory was completed in Berlin in late November 1915 in a frenzied race with Göttingen mathematician David Hilbert. Beginning in 1907, his last year working as a patent clerk in Bern, Einstein sought to extend the principle of relativity to non-inertial systems, i.e., systems in non-uniform motion. Persuaded by Ernst Mach’s rejection of Newton’s absolute space as the agency responsible for the effects of inertia, Einstein attempted to construct a theory of relativity applicable to all motions, hence the name general relativity. Straightforward analogy to the principle of special relativity (asserting the equivalence of all inertial frames of reference for the non-gravitational laws of physics) would have it that a principle of general relativity correspondingly affirms the equivalence of all reference systems (both inertial and non-inertial) for all laws of physics. That is, just as velocities are relative to an observer’s frame of reference within special relativity, so according to this putative principle, accelerations are to be relative to an observer’s frame of reference within general relativity. Yet the analogy cannot be correct, for it is tantamount to ignoring the press of inertial forces familiarly experienced by any rapidly accelerated observer (e.g., riding a roller coaster).

Too-close analogy to special relativity proved misleading in another crucial respect. Recall that a covariance requirement pertains merely to the form of the equations under consideration. In the theory of special relativity, satisfaction of a covariance requirement (Lorentz covariance – the laws of non-gravitational physics retain the same form under Lorentz transformations between inertial frames) implies a relativity principle; namely, that these laws are the same in all inertial frames of reference. After learning Riemannian geometry in Zurich in 1912–1913, Einstein was swayed by the power of analogy into thinking that the requirement of general covariance, i.e., that the laws of physics (now including gravitation) must be tensor expressions (see Box 6.1) that retain the same form in all coordinate systems, was mandated by a principle of relativity generalized to accelerated motions. The thought was this: to essentially eliminate any privileged roles for observers or coordinate systems, it was necessary to put all observers and coordinate systems on an equal footing, i.e., to impose the requirement of general covariance. From a contemporary point of view, it is easy to catalogue Einstein’s various conflations, of the relativity of inertial mass with relativity of frames of reference, of relativity principles with covariance requirements, and of “observers” (or “reference frames”) and “coordinate systems”.

To complete the general theory of relativity, Einstein would have to come to a new understanding of the requirement of general covariance. This did not completely sever the faulty implication from covariance condition to relativity principle that he retained more or less to the end of his life, but it did require him to “unlearn” the very lesson that guided him to the theory of special relativity, that by using postulated rigid rods and ubiquitous synchronized resting clocks, the resulting coordinates could be used by themselves to identify events in space and time, i.e., that space-time coordinates had an immediate physical significance. Much later, in 1946, he admitted this message to be the “main reason” for the lengthy gestation period of the general theory of relativity:

How in fact Einstein freed himself from this idea is one of the more storied episodes in history and philosophy of physics, for it involves entering into the meaning, and philosophical significance, of general covariance, a discussion that continues today.⁴

Milestones on the way

At first, progress was immediate: Sitting at his desk in the patent office in Bern (on January 7, 1907, to be precise), Einstein had what he later described as “the most fortunate thought of my life”, that “the gravitational field has only a relative existence, since for an observer freely falling from the roof of a house no gravitational field exists while he is falling”.⁵ If an unfortunate roofer drops a hammer while he falls, the tool descends at the same rate of acceleration (at sea level, 9.8 meters/sec²) and so remains – relative to him – at rest. Since Galileo it was known that if air resistance is neglected, all bodies regardless of material composition fall in the same way with constant acceleration toward the Earth. But this largely ignored fact suggested to Einstein that a freely falling observer would be justified in assuming that he occupied a mini-inertial system. Alternatively, an observer enclosed within a box (unbeknowst to him, at sea level on Earth) could consider objects released from shoulder height to be inertial bodies whilst he himself was accelerating, the elevator floor pressing against his feet with a force of 9.8 m/sec². Just this glimpse of recognition that acceleration (relative to one observer) might cancel freefall (according to another) and so be in some sense equivalent to gravitation, led Einstein to infer that a gravitational field, if only in the vicinity of the freely falling body, has a relative existence. That insight, deemed the principle of equivalence (Figure 6.1), proved the invaluable heuristic on which the development of the new theory would depend. For if the principle of relativity is to be extended to systems whose relative motion is non-uniform, even systems in relative uniform acceleration, then the phenomenon of gravitation must be taken into account.

Other guidelines also were clear from the beginning. Any relativistic theory of gravitation had to be compatible with the abolition of Newtonian absolute time and simultaneity already demanded by special relativity. Moreover, unlike the theoretically instantaneous gravitational attraction between any two bodies in Newtonian theory, special relativity mandated that the effects of gravity propagate no faster than the speed of light. On the other hand, Newtonian planetary astronomy is highly accurate, so some close approximation of Newtonian gravity must be recoverable within the relativistic theory. Indeed, the enormous empirical success of Newtonian gravitational theory deterred nearly everyone else from seeking another. With but a single known observational anomaly, the tiny 43’ of arc per century unaccounted for in the precession of Mercury’s orbit, the Newtonian theory had long been the firmest arch supporting physicists’ world view. The understandable opinion, that where there is nothing really wrong there is nothing to rectify, was nearly universal. On traveling to Zurich in 1913 to learn whether Einstein would consider a move to Berlin, Max Planck used the opportunity to give Einstein a bit of advice. An initial and enthusiastic supporter of special relativity, Planck frankly urged Einstein against wasting his time on gravitational theory and to turn his talents elsewhere:

A crucial step occurred in 1912 when Einstein came to realize that a relativistic theory of gravitation would require a non-Euclidean geometry for the appropriate mathematical description of the gravitational field. The realization came from a very simple example prompted by the principle of equivalence, extending the principle of relativity from uniformly moving to uniformly accelerated systems. In place of systems accelerating uniformly in a rectilinear direction, Einstein considered a rigid circular disk uniformly rotating in a Minkowski space-time, a non-inertial system in a region free of gravitational fields. The disk is a circle in the x-y plane in an inertial “Galilean system” K, and it is a circle in the x′-y′ plane of a system of reference K′ rotating with uniform velocity with respect to K, where it is supposed that the origin and z-axis of two systems of reference coincide. The question then is: What is the geometry of the disk? An observer in K, measuring with rods at rest in K, will find that the ratio of the circumference of the disk to its diameter to be π, concluding the geometry of the disk to be Euclidean. But in the context of special relativity, rods at rest in K′ determine a quotient different from π. From the standpoint of the observer in K, measuring rods laid out along the periphery of the rotating disk appear to suffer Lorentz contraction, slightly shortening in the direction of their rotational motion, while measuring rods laid out along the diameter are not affected since these are perpendicular to the direction of motion. Since more rods are then needed to span the circumference while the diameter remains the same, the ratio of the former to the latter is greater than π when both distances are measured within the frame in which the disk is at rest. Einstein concluded that Euclidean geometry has to be abandoned in a relativistic gravitational theory. This led him to the study non-Euclidean geometries, a subject that came to fruition in the second half of the 19th century (see Chapter 7).

In 1913, Einstein, recently arrived at the ETH, collaborated with his friend from school days, Marcel Grossmann, now a mathematics professor also at the ETH, to produce an initial “Outline (Entwurf) of a Relativistic Theory of Gravitation”.⁷ On leaving Prague in 1912, Einstein immediately sought out Grossmann for mathematical assistance with geometries of curved spaces. Grossmann obliged by introducing Einstein to the most general of these, Riemannian geometry, in its natural mathematical setting, tensor calculus, and to the so-called Riemann (or Riemann-Christoffel) tensor, the most general expression for the curvature of a Riemannian space. The Entwurf has a “mathematical part” (under Grossmann’s name) and a “physical part” (under Einstein’s); Einstein’s main result is to have found a field equation for the gravitational field, showing how the field is generated from its sources, matter and energy. This equation featured an essential aspect of the final field equations formulated two years later in November 1915; it contains a tensor (the Ricci tensor, see below) representing the curvature of space-time, derived from the fundamental curvature tensor of Riemannian geometry. The Entwurf field equation is inherently generally covariant by this fact. Einstein, however, erroneously believed the covariance of the equation had to be restricted (by imposing special coordinate conditions) for two distinct reasons: to arrive as a first approximation at the Newtonian theory of planetary orbits (the Newtonian limit) and additionally to satisfy the conservation laws of energy and momentum. He subsequently concocted a spurious argument (the “hole argument”, considered below) purportedly showing that the field equations in any case could not be generally covariant, if they were to be deterministic (in the manner of Maxwell’s equations) by enabling univocal prediction of the values of field strengths from the initial data of matter and energy sources. As Einstein also erroneously thought, the principle of general covariance was required by a principle of general relativity, i.e., the extension of the principle of relativity to accelerating frames of reference, this came as a heavy blow. For some two years he believed that a relativistic theory of gravitation could not be generally covariant, i.e., that the field equations could not keep the same form in all coordinate systems.

Breakthrough

In 1915 Hilbert invited Einstein to lecture in Göttingen on his progress, and in late June and early July he did so, entering into extensive discussions with Hilbert. Champion of the axiomatic method, Hilbert may well have seen that general covariance was an achievable goal, ideal from the axiomatic standpoint since in accordance with the spirit of Riemannian geometry it removed any metrical significance from space and time coordinates. In any case, by autumn Einstein was engaged in a “race” with Hilbert, now similarly engaged in the project of finding generally covariant field equations of gravitation.⁸ By November, Einstein had returned to the Riemann tensor, giving four presentations to the Prussian Academy of Sciences in Berlin; one at each of the weekly Thursday sessions of the section for mathematics and physics (Nov. 4, 11, 18, and 25).⁹ On November 4, he presented a variation of the 1913 Entwurf theory, with a coordinate restriction to ensure energy conservation. On November 11, he introduced into the previous week’s theory the hypothesis that the only matter sources of gravitation are electromagnetic in origin, essentially the electromagnetic theory of matter that Hilbert was concurrently attempting to couple with gravitation. In the third presentation on November 18, he used the not-yet completely covariant theory to calculate the observed tiny anomaly in Mercury’s orbit by which the orbit did not form a closed ellipse; the result differed from the theoretical Newton value by exactly the observed anomaly, 43 seconds of arc per century.¹⁰ Though not yet the final theory, the calculation relied only on assumptions that would be taken over by the generally covariant theory of the following week, and it showed that space-time is curved in the presence of a strong gravitational field, such as that of the sun. Finally on November 25, Einstein presented the field equations of general relativity in a form equivalent to the one widely employed today. (see Box 6.2). On the left-hand side are the Ricci tensor (R_uν) and the Riemann curvature scalar (R), both constructed from the metric tensor (g_μν) as are all invariant expressions within Riemannian geometry. On the right-hand side is a coupling term and T_μν, the so-called stress-energy-momentum tensor representing the mass and energy sources of the gravitational field in merely phenomenological fashion. The left-hand side represents space-time curvature (which is responsible for the effects of gravitation and inertia) and the right-hand side the matter-energy sources of this curvature. According to Einstein in 1936, the left-hand side is “a palace of fine marble” while the right-hand side is but “a house of cards”, since the T_μν is a crude skeleton, the fleshed out more particular form needed for any application depending on the type of matter and the specific interactions chosen for representation within a particular model.¹¹ Following John Wheeler, it is customary to read the expression from right to left, as “matter-energy tells space-time how to curve”, and from left to right, as “space-time tells matter how to move”.

Overview

General relativity’s two core ideas are briefly summarized: 1) distributions of matter and energy determine the geometry of space-time encoded in the metric tensor g_μν; 2) the trajectory (world line) of a freely falling uncharged “test” particle is a geodesic of this geometry, essentially the longest world line (an extremal curve) connecting two space-time events. Such world lines define a “local inertial frame” in general relativity, a small region of locally curved space-time indistinguishable from flat Minkowski space-time. With respect to the freely falling test body, in this region accelerations due to other (non-gravitational) forces may be defined. Taken together, these statements affirm that there is no prior geometry; rather the metric of space-time geometry is itself a dynamical inertio-gravitational field accounting for the dependence of spatio-temporal relations between events on the surrounding distributions and motion of matter and energy. In this book we cannot presuppose the requisite mathematical and physical background required for full comprehension of the theory of general relativity. Nonetheless it is possible to come to a qualitative understanding of the theory, and of how it was crafted to pursue the philosophical objective Einstein set for it, of ridding physics of fixed inertial systems, and of any residue of the old Newtonian concept of “absolute space”. The theory can be illuminated by reference to the three principles Einstein in 1918 considered undergirding the theory. These are the principle of equivalence, the principle of general relativity (and associated requirement of general covariance), and Mach’s principle.

Three principles

In the spring of 1918, in response to critics and others who sought clarification of the conceptual underpinnings of the newly completed theory of general relativity, Einstein published a brief discussion of three assumptions on which, he then claimed, the theory rested.¹² This short paper has been extensively discussed, since in it Einstein conceded, in response to a pointed criticism by Kretschmann, that the assumption of general covariance (as noted above; Einstein believed it was required by the principle of general relativity) possesses by itself no physical content but had a “heuristic significance” if conjoined with a principle of simplicity (see below). The concession lends itself to various interpretations, with the result that disputes over the physical meaning, if any, of the mathematical requirement of general covariance continue to this day.¹³ But the paper merits scrutiny for another reason, for it brings out complex interconnections between the philosophical requirements Einstein placed on the theory. These assumptions, each denominated a principle, are:

The 1918 formulations of A and C are the upshot of recent controversies with critics. But prefacing the list is a frank admission that previous failure to distinguish between A and C, statements ostensibly having little to do with one another, had been a confusion. The relations among the shifting content of all three principles is highly intricate; it is not difficult to find passages in Einstein’s writings that deem B to be the foundation of A or that A is a special case of B [A → B], but also that B requires A [B → A]. There are also statements to the effect that what is now distinguished separately as C is an extension or generalization of B [C → B] or that satisfaction of C requires A [C → A]. Scrutiny of earlier writings on the problem of gravitation, both published and in private correspondence, shows that the three principles took on various non-synonymous meanings during the gestational years of the theory from 1911–1915. Inadvertent confusion or lack of clarity regarding the meaning of each was the result, so this paper seems an analytical attempt to demarcate their respective boundaries, and thus to isolate the distinctive requirement each should impose on the completed theory in Einstein’s retrospective and presumably re-considered opinion.

Two general but related comments might be made before proceeding to a closer discussion of the three principles. First, a reader of this 1918 paper is bound to be struck by a notable admission that by encapsulating general relativity under the above three principles, the theory has been assessed “as it now appears to me”, seemingly suggesting that the theory, or at least its underlying principles, appeared to Einstein somewhat differently in the past. But how can a successful theory appear differently to its own creator within just a few years of its inception? To make sense of this, it is helpful to distinguish two distinct meanings of what might be meant in referring to “the theory”, distinguishing philosophical underpinnings from the theory narrowly construed, understood in contemporary terms as the set of all the theory’s models – here, models of general relativistic space-times. The latter, e.g., space-time descriptions of particular gravitational fields (such as the sun’s) did not fundamentally change between 1916–1918, nor have they since, although today more solutions of Einstein’s equations are known and studied. Rather, the changed appearance had to do with Einstein’s reappraisal of the theory’s motivating underlying assumptions.

The distinction between theory, as formalism, and underlying assumptions recalls Heinrich Hertz’s famous characterization of Maxwell’s theory of electromagnetism merely as Maxwell’s equations,¹⁴ jettisoning the attempts of Maxwell and others who sought to understand the theory’s central theoretical structures, i.e., the electric and magnetic fields, in terms of mechanical disturbances propagating through an ethereal substance supposedly required to support these processes. In this regard, Hertz is reported to have quipped that the formalism of the theory is wiser than its creator.¹⁵ Something like this is also true of the equations of general relativity, as it has been subsequently recognized that each of the supposed “core” principles of the theory either does not have, or cannot have, quite the meaning Einstein intended. First up is the principle of equivalence.

The principle of equivalence

Only formulated as a principle in 1912, the principle of equivalence is commonly illustrated by the “elevator” thought experiment used by Einstein, initially in a lecture in Vienna in December 1913. Imagine observers (“physicists awakening from a drugged sleep”), enclosed in a windowless elevator (“chest”) who are attempting to determine whether the elevator is at rest on a planet possessing a gravitational field (of a very special homogeneous kind, see below), or is undergoing a uniform acceleration upwards (in the opposite direction) in a region of space far from any planet or other massive body (for all intended purposes, a gravity-free region). Conceptually, the two cases can be distinguished as follows. In a gravitational field, an unsupported test object is observed to freely fall with an acceleration g = −a. Its mass will be regarded as weight, measurable in a scale balance. In the other, the test object, seemingly weightless and at rest, is met by the up-rushing floor of the elevator, rising with an acceleration a. Its inertial mass is manifested as resistance to acceleration, determinable by the magnitude of the required uniformly accelerating force. In both cases the object encounters the same location on the floor with the same absolute magnitude of acceleration. While there remains a conceptual distinction, the test object’s behavior is empirically indistinguishable in the two cases, indicating that its gravitational mass can be considered as identical to its inertial mass, m_g = m_i. This identity states the so-called Galilean, or “weak” principle of equivalence. As gravitational freefall and uniform upward acceleration are both non-inertial motions, the equivalence between them suggests that the principle of relativity extends beyond inertial frames of reference.

“Einstein’s elevator” has been employed at times (by Einstein himself¹⁶) to draw a stronger conclusion: that the laws of physics are the same in a laboratory in a gravitational field and in an accelerating laboratory in “empty space”, far from any source of gravitational force. That statement – known as the “strong” principle of equivalence – requires qualification, for it is correct only within the accuracy of measurements (of distances, speeds, angles) that a freely falling observer might make in his “local” region of space-time. Outside this limited region it is not difficult to detect the presence of a true gravitational field by showing the effects of “tidal forces” generated by the non-uniformity of the field. These forces, manifested in the different strengths of the gravitational field at different space-time points, are responsible for the relative accelerations of two test objects placed at a small separation. In an unchanging (static) and completely homogeneous gravitational field, a highly artificial situation almost never to be found in nature, these relative differences – by definition – do not exist. But generically they do, and they can be measured. For example, if two identical rubber balls are placed at horizontal separation high above the Earth and then released, then, on account of the inhomogeneity of the Earth’s gravitational field, their separation will decrease as they fall toward the center of the Earth, the greater the initial separation, the greater their relative acceleration toward each other. The non-zero mathematical expression (a tensor, see Box 6.1) linking the vectors of relative acceleration of the two balls to their separation vector is an unmistakable signature of a true gravitational field. As this phenomenon does not appear in an accelerated frame far away from all gravitational sources, the “strong” principle of equivalence is not generally valid. This has led some leading relativity physicists to question the need for the principle of equivalence.¹⁷

Correctly interpreted, the principle of equivalence essentially consists of four assertions:

So understood, the principle of equivalence is the sine qua non of the theory of general relativity. Equally importantly, it furnishes a necessary basis for the physical interpretation of the purely mathematical (pseudo-) Riemannian geometry of space-time on which the theory is based for it permits measuring rods and clocks to be coordinated with the purely mathematical quantity g_μν, the metric tensor.¹⁸ Without such direct linkage to a conceptually simple observable physical process, the Riemannian theory of manifolds arguably has no claim to be descriptive of the phenomena of physical space-time (see Chapter 7). In this regard, the principle of equivalence is fundamental in characterizing the transition between the four-dimensional globally flat space-times of special relativity and the (generically) curved four-dimensional space-times of general relativity. In particular, according to the “for all practical purposes” absence of gravity allowed locally by the principle of equivalence, a local inertial frame may be arbitrarily extended, becoming an inertial frame of special relativity. As a result, the principle, however stated, does not possess rigorous mathematical validity but has the character of a heuristic approximation.

On the other hand, on account of the principle of equivalence, manifestations of a body’s inertial behavior (its resistance to acceleration, or its uniform straight-line motion) can no longer be accounted as the effects of its situation with respect to a space (or space-time) of fixed predetermined structure. Rather, this behavior can be considered the result of the body’s dynamical interaction with the combined inertial-gravitational field generated by all surrounding masses and energies, changing as these change according to the Einstein field equations. That the effects of inertia and gravitation have a common origin and a common explanation as a dynamical interaction between given bodies and their material surrounding, is a core Machian idea that motivated the general theory of relativity (see below).

Finally, two subtle but striking predictions can be drawn from the principle of equivalence alone. The first is gravitational redshift. In the frame of a freely falling observer, light should behave just as if there were no gravity, i.e., traveling at fixed speed c with unchanging frequency (color, in the visible spectrum). However, frequency is not an intrinsic property of light. This is shown by the Doppler effect due to the relative motion of an observer and the source of a light wave or acoustic wave, most familiarly in the suddenly lowered pitch of the horn of a train engine rushing by. According to the principle of equivalence, if a flashlight pointing upwards on the floor of the enclosed elevator is switched on during the elevator’s upward acceleration (in the absence of gravity), the frequency of the light wave arriving at the ceiling will be slightly lower than its frequency on emission because the speed of the ceiling at the reception of the signal is greater than the speed of the floor when the flashlight was switched on. A gravitational redshift is then equivalent to a Doppler shift between two accelerating frames. But gravitational redshift is also the most direct observational test of the curvature of space-time. If one considers clocks as periodic emitters of light (e.g., the spectral lines of atoms), then the spectral lines of atoms near a large mass will exhibit somewhat lower frequencies than the lines emitted by the same kind of atoms elsewhere. In this regard, gravitational redshift is a special case of a more general effect: all processes occurring in a gravitational field surrounding a large mass appear “slowed down”, as now will be seen.

Ignoring the motion of the Earth in its orbit, Einstein believed it possible to confirm general relativity by measuring the redshift of light emitted from the surface of the sun when seen from Earth. But measurements did not at all agree with the predicted value of the shift, in large measure because the surface of the sun is a highly complicated and violent gaseous region, though this was not well understood until some decades later. As a result, for many years not a few critics of the theory of general relativity pointed to the failed prediction of gravitational redshift as sufficient reason to reject the theory. The first confirmation of gravitational redshift occurred in the period 1960–1965 (after Einstein’s death) in an experiment using the 73.8-foot tower of the Jefferson Physics laboratory at Harvard. R. V. Pound, G.A. Rebka, and J.L. Snider were able to detect the tiny effect expected in the Earth’s modest gravity by exploiting a subtle effect of atomic crystals previously demonstrated by Mössbauer in Heidelberg. Placing an emitting crystal at the top of the tower, and a detecting crystal in the basement and monitoring the detection rates, Pound, Rebka, and Snider were able to displace the emitter with just the right motion relative to the detector that the velocity difference between emitter and detector compensated for the gravitationally induced blueshift of the emitted photon falling in the Earth’s gravitational field to the detector. Although the predicted frequency shift was merely two parts in a thousand trillion (10¹³), the initial experiments produced a result to within 10 percent of predicted value. Subsequent experiments narrowed the gap between prediction and observation to 1 percent.

From the gravitational redshift of light, another implication of the principle of equivalence is immediate: time dilation in a gravitational field. This is most clearly seen using atomic clocks based on the oscillations of a source (e.g., cesium atom) having a constant, stable, and well-defined frequency. Suppose that at a given location two such identically built clocks are synchronized so that at one moment they tick at the same rate and read the same time. Then carry one to the top of a skyscraper while placing the other at the bottom, leaving them in place for a few days. On bringing them together again, it will be found that they are no longer synchronized, but that the clock from the top of the skyscraper is running faster than the clock left at the bottom; it will have ticked more times in the interval during which they were separated. Both atoms and clocks (whether mechanical, atomic, or biological) perform their functions more slowly where gravitational field strengths are stronger. This is a relative effect: the frequency of the light emitted by the atom at the bottom looks normal to an observer there, but appears redshifted to the observer at the top, whose clock is running faster. Recent advances in navigation using Earth-orbiting atomic clocks and highly accurate time-transfer technology routinely take gravitational redshift and time-dilation effects into account. The accuracy of Global Positioning System (GPS) devices rests on the difference in rate between satellite and ground clocks as a result of relativistic effects amounting to 39 microseconds per day (46 µs from the gravitational redshift, and −7 µs from time dilation).

The principle of general relativity

The 1918 formulation of this principle cited above requires considerable explanation. It would seem that some such principle is implicated in the very name of Einstein’s theory. In his first comprehensive review of relativity theory, written in 1907, Einstein raised the question, “Is it conceivable that the principle of relativity also holds for systems that are relatively accelerated with respect to each other?” Such an extension of the principle of relativity appeared possible on the grounds of the thought experiment concerning the freely falling observer described above. In 1907, Einstein spoke not of a principle, but of the assumption (Annahme) of equivalence between an inertial system containing a homogeneous gravitational field and uniformly accelerated system. By 1911, the supposition of equivalence between an inertial system K in a homogeneous gravitational field and a uniformly accelerated system K′ in gravitation-free space came to serve as the basis for a much stronger claim, of the relativity of accelerated motions:

In more cautious moments, as in a lecture in Vienna in September 1913, Einstein frankly denied that the relativity principle extended to nonuniform motions.²⁰ Yet the postulate of general relativity is given pride of place in the early sections of the first complete exposition of the final theory in April 1916. In §2 (“On the reasons suggesting an extension of the postulate of relativity”) of that paper, Einstein presented a thought experiment hypothetically involving an otherwise empty universe containing only two fluid spheres in relative rotation. Observers on each can find by local measurements which sphere is actually rotating: it has a bulging equator. The explanation of this fact given by Newtonian mechanics, singling out the “true” inertial frame of the non-rotating sphere as the legitimate space for the laws of mechanics, is criticized for its appeal to a “factitious cause”, an unobservable absolute space, rotations in respect to which occasion the observed inertial effects. In contrast, Einstein demanded that any epistemologically satisfactory account of the relative rotations appeal only to observable causes, a restriction that ostensibly appears vintage Machian positivism. But this demand is followed by another having only tenuous relation to epistemological strictures requiring observability.

A few paragraphs later in §3 (“The space-time continuum. Requirement of general covariance for expressing the equations of the general laws of nature”), the implicit inference of §2 is made explicit: the postulate of general covariance (“the general laws of nature are to be expressed through equations which are valid for all coordinate systems, that is, are covariant with respect to arbitrary substitutions”) is implied by a “general postulate of relativity”.

This is precisely the inference seen above to be false. The principle of general relativity is not a generalization of the principle of special relativity in the sense that “all motions are relative” since uniform and accelerated motions are not (outside of highly contrived situations) physically equivalent. But then what is it? From a contemporary perspective, it is essentially the requirement that the laws of nature must be formulated in terms of geometrical relationships between physical quantities, the latter expressed in coordinate-free geometric form (i.e., represented as tensor quantities).

As Hilbert surely knew, and as Erich Kretschmann (a Planck student) would observe in a critical article in 1917, the requirement of general covariance is purely formal: it amounts to no more than a stipulation that the laws of nature are to be stated in tensor form; after all, tensors are expressions that have the same form in all coordinate systems.²² Both gravitation-free special relativity and Newtonian gravitational theory can be formulated in a generally covariant manner using only tensor equations. In his 1918 response, Einstein had to agree but countered that Newtonian gravitational theory, if expressed in tensor form, would be considerably more complicated than general relativity. In short, the principle of general covariance together with the constraint of simplicity can have physical significance, at least if it is assumed that the simpler of two theories in generally covariant formulation is the more likely to be true. This kind of defense of general covariance as a physical principle was paraphrased much later in what remains the lengthiest textbook of general relativity, affectionately known (in its day) as “the telephone book”:

Indeed, Newtonian theory in tensor form appears horribly complex when expressed as (now) ten differential field equations for the metric coefficients in a specific coordinate system. Still it turned out to be a reasonably straightforward matter in the 1920s to write down Newtonian theory (and its absolute motions) in coordinate-free geometric form. This is just what careful contemporary comparisons of Einstein’s gravitational theory with Newton’s do, employing an empirically equivalent geometric version of Newton’s theory.²⁴ Furthermore, and more significantly, Kretschmann observed that formulated in the geometry of Minkowski space-time, the relativity postulate of special relativity, i.e., Lorentz transformations carrying inertial states (geodesic trajectories) into inertial states (geodesic trajectories), is a symmetry of the space-time and so a physical, not a merely formal, postulate. But in general relativity the local inertial (geodesic) structure about a freely falling test body depends on surrounding (and variable) mass-energy distributions; there are as many local inertial frames as there are variable distributions of matter with no symmetries between them. In brief, in general relativity there are no (non-trivial) symmetries of space-time. Hence, general covariance, even as a purely formal requirement, cannot be associated with any relativity of motion principle. On the other hand, while Newtonian gravity need not be expressed in tensor form, Einsteinian gravity must be because of the relativity of the gravitational field established by the principle of equivalence. In general relativity, there is no invariant (tensorial) way of capturing the core Newtonian distinction between gravitation and inertia; in general relativity, there is only a common inertio-gravitational field, the distinction between them being observer-dependent.²⁵

Even though he repeated the inference in 1916 and even later on, in order to find the final form of the general theory of relativity in late 1915, Einstein had to sever the connection between relativity principle and covariance condition. In returning to general covariance, he effectively had to reject the default assumption that space-time coordinates have an immediate metrical meaning. The path along which he traveled in doing this, extending over more than two years, was for a long time considered by historians of relativity and physicists alike as a gradual realization of a mere mathematical confusion.²⁶ It is now one of the more storied episodes in history and philosophy of physics since 1979 when physicist and Einstein scholar John Stachel – making essential use of Einstein’s correspondence in late 1915 and early 1916 with Paul Ehrenfest, H. A. Lorentz, Michele Besso, and others – revealed the full extent of Einstein’s struggles with general covariance in 1913–1915.²⁷ It is again a story of a misleading analogy to the special theory of relativity. There, the coordinate system of any inertial frame constructed using rigid rods and synchronized clocks located at each space-time point provides an immediate metrical meaning to these points. The attempt to carry this out in the context of gravitational theory led Einstein to fall into the clutches of what he called the “hole consideration” (die Lochbetrachtung), widely known today as the “hole argument”. The argument convinced Einstein that his relativistic gravitational theory could not be generally covariant, on pain of a failure of the determinism required of all satisfactory field theories: from given boundary conditions and sources, temporal evolution of the field equations should completely determine field strengths at any point in a given region. Until autumn 1915, he remained persuaded of its faulty conclusion, a bitter pill to swallow given the erroneous association of the principle of general covariance with a generalization of the relativity principle to accelerated systems.

The hole argument and the point coincidence argument

Einstein came under the sway of the hole argument in the summer of 1913; it appears in several papers of 1913–1914. Recall that in the 1913 Entwurf theory, Einstein erroneously believed that special coordinate conditions (restricting the theory’s covariance) were required in order to recover the Newtonian limit and to satisfy the laws of conservation of energy and momentum. Since he believed in the faulty implication above, this, as he wrote to H.A. Lorentz, was an “ugly dark spot” on the theory.²⁸ He then invented an argument to purportedly show that this blemish was in any case a necessary one as it was impossible to reconcile the requirement of general covariance with the determinism required of all reasonable physical theories. For the next two years, Einstein chose to restrict the covariance of his theory in order to preserve determinism, arguing, as in this review paper from 1914:

The hole argument is rather intricate; to properly extract Einstein from its clutches requires invoking machinery of modern differential geometry not developed until around 1950.³⁰ But it can be stated in simplified way. Consider, as Einstein did, a limited region of space-time where no matter is present, a “hole” H, completely surrounded by a matter-filled region. Using the notation of Einstein’s equation (Box 6.2) inside H, T_μν = 0, whereas in the containing region, T_μν ≠ 0. If that equation is read from right to left (“matter-energy tells space-time how to curve”), the space-time geometry (the values g_μν) inside H are determined by matter-energy sources outside H (where T_μν ≠ 0). As noted, the g_μν are functions of the space-time coordinates x^σ = x¹, x², x³, x⁴. Einstein presupposed that space-time coordinates suffice to individuate physical events of space-time. After all, within special relativity differences between space-time point-events in every inertial system are understood as marked off by rigid rods of unit length and perfectly synchronized clocks at each point of the system; these differences have an immediate and fixed metrical significance. He then argued that if one requires general covariance of the field equations, then a specific matter distribution outside H does not uniquely determine the g_μν within H.

Einstein’s response to the hole argument for more than two years was to abandon the requirement of general covariance since it led to a failure of determinism (in the above sense). Only in late 1915 did Einstein see that the argument’s conclusion rested on a questionable philosophical presupposition: that space-time has a physical existence independent of the presence of physical fields (i.e., actual physical events corresponding to values of the metrical g_μν field or other matter fields). In particular, this amounts to the claim that the space-time points (and so those within H, where the solutions differ) have an identity fixed prior to the specification of the metric g_μν; that is, prior to solving the field equations. Resolution of the hole conundrum lay in seeing that this assumption is by no means necessary and might be denied. And this is to say that space-time points cannot be individuated, or identified, without the physical properties g_μν specified there by the gravitational field equations. In general relativity, the geometrical g_μν is a dynamical variable to be determined at all points of a four-dimensional manifold. Any physical properties attributed to these points – indeed, their physical reality as space-time point-events – are consequences of that determination. Then the conclusion that there are two physically distinct solutions g_μν at some given point p within H is faulty since it relies on the assumption that the space-time coordinates alone give p an independent existence, that it can be the bearer of two physically distinct values of g_μν. Under the coordinate transformation, g′_μν (x) ≠ g_μν (x) do differ but they are only different mathematical descriptions of the same physical geometry at p, just different tensorial characterizations (eliminating any special role for coordinate systems) of the same event.

Precisely when in the autumn of 1915 Einstein received this illumination, in many ways the most novel philosophical aspect of general relativity, is not known. In an unpublished account written in 1934, Einstein reports specifically on how he was troubled by having to “unlearn” the lesson of special relativity:

When did he realize that coordinates in general relativity are simply parameters? An elliptical passage in §3 of the canonical April 1916 presentation of the theory accounts for the specific 1918 formulation of the “principle of general relativity” cited above. This is the so-called “point coincidence argument”:

Positivists and empiricists took special notice of this passage that seems to rest the observational basis of physics on the verification of “point coincidences”. Max Born read Einstein as identifying “das physikalische Urphänomen”, the primitive phenomena to which the entire empirical content of physics can be reduced; Phillip Frank went further, assimilating Einstein to Machian phenomenalism:

The cryptic remarks on point coincidences are in fact the conclusion of an enthymematic argument by which Einstein extracted himself from the hole argument. Their meaning could hardly be recovered from the cited passage except by the few of Einstein’s correspondents or confidants for whom the missing context was understood. They do not intend what they appear to say: that in locating the criterion of what is in principle observable in the coincidence of points (intersections of world lines), only such observable phenomena are real. Rather, in the space-time setting of general relativity, “coincidences” are any physical event, a physical occurrence in a small localized region: the pop of a firecracker, the intersection of the trajectories of particles (i.e., a meeting of the world lines of two bodies), an atom’s emission of a photon. All observers will agree whether such an event occurs, while coordinates are just arbitrary labels of the event or, as it is sometimes called, “world point”. The totality of such events – past, actual, and future – is then the physical reality of space-time. This is the sense in which the requirement of general covariance removes “the last remnant of physical objectivity” from the old Newtonian idea of an absolute fixed background structure to space. Relations of connection and succession suffice to establish the essential meaning of space and time as forms of the coexistence, connection, and order of events.³⁴

can be understood as giving rhetorical emphasis to the now well-understood fact that coordinates have no direct metrical meaning, that they are only a means of labeling the points of a four-dimensional manifold in such a way that differential calculus can be performed on a mathematical representation of space-time. The physical meaning of general covariance is that only space-time quantities that legitimately appear in the equations of physics are the metric tensor g_μν and quantities derivable from it.³⁶ Mere points of the underlying mathematical manifold then have no physical meaning. In the absence of fields with their physical sources, manifold points are not points of space-time; i.e., they have only mathematical, not physical, meaning.³⁷ Late in his life, Einstein made several attempts to elucidate this point of view. These efforts to clarify the significance of the principle of general covariance notably cluster around Einstein’s reliance on the principle as a heuristic guide for the futile unified field theory program.³⁸ The fundamental meaning of general covariance is then expressed by the statement that there can be no principled distinction between the structure of space-time and its “contents”; in brief, “no metric, no space-time”. In this way, space and time by themselves have indeed lost “the last remnant of physical objectivity”. In this broadened heuristic sense, the meaning of general covariance encompasses the purely formal requirement of general covariance (requiring use of tensor expressions and freedom to make “arbitrary”, including nonlinear, transformations of the space-time coordinates). The heuristic meaning sets a programmatic agenda within which the distasteful notion of a fixed inertial system does not appear; it has the latter as an implication. Any theory in which the space-time metric is dynamical must be (like general relativity) formulated in a generally covariant way.

Late in life Einstein thought the matter important enough to underscore once again. Declining, on grounds of illness, an invitation to attend the 50th anniversary conference of relativity at Bern in 1955, he noted that for him, the “most essential thing” about the general theory of relativity is its ambition to remove from physical theory, once and for all, the notion of a privileged frame of reference, background space-time structures that act (in the explanation of the inertial behavior of a body) but which in turn are not acted upon.³⁹ A core motivation underlying the unsuccessful unified field theory program dominating the last three decades of Einstein’s life was the recognition that the theory of general relativity, as a theory of the gravitational interaction alone, could not adequately accomplish this goal but that a “theory of the total field” encompassing the other matter/energy fields might. It is general covariance in this wider meaning (sometimes also called, “the general principle of relativity”) that served as a “limiting heuristic principle” in the search for such a unified theory.⁴⁰

Mach’s principle

The final fundamental principle on Einstein’s 1918 list while bearing Mach’s name is informally paraphrased as “a generalization of Mach’s demand that inertia be explained by the interaction of bodies”. The demand on the kind of explanation considered satisfactory leaves wide open the nitty-gritty details of how it might be carried out. In particular, an explanation of this kind requires showing how other masses influence the value of a given body’s inertial mass, as well as how they exercise their influence on each local segment of a body’s inertial trajectory. Mach himself provided only the murkiest idea of how the forces that Newton sought to explain in terms of acceleration relative to absolute space might be better explained in terms of accelerative motions with respect to other masses. Nonetheless “Mach’s principle” pays tribute to the inspiration Einstein received from the familiar passages in Mach’s Mechanics criticizing Newton’s purported demonstration of an effect of absolute motion, and hence the existence of absolute motions, in the famous bucket experiment.⁴¹ Mach’s tacit appeal to an “interaction of bodies” in accounting for the manifestations and effects of inertia – the straight-line uniform motion of a body on which no net forces act, the body’s resistance to acceleration, the appearance of centrifugal forces in accelerating bodies, etc. – comprised, in Einstein’s terms, at least a template for the “relativization inertia”, a phrase not found in Mach. In general relativity the problem takes the form of attempting to show that the matter content of the universe should determine a body’s freely falling local inertial frame. Einstein’s difficulties in trying to carry out such a program within general relativity occasioned a shift in explanatory emphasis on “an interaction of bodies” to the purely negative injunction that no physical properties be ascribed to “empty space”.

Mach deemed it necessary to purge non-empirical metaphysical elements from the foundation of mechanics as established by Newton. The essential flaws lay in Newton’s unsatisfactory definition of mass as “quantity of matter” and his statement of the first law, the law of inertia. Mach reformulated the definition of mass as a ratio of masses, so that the second law appears as a mere definition and the third law, the law of action and reaction, as a consequence of the new definition of mass. To Mach, Newton’s mechanics entirely rested on the first law. Mach recognized that Newton’s tacit appeal to absolute space in the statement of the law of inertia was occasioned by doubts whether any given fixed star is truly or only apparently at rest. As “we have knowledge only of relative spaces and motions”,⁴² the first law is then to be understood so that uniform motion is with respect to the observable distant stars. If perchance these are discovered to be not at rest, then the fixed reference system presupposed by the law of inertia remains “still to be found”. Similarly, Mach sought to explain the appearance of inertial effects in accelerating systems, such as water rising up the sides of Newton’s rotating bucket, as due to the influence of observable cosmic masses. In particular, Mach hinted that a body’s inertia is not a resistance to acceleration per se but to acceleration with respect to the distant cosmic masses. Mach’s own concrete suggestions regarding how the principles of mechanics might be reformulated so that inertial forces (as in Newton’s rotating bucket) appear in rotations with respect to other masses were left perhaps intentionally vague, with the consequence that they were largely ignored. Much more important was Mach’s assertion that it is unnecessary to refer the law of inertia to absolute space. This resonated with Einstein who – according to the “principle of general relativity” – sought to eliminate altogether the notion of inertial frame. Such an elimination points to a “relativization of inertia” in the following sense: without a distinguished class of reference frames, Einstein could conjecture à la Mach in 1912 that “the entire inertia of a mass point is an effect of the existence of all other masses, resting on a kind of interaction (Wechselwirkung) with the latter”.⁴³

Immediately on publication of the first attempt to formulate a relativistic theory of gravitation, the Entwurf theory completed in Zurich in June 1913 with the collaboration of Grossmann, Einstein posted an offprint to Mach on June 25, acknowledging in an accompanying letter the salutary influence of the Mach’s “brilliant (genialen) investigations on the foundations of mechanics”.⁴⁴ Boasting that the Entwurf theory “establishes with necessity that inertia has its origin in a kind of interaction (Wechselwirkung) of bodies, entirely in the sense of your considerations on Newton’s bucket experiment”, Einstein listed several suggestive results of the theory (described below) that yield only partial realization of the goal of a complete “relativization of inertia”. A further postcard to Mach, written in the second half of December that year, contains a declaration that might have been penned by Mach himself, “For me it is absurd to ascribe physical properties to empty space”.⁴⁵ Einstein must have known that old age, a stroke that paralyzed his right side, and failing health had not diminished Mach’s anti-metaphysical ardor. Just a year before, writing in the Preface to the seventh German edition of his Mechanics, Mach had described himself as an old man “struck down by a grave malady (who) shall not cause any more revolutions”. Nonetheless he could not resist one last expression of scorn for “the monstrous conceptions of absolute space and absolute time”. Though Einstein and Mach were not personally close, and may have only met once or at most twice, Einstein clearly felt that it was important to keep Mach apprised of progress on carrying out what Einstein considered Mach’s agenda.

Einstein’s most comprehensive early treatment of the problem of “relativization of inertia” appeared in a September 1913 lecture “On the Present State of the Problem of Gravitation” delivered, fittingly, in Mach’s Vienna.⁴⁶ Section 9, bearing the heading “On the Relativity of Inertia”, describes more fully the results mentioned in the letter to Mach. Einstein asserts that according to the Entwurf theory, he can demonstrate three effects suggesting a “relativitization of inertia”: first, the inertia of a body can be shown to vary with the proximity of neighboring masses; second, he can show that there is a precession of the plane of oscillation of a pendulum contained within a rotating mass shell (see Chapter 11); finally, he argues that a rotating mass shell generates a so-called Coriolis force, whereby objects actually traveling in a straight line appear to an observer rotating with the shell to be deflected. The three Machian predictions are repeated in Einstein’s 1921 Princeton lectures, after the completion of general relativity.⁴⁷ However, the main Machian result of this 1913 paper is presentation of a two-termed expression for the energy of a slowly moving mass point in a Newtonian gravitational field. Einstein argued that according to the first term, the energy of the body diminishes if the surrounding masses are augmented, corresponding to a greater inertial mass of the point. He then elaborated:

In this lengthy exposition, noticeable leeway is given to the distinction between an interaction with other masses determining a given body’s inertia, as opposed to merely influencing or conditioning it. As subsequent events will show, this will prove to be a necessary qualification. The extended discussion reveals also a distinction between a purely “kinematic” principle of relativity of motion – only the relative motion of a body with respect to other bodies is observable; the motion of a body with respect to absolute space is unobservable – and a “dynamical” principle of relativity of motion – the inertial motion of a body is influenced by surrounding masses. While relationist accounts of motion from Leibniz onward criticized the unobservability of absolute motion, it is clear that Einstein considered Mach’s unique contribution to relationist/absolutist debate to have posed the explanatory requirement of a functional dependence (causal, see below) account of inertia as a criterion to be met by any satisfactory account of mechanics.

Nonetheless, a Machian insistence on “observable empirical fact” becomes the centerpiece of an argument in §2 of the above-mentioned first full synthetic exposition of general relativity in April 1916. Entitled “On the Reasons That Suggest an Extension of the Relativity Postulate”, Einstein presented a thought experiment involving a relative rotation. He considered two fluid spheres “hovering” in space at a constant but considerable distance from one another and at such a great distance from all other bodies that external influences can be ignored. On each is an observer who judges his own sphere at rest but the other sphere to be rotating with constant angular velocity about the line joining the bodies. However, using local measures, one body is found not to be a sphere but an ellipsoid of rotation, i.e., it bulges along its equator as does the Earth. A variant of Newton’s familiar thought experiment of two rotating globes, Einstein’s purpose is to identify an “inherent epistemological defect” in the Newtonian account of the situation that “perhaps for the first time was pointed out by Ernst Mach”. The defect is this: the Newtonian explanation – that the bulging body is “truly” rotating, i.e., rotating with respect to the preferred inertial frame of absolute space – appeals to a “factitious cause”, absolute space, as the reason for the observable difference between the two bodies. This charge is coupled with a highly restrictive codicil governing causality, namely, that application of the “law of causality” to experience has meaning only if it pertains to observable causes and effects.⁴⁹ Einstein insisted that an observable effect must have an observable cause. But this demand unfortunately sidetracks, even distorts, the salient epistemological issue.

Mach himself regarded the very use of the concepts of cause and effect in physics to be merely picturesque description, a holdover from everyday experience, but liable to give rise to metaphysical fetishism (see Chapter 8). In its place, he urged substitution of the language of functional dependence of (observable) quantities on one another, e.g., the variation of position with differential increments of time as expressed by the equations of mechanics. In a 1912 paper “On the Notion of Cause”, well-known to philosophers, Bertrand Russell, following Mach, had urged just this understanding of functional dependence as replacement for talk of causation in physics.⁵⁰ Secondly, these observational constraints on causal attributions were read by subsequent philosophers of science as providing yet further confirmation that Einstein’s philosophy and methodology of physics fully conformed to logical positivist strictures, which indeed had been fashioned on this understanding of Einstein. But consistency with the overall non-positivist character of general relativity, the archetype of a “principle theory”, suggests that criticism of the Newtonian invocation of absolute space as appeal to a “factitious cause” is really a methodological indictment that the alleged cause is not independently discoverable. As such, absolute space is an unexplained explainer invoked only, and conveniently, in the Newtonian story about absolute motion. If, independently of that familiar story, absolute space played an explanatory role in accounting for other mechanical phenomena, its causal standing in the explanation of the observable difference between the two spheres would no longer merit the epithet “factitious”, despite the fact that absolute space remains unobservable. However, it plays no other such role in Newtonian mechanics.

This criticism of absolute space can be pushed yet further, from methodology to metaphysics. The dismissal of the Newtonian appeal to absolute space as a “factitious cause”, though packaged in positivist window-dressing for publication, is a complaint that absolute space is an artificial, contrived, and unnatural posit. And this is because absolute space has the metaphysical attributes of absolute substance, i.e., something standing outside the nexus of causal relations because it acts, but is not in turn acted upon. From the incipient beginnings of general relativity to the end of his life, Einstein returned again and again to lodge this objection to absolute space. For example, in his Princeton lectures of 1921:

In recent times, Einstein’s objection to unidirectional causal agency in physics has been understood to mean that in general relativity there can be no non-dynamical objects, though there continues to be disagreement upon precisely what counts as dynamical and as an absolute, or non-dynamical, object. The broad metaphysical prohibition against the one-way causal isolation of substance long antedates these discussions.

Mach’s principle and relativistic cosmology

During the long months of struggle between 1913 and the completion of general relativity in November 1915, Einstein sustained his faith that such a theory was possible with the idea that a body’s inertia was due to its interaction with all other matter in the universe. The Dutch astronomer Willem de Sitter proved the sharpest critic of Einstein’s Machian ambitions with general relativity. To de Sitter, Einstein confessed that Mach’s conception was “psychologically important because it gave me courage to continue work on the problem when I absolutely could not find covariant field equations”.⁵³ Indeed, the 1918 formulation of “Mach’s principle” above originated in a dispute with de Sitter about conflicting cosmological models. In discussions with Einstein in late 1916 in Leiden, de Sitter emphasized that determination of the inertial-gravitational field requires reference not only to distant masses but also to boundary conditions. Since these were generally assumed to be fixed “at infinity” with the fixed, non-dynamical values of the metric of special relativity, they are effectively a priori unilateral causal agents. To the extent that they play a role in the explanation of motions in general relativity (e.g., that a freely falling test particle moves on a geodesic), those motions are accordingly absolute in the sense forbidden by Mach.

Having completed the theory of general relativity, Einstein sought to eliminate such “un-Machian” solutions to the field equations of gravitation. In doing so, he created relativistic cosmology in 1917 in a paper declaring:

Initially considering the cosmological problem, Einstein assumed a universe infinite in both space and time. Its mass was assumed finite and essentially static, with no large-scale motions between cosmic bodies; the relative distances between stars and “nebulae” (galaxies) remained fixed. These assumptions presumably were made to render cosmology a scientific and not a theological question, and so to avoid the problem of origin. As de Sitter emphasized, solutions of the gravitational field equations in this model required fixing boundary conditions at spatial infinity. For Machian reasons (mass-energy completely determines inertia, see below), Einstein assumed that space (the spatial components of the metric tensor) should vanish at infinity. However, he was unable to find any such solutions, even approximately.

Einstein then effectively “abolished infinity”.⁵⁵ Under the above requirement on a “consistent theory of relativity” Einstein projected the first relativistic cosmological model, a spatially closed cylindrical world positively curved in the three spatial dimensions hence spatially finite, but indefinitely extended in the time dimension. It is also a quasi-static universe with a uniform distribution of stars. He immediately recognized, however, that such a universe would collapse under the mutual gravitational attraction of cosmic masses. In order to ensure the model’s stability, he inserted into his field equation a new “cosmological term” that would be effectively zero at scales smaller than the solar system but at larger scales acted as a repulsive force (a “negative pressure”) to exactly balance against gravitational pull:

While it detracts from the simplicity of the equation, the Λ term (‘weighted’ by the metric g_μν) is the only type of term that could be added in a way consistent with local conservation of energy and momentum of T_μν.⁵⁶ The new term sought to ensure that at the scale of galaxies, there are no secular (non-periodic) motions. Opposing gravitational attraction with a tendency to increase the spatial separation between objects, Einstein deemed the term

Yet if one considers the matter for a moment, it is easy to see that the cosmological term can provide at most only an extremely fragile balance against gravitational collapse. In order to do its job, it has to be “fine-tuned” to the detailed specifics of actual distances between stars and galaxies. And since the stars do have small relative velocities resulting in changing local condensations of matter, to maintain balance, the cosmological term must continually re-adapt to the varying gravitational pull between them. In short, the Einstein universe is highly unstable even for tiny perturbations in the distribution of cosmic masses, a result that some years later would be shown rigorously by Lemaître and Eddington.

Within just two weeks, Dutch astronomer Willem de Sitter produced an exact solution of the now-modified field equations for a matter-free universe, inconveniently demonstrating just how elusive is the goal of a complete relativization of inertia. De Sitter’s universe contains just the repulsive energy density of empty space represented in the new cosmological term. The inertial-gravitational field in such a world, having no material sources, is essentially indistinguishable from absolute space; accordingly, from a Machian point of view, it is completely unacceptable. Einstein could only hold his nose while hoping to find some flaw in de Sitter’s argument. In this he was unsuccessful, and an ensuing correspondence with de Sitter centered precisely on the question of whether the theory of general relativity satisfied any version of a Machian requirement to relativize inertia. In a letter to de Sitter in March 1917, Einstein made is his philosophical position fully explicit:

De Sitter’s model universe turned out to be non-static and expanding, though this was not clear for some time (today, de Sitter’s model is regarded as a standard early universe vacuum solution to Einstein’s field equations). In 1922 Russian mathematician Alexander Friedmann showed that a closed universe of roughly uniform density will inevitably expand to a maximum dimension, then contract in complete gravitational collapse. The most natural solutions of Einstein’s unmodified field equations resulted in a dynamic (expanding or contracting) universe, not a static one. Without knowledge of Friedmann’s solutions, Belgian Jesuit priest Georges Lemaître rediscovered them and in 1927 formulated a general relativistic model of an expanding universe to accommodate new astronomical data showing an apparent radial velocity of distant galaxies in the line of sight from Earth. Lemaître’s model, which crucially contained the cosmological term, was a “fireworks theory” (Eddington),⁵⁹ and a precursor of today’s hot big bang models. The universe has a beginning, a very small highly compact mass (an initial singularity) and subsequently expands due to radiation pressure. In March 1929 American Edwin Hubble reported data from the 100-inch telescope at Mt. Wilson in California indicating a roughly linear relation between velocities and distance, as measured by redshifts from receding galaxies. Theory and observation had coalesced, agreeing that the universe is described by general relativity and is expanding.

From a philosophical point of view, Einstein was far more comfortable with the notion of an eternal, static, bounded universe. His initial reactions to the possibility of dynamic universes by Friedmann, then Lemaître, were accordingly critical; with Friedmann, he was (wrongly) convinced a mathematical error had crept in, and while he could find no errors in Lemaître, he wrote that “from the physical point of view it was “tout à fait abominable”.⁶⁰ Later on, after Hubble’s measurements (together with his assistant Humason) of stellar redshifts convincingly demonstrated the recession of stellar systems from one another, Einstein had to capitulate. On February 5, 1931, the New York Times reported that, during a current visit to Caltech, conversations with Hubble and with physicist Richard C. Tolman had convinced Einstein that his original 1917 cosmology, assuming a static and uniform closed universe, “would have to be modified in accordance with more recent data”.⁶¹ Since the term had been introduced to ensure the static, fixed character of the universe, Einstein is reported (by George Gamow) to have called its introduction into the field equations of gravitation his “biggest blunder”. Whether he in fact said this is unlikely.⁶² Today, however, the so-called cosmological constant is a fundamental link, and glaring problem, between quantum field theory and general relativity (see Chapter 11).

The Schwarzschild solutions

Early in January 1916, soon after completion of general relativity the previous November, German physicist and astronomer Karl Schwarzschild (1873–1916), serving as an artillery officer on the Russian front, sent Einstein the first exact solution of the new gravitational field equations. Schwarzschild’s solution pertained to the vacuum space-time exterior to the surface of a star or other massive object, idealized as a static (time-independent), non-rotating spherically symmetric body. The so-called Schwarzschild exterior solution describes how the gravitational field of a compact object such as the sun determines the paths of particles and light rays in its vicinity; as with Newton’s law, the influence of the localized source of gravity falls off with distance. While congratulating Schwarzschild on the simplicity and elegance of the solution, Einstein was dismayed by its apparent anti-Machian character, for it could be taken to be a solution of his field equations corresponding to the presence of a single body in an otherwise empty universe. On the other hand, it also defined what is now called the gravitational or Schwarzschild radius of the body: where r is the body’s radius, this is r = 2GM/c² (where G is the Newton gravitational constant and M is the body’s mass; here we follow custom and use units where G = c = 1, so r = 2M). For a star of the sun’s mass, this is about 3 km, whereas the Sun’s actual radius is 696,000 km. At the time, and for years afterward, it was generally thought that no actual body would ever become so compressed as to lie within its Schwarzschild radius.⁶³ Einstein reasoned similarly that a clock kept at this radius would cease to tick (as the g₄₄ component vanishes) and so both light rays and material particles would take an infinitely long time (in “coordinate time”) to reach it when coming from outside. He therefore considered that a sphere of radius r = 2M to be a place where the gravitational field is singular, and hence unphysical. Gradually (over more than forty years) it become clear that the supposed “singularity” at r = 2M is merely a coordinate effect (similar to the singularity of polar coordinates at the origin), due to a breakdown of the Schwarzschild time coordinate here. Not until the 1960s was it generally accepted that should a massive body collapse under gravitation (e.g., a star that has exhausted its fuel) to less than the Schwarzschild radius, the escape velocity from the body becomes equal to the speed of light, and so the object becomes a “black hole” (a term coined by John Wheeler in 1968). From a general relativistic perspective, the Schwarzschild radius is indeed physical; it defines the radius of the horizon of a Schwarzschild (non-rotating) black hole, the surrounding space-time region into which things may enter from without but from which nothing – not even light – can escape.

That light could not escape the critical circumference of a massive compact “dark star” was not at all a new notion. Though without the concept of space-time curvature, and still assuming the Newtonian corpuscular theory of light, that light could not escape the gravitational pull of a very massive small star had been theoretically considered already in Newtonian gravity in the late 18th century.⁶⁴ A few days later in January, however, Schwarzschild sent Einstein an exact computation of the space-time geometry extending the solution to also within a massive spherical body, idealized as a incompressible fluid with a definite radius r. In addition to the Schwarzschild radius, this “interior” solution had the peculiar property that there was a “singularity” at the center (r = 0) where the space-time curvature became infinite.⁶⁵ While Einstein had to agreed that the exterior Schwarzschild solution indeed showed that the more compact the star, the greater the curvature of space-time around it, as well as the larger the gravitational redshift of light from its surface, on account of the singularity at r = 0, Einstein completely rejected the interior Schwarzschild solution as illogical (on grounds that the concept of an incompressible fluid is not compatible with relativity theory: elastic waves propagating in it would have to travel at infinite velocity). As late as 1939, he wrote a paper whose principal conclusion was to show that “the ‘Schwarzschild singularities’ do not exist in physical reality”.⁶⁶ Despite Einstein’s antipathy to singularities, general relativity indeed predicts that an r = 0 singularity exists at the center of a black hole where the density and curvature become infinite, and the known laws of physics break down. Some contemporary relativists assume that such singularities must be “hidden” behind a black hole horizon, and so cannot influence what happens outside, where the normal laws of physics apply (see Chapter 11).

The return of the ether

Many were taken aback when Einstein, in a 1920 lecture at Leiden, rehabilitated the old concept in a discussion of the problem of space. In the presence of the august Lorentz, who never surrendered his belief that an etherial medium is explanatorily preferred in electrodynamics, Einstein declared the “denial of the ether is not necessarily required by the special principle of relativity”.⁶⁷ A surprising admission, surely, since fifteen years before, in his epochal paper on the theory of special relativity, Einstein had scornfully deemed the ether a “superfluous” theoretical posit. Yet, as he went on to explain both here and in a similar lecture four years later, the general theory of relativity has given a new and distinct sense to the term, bereft of every mechanical and kinematical property ever accorded it.

In Leiden Einstein had urged that “to deny the ether is ultimately to assume that empty space has no physical properties whatever”.⁶⁸ According to this criterion, Newton might just as well have used the term “ether” for he accorded some kind of physical reality to absolute space, in his explanation of the presence of inertial effects in rotating bodies. In this sense one can speak of an “ether of mechanics”, but it is justifiably called “absolute” since it plays only the asymmetrical role of causal actor, the nexus of causation is not closed. Einstein returned to this point in 1924:

Nor is it true of Lorentz’s ether. Lorentz posited an ether to explain the transmission of force and radiation across empty space. But his immobile ether is a substance that dynamically acts (contracting lengths and dilating times of bodies and clocks in motion), but is not acted upon by “ponderable matter”. As Poincaré forcefully objected, Lorentz’s ether failed to obey Newton’s third law of action and reaction. Such a failure similarly afflicts the Minkowski metric of special relativity that stands aloof from matter and energy, and so from this point of view the claim to have completely abolished the ether must be seen as over-reaching. In the aftermath of general relativity, Einstein spoke of the “ether of special relativity” which is “absolute, because its influence on inertia and light propagation was thought to be independent of physical influence of any kind”.⁷⁰ At the end of his life, he even characterized Minkowski space-time as “a four-dimensional analogue of H.A. Lorentz’s three-dimensional ether”, for failing to eliminate “the a priori existence of “empty space”.⁷¹

Summary

In the new conception afforded by general relativity, brought painfully home when Einstein extracted himself from the clutches of the hole argument, empty space-time is not really “empty” but “occupied” or, rather, constituted by the metric inertial-gravitational field. The field is not a quiescent container but has the properties of a dynamical system, interacting with other systems. The shift of emphasis allows the achievement of general relativity, stated in terms of Einstein’s cosmological model, to be characterized in terms of the ether:

Einstein’s rehabilitation of ether terminology stems from the continued attempt to underscore the Machian ambitions of the theory of general relativity. It is not Machian à la lettre since general relativity “excludes direct distant action”,⁷³ perhaps a supposition that Mach actually put forward as a putative dynamical explanation of inertia that may not be justified. It is highly implausible that Mach advanced or even intended to suggest an explanation of inertia in terms of the mysterious action at a distance of cosmic masses. More charitably, Einstein may be seen as claiming that general relativity gives a field-theoretic implementation of an attempted explanation of inertia along Machian lines. That Mach has been Einstein’s inspiration for the change in viewpoint regarding the ether is clear, so much so, that the ether of general relativity is also called “Mach’s ether”.

The ether stands in place of “empty space” in Einstein’s conception of general relativity. Mach’s idée fixe, that acceleration should be defined relative to a frame of reference determined by the configuration of the entire universe and so eliminating the concept “absolute space” from physics, proved to be the guiding and perhaps strongest motivation in Einstein’s pursuit of a generalized theory of relativity.

Notes

1Albert Einstein-Max Born Briefwechsel 1916–1955. München: Nymphenburger Verlagshandlung, 1969, p. 251; The Born-Einstein Letters, translated by Irene Born. New York: Walker and Co., 1971, p. 192.

2 Dirac, Paul, “Methods in Theoretical Physics”, in From a Life in Physics: Evening Lectures at the International Centre for Theoretical Physics. Trieste, Italy, June 1968. The IAEA Bulletin, Special Supplement, p. 24; Landau, Lev and Evgeny Lifshitz, The Classical Theory of Fields. Fourth Revised English Edition. Translated by Morton Hamermesh. Oxford, UK and New York: Pergamon Press, 1975, p. 228.

3“Autobiographical Notes”, pp. 66–7.

4E.g., Norton, John D., “Did Einstein Stumble? The Debate Over General Covariance”, Erkenntnis v. 42 (1995), pp. 223–45.

5“Fundamental Idea and Methods of the Theory of Relativity, Presented in Their Development” (“after 22 January 1920”), reprinted in CPAE 7 (2002), Doc. 31, pp. 245–81; p. 265.

6Remark related by Einstein to a research assistant in Princeton, Ernst G. Straus; reported in a letter of Straus to A. Pais, October 1979, quoted in Pais, Abraham, “Subtle Is the Lord …” The Science and the Life of Albert Einstein. New York: Oxford University Press, 1982, p. 239.

7Einstein, Albert and M. Grossmann, Outline of a Generalized Theory of Relativity and of a Theory of Gravitation. Leipzig: B. Teubner Verlag, 1913; as reprinted in CPAE 4 (1995), Doc. 13, pp. 302–44.

8In Göttingen on November 20, and so five days before Einstein’s final presentation, Hilbert presented an axiomatic but highly schematic derivation of the Einstein field equations using a variational method. One of Hilbert’s two axioms is the stipulation of a generally invariant (i.e., generally covariant) “world function”, the object that subjected to variations by “Hamilton’s principle”, yields the sought-for field equations. The other axiom concerns the form of the world function, coupling its gravitational part to a specific source term T_μv (rather than the unspecified T_μv of Einstein); this is a nonlinear (and of course, non-quantum) generalization of Maxwell’s electromagnetic theory due to German physicist Gustav Mie that in 1915 purported to be a complete theory of matter (i.e., matter comprised solely of gravitation and electromagnetism). In this sense, Hilbert’s ambition was broader than Einstein’s; rather than only a gravitational theory, Hilbert proposed a “theory of everything”. The importance of general covariance is underscored by its axiomatic status while the gravitational part of Hilbert’s “world function” is today known as the Hilbert-Einstein action. Hilbert always maintained that general relativity was Einstein’s theory; see K.A. Brading and T.A. Ryckman, “Hilbert’s ‘Foundations of Physics’: Gravitation and Electromagnetism within the Axiomatic Method”, Studies in History and Philosophy of Modern Physics v. 39 (2008), pp. 102–53.

9“On the General Theory of Relativity” (November 4, 1915); “On the General Theory of Relativity (Addendum)” (November 11, 1915); “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity” (November 18, 1915); “The Field Equations of Gravitation” (November 25, 1915), as reprinted in CPAE 6 (1996), Docs. 21, 22, 24, 25.

10Observations in the 19th century showed that Mercury’s perihelion precesses (the orbits do not close at the point nearest the sun) at a rate of 574 arcseconds (0.159 degree; 1 degree is 3,600 arcseconds) per Earth century. By taking into account the orbits of the other planets, Newtonian mechanics (as computed Urbain Le Verrier, in 1859, and refined by Simon Newcomb in 1895) could account for 531 seconds of arc per century. Mercury’s orbit is, however, a highly nonlinear problem involving the gravitational equations of the Sun and at least five other planets; no exact Newtonian solutions are known and approximations are required. The residual 43 arcseconds was derived in Einstein’s paper of November 18. Le Verrier believed the discrepancy due to an unknown small planet closer to the Sun, which he termed Vulcan. For a concise modern treatment of Einstein’s result, see Robert Wald, General Relativity. Chicago and London: University of Chicago Press, 1984, pp. 142–3.

11Einstein, “Physik und Realität”, The Journal of the Franklin Institute v. 221, no. 3 (March 1936), pp. 313–47; p. 335; English translation, pp. 349–82. Translation reprinted in Einstein, Ideas and Opinions. New York: Crown Publishers, 1954, pp. 290–323, p. 311. In particular, there may be a distinct T_μv for every conceivable matter Lagrangian. To make the situation manageable, one seeks to formulate a set of generic conditions that all “reasonable” T_μv must satisfy, for example, energy densities must always be positive”.

12“On the Foundations of the General Theory of Relativity”, Annalen der Physik (1918); reprinted in CPAE 7 (2002), Doc. 4, pp. 33–6.

13See Norton, John, “General Covariance and the Foundations of General Relativity: Eight Decades of Dispute”, Reports on Progress in Physics v. 56 (1993), pp. 791–858.

14Hertz, Heinrich, “Introduction”, in D.E. Jones (trans.), Electric Waves, Being Researches on the Propagation of Electric Action with Finite Velocity Through Space. London and New York: Macmillan and Co., 1893, p. 21.

15“One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discovers, that we get more out of them than was originally put into them”. Quoted in Freeman Dyson, “Mathematics in the Physical Sciences”, in National Research Council’s COSRIMS (ed.), The Mathematical Sciences. Cambridge, MA: MIT Press, 1969, pp. 97–115; p. 99.

16Einstein, Albert and Leopold Infeld, The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. New York: Simon and Schuster, 1938, pp. 226–35.

17See, e.g., Eddington, Arthur S., The Mathematical Theory of Relativity. Second edition. Cambridge, UK: Cambridge University Press, 1924, §17.

18In particular, the space-time equivalent of “distance” between two point-events p and q is given by the space-time interval $d s^{2} = \sum_{μ, ν = 1}^{4} g_{μ ν} d x^{μ} d x^{ν}$ , an expression for the Pythagorean theorem holding only for “nearby” space-time events (see Box 6.2).

19“On the Influence of Gravitation on the Propagation of Light” (1911), in CPAE v. 3 (1993), Doc. 23, pp. 485–97; p. 487.

20“On the Present State of the Problem of Gravitation”, Physikalische Zeitschrift Bd. 14 (1913), pp. 1249–62, p. 1254; reprinted in CPAE 4 (1995), Doc. 17, pp. 486–503, p. 492: “Abstrakt gesprochen: Es gibt kein Relativitätsprinzip der ungleichförmigen Bewegung”.

21“The Foundation of the General Theory of Relativity” (Die Grundlage der allgemeinen Relativitätstheorie), CPAE 6 (1996), Doc. 30, pp. 283–339; p. 287. Emphasis added.

22For discussion, see Janssen, Michel, “ ‘No Success Like Failure …’: Einstein’s Quest for General Relativity”, in Michel Janssen and Christoph Lehner (eds.), The Cambridge Companion to Einstein. New York: Cambridge University Press, 2014, pp. 167–227; pp. 186–7.

23Misner, Charles W., Kip S. Thorne, and John A. Wheeler, Gravitation. New York: W.H. Freeman and Co., 1973, pp. 302–3.

24See Friedman, Michael, Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science. Princeton, NJ: Princeton University Press, 1983.

25This is seen in representing the gravitational field strengths by Christoffel symbols. Since these expressions are not tensors, inertio-gravitational forces attributed by one observer to gravitation may be regarded by another observer as due to inertia. In general relativity, there is no invariant way of distinguishing the two.

26For example, Banesh Hoffmann, one of Einstein’s research assistants in Princeton in the 1930s, and later Professor of Mathematics at Queen’s College in New York, commented,

As for the principle of general covariance, Einstein’s belief that it expressed the relativity of all motion was erroneous…. Worse, as was quickly pointed out, the principle of general covariance is, in a sense, devoid of content since practically any physical theory expressible mathematically can be put into tensor form.

Albert Einstein: Creator and Rebel. New York: Viking Press, 1972, p. 127

27Stachel, John, “Einstein’s Search for General Covariance, 1912–1915”, in D. Howard and J. Stachel (eds.), Einstein and the History of General Relativity (Einstein Studies v. 1), Basel, Boston, and Berlin: Birkhäuser, 1989, pp. 63–100. This paper is based on the written version of a talk circulated since 1980.

28Einstein, letter to H.A. Lorentz, August 16, 1913; CPAE 5 (1993), Doc. 470, pp. 352–3.

29“The Formal Foundation of the General Theory of Relativity”, reprinted in CPAE 6 (1996), English translation supplement, Doc. 9, pp. 72–130; p. 110 (emphasis added).

30In particular, Einstein used the fact that there is a 1:1 correspondence between “passive” and “active” diffeomorphisms; a coordinate transformation at a given point p in the hole is used to construct a tensor field at p that is a “carry-along” from another point q in the hole with the same coordinates as p but in another chart. The result is two metrics g_μv (x) and g'_μv (x) at the same point p in the same coordinate system x^σ (σ = 1, 2, 3, 4) from the same sources T_μv outside the hole, an apparent failure of determinism.

31“Einiges über die Entstehung der allgemeinen Relativitätstheorie”, notes published in C. Seelig (ed.), Mein Weltbild. Amsterdam: Querido Verlag, 1934, pp. 134–8; p. 135; translation in Ideas and Opinions, 1954, pp. 285–90; p. 288.

32“The Foundation of the General Theory of Relativity”, Annalen der Physik Bd. 49 (1916), pp. 769–822; reprinted in CPAE 6 (1996), Doc. 30, pp. 293–339; pp. 290–2.

33Born, Max, Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen. Berlin: J. Springer, 1920, p. 223; Phillip Frank, “Die Bedeutung der physikalischen Erkenntnistheorie Machs für die Geisteleben der Gegenwart”, Die Naturwissenschaften Bd. 5 (1917), pp. 65–71; reprinted in Frank, Modern Science and Its Philosophy. Cambridge, MA: Harvard University Press, p. 73.

34See Ryckman, Thomas, “ ‘P(oint)-C(oincidence) Thinking’: The Ironical Attachment of Logical Empiricism to General Relativity (and Some Lingering Consequences)”, Studies in History and Philosophy of Science v. 23 (1992), pp. 471–97.

35Einstein, “On the Foundations of the General Theory of Relativity”, Annalen der Physik (1918); CPAE 7 (2002), English translation supplement, Doc. 4, pp. 33–36, p. 33.

36The formulation of Wald Robert, General Relativity, Chicago: University of Chicago Press, 1984, p. 68.

37See Stachel (1989), note 26 and John Norton, “How Einstein Found His Field Equations, 1912–1915”, also in D. Howard and J. Stachel (eds.), Einstein and the History of General Relativity (Einstein Studies v. 1), Basel, Boston, and Berlin: Birkhäuser, 1989, pp. 101–59.

38The clearest one is “Relativity Theory and the Problem of Space”, Appendix V. (1952). Relativity: The Special and the General Theory: A Popular Exposition. New York: Crown Publishers, 1961, pp. 135–57; p. 152: “On the basis of the general theory of relativity … space as opposed to ‘what fills space’ … has no separate existence…. If we imagine the gravitational field, i.e., the functions g_μv to be removed, there does not remain a space of the type (of special relativity), but absolutely nothing, and also not ‘topological space’. For the functions g_μv describe not only the field, but at the same time also the topological and metrical structural properties of the manifold…. There is no such thing as an empty space; i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field”.

39Einstein to Andre Mercier, November 9, 1953 (EA 41–884).

40On the pro tem character of the general theory of relativity, see “Autobiographical Notes”, p. 75: “Not for a moment, of course, did I doubt that this formulation (the field equations) was only a makeshift in order to give the principal of general relativity a preliminary closed expression. Certainly, it was not anything more than a theory of the gravitational field that was rather artificially isolated from a total field of yet unknown structure”.

41Mach, Ernst, The Science of Mechanics: A Critical and Historical Account of Its Development. First German edition, 1883. Translated by T.J. McCormack with revisions through the ninth (1933) German edition. LaSalle, IL: Open Court Publishing Co., 1960, pp. 276–88.

42Ibid., p. 283.

43Einstein, “Is There a Gravitational Effect Which Is Analogous to Electrodynamic Induction?” (1912), as reprinted in CPAE 4 (1995), Doc. 7, pp. 174–9; p. 178.

44Einstein, letter to Mach, June 25, 1913, CPAE 5 (1993), Doc. 448, pp. 531–2.

45Einstein, postcard to Mach, December 1913, CPAE 5 (1993), Doc. 495, pp. 584–5.

46“On the Present State of the Problem of Gravitation”, Physikalische Zeitschrift Bd. 14 (1913), pp. 1249–62; reprinted in CPAE 4 (1995), Doc. 17, pp. 486–503.

47“Four Lectures on the Theory of Relativity Held at Princeton University in May 1921”, CPAE 7 (2002), Doc. 71, pp. 496–577; p. 563; translation in Einstein, The Meaning of Relativity. Fifth edition. Princeton: Princeton University Press, 1956, p. 100.

48Ibid., pp. 498–9.

49“The Foundation of the General Theory of Relativity”, Annalen der Physik Bd. 49 (1916), pp. 769–822; reprinted in CPAE 6 (1996), Doc. 30, pp. 293–339; pp. 286–88.

50Russell, Bertrand, “On the Notion of Cause”, Proceedings of the Aristotelian Society v. 13 (1912–13), pp. 1–25.

51“Four Lectures on the Theory of Relativity Held at Princeton University in May 1921”, CPAE 7 (2002), Doc. 71, pp. 496–577; p. 535; translation in Einstein, The Meaning of Relativity. Fifth edition. Princeton, NJ: Princeton University Press, 1956, pp. 55–6.

52Einstein, letter to Georg Jaffe, January 19 1954; cited by J. Stachel, “What a Physicist Can Learn From the Discovery of General Relativity”, in R. Ruffini (ed.), Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity. Amsterdam: Elsevier Science Publishers, 1986, pp. 1857–62; p. 1858.

53Einstein, letter to de Sitter, November 4, 1916 CPAE 8, Doc. 273, pp. 359–61.

54“Cosmological Considerations on the General Theory of Relativity”, reprinted in CPAE 6 (1996), Doc. 43, pp. 540–52; p. 544.

55Eddington, Arthur S., The Expanding Universe. Cambridge, UK: Cambridge University Press, 1933, p. 21.

56 $Λ = \frac{8 π G ρ}{c^{2}} .$

where G is Newton’s gravitational constant and ρ is the energy density of empty space; see Chapter 11.

57“Cosmological Considerations on the General Theory of Relativity”, reprinted in CPAE 6 (1996), Doc. 43, pp. 540–52; p. 551.

58Einstein, letter to de Sitter, 24 March 1917, CPAE 8, Doc. 317, pp. 421–23; p. 422.

59Eddington, Arthur S. The Expanding Universe. Cambridge, UK: Cambridge University Press, 1933, p. 59.

60As quoted in Helge S. Kragh, Conceptions of the Cosmos: From Myths to the Accelerating Universe: A History of Cosmology. New York: Oxford University Press, 2007, p. 141.

61The New York Times, February 5, 1931, p. 17.

62See Weinstein, Galina, “George Gamow and Albert Einstein: Did Einstein Say the Cosmological Constant Was the ‘Biggest Blunder’ He Ever Made in Life?”, Ms. Ben Gurion University, October 3, 2013.

63Contemplating in 1926 the hypothetical example of a star with a density of 61,000 gm/cm³ (the mass of the sun within a radius much less than Uranus), A.S. Eddington’s verdict may be taken as authoritative: “I think it has generally been considered proper to add the conclusion ‘which is absurd’ ”. The Internal Constitution of the Stars. Cambridge, UK: Cambridge University Press, 1926, p. 171.

64See Israel, Werner, “Dark Stars: The Evolution of an Idea”, in Stephen Hawking and Werner Israel (eds.), 300 Years of Gravitation. Cambridge, UK and New York: Cambridge University Press, 1987, pp. 199–276.

65More carefully, since the components of the Riemann curvature tensor are coordinate-dependent but its scalars are coordinate-independent, the scalars formed from the Riemann curvature tensor become infinite. In particular, the so-called Kretschmann scalar K = R_μνστ R^μνστ blows up as the singularity at r = 0 is approached. However, a singularity is better defined as a space-time point beyond which a geodesic is not well behaved. See John Earman, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. New York: Oxford University Press, 1995.

66Einsten, Albert“On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses”, Annals of Mathematics, Second Series v. 40, no. 4 (October 1939), pp. 922–36; p. 936.

67“Ether and the Theory of Relativity”, reprinted in CPAE 7 (2002), Doc. 38, 305–23; p. 314.

68Ibid., p. 316.

69“Über den Äther”, Schweizerische naturforschende Gesellschaft Verhanflungen v. 105 (1924), pp. 85–93; translated in Simon Saunders and Harvey R. Brown (eds.), The Philosophy of Vacuum. Oxford, UK: Clarendon Press, 1991, pp. 13–20; p. 15.

70Ibid., p. 17.

71“Relativity and the Problem of Space”, Appendix V. (1952). Relativity: The Special and the General Theory: A Popular Exposition, pp. 135–57; p. 151.

72“Über den Äther”, Schweizerische naturforschende Gesellschaft Verhanflungen; as translated by Simon Saunders in Harvey R. Brown and Simon Saunders (eds.), The Philosophy of Vacuum, Oxford, UK: Clarendon Press, 1991, pp. 13–17.

73Ibid., p. 20.

74“Ether and the Theory of Relativity”, reprinted in CPAE 7 (2002), Doc. 38, 305–23; p. 316.

Step 1:	Everyone will agree that under the arbitrary coordinate transformation x → x′ permitted by general covariance, if g_μν(x) is a solution at some point of space-time for sources at the point T_μν(x), then so is g′_{_µv} (x′) at that same point for sources T′_μν (x′).
Step 2:	According to general covariance, there is a permissible (nonlinear) coordinate transformation over the entire region that is the identity outside H (i.e., leaving the coordinates of the space-time points in this region unchanged) but changes coordinates at points inside H, going smoothly from one to the other on H’s boundaries. Then in particular, T_μν (x) = T′_μν (x′) ≠ 0 inside H.
Step 3:	Using this coordinate transformation, Einstein was able to construct (for details, see Note 29) a second solution g′_μν (x) ≠ g_μν (x) in the same coordinate system x^σ (σ = 1, 2, 3, 4) at the same point p inside the hole from the same sources T_μν (x) = T′_μν (x′) = 0 outside the hole.
	Conclusion: the matter distribution outside H does not uniquely determine the g_μν at points within, as should be the case if the theory is deterministic. For under the above coordinate transformation, the same sources T_μν will assign different values g_μν to the same point within H. It is an apparent failure of determinism.

Six

General relativity

Introduction