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120-494: The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza , who sent his results to Albert Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called

240-399: A b {\displaystyle {\widetilde {R}}_{ab}} is calculated from the 5D connections . The classic results of Thiry and other authors presume the cylinder condition: Without this assumption, the field equations become much more complex, providing many more degrees of freedom that can be identified with various new fields. Paul Wesson and colleagues have pursued relaxation of

360-436: A b {\displaystyle {\widetilde {g}}_{ab}} , where Latin indices span five dimensions. Let one also introduce the four-dimensional spacetime metric g μ ν {\displaystyle {g}_{\mu \nu }} , where Greek indices span the usual four dimensions of space and time; a 4-vector A μ {\displaystyle A^{\mu }} identified with

480-401: A gauge theory on a fiber bundle , the circle bundle , with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions. Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group . Such generalizations are often called Yang–Mills theories . If a distinction

600-543: A pair of black holes merging . The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear , arbitrarily strong gravitational waves do not obey linear superposition , making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe

720-546: A ( Privatdozent ) at Königsberg until 1929, when he was appointed as professor at the University of Kiel . In 1935, he became a full professor at the University of Göttingen , where he remained until his death in 1954. Perhaps his finest mathematical work is the textbook Höhere Mathematik für den Praktiker , which was written jointly with Georg Joos . Kaluza was extraordinarily versatile. He spoke or wrote 17 languages. He also had an unusually modest personality. He refused

840-570: A body in accordance with Newton's second law of motion , which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration . The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics , straight world lines in curved spacetime . Conversely, one might expect that inertial motions, once identified by observing

960-604: A complete set of 5D curvature tensors under the cylinder condition, evaluated using tensor-algebra software. The equations of motion are obtained from the five-dimensional geodesic hypothesis in terms of a 5-velocity U ~ a ≡ d x a / d s {\displaystyle {\widetilde {U}}^{a}\equiv dx^{a}/ds} : This equation can be recast in several ways, and it has been studied in various forms by authors including Kaluza, Pauli, Gross & Perry, Gegenberg & Kunstatter, and Wesson & Ponce de Leon, but it

1080-560: A computer, or by considering small perturbations of exact solutions. In the field of numerical relativity , powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities . Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization,

1200-508: A curiosity among physical theories. It was clearly superior to Newtonian gravity , being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters (" fudge factors "), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for

1320-530: A curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection , and this is, in fact,

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1440-539: A curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler , spacetime tells matter how to move; matter tells spacetime how to curve. While general relativity replaces

1560-395: A fifth dimension irrespective of the cylinder condition. Most authors have therefore employed the cylinder condition in deriving the field equations. Furthermore, vacuum equations are typically assumed for which where and The vacuum field equations obtained in this way by Thiry and Jordan's group are as follows. The field equation for ϕ {\displaystyle \phi }

1680-597: A gravitational field (cf. below ). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle , a crucial guiding principle for generalizing special-relativistic physics to include gravity. The same experimental data shows that time as measured by clocks in

1800-471: A gravitational field— proper time , to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric . As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with

1920-450: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian gravity , which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect. There are alternatives to general relativity built upon

2040-779: A number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution , the Reissner–Nordström solution and the Kerr metric , each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes , each describing an expanding cosmos. Exact solutions of great theoretical interest include

2160-442: A prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics . These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall , and the propagation of light, and include gravitational time dilation , gravitational lensing ,

2280-453: A short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set , and construction of this compact dimension is referred to as compactification . In modern geometry, the extra fifth dimension can be understood to be the circle group U(1) , as electromagnetism can essentially be formulated as

2400-470: A unified description of gravity as a geometric property of space and time , or four-dimensional spacetime . In particular, the curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as

2520-490: A university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

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2640-539: A wave train traveling through empty space or Gowdy universes , varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ( precession ) of planetary orbits, as well as orbital decay caused by

2760-526: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . The core concept of general-relativistic model-building is that of a solution of Einstein's equations . Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular,

2880-522: Is a problem: the term quadratic in U 5 {\displaystyle U^{5}} , If there is no gradient in the scalar field, the term quadratic in U 5 {\displaystyle U^{5}} vanishes. But otherwise the expression above implies For elementary particles, U 5 > 10 20 c {\displaystyle U^{5}>10^{20}c} . The term quadratic in U 5 {\displaystyle U^{5}} should dominate

3000-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and

3120-404: Is a standard, 4D covariant derivative . It shows that the electromagnetic field is a source for the scalar field . Note that the scalar field cannot be set to a constant without constraining the electromagnetic field. The earlier treatments by Kaluza and Klein did not have an adequate description of the scalar field and did not realize the implied constraint on the electromagnetic field by assuming

3240-445: Is a universality of free fall (also known as the weak equivalence principle , or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment , illustrated in the figure on the right: for an observer in an enclosed room, it

3360-402: Is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames . But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through

3480-499: Is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects —the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

3600-405: Is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames . Instead there are approximate inertial frames moving alongside freely falling particles. Translated into

3720-489: Is drawn, then it is that Yang–Mills theories occur on a flat spacetime, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional spacetime; it can be any ( pseudo- ) Riemannian manifold , or even a supersymmetric manifold or orbifold or even a noncommutative space . Theodor Kaluza Theodor Franz Eduard Kaluza ( German: [kaˈluːt͡sa] ; 9 November 1885 – 19 January 1954)

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3840-445: Is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

3960-471: Is instructive to convert it back to the usual 4-dimensional length element c 2 d τ 2 ≡ g μ ν d x μ d x ν {\displaystyle c^{2}\,d\tau ^{2}\equiv g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }} , which is related to the 5-dimensional length element d s {\displaystyle ds} as given above: Then

4080-560: Is known as gravitational time dilation. Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks , while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by

4200-404: Is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor , which includes both energy and momentum densities as well as stress : pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that

4320-456: Is merely a limiting case of (special) relativistic mechanics. In the language of symmetry : where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group , which includes translations, rotations, boosts and reflections.) The differences between

4440-430: Is now associated with electrically charged black holes . In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. By 1929, however, the work of Hubble and others had shown that

4560-653: Is obtained from where F α β ≡ ∂ α A β − ∂ β A α , {\displaystyle F_{\alpha \beta }\equiv \partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha },} ◻ ≡ g μ ν ∇ μ ∇ ν , {\displaystyle \Box \equiv g^{\mu \nu }\nabla _{\mu }\nabla _{\nu },} and ∇ μ {\displaystyle \nabla _{\mu }}

4680-481: Is particle mass, and q {\displaystyle q} is particle electric charge. Thus electric charge is understood as motion along the fifth dimension. The fact that the Lorentz force law could be understood as a geodesic in five dimensions was to Kaluza a primary motivation for considering the five-dimensional hypothesis, even in the presence of the aesthetically unpleasing cylinder condition. Yet there

4800-504: Is the Planck constant . Klein found that λ 5 ∼ 10 − 30 {\displaystyle \lambda ^{5}\sim 10^{-30}}  cm, and thereby an explanation for the cylinder condition in this small value. Klein's Zeitschrift für Physik article of the same year, gave a more detailed treatment that explicitly invoked the techniques of Schrödinger and de Broglie. It recapitulated much of

4920-518: Is the permeability of free space . In the Kaluza theory, the gravitational constant can be understood as an electromagnetic coupling constant in the metric. There is also a stress–energy tensor for the scalar field. The scalar field behaves like a variable gravitational constant, in terms of modulating the coupling of electromagnetic stress–energy to spacetime curvature. The sign of ϕ 2 {\displaystyle \phi ^{2}} in

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5040-495: Is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on

5160-409: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Assuming that

5280-471: Is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity. At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free (or inertial ) motion, and deviations from this free motion. Such deviations are caused by external forces acting on

5400-401: Is the standard 4D Ricci scalar. This equation shows the remarkable result, called the "Kaluza miracle", that the precise form for the electromagnetic stress–energy tensor emerges from the 5D vacuum equations as a source in the 4D equations: field from the vacuum. This relation allows the definitive identification of A μ {\displaystyle A^{\mu }} with

5520-432: The Einstein notation , meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

5640-609: The Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous , but anisotropic ), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

5760-539: The Nazi ideology , and his appointment to the Göttingen professorship was possible only with difficulties and by assistance of his colleague Helmut Hasse . Strange stories were told of his private life, for example, that he taught himself to swim during his thirties by reading a book about it and succeeded at his first attempt in the water. Kaluza had a son (1910-1994), also named Theodor Kaluza  [ de ] , who

5880-597: The Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry , called Riemannian geometry , enabled Einstein to develop general relativity by providing

6000-682: The field equation for gravity relates this tensor and the Ricci tensor , which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy –momentum corresponds to the statement that the energy–momentum tensor is divergence -free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of

6120-561: The gravitational redshift of light, the Shapiro time delay and singularities / black holes . So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology , thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite

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6240-473: The post-Newtonian expansion , both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion

6360-454: The scalar gravitational potential of classical physics by a symmetric rank -two tensor , the latter reduces to the former in certain limiting cases . For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

6480-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs

6600-481: The " radion " or the "dilaton". Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations , the Maxwell equations for the electromagnetic field , and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis, that no component of the five-dimensional metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of

6720-424: The "Kaluza miracle". The same hypothesis for the 5D metric that provides electromagnetic stress–energy in the Einstein equations, also provides the Lorentz force law in the equation of motions along with the 4D geodesic equation. Yet correspondence with the Lorentz force law requires that we identify the component of 5-velocity along the fifth dimension with electric charge: where m {\displaystyle m}

6840-469: The 5D geodesic equation can be written for the spacetime components of the 4-velocity: The term quadratic in U ν {\displaystyle U^{\nu }} provides the 4D geodesic equation plus some electromagnetic terms: The term linear in U ν {\displaystyle U^{\nu }} provides the Lorentz force law : This is another expression of

6960-413: The actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction ), can be used to define the geometry of space, as well as a time coordinate . However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment ), there

7080-781: The classical theory by providing a properly normalized 5D metric. Work continued on the Kaluza field theory during the 1930s by Einstein and colleagues at Princeton University . In the 1940s, the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups: Yves Thiry, working in France on his dissertation under André Lichnerowicz ; Pascual Jordan , Günther Ludwig, and Claus Müller in Germany, with critical input from Wolfgang Pauli and Markus Fierz ; and Paul Scherrer working alone in Switzerland. Jordan's work led to

7200-399: The classical theory of Kaluza described above, and then departed into Klein's quantum interpretation. Klein solved a Schrödinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension. In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius , so that a particle moving

7320-406: The connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates , the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

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7440-407: The covariant 5-velocity: This means that under the cylinder condition, U ~ 5 {\displaystyle {\widetilde {U}}_{5}} is a constant of the five-dimensional motion: Kaluza proposed a five-dimensional matter stress tensor T ~ M a b {\displaystyle {\widetilde {T}}_{M}^{ab}} of

7560-408: The cylinder condition to gain extra terms that can be identified with the matter fields, for which Kaluza otherwise inserted a stress–energy tensor by hand. It has been an objection to the original Kaluza hypothesis to invoke the fifth dimension only to negate its dynamics. But Thiry argued that the interpretation of the Lorentz force law in terms of a five-dimensional geodesic militates strongly for

7680-560: The deflection of starlight by the Sun during the total solar eclipse of 29 May 1919 , instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity . Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed

7800-480: The electromagnetic stress–energy tensor above. The 5D curvature tensors are complex, and most English-language reviews contain errors in either G ~ a b {\displaystyle {\widetilde {G}}_{ab}} or R ~ a b {\displaystyle {\widetilde {R}}_{ab}} , as does the English translation of Thiry. See Williams for

7920-444: The electromagnetic vector potential, the five-dimensional stress–energy tensor comprises the four-dimensional stress–energy tensor framed by the vector 4-current. Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. By the time of Klein's contribution, the discoveries of Heisenberg, Schrödinger, and Louis de Broglie were receiving a lot of attention. Klein's Nature article suggested that

8040-496: The electromagnetic vector potential. Therefore, the field needs to be rescaled with a conversion constant k {\displaystyle k} such that A μ → k A μ {\displaystyle A^{\mu }\to kA^{\mu }} . The relation above shows that we must have where G {\displaystyle G} is the gravitational constant , and μ 0 {\displaystyle \mu _{0}}

8160-441: The electromagnetic vector potential; and a scalar field ϕ {\displaystyle \phi } . Then decompose the 5D metric so that the 4D metric is framed by the electromagnetic vector potential, with the scalar field at the fifth diagonal. This can be visualized as One can write more precisely where the index 5 {\displaystyle 5} indicates the fifth coordinate by convention, even though

8280-452: The emission of gravitational waves and effects related to the relativity of direction. In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass ) will precess ; the orbit is not an ellipse , but akin to an ellipse that rotates on its focus, resulting in a rose curve -like shape (see image). Einstein first derived this result by using an approximate metric representing

8400-458: The energy–momentum source terms is treated by L. L. Williams. In his 1921 article, Kaluza established all the elements of the classical five-dimensional theory: the metric, the field equations, the equations of motion, the stress–energy tensor, and the cylinder condition. With no free parameters , it merely extends general relativity to five dimensions. One starts by hypothesizing a form of the five-dimensional metric g ~

8520-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

8640-400: The equation, perhaps in contradiction to experience. This was the main shortfall of the five-dimensional theory as Kaluza saw it, and he gives it some discussion in his original article. The equation of motion for U 5 {\displaystyle U^{5}} is particularly simple under the cylinder condition. Start with the alternate form of the geodesic equation, written for

8760-411: The equations of general relativity and of electrodynamics; the equations of motion provide the four-dimensional geodesic equation and the Lorentz force law , and one finds that electric charge is identified with motion in the fifth dimension. The hypothesis for the metric implies an invariant five-dimensional length element d s {\displaystyle ds} : The field equations of

8880-446: The equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted , whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted ; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect

9000-456: The exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10 − 21 {\displaystyle 10^{-21}} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed . Some exact solutions describe gravitational waves without any approximation, e.g.,

9120-408: The exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity. Closely related to light deflection

9240-510: The field equations in the 1940s and earlier. Thiry is perhaps best known only because an English translation was provided by Applequist, Chodos, & Freund in their review book. Applequist et al. also provided an English translation of Kaluza's article. Translations of the three (1946, 1947, 1948) Jordan articles can be found on the ResearchGate and Academia.edu archives. The first correct English-language Kaluza field equations, including

9360-531: The fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics. In 1926, Oskar Klein gave Kaluza's classical five-dimensional theory a quantum interpretation, to accord with the then-recent discoveries of Werner Heisenberg and Erwin Schrödinger . Klein introduced

9480-518: The fifth dimension is closed and periodic, and that the identification of electric charge with motion in the fifth dimension can be interpreted as standing waves of wavelength λ 5 {\displaystyle \lambda ^{5}} , much like the electrons around a nucleus in the Bohr model of the atom. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum. Combining

9600-442: The first four coordinates are indexed with 0, 1, 2, and 3. The associated inverse metric is This decomposition is quite general, and all terms are dimensionless. Kaluza then applies the machinery of standard general relativity to this metric. The field equations are obtained from five-dimensional Einstein equations , and the equations of motion from the five-dimensional geodesic hypothesis. The resulting field equations provide both

9720-433: The first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric . This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution , which

9840-419: The five-dimensional theory were never adequately provided by Kaluza or Klein because they ignored the scalar field . The full Kaluza field equations are generally attributed to Thiry, who obtained vacuum field equations, although Kaluza originally provided a stress–energy tensor for his theory, and Thiry included a stress–energy tensor in his thesis. But as described by Gonner, several independent groups worked on

9960-480: The form where ρ {\displaystyle \rho } is a density, and the length element d s {\displaystyle ds} is as defined above. Then the spacetime component gives a typical "dust" stress–energy tensor: The mixed component provides a 4-current source for the Maxwell equations: Just as the five-dimensional metric comprises the four-dimensional metric framed by

10080-412: The general relativistic framework—take on the same form in all coordinate systems . Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent . It thus satisfies a more stringent general principle of relativity , namely that the laws of physics are the same for all observers. Locally , as expressed in the equivalence principle, spacetime

10200-484: The geometry of space. Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves . On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from

10320-401: The hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein suggested that the geometry of the extra fifth dimension could take the form of a circle, with the radius of 10 cm . More precisely, the radius of the circular dimension is 23 times the Planck length , which in turn is of the order of 10 cm . Klein also made a contribution to

10440-441: The image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer -independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry. Special relativity

10560-446: The influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential . Space, in this construction, still has

10680-512: The introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity . It is not yet known how gravity can be unified with the three non-gravitational forces: strong , weak and electromagnetic . Einstein's theory has astrophysical implications, including

10800-417: The key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913. The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found

10920-410: The language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry. A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory

11040-457: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

11160-477: The light of stars or distant quasars being deflected as it passes the Sun . This and related predictions follow from the fact that light follows what is called a light-like or null geodesic —a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either

11280-455: The matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

11400-403: The metric is fixed by correspondence with 4D theory so that electromagnetic energy densities are positive. It is often assumed that the fifth coordinate is spacelike in its signature in the metric. In the presence of matter, the 5D vacuum condition cannot be assumed. Indeed, Kaluza did not assume it. The full field equations require evaluation of the 5D Einstein tensor as seen in the recovery of

11520-442: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} is the energy–momentum tensor. All tensors are written in abstract index notation . Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that

11640-424: The most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at

11760-432: The observation of binary pulsars . All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing

11880-459: The ordinary Euclidean geometry . However, space time as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable . From this, one can deduce that spacetime

12000-421: The prediction of black holes —regions of space in which space and time are distorted in such a way that nothing, not even light , can escape from them. Black holes are the end-state for massive stars . Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes . It also predicts gravitational lensing , where the bending of light results in multiple images of

12120-511: The preface to Relativity: The Special and the General Theory , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of

12240-394: The previous Kaluza result for U 5 {\displaystyle U^{5}} in terms of electric charge, and a de Broglie relation for momentum p 5 = h / λ 5 {\displaystyle p^{5}=h/\lambda ^{5}} , Klein obtained an expression for the 0th mode of such waves: where h {\displaystyle h}

12360-430: The principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency. In

12480-419: The same distant astronomical phenomenon. Other predictions include the existence of gravitational waves , which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as

12600-446: The same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory , Brans–Dicke theory , teleparallelism , f ( R ) gravity and Einstein–Cartan theory . The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how

12720-430: The scalar field to be constant. The field equation for A ν {\displaystyle A^{\nu }} is obtained from It has the form of the vacuum Maxwell equations if the scalar field is constant. The field equation for the 4D Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} is obtained from where R {\displaystyle R}

12840-438: The scalar field, were provided by Williams. To obtain the 5D field equations, the 5D connections Γ ~ b c a {\displaystyle {\widetilde {\Gamma }}_{bc}^{a}} are calculated from the 5D metric g ~ a b {\displaystyle {\widetilde {g}}_{ab}} , and the 5D Ricci tensor R ~

12960-518: The scalar–tensor theory of Brans–Dicke ; Carl H. Brans and Robert H. Dicke were apparently unaware of Thiry or Scherrer. The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews, as well as the English translations of Thiry, contain some errors. The curvature tensors for the complete Kaluza equations were evaluated using tensor-algebra software in 2015, verifying results of J. A. Ferrari and R. Coquereaux & G. Esposito-Farese. The 5D covariant form of

13080-472: The speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The geodesic equation is: where s {\displaystyle s}

13200-399: The speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to

13320-518: The theory can be used for model-building. General relativity is a metric theory of gravitation. At its core are Einstein's equations , which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall , orbital motion, and spacecraft trajectories ), correspond to inertial motion within

13440-644: The theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ( i.e . elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were

13560-487: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

13680-489: The universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of

13800-494: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

13920-619: Was a German mathematician and physicist known for the Kaluza–Klein theory , involving field equations in five-dimensional space-time. His idea that fundamental forces can be unified by introducing additional dimensions was reused much later for string theory . Kaluza was born to a Roman Catholic family from the town of Ratibor (present-day Racibórz in Poland) in the German Empire 's Prussian Province of Silesia . Kaluza himself

14040-442: Was a notable mathematician . General relativity General relativity , also known as the general theory of relativity , and as Einstein's theory of gravity , is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics . General relativity generalizes special relativity and refines Newton's law of universal gravitation , providing

14160-833: Was born in Wilhelmsthal (a village that was incorporated into Oppeln (presently Opole ) in 1899). He spent his youth in Königsberg , where his father, Maximilian "Max" Kaluza , was a professor of the English language. He entered the University of Königsberg to study mathematics and gained his doctorate with a thesis on Tschirnhaus transformations . Kaluza was primarily a mathematician but began studying relativity . In April 1919 Kaluza noticed that when he solved Albert Einstein 's equations for general relativity using five dimensions, then Maxwellian equations for electromagnetism resulted spontaneously. Kaluza wrote to Einstein who, in turn, encouraged him to publish. Kaluza's theory

14280-441: Was not reused until string theory was developed. It is, however, also notable that many of the aspects of this body of work were already published in 1914 by Gunnar Nordström , but his work also went unnoticed and was not recognized when the ideas were reused. For the rest of his career Kaluza continued to produce ideas about relativity and about models of the atomic nucleus . Despite Einstein's encouragement, Kaluza remained only

14400-542: Was published in 1921 in a paper "Zum Unitätsproblem der Physik" with Einstein's support in Sitzungsberichte Preußische Akademie der Wissenschaften 966–972 (1921). Kaluza's insight is remembered as the Kaluza–Klein theory (named also after physicist Oskar Klein ). However, the work was neglected for many years, as attention was directed towards quantum mechanics . His idea that fundamental forces can be explained by additional dimensions

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