Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein's Theory of Gravitation in Its Centennial Year
aa r X i v : . [ g r- q c ] A p r Universe , , 38-82; doi:10.3390 / universe1010038 OPEN ACCESS universe
ISSN 2218-1997 / journal / universe Article
Editorial for the Special Issue 100 Years ofChronogeometrodynamics: The Status of the Einstein’sTheory of Gravitation in Its Centennial Year
Lorenzo Iorio
Ministero dell’ Istruzione dell’ Università e della Ricerca (M.I.U.R), Fellow of the Royal AstronomicalSociety (F.R.A.S.), Viale Unità di Italia, 68 70125 Bari, Italy; E-Mail: [email protected];Tel.: + Received: 6 March 2015 / Accepted: 17 April 2015 / Published: 24 April 2015
The present Editorial introduces the Special Issue dedicated by the journal
Universe to theGeneral Theory of Relativity, the beautiful theory of gravitation of Einstein, a century afterits birth. It reviews some of its key features in a historical perspective, and, in welcomingdistinguished researchers from all over the world to contribute it, some of the main topics atthe forefront of the current research are outlined.general relativity and gravitation; classical general relativity; gravitational waves; quantumgravity; cosmology; experimental studies of gravity
PACS classifications:
1. Introduction
This year marks the centenary of the publication of the seminal papers [1–3] in which AlbertEinstein laid down the foundations of his theory of gravitation, one of the grandest achievements of thehuman thought which is the best description currently at our disposal of such a fundamental interactionshaping the fabric of the natural world. It is usually termed “General Theory of Relativity” (GTR,from
Allgemeine Relativitätstheorie ), often abbreviated as “General Relativity” (GR). It replaced theNewtonian concept of “gravitational force” with the notion of “deformation of the chronogeometricstructure of spacetime” [4] due to all forms of energy weighing it; as such, it can be defined as achronogeometrodynamic theory of gravitation [5].GTR is connected, in a well specific sense, to another creature of Einstein himself, with Lorentz [6]and Poincaré [7,8] as notable predecessors, published in 1905 [9]: the so-called Special (or Restricted) niverse , Theory of Relativity (STR). The latter is a physical theory whose cornerstone is the requirement ofcovariance of the di ff erential equations expressing the laws of physics (originally only mechanics andelectromagnetism) under Lorentz transformations of the spacetime coordinates connecting di ff erentinertial reference frames, in each of which they must retain the same mathematical form. Moreprecisely, if A ( x , y , z , t ) , B ( x , y , z , t ) , C ( x , y , z , t ) , . . . (1)represent the state variables of a given theory depending on spacetime coordinates x , y , z , t and aremutually connected by some mathematical relations f ( A , B , C , . . . ) = f ′ (cid:16) A ′ (cid:16) x ′ , y ′ , z ′ , t ′ (cid:17) , B ′ (cid:16) x ′ , y ′ , z ′ , t ′ (cid:17) , C ′ (cid:16) x ′ , y ′ , z ′ , t ′ (cid:17) , . . . (cid:17) = f ′ connecting the transformed state variables A ′ , B ′ , C ′ , . . . aredi ff erent from the ones of f . If, as for the Lorentz transformations, it turns out f ′ = f (4)which does not necessarily implies that also the state variables A , B , C , . . . remain unchanged, then it issaid that the equations of the theory retain the same form. It is just the case of the Maxwell equations,in which the electric and magnetic fields E , B transform in a given way under a Lorentz transformationin order to keep the form of the equations connecting them identical, which, instead, is not retainedunder Galilean transformations [10]. In the limiting case of the Galilean transformations applied to theNewtonian mechanics, it turns out that the theory’s equations are even invariant in the sense that also thestate variables remain unchanged, i.e ., it is F ′ − m a ′ = F ′ = F (6) a ′ = a As such, strictly speaking, the key message of STR is actually far from being: “everything is relative”,as it might be seemingly suggested by its rather unfortunate name which, proposed for the first timeby Planck [11] (
Relativtheorie ) and Bucherer [12] (
Relativitätstheorie ), became soon overwhelminglypopular (see also [13]). Su ffi ce it to say that, in informal correspondence, Einstein himself wouldhave preferred that its creature was named as Invariantentheorie (Theory of invariants) [14], as alsoexplicitly proposed-unsuccessfully-by Klein [15]. Note that, here, the adjective “invariant” is used, ina looser sense, to indicate the identity of the mathematical functional form connecting the transformedstate variables. niverse , Notably, if the term “relativity” is, instead, meant as the identity of all physical processes inreference frames in reciprocal translational uniform motion connected by Lorentz transformations,then, as remarked by Fock [16], a name such as “Theory of Relativity” can, to some extent, bejustified. In this specific sense , relativity geometrically corresponds to the maximal uniformity of thepseudo-Euclidean spacetime of Poincaré and Minkowski in which it is formulated. Indeed, given a N − dimensional manifold, which can have constant curvature, or, if with zero curvature, can be Euclideanor pseudo-Euclidean, the group of transformations which leave identical the expression for the squareddistance between two nearby points can contain at most (1 / N ( N +
1) parameters. If there is a groupinvolving all the (1 / N ( N +
1) parameters, then the manifold is said to have maximal uniformity. Themost general Lorentz transformations, which leave unchanged the coe ffi cients of the expression of the4-dimensional distance between two nearby spacetime events, involve just 10 parameters. Now, in thepseudo-Riemannian spacetime of GTR the situation is di ff erent because, in general, it is not uniform atall in the geometric sense previously discussed. Following Fock [16], it can be e ff ectively illustratedby a simple example whose conclusion remains valid also for the geometry of the 4-dimensionalspacetime manifold. Let us think about the surface of a sphere, which is a 2-dimensional manifoldof a very particular form. It is maximally uniform since it can be transformed into itself by means ofrotations by any angle about an arbitrary axis passing through the centre, so that the associated groupof transformations has just three parameters. As a result, on a surface of a sphere there are neitherpreferred points nor preferred directions. A more general non-spherical surface of revolution has onlypartial uniformity since it can be transformed into itself by rotation about an axis which is now fixed, sothat the rotation angle is the only arbitrary parameter left. There are privileged points and lines: the polesthrough which the axis passes, meridians, and latitude circles. Finally, if we consider a surface of generalform, there will be no transformations taking it into itself, and it will possess no uniformity whatsoever.Thus, it should be clear that the generality of the form of the surface is a concept antagonistic to theconcept of uniformity. Returning now to the concept of relativity in the aforementioned specified sense ,it is related to uniformity in all those cases in which the spacetime metric can be considered fixed. Thisoccurs not only in the Minkowskian spacetime, but also in the Einsteinian one, provided only that thephysical processes one considers have no practical influence on the metric. Otherwise, it turns out thatrelativity can, to a certain extent, still be retained only if the non-uniformity generated by heavy massesmay be treated as a local perturbation in infinite Minkowskian spacetime. To this aim, let us think abouta laboratory on the Earth’s surface [16]. If it was turned upside down, relativity would be lost since thephysical processes in it would be altered. But, if the upset down laboratory was also parallel transportedto the antipodes, relativity would be restored since the course of all the processes would be the same asat the beginning. In this example, a certain degree of relativity was preserved, even in a non-uniformspacetime, because the transformed gravitational field g ′ in the new coordinate system n x ′ o has the sameform as the old field g in the old coordinates { x } , i.e ., { x } 7→ n x ′ o (7) g ( x ) g ′ (cid:16) x ′ (cid:17) = g ( x ) (8)Such considerations should have clarified that relativity, in the previously specified sense, either doesno exist at all in a non-uniform spacetime like the Einsteinian one, or else it does exist, but does not go niverse , beyond the relativity of the Minkowskian spacetime. In this sense , the gravitational theory of Einstein cannot be a generalization of his theory of space and time of 1905 , and its notion of relativity alongwith its related concept of maximal uniformity was not among the concepts subjected to generalization .Since the greatest possible uniformity is expressed by Lorentz transformations, there cannot be a moregeneral principle of relativity than that discussed in the theory of 1905. All the more, there cannot bea general principle of relativity having physical meaning which would hold with respect to arbitraryframes of references. As such, both the denominations of “General Relativity” and “General Theory ofRelativity” are confusing and lead to misunderstandings. Furthermore, such adjectives reflect also anincorrect understanding of the theory itself since they were adopted referring to the covariance of theequations with respect to arbitrary transformations of coordinates accompanied by the transformationsof the coe ffi cients of the distance between two events in the 4-dimensional spacetime. But it turnedout that such kind of covariance has actually nothing to do with the uniformity or non-uniformity ofspacetime [16,17]. Covariance of equations per se is just a merely mathematical property which in noway is expression of any kind of physical law. Su ffi ce it to think about the Newtonian mechanics and thephysically equivalent Lagrange equations of second kind which are covariant with respect to arbitrarytransformations of the coordinates. Certainly, nobody would state that Newtonian mechanics containsin itself “general” relativity. A principle of relativity-Galilean or Einsteinian-implies a covariance ofequations, but the converse is not true: covariance of di ff erential equations is possible also when noprinciple of relativity is satisfied. Incidentally, also the the adjective “Special” attached to the theory of1905 seems improper in that it purports to indicate that it is a special case of “General” Relativity.In the following, for the sake of readability, we will adhere to the time-honored conventions by usingSTR and GTR (or GR) for the Einsteinian theory of space and time of 1905 and for his gravitationaltheory of 1915, respectively.Of course, the previous somewhat “philosophical” considerations are, by no means, intended toundermine the credibility and the reliability of the majestic theory of gravitation by Einstein, whoseconcordance with experiments and observations has been growing more and more over the latestdecades [18].Below, some key features of GTR, to which the present Special Issue is meritoriously and timelydedicated, are resumed in a historical perspective [19–21], without any pretence of completeness. It ishoped that the distinguished researchers who will kindly want to contribute it will provide the communityof interested readers with the latest developments at the forefront of the research in this fascinating andnever stagnant field.In the following, Greek letters µ, ν, ̺ . . . denote 4-dimensional spacetime indexes running over0 , , ,
3, while Latin ones i , j , k , . . . , taking the values 1 , ,
3, are for the 3-dimensional space.
2. The Incompatibility of the Newtonian Theory of Gravitation with STR
In the framework of the Newtonian theory of universal gravitation [22], the venerable force-lawyielding the acceleration a imparted on a test particle by a mass distribution of density ρ could be niverse , formally reformulated in the language of the di ff erential equations governing a field-type state variable Φ , known as potential, through the Poisson equation [23] ∇ Φ = π G ρ (9)where G is the Newtonian constant of gravitation, so that a = − ∇ Φ (10)Nonetheless, although useful from a mathematical point of view, such a field was just a non-dynamicalentity, deprived of any physical autonomous meaning: it was just a di ff erent, mathematical way of tellingthe same thing as the force law actually did [20]. It is so because, retrospectively, in the light of STR, itwas as if, in the Newtonian picture, the gravitational interaction among bodies would take place de facto instantaneously, irrespectively of the actual distance separating them, or as if gravity would be somesort of occult, intrinsic property of matter itself. Remarkably, such a conception was opposed by Newtonhimself who, in the fourth letter to R. Bentley in 1692, explicitly wrote [24]: “[. . . ] Tis inconceivable thatinanimate brute matter should (without the mediation of something else which is not material) operateupon & a ff ect other matter without mutual contact; as it must if gravitation in the sense of Epicurus beessential & inherent in it. And this is one reason why I desired you would not ascribe {innate} gravity tome. That gravity should be innate inherent & {essential} to matter so that one body may act upon anotherat a distance through a vacuum without the mediation of any thing else by & through which their actionor force {may} be conveyed from one to another is to me so great an absurdity that I believe no manwho has in philosophical matters any competent faculty of thinking can ever fall into it. Gravity must becaused by an agent {acting} consta{ntl}y according to certain laws, but whether this agent be materialor immaterial is a question I have left to the consideration of my readers.”. In the previous quotation, thetext in curly brackets { . . . } is unclear in the manuscript, but the editor of the original document is highlyconfident of the reading.In the second half of the nineteenth century, with the advent of the Maxwellian field theory ofelectromagnetism [25] scientists had at disposal a mathematically coherent and empirically well testedmodel of a physical interaction among truly dynamical fields which propagate as waves even in vacuo atthe finite speed of light c transferring energy, momentum and angular momentum from a point in spaceto another. Now, STR is based on two postulates: The Principle of Relativity, extended by Einsteinto all physical interactions, and another principle that states that the speed of light is independent ofthe velocity of the source. In this form, it retains its validity also in GTR. The latter is an immediateconsequence of the law of propagation of an electromagnetic wave front which is straightforwardlyobtained from the Maxwell equations obeying, by construction, the Principle of Relativity itself sincethey turned out to be covariant under Lorentz transformations. It necessarily follows [16] that there existsa maximum speed for the propagation of any kind of physical action. This is numerically equal just to thespeed of light in vacuo . If there was no single limiting velocity but instead di ff erent agents, e.g., light andgravitation, propagated in vacuo with di ff erent speeds, then the Principle of Relativity would necessarilybe violated as regards at least one of the the agents. Indeed, it would be possible to choose an inertialframe traveling just at the speed of the slower agent in which the di ff erential equations governing itscourse would take a particular form with respect to that assumed in all the other frames, thus predicting niverse , spurious, unphysical phenomena. It is reminiscent of the famous first gedankenexperiment made byEinstein about STR around 1895-1896 described by himself as follows [26]: “[. . . ] Wenn man einerLichtwelle mit Lichtgeschwindigkeit nachläuft, so würde man ein zeitunabhängiges Wellenfeld vor sichhaben. So etwas scheint es aber doch nicht zu geben!” [“If one goes after a light wave with light velocity,then one would have a time-independent wavefield in front of him. However, something like that does notseem to exist!”] Indeed, the Maxwell equations in vacuo , in their known form, do not predict stationarysolutions. That posed severe challenges to the Newtonian gravitational theory [8], which necessarilywould have had to abandon its strict force-law aspect in favor of a genuine field-type framework makingthe Poisson equation covariant under Lorentz transformations [19,27].Furthermore, as pointed out by Einstein himself [28], Newtonian universal gravitation did not fit intothe framework of the maximally uniform spacetime of SRT for the deepest reason that [16], while inSRT the inertial mass m i of a material system had turned out to be dependent on its total energy, in theNewtonian picture the gravitational mass m g , did not. At high speeds, when the change in the inertia of abody becomes notable, this would imply a breakdown of the law of free fall, whose validity was actuallywell tested, although only at non-relativistic regimes (see Section 3).Finally, it can be remarked also that the required Lorentz covariance would have imposed, in principle,also the existence of a new, magnetic-type component of the gravitational field so to yield some sort ofgravitational inductive phenomena and travelling waves propagating at the finite speed of light in vacuo .Unfortunately, at the dawn of the twentieth century, there were neither experimental nor observationalevidence of such postulated manifestations of a somehow relativistic theory of gravitation.
3. The Equivalence Principle and Its Consequences
Luckily, at that time, Einstein was pressed also by another need: The quest for a coherent frameworkto consistently write down the laws of physics in arbitrary frames of references moving according tomore complicated kinematical laws than the simple uniform translation. In 1907 [29], Einstein realizedthat the bridge across such two apparently distinct aspects could have been represented by the equalityof the inertial and gravitational masses, known at that time to a 5 × − accuracy level thanks to theEötvös experiment [30].That was an empirical fact well known since the times of Galilei thanks to the (likely) fictional [31–33]tales of his evocative free fall experiments [34] allegedly performed from the leaning tower of Pisaaround 1590. Newton himself was aware of the results by Galilei, and made his own experiments withpendulums of various materials obtaining an equality of inertial and gravitating masses to a 10 − levelof relative accuracy. Indeed, in the Proposition VI, Theorem VI, Book III of his Principia [22] Newtonwrote [35]: “It has been, now for a long time, observed by others, that all sorts of heavy bodies [. . . ]descend to the Earth from equal heights in equal times; and that equality of times we may distinguishto a great accuracy, by the help of pendulums. I tried experiments with gold, silver, lead, glass, sand,common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one niverse , with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation ofthe other. The boxes, hanging by equal threads of 11 feet, made a couple of pendulums perfectly equalin weight and figure, and equally receiving the resistance of the air. And, placing the one by the other,I observed them to play together forwards and backwards, for a long time, with equal vibrations. Andtherefore the quantity of matter in the gold (by Cors. I and VI, Prop. XXIV, Book II) was to the quantityof matter in the wood as the action of the motive force (or vis motrix ) upon all the gold to the actionof the same upon all the wood; that is, as the weight of the one to the weight of the other: and the likehappened in the other bodies. By these experiments, in bodies of the same weight, I could manifestlyhave discovered a di ff erence of matter less than the thousandth part of the whole, had any such been.”Interestingly, in the Proposition VI, Theorem VI, Book III of his Principia [22], Newton looked alsothe known motions of the natural satellites of Jupiter to make-from a phenomenological point of view -afurther convincing case for the equality of the inertial and gravitational masses. Indeed, if the ratios ofthe gravitational to the inertial mass of Jupiter and of its satellites were di ff erent, the orbits of the Jovianmoons about their parent planet would be unstable because of an imperfect balancing of the centrifugalacceleration and the Jupiter centripetal attraction caused by a residual, uncancelled force due to theSun’s attractions on either Jupiter and its moons themselves. Indeed, Newton wrote [36]: “[. . . ] that theweights of Jupiter and of his satellites towards the Sun are proportional to the several quantities of theirmatter, appears from the exceedingly regular motions of the satellites (by Cor. III, Prop. LXV, Book I).For if some of those bodies were more strongly attracted to the Sun in proportion to their quantity ofmatter than others, the motions of the satellites would be disturbed by that inequality of attraction (byCor. II, Prop. LXV, Book I). If, at equal distances from the Sun, any satellite, in proportion to thequantity of its matter, did gravitate towards the Sun with a force greater than Jupiter in proportion tohis, according to any given proportion, suppose of d to e ; then the distance between the centres of theSun and of the satellite’s orbit would be always greater than the distance between the centres of theSun and of Jupiter, nearly as the square root of that proportion: as by some computations I have found.[. . . ]” In principle, the Newtonian gravitational theory would have not lost its formal consistency evenif experiments-all conducted at low speeds with respect to c -would have returned a di ff erent verdictabout m i / m g . Nonetheless, one cannot help but notice as the very same name chosen by Newton for theuniversally attractive force regulating the courses of the heavens, i.e ., gravitation, may point, somehow,towards a not so accidental nature of the equality of inertial and gravitating masses. Indeed, it comesfrom the Latin word gravis (‘heavy’) with several Indoeuropean cognates [37], all with approximatelythe same meaning related to the weight of common objects on the Earth’s surface: Sanskrit, guruh. (‘heavy, weighty, venerable’); Greek, βάρος (‘weight’) and βαρύς (‘heavy in weight’); Gothic, kaurus (‘heavy’); Lettish, gruts (‘heavy’). It is tempting to speculate that, perhaps, Newton had some sort ofawareness of the fundamental nature of that otherwise merely accidental fact. It seems not far fromthe position by Chandrasekhar who wrote [38]: “There can be no doubt that Newton held the accurateproportionality of the weight ‘to the masses of matter which they contain’ as inviolable”.Whatever the case, Einstein promoted it to a truly fundamental cornerstone on which he erected hisbeautiful theoretical building: the Equivalence Principle (EP). Indeed, the postulated exact equality of theinertial and gravitational mass implies that, in a given constant and uniform gravitational field, all bodiesmove with the same acceleration in exactly the same way as they do in an uniformly accelerated reference niverse , frame removed from any external gravitational influence. In this sense, an uniformly accelerated framein absence of gravity is equivalent to an inertial frame in which a constant and uniform gravitationalfield is present. It is important to stress that the need of making the universality of the free fall, uponwhich the EP relies, compatible with the dictates of the SRT was not at all a trivial matter [21] (cfr.Section 1), and the merit of keeping the law of free fall as a fundamental principle of a viable relativistictheory of gravitation which could not reduce to a mere extension of the Newtonian theory to the SRTmust be fully ascribed to Einstein. To better grasp the di ffi culties posed by such a delicate conceptualoperation, let us think about an inertial reference frame K in which two stones, di ff ering by shape andcomposition, move under the action of a uniform gravitational field starting from the same height butwith di ff erent initial velocities; for the sake of simplicity, let us assume that, while one of the two stonesis thrown horizontally with an initial velocity with respect to K , the other one falls vertically startingat rest [21]. Due to the universality of free fall, both the stones reach the ground simultaneously. Letus, now, consider an inertial frame K ′ moving uniformly at a speed equal to the horizontal componentof the velocity of the projectile; in this frame, the kinematics of the two objects gets interchanged: theprojectile has no horizontal velocity so that now it falls vertically, while the stone at rest acquires anhorizontal velocity making it move parabolically in the opposite direction with respect to K ′ . Accordingto the universality of the free fall, also in this case they should come to the rest at the same time. Butthis is in disagreement with the relativity of the simultaneity of the SRT. Moreover, another source ofpotential tension between the universality of the free fall and the SRT is as follows [21]. According tothe latter one, a change in the energy of a body corresponds to a change also in its inertial mass, whichacts as a “brake”. On the other hand, since the inertial mass is equivalent to the gravitational mass,which, instead, plays the role of “accelerator”, the correct relativistic theory of gravitation necessarilyimplies that also the gravitational mass should depend in an exactly known way from the total energyof the body. Actually, other scientists like, e.g., Abraham [27] and Mie [39] were willing to discard theGalileo’s law of universality of free fall to obtain a relativistic theory of gravitation.The heuristic significance of the original form of the EP unfolded in the findings by Einstein thatidentical clocks ticks at di ff erent rates if placed at di ff erent points in a gravitational potential, a featurewhich was measured in a laboratory on the Earth’s surface in 1960 [40] by means of the Mössbauere ff ect which has recently received a general relativistic interpretation [41], and the gravitational redshiftof the spectral lines emitted at the Sun’s surface with respect to those on the Earth, which was measuredonly in the sixties of the last century [42] following the 1925 measurement with the spectral lines in thecompanion of Sirius [43]. Furthermore, it turned out that the speed of light in a gravitational field isvariable, and thus light rays are deflected, as if not only an inertial mass but also a gravitational masswould correspond to any form of energy. Einstein [28] was also able to calculate the deflection of theapparent position of background stars due to the Sun’s gravitational potential, although the value hefound at that time was only half of the correct one later predicted with the final form of his GTR [44] andmeasured in 1919 [45,46] (see Section 4). In 1912, he [47,48] explored the possibility of gravitationallensing deriving the basic features of the lensing e ff ect, which will be measured for the first time notuntil 1979 [49]. It must be noted [19,21] that this theory of the constant and uniform gravitational fieldwent already beyond STR. Indeed, because of the dependence of the speed of light and the clock rateson the gravitational potential, STR definition of simultaneity and the Lorentz transformation themselves niverse , lost their significance (cfr. Section 1). In this specific sense , it can be said that STR can hold only inabsence of a gravitational field.The existence of non-uniformly accelerated reference frames like, e.g., those rotating with atime-dependent angular velocity Ω ( t ), naturally posed the quest for a further generalization of the EPable to account for spatially and temporally varying gravitational fields as well. The extension of the EPto arbitrarily accelerating frames necessarily implies, in principle, the existence of further, non-uniform,non-static (either stationary and non-stationary) and velocity-dependent gravitational e ff ects, as guessedby Einstein [50–52]. They were later fully calculated by Einstein himself [53] and others [54–60] withthe final form of the GTR (see Section 4 and [61–63] for critical analyses of the seminal works), whichcould not be encompassed by the gravito-static Newtonian framework. Indeed, it must be recalled thatthe inertial acceleration experienced by a body (slowly) moving with velocity v ′ with respect to a rotatingframe K ′ is a ′ Ω = Ω × v ′ + ˙ Ω × r ′ + Ω × (cid:16) Ω × r ′ (cid:17) (11)At least to a certain extent, such new gravitational e ff ects, some of which have been measured only afew years ago [64–67], might be considered as reminiscent of the Machian relational conceptions ofmechanics [68–71].Such a generalization of the EP to arbitrary gravitational fields lead Einstein to reformulate it asfollows: in any infinitesimal spacetime region ( i.e ., su ffi ciently small to neglect either spatial andtemporal variations of gravity throughout it), it is always possible to find a suitable non-rotatingcoordinate system K in which any e ff ect of gravity on either test particle motions and any other physicalphenomena is absent. Such a local coordinate system can ideally be realized by a su ffi ciently small boxmoving in the gravitational field freely of any external force of non-gravitational nature. Obviously,it appeared natural to assume the validity of STR in K in such a way that all the reference framesconnected to it by a Lorentz transformation are physically equivalent. In this specific sense , it could besaid that the Lorentz covariance of all physical laws is still valid in the infinitely small.At this point, still relying upon the EP, it remained to construct a theory valid also for arbitrarilyvarying gravitational fields by writing down the di ff erential equations connecting the gravitationalpotential, assumed as state variable, with the matter-energy sources and requiring their covariance withrespect to a fully general group of transformations of the spacetime coordinates. A step forward was done in 1914 when, in collaboration with Grossmann, Einstein [72], on the basisof the Riemannian theory of curved manifolds, was able to introduce the ten coe ffi cients g µν of thesymmetric metric tensor g by writing down the square of the spacetime line element ( ds ) between twoinfinitely near events in arbitrary curvilinear coordinates x µ as( ds ) = g µν dx µ dx ν (12)As a consequence, the equations of motion of a test particle, the energy-momentum theorem andthe equations of the electromagnetism in vacuo were simultaneously written in their generally covariantultimate form. In particular, from the right-hand-side of the geodesic equation of motion of a test particle niverse , d x α ds = − Γ αβ̺ dx β ds dx ̺ ds (13)where the Christo ff el symbols Γ αβ̺ (cid:17) g ασ ∂ g σβ ∂ x ̺ + ∂ g σ̺ ∂ x β − ∂ g β̺ ∂ x σ ! (14)are constructed with the first derivatives of g µν , it was possible to straightforwardly identify thecomponents of g as the correct state variables playing the role of the Newtonian scalar potential Φ .Indeed, to a first-order level of approximation characterized by neglecting terms quadratic in v / c and thesquares of the deviations of the g µν from their STR values η = + η i j = − δ i j the geodesic equations of motion for the spatial coordinates become d x i dt = − c Γ i (16)Furthermore, if the gravitational field is assumed static or quasi-static and the time derivatives can beneglected, the previous equations reduce to d x i dt = c ∂ g ∂ x i (17)By posing Φ (cid:17) − c ( g −
1) (18)so that g = − Φ c (19)the Newtonian acceleration is obtained. The additive constant up to which the potential is defined isfixed in such a way that Φ vanishes when g assumes its STR value η . It is worthwhile remarking that,to the level of approximation adopted, only g enters the equations of motion, although the deviationsof the other metric coe ffi cients from their STR values may be of the same order of magnitude. It is thiscircumstance that allows to describe, to a first order approximation, the gravitational field by means of asingle scalar potential.In analogy with the geodesic equations of motion for a test particle, also those for the propagationof electromagnetic waves followed. Indeed, the worldlines of light rays are, thus, geodesics curves ofnull length ( ds ) = d x α d λ = − Γ αβ̺ dx β d λ dx ̺ d λ where λ is some a ffi ne parameter. niverse , The components of the metric tensor g are not assigned independently of the matter-energydistributions, being determined by field equations.A further consequence of EP and the fact that, to the lowest order of approximation, g is proportionalto the Newtonian potential Φ is that, in general, it is possible to predict the influence of the gravitationalfield on clocks even without knowing all the coe ffi cients g µν ; such an influence is actually determined by g through d τ = √ g dt (21)where τ is the reading of a clock at rest. Instead, it is possible to predict the behaviour of measuringrods only knowing all the other coe ffi cients g i , g ik . Indeed, it turns out that the square of the distance dl between two nearby points in the 3-dimensional space is given by [73]( dl ) = − g jh + g j g h g ! dx j dk h (22)Thus, the field g determines not only the gravitational field, but also the behaviour of clocks andmeasuring rods, i.e ., the chronogeometry of the 4-dimensional spacetime which contains the geometry ofthe ordinary 3-dimensional space as a particular case. Such a fusion of two fields until then completelyseparated-metric and gravitation-should be regarded as a major result of GTR, allowing, in principle, todetermine the gravitational field just from local measurements of distances and time intervals.
4. The Field Equations for the Metric Tensor and Their Physical Consequences
The di ff erential equations for the g tensor itself followed in 1915 [1–3].The tortuous path [21] which lead to them can be sketchily summarized as follows [19]. According tothe EP, the gravitational mass of a body is exactly equal to its inertial mass and, as such, it is proportionalto the total energy content of the body. The same must, then, hold also in a given gravitational field for theforce experienced by a body which is proportional to its (passive) gravitational mass. It is, thus, natural toassume that, conversely, only the energy possessed by a material system does matter, through its (active)gravitational mass, as for as its gravitational field is concerned. Nonetheless, in STR the energy densityis not characterized by a scalar quantity, being, instead, the 00 component of the so-called stress-energytensor T . It follows that also momentum and stresses intervene on the same footing as energy itself.These considerations lead to the assumption that no other material state variables than the components T µν of T must enter the gravitational field equations. Moreover, in analogy with the Poisson equation, T must be proportional to a di ff erential expression G of the second order containing only the statevariables of the gravitational field, i.e ., the components of the metric tensor g ; because of the requiredgeneral covariance, G must be a tensor as well. The most general expression for it turned out to be G µν = c R µν + c g µν R + c g µν (23)where R is the contracted curvature tensor whose components are R µν = ∂Γ αµα ∂ x ν − ∂Γ αµν ∂ x α + Γ βµα Γ ανβ − Γ αµν Γ βαβ (24) niverse , and R is its invariant trace. The coe ffi cients c , c , c were determined by imposing that the stress-energytensor satisfies the energy-momentum conservation theorem. By neglecting the third term in G , whichusually plays a negligible role in the e ff ects which will be discussed in this Section (see Section 5 forphenomena in which it may become relevant), the Einstein field equations became [1,2] G = − κ T (25)with G µν = R µν − g µν R (26)and κ is a constant which is determined by comparison with the Newtonian Poisson equation. Bycontraction, one gets R = κ T (27)where T is the trace of T , so that R µν = − κ T µν − g µν T ! (28)This is the generally covariant form of the gravitational field equations to which, after many attempts,Einstein came in 1915 [3].The same field equations were obtained elegantly by Hilbert through a variational principle [74].On the reciprocal influences between Einstein and Hilbert in the process of obtaining the GTR fieldequations and an alleged priority dispute about their publication, see [75].It should be noted [19] that GTR, per se , yields neither the magnitude nor the sign (attraction orrepulsion of the gravitational interaction) of κ which are, instead, retrieved from the observations. Forweak and quasi-static fields generated by pressureless, extremely slowly moving matter of density ρ , theright-hand-side of the field equation for the 00 component becomes − κ c ρ (29)indeed, the only non-vanishing component of the matter stress-energy tensor is T = ρ c (30)so that T = − ρ c (31)Since the time derivatives and the products of the Christo ff el symbols can be neglected, the 00 componentof the Ricci tensor reduces to R = ∇ g = − ∇ Φ c (32)Thus, it is ∇ Φ = κ c ρ (33)the Poisson equation really holds. A comparison with the Newtonian equation tells that κ is positive,being equal to κ = π Gc = × − kg − m − s (34)the spacetime can, thus, be assimilated to an extremely rigid elsatic medium. niverse , In the same year [44], Einstein readily employed his newborn theory to successfully explain thelong-standing issue of the anomalous perihelion precession of Mercury [76]. To this aim, and also inorder to derive the correct value of the deflection of a light ray grazing the Sun’s limb [44] throughthe Fermat principle, it was necessary to know not only the coe ffi cient g of the gravitational field ofa point mass, as in the Newtonian approximation, but also the other metric coe ffi cients g i j . Since thespacetime outside a spherical body is isotropic, the o ff -diagonal metric coe ffi cients g i are identicallyzero: otherwise, they would induce observable e ff ects capable of distinguishing between, e.g., twoopposite spatial directions (see Section 3.2). Moreover, it was also required to approximate g itselfto a higher order. Einstein [44] solved that problem by successive approximations. The exact vacuumsolution was obtained one year later by Schwarzschild [77] and, independently, Droste [78]; their resultsare virtually indistinguishable from those of Einstein. Relevant simplifications were introduced one yearlater by Weyl [79], who used cartesian coordinates instead of spherical ones, and worked on the basis ofthe action principle instead of recurring to the di ff erential equations for the field g . Schwarzschild [80]extended the validity of his solution also to the interior of a material body modelled as a sphere ofincompressible fluid. Having in hand this exact solution of the Einstein field equations revolutionizedthe successive development of GTR. Indeed, instead of dealing only with small weak-field correctionsto Newtonian gravity, as Einstein had initially imagined would be the case, fully nonlinear featuresof the theory such as gravitational collapse and singularity formation could be studied, as it becameclear decades later. About the Schwarzschild solution, the Birkho ff ’s Theorem [81] was proved in1923. According to it, even without the assumption of staticity, the Schwarzschild metric is the unique vacuum solution endowed with spherically symmetry. As a consequence, the external field of a sphericalbody radially pulsating or radially imploding / exploding is not influenced at all by such modifications ofits source.The successful explanation of the anomalous perihelion precession of Mercury was a landmark forthe validity of GTR since, as remarked in [82,83], it was a successful retrodiction of an e ff ect whichwas known for decades. In particular, Weinberg wrote [83]: “It is widely supposed that the true test ofa theory is in the comparison of its predictions with the results of experiment. Yet, with the benefit ofhindsight, one can say today that Einstein’s successful explanation in 1915 of the previously measuredanomaly in Mercury’s orbit was a far more solid test of general relativity than the verification of hiscalculation of the deflection of light by the sun in observations of the eclipse of 1919 or in later eclipses.That is, in the case of general relativity a retrodiction , the calculation of the already-known anomalousmotion of Mercury, in fact provided a more reliable test of the theory than a true prediction of a newe ff ect, the deflection of light by gravitational fields.I think that people emphasize prediction in validating scientific theories because the classic attitude ofcommentators on science is not to trust the theorist. The fear is that the theorist adjusts his or her theoryto fit whatever experimental facts are already known, so that for the theory to fit to these facts is not areliable test of the theory.But [. . . ] no one who knows anything about how general relativity was developed by Einstein, whoat all follows Einstein’s logic, could possibly think that Einstein developed general relativity in order to niverse , explain this precession. [. . . ] Often it is a successful prediction that one should really distrust. In thecase of a true prediction, like Einstein’s prediction of the bending of light by the sun, it is true that thetheorist does not know the experimental result when she develops the theory, but on the other hand theexperimentalist does know about the theoretical result when he does the experiment. And that can lead,and historically has led, to as many wrong turns as overreliance on successful retrodictions. I repeat: itis not that experimentalists falsify their data. [. . . ] But experimentalists who know the result that theyare theoretically supposed to get naturally find it di ffi cult to stop looking for observational errors whenthey do not get that result or to go on looking for errors when they do. It is a testimonial to the strengthof character of experimentalists that they do not always get the results they expect”.The final work of Einstein on the foundations of GTR appeared in 1916 [84].In the same year, de Sitter [85] was able to derive a further consequence of the static, sphericallysymmetric spacetime of the Schwarzschild solution: the precession of the orbital angular momentumof a binary system, thought as a giant gyroscope, orbiting a non-rotating, spherical body such as in thecase of the Earth-Moon system in the Sun’s field. Some years later, Schouten [86] and Fokker [87]independently obtained the same e ff ect by extending it also to spin angular momenta of rotating bodies.Such an e ff ect is mainly known as de Sitter or geodetic precession. It was measured decades later inthe field of the Sun by accurately tracking the orbit of the Earth-Moon system with the Lunar LaserRanging technique [88,89], and in the field of the Earth itself with the dedicated Gravity Probe B (GP-B)space-based experiment [64] and its spaceborne gyroscopes.In 1964 [90], Shapiro calculated a further prediction of the static Schwarzschild spacetime: Thetemporal delay, which since then bears his name, experienced by travelling electromagnetic waves whichgraze the limb of a massive body as the Sun in a back-and-forth path to and from a terrestrial station afterhaving been sent back by a natural or artificial body at the superior conjunction with our planet. In itsfirst successful test performed with radar signals [91], Mercury and Venus were used as reflectors. Latestaccurate results [92] relied upon the Cassini spacecraft en route to Saturn. In 1916 [93], Einstein, with a suitable approximation method, was able to derive the field generatedby bodies moving with arbitrary speeds, provided that their masses are small enough. In this case, the g µν di ff er slightly from the STR values η µν , so that the squares of their deviations h µν with respect to thelatter ones can be neglected, and it is possible to keep just the linear part of the field equations. Startingfrom their form [1,2] R µν − g µν R = − κ T µν (35)working in the desired approximation, they can be cast into a linearized form in terms of the auxiliarystate variables h µν (cid:17) h µν − δ νµ h (36)where δ νµ is the Kronecker delta, and h is the trace of h which is a tensor only with respect to the Lorentztransformations. A further simplification can be obtained if suitable spacetime coordinates, satisfyingthe gauge condition niverse , ∂ h αβ ∂ x β = ff erential equations for the state variables h µν are (cid:3) h µν = − κ T µν (38)which is the inhomogeneous wave equation; (cid:3) is the STR form of the d’Alembertian operator. The usualmethod of the retarded potentials allows to obtain h µν = κ π Z T µν (cid:16) x ′ , y ′ , z ′ , t − r / c (cid:17) r dx ′ dy ′ dz ′ (39)Among other things, it implies that the action of gravity propagates to the speed of light: a quite importantresults which, some years ago, was the subject of dispute [94–96] boosted by the interpretation of certainVLBI measurements of the time delay su ff ered at the limb of Jupiter by electromagnetic waves fromdistant quasars [97,98].4.3.1. Gravitational WavesThe form of the gravitational waves in empty regions follows from the Lorentz gauge condition andthe inhomogeneous wave equation by posing T =
0: it was studied by Einstein in [99], where he alsocalculated the emission and the absorption of gravitational waves. It turned out that, when oscillationsor other movements take place in a material system, it emits gravitational radiation in such a way thatthe total power emitted along all spatial directions is determined by the third temporal derivatives of thesystem’s moment of inertia I i j = Z ρ x i x j dx dx dx (40)Instead, when a gravitational wave impinges on a material system whose size is smaller than the wave’swavelength, the total power absorbed is determined by the second temporal derivatives of its moment ofinertia [99].Gravitational waves were indirectly revealed for the first time [100–102] in the celebratedHulse-Taylor binary pulsar PSR B1913 +
16 [103,104].
Direct detection (some of) their predictede ff ects in both terrestrial [105–110] and space-based laboratories [111–117] from a variety of di ff erentastronomical and astrophysical sources [118], relentlessly chased by at least fifty years since the firstproposals by Gertsenshtein and Pustovoit [119] of using interferometers and the pioneering attempts byJ. Weber [120] with its resonant bars [121], is one of the major challenges of the current research inrelativistic physics [122,123].Conversely, by assuming their existence, they could be used, in principle, to determine keyparameters of several extreme astrophysical and cosmological scenarios which, otherwise, wouldremain unaccessible to us because of lack of electromagnetic waves from them [124] by establishingan entirely new “Gravitational Wave Astronomy” [122,125]. A recent example [126] is given bythe possibility that the existence of primordial gravitational waves may a ff ect the polarization ofthe electromagnetic radiation which constitutes the so-called Cosmic Microwave Background (CMB), niverse , discovered in 1965 [127]. In this case, the polarizing e ff ect of gravity is indirect since the field of thegravitational waves does not directly impact the polarization of CMB, a ff ecting, instead, the anisotropyof the spatial distribution of CMB itself. Indeed, the polarization of CMB is a direct consequenceof the scattering of the photons of the radiation with the electrons and positrons which formed theprimordial plasma, existing in the primordial Universe at the so-called decoupling era [128]. At laterepochs, when the temperature fell below 3000 K ◦ , the radiation decoupled from matter, photons andelectrons started to interact negligibly, and the polarization got “frozen” to the values reached at theinstant of decoupling. Thus, mapping the current CMB’s polarization state has the potential of providingus with direct information of the primordial Universe, not contaminated by the dynamics of successiveevolutionary stages. In particular, it turns out that the presence of metric fluctuations of tensorial type, i.e ., of gravitational waves, at the epoch in which the CMB radiation interacted with the electrons of thecosmic matter getting polarized, may have left traces in terms of polarization modes of B type [129,130].They could be currently measurable, provided that the intensity of the cosmic background of gravitationalwaves is strong enough. An example of cosmic gravitational background able to produce, in principle,such an e ff ect is represented by the relic gravitational radiation produced during the inflationary epochs.The gravitational waves produced in this way are distributed over a very wide frequency band ∆ω ( t )which is generally time-dependent. In order to characterize the intensity of such relic gravitationalwaves, it turns out convenient to adopt the spectral energy density ε h ( ω, t ) (cid:17) d ε ( t ) d ln ω (41)defined as the energy density ε ( t ) per logarithmic interval of frequency, normalized to the critical energydensity ε crit (see Section 5.4), i.e ., the dimensionless variable Ω h ( ω, t ) (cid:17) ε crit d ε d ln ω (42)The simplest inflationary models yield power-law signatures for it. In 2014 [131], the BICEP2experiment at the South Pole seemed to have successfully revealed the existence of the B modes;the measured values seemed approximately in agreement-at least in the frequency band exploredby BICEP2-with a cosmic gravitational radiation background corresponding to the aforementionedpower-law models. More recently [132], a joint analysis of data from ESA’s Planck satellite and theground-based BICEP2 and Keck Array experiments did not confirm such a finding.4.3.2. The E ff ect of Rotating MassesThe previously mentioned solution h µν of the inhomogenous wave equation in terms of the retardedpotentials was used by Thirring [56,57,61] to investigate, to a certain extent, the relative nature of thecentrifugal and Coriolis fictitious forces arising in a rotating coordinate system with respect to anotherone connected with the static background of the fixed stars. Indeed, according to a fully relativisticpoint of view, they should also be viewed as gravitational e ff ects caused by the rotation of the distantstars with respect to a fixed coordinate system. At first sight, it may seem that such a possibility isalready included in the theory itself in view of the covariance of the field equations. Actually, it is notso because the boundary conditions at infinite distance play an essential role in selecting, de facto , some niverse , privileged coordinate systems, in spite of a truly “relativistic” spirit with which the theory should beinformed. In other words, although the equations of the theory are covariant, the choice of the boundaryconditions at spatial infinity, which are distinct from and independent of the field equations themselves,would pick up certain coordinate systems with respect to others, which is a conceptual weakness of analleged “generally relativistic” theory. Thus, Thirring [56] did not aim to check the full equivalence ofthe gravitational e ff ects of the rotation of the whole of the distant stars of the Universe with those dueto the rotation of the coordinate system with respect to them, assumed fixed. Indeed, he consideredjust a rotating hollow shell of finite radius D and mass M , so to circumvent the issue of the boundaryconditions at infinite distance by setting the spacetime metric tensor there equal to the Minkowskian one.By assuming M small with respect to the whole of the fixed stars, so to consider the departures of the g µν coe ffi cients from their STR values η µν small inside the shell, the application of the previously obtainedEinsteinian expression for h µν to the shell yielded that a test particle inside the hollow space inside it isa ff ected by accelerations which are formally identical to the centrifugal and Coriolis ones, apart from amultiplicative scaling dimensionless factor as little as GMc D (43)This explains the failures by Newton [133] in attributing the centrifugal curvature of the free surface ofwater in his swirling bucket to the relative rotation of the bucket itself and the water, and the Friedländerbrothers [134] who unsuccessfully attempted to detect centrifugal forces inside a heavy rotating flywheel.Another application of the approximate solution h µν of the inhomogeneous wave equation allowed todiscover that, while in either GTR and the Newtonian theory the gravitational field of a static, sphericalbody is identical to that of a point mass [81], it is not so-in GTR-if the body rotates. Indeed, Einstein [53],Thirring and Lense [54] calculated the (tiny) precessions a ff ecting the orbits of test particles as naturalsatellites and planets moving in the field of rotating astronomical bodies such as the Sun and someof its planets. Such a peculiarity of the motion about mass-energy currents, universally known as“Lense-Thirring e ff ect” by historical tradition (cfr. [62] for a critical historical analysis of its genesis),was subjected to deep experimental scrutiny in the last decades [65–67].In the sixties of the twentieth century, another consequence of the rotation of an astronomical bodywas calculated within GTR: the precession of an orbiting gyroscope [135,136], sometimes dubbed as“Pugh-Schi ff e ff ect”. The GP-B experiment [137], aimed to directly measure also such an e ff ect in thefield of the Earth, was successfully completed a few years ago [64], although the final accuracy obtained( ∼ ∼
1% or better) . Q . One yearlater, Weyl [79] obtained the same metric from a variational action principle. In 1918, Nordström [139],generalized it to the case of a spherically symmetric charged body. The metric for a non-rotating charge niverse , distribution is nowadays known as the Reissner-Nordström metric; in the limit Q →
0, it reduces to theSchwarzschild solution.The physical relevance of the Reissner-Nordström metric in astronomical and astrophysical scenariosdepends on the existence of macroscopic bodies stably endowed with net electric charges.4.4.2. Black HolesOne of the consequences of the vacuum Schwarzschild solution was that it predicts the existence of asurface of infinite red-shift at r = r g (cid:17) GMc (44)Thus, if, for some reasons, a body could shrink so much to reduce to such a size, it would disappearfrom the direct view of distant observers, who would not be anymore able to receive any electromagneticradiation from such a surface, later interpreted as a spatial section of an “event horizon” [140–142]. A“frozen star”, a name common among Soviet scientists from 1958 to 1968 [143,144], would have, then,formed, at least from the point of view of an external observer. In 1968 [143,145], Wheeler renamedsuch objects with their nowadays familiar appellative of “black holes” [146].In fact, both Eddington in 1926 [147] and Einstein in 1939 [148], although with arguments at di ff erentlevels of soundness, were firmly convinced that such bizarre objects could not form in the real world.Instead, in 1939 [149], Oppenheimer and Snyder demonstrated that, when all the thermonuclear sourcesof energy are exhausted, a su ffi ciently heavy star will unstoppably collapse beyond its Schwarzschildradius to end in a spacetime singularity. The latter one is not to be confused with the so-called“Schwarschild singularity” occurring in the Schwarzschild metric at r = r g , which was proven in1924 [150] to be unphysical, being a mere coordinate artifact; nonetheless, it took until 1933 for Lemaîtreto realize it [151]. In 1965, Penrose [152], in his first black hole singularity theorem, demonstrated thatthe formation of a singularity at the end of a gravitational collapse was an inevitable result, and notjust some special feature of spherical symmetry. A black hole is the 4-dimensional spacetime regionwhich represents the future of an imploding star: it insists on the 2-dimensional spatial critical surfacedetermined by the star’s Schwarzschild radius. The 3-dimensional spacetime hypersurface delimitingthe black hole, i.e ., its event horizon, is located in correspondence of the critical surface [142].4.4.3. The Kerr MetricIn 1963, the third physically relevant exact vacuum solution of the Einstein field equations was foundby Kerr [153]. It describes the spacetime metric outside a rotating source endowed with mass M andproper angular momentum J . It was later put in a very convenient form by Boyer and Lindquist [154]. Atthat time, it was generally accepted that a spherical star would collapse to a black hole described by theSchwarzschild metric. Nonetheless, people was wondering if such a dramatic fate of a star undergoinggravitational collapse was merely an artifact of the assumed perfect spherical symmetry. Perhaps, theslightest angular momentum would halt the collapse before the formation of an event horizon, or at leastbefore the formation of a singularity. In this respect, finding a metric for a rotating star would have beenquite valuable. niverse , Contrary to the Schwarzschild solution [80], the Kerr one has not yet been satisfactorily extendedto the interior of any realistic matter-energy distribution, despite several attempts over the years [155].Notably, according to some researchers [156–158], this limit may have no real physical consequencessince the exterior spacetime of a rotating physically likely source is not described by the Kerr metricwhose higher multipoles, according to the so-called “no-hair” conjecture [159,160], can all be expressedin terms of M and J [161,162], which is not the case for a generic rotating star [163]. Moreover, theKerr solution does not represent the metric during any realistic gravitational collapse; rather, it yieldsthe asymptotic metric at late times as whatever dynamical process produced the black hole settlesdown, contrary to the case of a non-rotating collapsing star whose exterior metric is described by theSchwarzschild metric at all times. The Birkho ff ’s Theorem [81] does not hold for the Kerr metric.The enormous impact that the discovery by Kerr has had in the subsequent fifty years on everysubfield of GTR and astrophysics as well is examined in [158]; just as an example, it should be recalledthat, at the time of the Kerr’s discovery, the gravitational collapse to a Schwarzschild black hole haddi ffi culty in explaining the impressive energy output of quasars, discovered and characterized just inthose years [164,165], because of the “frozen star” behavior for distant observers. Instead, the propertiesof the event horizon were di ff erent with rotation taken into account. A comparison of the peculiarfeatures of the Schwarzschild and the Kerr solutions can be found in [166].4.4.4. The Kerr-Newman MetricIn 1965 [167], a new exact vacuum solution of the Einstein-Maxwell equations of GTR appeared:the Kerr-Newman metric [168]. It was obtained from the Reissner-Nordström metric by a complextransformation algorithm [169] without integrating the field equations, and is both the spinninggeneralization of Reissner-Nordström and the electrically charged version of the Kerr metric. Suchsolutions point towards the possibility that charged and rotating bodies can undergo gravitational collapseto form black holes just as in the uncharged, static case of the Schwarzschild metric.Leaving the issue of its physical relevance for astrophysics applications out of consideration, theKerr-Newman metric is the most general static / stationary black hole solution to the Einstein-Maxwellequations. Thus, it is of great importance for theoretical considerations within the mathematicalframework of GTR and beyond. Furthermore, understanding this solution also provides valuable insightsinto the other black hole solutions, in particular the Kerr metric.
5. Application to Cosmology ffi culties of Newtonian Cosmologies The birth of modern cosmology might be dated back to the correspondence between Newton andBentley in the last decade of the seventieth century [170], when the issue of the applicability ofNewtonian gravitational theory to a static, spatially infinite (Euclidean) Universe uniformly filled withmatter was tackled. In four letters to R. Bentley, Newton explored the possibility that matter might bespread uniformly throughout an infinite space. To the Bentley’ s suggestion that such an even distributionmight be stable, Newton replied that, actually, matter would tend to collapse into large massive bodies. niverse , However, he apparently also thought that they could be stably spread throughout all the space. Inparticular, in his letter of 10 December 1692, Newton wrote [171]: “it seems to me that if [. . . ] allthe matter in the Vniverse was eavenly scattered throughout all the heavens, & every particle had aninnate gravity towards all the rest & the whole space throughout which this matter was scattered, wasbut finite: the matter on the outside of this space would by its gravity tend towards all the matter on theinside & by consequence fall down to the middle of the whole space & there compose one great sphericalmass But if the matter was eavenly di ff used through an infinite space, it would never convene into onemass but some of it convene into one mass & some into another so as to make an infinite number of greatmasses scattered at great distances from one another throughout all that infinite space.”Connected with the possibility that matter would fill uniformly an infinite space, and, thus,indirectly with the application of Newtonian gravitation to cosmology, there was also the so-calledOlbers paradox [172], some aspects of which had been previously studied also by Kepler [173],Halley [174,175] and de Chéseaux [176]. According to it, although the light from stars diminishesas the square of the distance to the star, the number of stars in spherical shells increases as the squareof the shell’s radius. As a result, the accumulated e ff ect of the light intensity should make the night skyas bright as the surface of the Sun. In passing, the Olbers paradox touched also other topics which willbecome crucial in contemporary cosmology like the temporal infinity of the Universe and its materialcontent, and its spatial infinity as well. At the end of the nineteenth century, Seeliger [177] showed that,in the framework of the standard Newtonian theory, matter cannot be distributed uniformly throughoutan infinite Universe. Instead, its density should go to zero at spatial infinity faster than r − ; otherwise,the force exerted on a point mass by all the other bodies of the Universe would be undeterminate becauseit would be given by a non-convergent, oscillating series. Later, Einstein [178] critically remarked that,if the potential was finite at large distances as envisaged by Seeliger [177] to save the Newtonian law,statistical considerations would imply a depopulation of the fixed stars ensemble, assumed initially instatistical equilibrium. The possibility of an infinite potential at large distances, corresponding to a finiteor vanishing not su ffi ciently fast matter density, already ruled out by Seeliger himself, was excluded alsoby Einstein [178] because it would yield unrealistically fast speeds of the distant stars. Seeliger [179]demonstrated also that matter density could be di ff erent from zero at arbitrary distances if the standardPoisson equation was modified as ∇ Φ − ΛΦ = π G ρ (45)It admits Φ = − π G ρΛ (46)as a viable solution for a uniform matter density, thus making an evenly filled Universe stable. Fora discussion of the problems encountered by the Newtonian theory of gravitation to cosmology, see,e.g., [180].The inadequacy of Newtonian gravitation to cosmological problems can be also inferred in view ofthe modern discoveries concerning the expansion of the Universe over the eons (see Section 5.2) which,in conjunction with the finite value of c , yielded to the notion of observable Universe. As previouslyrecalled in Section 3.2, the gravitational interaction among macroscopic bodies can be adequatelydescribed, to the first approximation, by the non-relativistic Newtonian model. Such an approximationis applicable over spatial scales ranging from laboratory to planetary, stellar, and galactic systems. On niverse , the other hand [181], the Newtonian model cannot be applied, not even to the first approximation, tocorrectly describe gravity over cosmological distances of the order the Hubble distance D H (cid:17) cH ∼ m (47)where [182] H = (67 . ± .
2) km s − Mpc − (48)is the current value of the Hubble parameter (see Section 5.4), which fixes the maximum spatial distanceaccessible to current observations (the radius of the observable universe is proportional to D H througha numerical coe ffi cient which, according to the present-day cosmological parameters, is equal to 3 . ε enclosed in aspherical volume of radius ∼ D H is | Φ H | = π G ε H (49)The condition of validity of the Newtonian approximation is that, for any test particle of mass m , thegravitational potential energy m | Φ H | resulting from the interaction with the cosmological mass of theobservable Universe is much smaller than its rest energy mc . Instead, it turns out [181]43 π G ε H c ∼ GTR, applied to cosmology for the first time in 1917 by Einstein himself [178], was able to put sucha fundamental branch of our knowledge on the firm grounds of empirical science.In the following, we will try to follow the following terminological stipulations [183]. We willgenerally use the word “Universe” to denote a model of the cosmological spacetime along with itsoverall matter-energy content; as we will see, the relativistic Universe is the space woven by timeand weighed by all forms of energy (matter-either baryonic and non-baryonic-, radiation, cosmologicalconstant). As such, the Universe has neither center nor borders, neither inside nor outside. Instead,by means of “universe” we will denote the observable portion of the cosmological spacetime delimitedby a cosmological horizon unavoidably set by the fact that all the physical means (electromagnetic andgravitational radiation, neutrinos, cosmic rays) by which we collect information from objects around ustravel at finite speeds. Its spatial section is a centered on the Earth-based observer with a radius equal to3 . D H = . = . niverse , depopulation and the observed small stellar velocities issues. Instead, in principle, it is mathematicallypossible to modify them in as much as the same way as it was doable with the Poisson equation byintroducing a Λ term which yielded R µν + Λ g µν = − κ T µν − g µν T ! (52)Some years later, Cartan [184] demonstrated that the most general form of the Einstein field equationsnecessarily implies the Λ term. It turned out that a Universe uniformly filled with constant matter density ρ and non-vanishing Λ E = π G ρ c (53)would rest in equilibrium. Moreover, since it would be spatially closed with g = g i = g i j = − δ i j + x i x j S − (cid:16) x + x + x (cid:17) and radius S connected with Λ by Λ = S (55)there would not be the need of choosing suitable boundary conditions at infinity, thus removing theaforementioned “non-relativistic” drawback of the theory (see Section 4). It should be noted that ifsuch a 4 − dimensional cylindrical Universe did not contain matter, there would not be any gravitationalfield, i.e ., T µν = g µν = / him, itfirst would diminish in size but then would come back beginning to magnify again. Thirteen years later,Eddington [185] showed that the static Einsteinian model is, actually, unstable.It may be interesting to note [20] how the Einstein’s Universe is, in fact, no less liable to the Olbersparadox than the Newtonian one; indeed, the light emitted by a star would endlessly circumnavigate thestatic spherical space until obstructed by another star.5.2.2. The de Sitter ModelIn 1917, de Sitter [186,187] found a solution for the modified Einstein field equations with Λ , g µν = η µν (cid:16) − Λ η αβ x α x β (cid:17) (58) niverse , Λ = S with non-zero gravitational field even in absence of matter, thus di ff ering from the Einstein model. Itallowed also a sort of spatial (and not material) origin of inertia, which would be relative to void space:a hypothetical single test particle existing in the otherwise empty de Sitter Universe would have inertiajust because of Λ .At the time of the Einstein and de Sitter models, there were not yet compelling means toobservationally discriminate between them [188], although their physical consequences were remarkablydi ff erent. Su ffi ce it to say that the spacetime geometry of the de Sitter Universe implied that, althoughstatic, test particles would have escaped far away because of the presence of the Λ term, unless theywere located at the origin. Such a recessional behaviour was known as “de Sitter e ff ect”. As said byEddington [189], “the de Sitter Universe contains motion without matter, while the Einstein Universecontains matter without motion”.After having lost appeal with the advent of the genuine non-stationary Fridman-Lemaître solutions(see Section 5.2.3), the de Sitter model was somewhat revamped in the framework of the inflationaryphase characterized by an ultrafast expansion that it is believed to have occurred in the early stages ofthe universe [190–192].5.2.3. The Fridman-Lemaître-Robertson-Walker Expanding ModelsIn the twenties of the last century, the first truly non-static theoretical models of the Universe wereproposed by Fridman [193,194]. Indeed, he found new solutions of the Einstein field equations with Λ representing spatially homogeneous and isotropic cosmological spacetimes filled with matter-energymodeled as a perfect fluid generally characterized only by time-varying density ρ ( t ), and endowed withan explicitly time-dependent universal scaling factor S ( t ) for the spatial metric having constant curvature k = , ± k = +
1, the 3-dimensional space is spherical and necessarily finite (asthe hypersphere); if k =
0, it is
Euclidean ; if k = −
1, it is hyperbolic . Euclidean and hyperbolic spacescan be either finite or infinite, depending on their topology [195,196] which, actually, is not determinedby the Einstein field equations governing only the dynamical evolution of ρ ( t ) , S ( t ). Importantly, viablesolutions exist also in absence of the cosmological Λ term for all the three admissible values of the spatialcurvature parameter k . The Einstein and de Sitter models turned out [193] to be merely limiting cases ofan infinite family of solutions of the Einstein field equations for a positive, time-varying matter density,any one of which would imply, at least for a certain time span, a general recession-or oncoming, sincethe solutions are symmetric with respect to time reversal-of test particle. According to their dynamicalbehaviour, the Fridman’s models are classified as closed if they recollapse, critical if they expand at anasymptotically zero rate, and open if they expand indefinitely. In this respect, a spherical universe can beopen if Λ is positive and large enough, but it cannot be infinite. Conversely, Euclidean or hyperbolicuniverses, generally open, can be closed if Λ <
0; their finiteness or infiniteness depends on theirtopology, not on their material content. Fridman’s simplifying assumptions were much weaker than thoseof either Einstein and de Sitter, so that they defined a much likelier idealization of the real world [20], asit turned out years later: indeed, the russian scientist was interested only in the mathematical aspects ofthe cosmological solutions of the Einstein equations. niverse , Approximately in the same years, a body of observational evidence pointing towardsmutual recessions of an increasingly growing number of extragalactic nebulae was steadilyaccumulating [197–201] from accurate red-shifts measurements, probably unbeknownst to Fridman. In1929, Hubble [200] made his momentous claim that the line-of-sight speeds of the receding galaxiesare proportional to their distances from the Earth. If, at first, the de Sitter model, notwithstanding itsmaterial emptiness, was regarded with more favor than the Einstein one as a possible explanation of theobserved red-shifts of distant nebulæ, despite the cautiousness by de Sitter himself [187], it would havebeen certainly superseded by the more realistic Fridman ones, if only they had been widely known at thattime (Fridman died in 1925). It may be that a role was played in that by the negative remark by Einsteinabout a claimed incompatibility of the non-stationary Fridman’s models with his field equations [202],later retracted by the father of GTR because of an own mathematical error in his criticism [203].At any rate, in 1927, Lemaître [204], who apparently did never hear of the Fridman’s solutions,rederived them and applied them to the physical universe with the explicit aim of founding a viableexplanation of the observed recessions of galaxies (the red-shifted nebulæ had been recognized asextra-galactic objects analogous to our own galaxy in 1925 by Hubble [205]). Lemaître [204] alsoshowed that the static solution by Einstein is unstable with respect to a temporal variation of matterdensity. Enlightened by the Hubble’s discovery [200], and, perhaps, also struck by the criticisms byLemaître [204] and Eddington [185] to his own static model, Einstein fully acknowledged the meritsof the non-static Fridman-type solutions rejecting outright his cosmological Λ term as unnecessary andunjustifiable [206].Interestingly, in 1931, Lemaître [207] did not appreciate the disown by Einstein of his cosmologicalconstant Λ , which, instead, was retained by the belgian cosmologist an essential ingredient of thephysical Universe for a number of reasons, one of which connected also with quantum mechanics, which,however, convinced neither Einstein nor the scientific community, at least until the end of the nineties ofthe last century [208]. His “hesitating” model was characterized by positive spatial curvature ( k = + g of standard expanding cosmologies is commonly namedas Fridman-Lemaître-Robertson-Walker (FLRW) metric (see Section 5.4). In 1932, Einstein and de Sitter [213] published a brief note of two pages whose aim was to simplifythe study of cosmology. About their work, as reported by Eddington [214], Einstein would have toldhim: “I did not think the paper very important myself, but de Sitter was keen on it”, while de Sitter wroteto him: “You will have seen the paper by Einstein and myself. I do not myself consider the result ofmuch importance, but Einstein seemed to think that it was”. At any rate, such an exceedingly simplifiedsolution, characterized by dust-like, pressureless matter, k = , Λ = niverse , expansion, served as “standard model” over about six decades, to the point of curb researches on othermodels. In it, mutual distances among test particles grow as t / . Such a behaviour is unstable in thesense that it can only occur if k = t =
23 1 H = . The assumptions of homogeneity and isotropy of the spatial sections of the FLRW models are ofcrucial importance. It must be stressed that they are, in general, distinct requirements. Homogeneitydoes not generally imply isotropy: for instance, think about a universe filled with galaxies whose axesof rotation are all aligned along some specific spatial direction, or a wheat field where the ears grow allin the same direction. Conversely, a space which is isotropic around a certain point, in the sense thatthe curvature is the same along all the directions departing from it, may well not be isotropic in otherpoints, or, if some other points of isotropy exist, the curvature there can be di ff erent from each other: anovoid surface is not homogenous since its curvature varies from point to point, but the space is isotropicaround its two “vertices”. Instead, the same vale of the curvature in all the directions, i.e ., the sameamount of isotropy, around all points of space implies homogeneity [218]. As far as our location isconcerned, it can be said phenomenologically that isotropy about us holds in several physical aspects toa high level of accuracy, as demonstrated, e.g., by the CMB which is isotropic at a 10 − level. In view ofthe Copernican spirit, it is commonly postulated that every other observer located everywhere would seethe same situation, thus assuring the homogeneity as well: it is the content of the so-called CosmologicalPrinciple. The fact that the curvature of the spatial parts of the FLRW models is the same everywhere,and that they are expanding over time, according to the Weyl principle [219], admit a peculiar foliationof the spacetime which allows for an unambiguous identification of the spatial sections of simultaneityand of the bundle of time-like worldlines orthogonal to them as worldlines of fundamental observers atrest marking a common, cosmic time. Thus, it is possible to describe the spacetime of the Universe as niverse , the mathematical product of a 3 − dimensional Riemannian space with the temporal axis. In comovingdimensionless spatial coordinates r , θ, φ , the line element can be written as( ds ) = c ( dt ) − S ( t ) " ( dr ) − kr + r ( d θ ) + r sin θ ( d φ ) (60)The Einstein field equations, applied to the FLRW metric with a pressureless cosmic fluid as standardsource with matter and radiation densities ρ m , ρ r , respectively, yield the Fridman equation˙ S = π G ( ρ m + ρ r ) S − kc + Λ c S (61)By defining the Hubble parameter as H (cid:17) ˙ SS (62)and the critical density as ρ crit (cid:17) H π G (63)it is possible to recast the Fridman equation in the form Ω m + Ω r + Ω Λ + Ω k = Ω tot = − Ω k (65)by posing Ω tot (cid:17) Ω m + Ω r + Ω Λ (66)with the dimensionless parameters entering Equation (66) defined as Ω m (cid:17) π G ρ m H > Ω r (cid:17) π G ρ r H > Ω Λ (cid:17) Λ c H S Ω k (cid:17) − kc S H S Ω r , ∼
0, so that the normalized Fridman equation reduces to Ω m , + Ω Λ, + Ω k , = Ω tot , = − Ω k , (69)Results collected in the last twenty years from a variety of observational techniques (e.g., SNeIa [220–222], Baryon acoustic oscillations [223], WMAP [224], Planck [182]), interpreted within aFRLW framework, point towards an observable universe whose spatial geometry is compatible withan Euclidean one (such a possibility, in view of the unavoidable error bars, is impossible to be provedexperimentally with certainty: on the contrary, it could be well excluded should the ranges of values niverse , for Ω tot , did not contain 1), and whose dynamical behaviour is characterized by a small positivecosmological constant Λ which makes it accelerating at late times. By assuming Ω tot , = exactly ,as allowed by the experimental data and predicted by the inflationary paradigm, the values for the othernormalized densities are inferred by finding [182,224] Ω m , ∼ . , Ω Λ, ∼ .
6. Summary
After its birth, GTR went to fertilize and seed, directly as well as indirectly, many branches of disparate sciences as mathematics [225–230], metrology [231–234], geodesy [236–238], geophysics [239–241],astronomy [242–247], astrophysics [248–252], cosmology [181,253–255], not to say about the exquisitetechnological spin-o ff [256–270] due to the long-lasting e ff orts required to put to the test various keypredictions of the theory [18,271]. Moreover, once some of them have been or will be successfullytested, they have or will become precious tools for determine various parameters characterizing severalnatural systems, often in extreme regimes unaccessible with other means: gravitational microlensingfor finding extrasolar planets, even of terrestrial size [272,273], weak and strong gravitational lensingto map otherwise undetectable matter distributions over galactic, extragalactic and cosmologicalscales [274–276], frame-dragging to measure angular momenta of spinning objects like stars andplanets [277–280], gravitational waves to probe, e.g., quantum gravity e ff ects [281], modified modelsof gravity [282,283] and cosmic inflationary scenarios [131,132], to characterize tight binary systemshosting compact astrophysical objects like white dwarves, neutron stars and black holes [284–289], andto investigate extremely energetic events like, e.g., supernovæ explosions [290].However, GTR has its own limits of validity, and presents open problems [291]. At certain regimes,singularities plague it [225,292–295]. Connected to this issue, there is also a major drawback of thetheory of gravitation of Einstein, i.e ., its lingering inability to merge with quantum mechanics yieldinga consistent theory of quantum gravity [296–302]. Moreover, in view of the discoveries made in thesecond half of the last century about the seemingly missing matter to explain the rotation curves ofgalaxies [303–305] and the accelerated expansion of the Universe [220–222], it might be that GTR needto be modified [306–311] also at astrophysical and cosmological scales in order to cope with the issueof the so-called “dark” [312] components of the matter-energy content of the Universe known as DarkMatter and Dark Energy.We consider it appropriate to stop here with our sketchy review. Now, we give the word to thedistinguished researchers who will want to contribute to this Special Issue by bringing us towards thelatest developments of the admirable and far-reaching theory of gravitation by Einstein. At a di ff erentlevel of coverage and completeness, the interested reader may also want to consult the recent two-volumebook [313,314]. References
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