General covariance
In theoretical physics, general covariance (also known as diffeomorphism covariance or general invariance) is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.
A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,[1] and is usually expressed in terms of tensor fields. The classical (non-quantum) theory of electrodynamics is one theory that has such a formulation.
Albert Einstein proposed this principle for his special theory of relativity; however, that theory was limited to space-time coordinate systems related to each other by uniform relative motions only, the so-called "inertial frames." Einstein recognized that the general principle of relativity should also apply to accelerated relative motions, and he used the newly developed tool of tensor calculus to extend the special theory's global Lorentz covariance (applying only to inertial frames) to the more general local Lorentz covariance (which applies to all frames), eventually producing his general theory of relativity. The local reduction of the general metric tensor to the Minkowski metriccorresponds to free-falling (geodesic) motion, in this theory, thus encompassing the phenomenon ofgravitation.
Much of the work on classical unified field theories consisted of attempts to further extend the general theory of relativity to interpret additional physical phenomena, particularly electromagnetism, within the framework of general covariance, and more specifically as purely geometric objects in the space-time continuum.
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[edit]Remarks
The relationship between general covariance and general relativity may be summarized by quoting a standard textbook:[2]
Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics.Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.
A more modern interpretation of the physical content of the original principle of general covariance is that the Lie group GL4(R) is a fundamental "external" symmetry of the world. Other symmetries, including "internal" symmetries based on compact groups, now play a major role in fundamental physical theories.
[edit]See also
- Coordinate conditions
- Coordinate-free
- Covariance and contravariance
- Diffeomorphism
- Fictitious force
- Galilean invariance
- Harmonic coordinate condition
- Inertial frame of reference
- Lorentz covariance
- Special relativity
- Symmetry in physics
[edit]Notes
- ^ More precisely, only coordinate systems related through sufficiently differentiable transformations are considered.
- ^ Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler (1973). Gravitation. Freeman. p. 431. ISBN 0-7167-0344-0.
[edit]References
- O'Hanian, Hans C.; & Ruffini, Remo (1994). Gravitation and Spacetime (2nd edition ed.). New York: W. W. Norton. ISBN 0-393-96501-5. See section 7.1.
[edit]External links
- General covariance and the foundations of general relativity: eight decades of dispute, by J. D. Norton (file size: 4 MB)
re-typeset version (file size: 460 KB)
Classical unified field theories
Since the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics. In particular, unification of gravitation and electromagnetism was actively pursued by several physicists and mathematicians in the years between World War I and World War II. This work spurred the purely mathematical development of differential geometry. Albert Einstein is the best known of the many physicists who attempted to develop a classical unified field theory.
This article describes various attempts at a classical (non-quantum), relativistic unified field theory. For a survey of classical relativistic field theories of gravitation that have been motivated by theoretical concerns other than unification, see Classical theories of gravitation. For a survey of current work toward creating a quantum theory of gravitation, see quantum gravity.
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[edit]Overview
The early attempts at creating a unified field theory began with the Riemannian geometry of general relativity, and attempted to incorporate electromagnetic fields into a more general geometry, since ordinary Riemannian geometry seemed incapable of expressing the properties of the electromagnetic field. Einstein was not alone in his attempts to unify electromagnetism and gravity; a large number of mathematicians and physicists, including Hermann Weyl, Arthur Eddington, Theodor Kaluza, Lancelot Law Whyte, and R. Bach also attempted to develop approaches that could unify these interactions.[1][2] These scientists pursued several avenues of generalization, including extending the foundations of geometry and adding an extra spatial dimension.
[edit]Early work
The first attempts to provide a unified theory were by G. Mie in 1912 and Ernst Reichenbacher in 1916.[3][4] However, these theories were unsatisfactory, as they did not incorporate general relativity – in the former case, because general relativity had yet to be formulated. These efforts, along with those of Forster, involved making the metric tensor (which had previously been assumed to be symmetric and real-valued) into an asymmetric and/or complex-valued tensor, and they also attempted to create a field theory for matter as well.
[edit]Differential geometry and field theory
From 1918 until 1923, there were four distinct approaches to field theory: the gauge theory of Weyl, Kaluza's five-dimensional theory, Lancelot Law Whyte's theory based on the Unitary Principle, and Eddington's development of affine geometry. Einstein corresponded with these researchers, and collaborated with Kaluza, but was not yet fully involved in the unification effort.
[edit]Weyl's infinitesimal geometry
In order to include electromagnetism into the geometry of general relativity, Hermann Weyl worked to generalize the Riemannian geometry upon which general relativity is based. His idea was to create a more general infinitesimal geometry. He noted that in addition to a metric field there could be additional degrees of freedom along a path between two points in a manifold, and he tried to exploit this by introducing a basic method for comparison of local size measures along such a path, in terms of a gauge field. This geometry generalized Riemannian geometry in that there was a vector field Q, in addition to the metric g, which together gave rise to both the electromagnetic and gravitational fields. This theory was mathematically sound, albeit complicated, resulting in difficult and high-order field equations. The critical mathematical ingredients in this theory, the Lagrangiansand curvature tensor, were worked out by Weyl and colleagues. Then Weyl carried out an extensive correspondence with Einstein and others as to its physical validity, and the theory was ultimately found to be physically unreasonable. However, Weyl's principle of gauge invariance was later applied in a modified form to quantum field theory.
[edit]Kaluza's fifth dimension
Kaluza's approach to unification was to embed space-time into a five-dimensional cylindrical world; one of four space dimensions and one of time. Unlike Weyl's approach, Riemannian geometry was maintained, and the extra dimension allowed for the incorporation of the electromagnetic field vector into the geometry. Despite the relative mathematical elegance of this approach, in collaboration with Einstein and Einstein's aide Grommer it was determined that this theory did not admit a non-singular, static, spherically symmetric solution. This theory did have some influence on Einstein's later work and was further developed later by Klein in an attempt to incorporate relativity into quantum theory, in what is now known as Kaluza-Klein theory.
[edit]Lancelot Law Whyte's unitary field theory
This theory was based on an organizing process called by Lancelot Law Whyte the "Unitary Principle". The history of this theoretical approach is: Michael Faraday and James Clerk Maxwellworked from Rudjer Boscovich's theory, which dealt with non-Euclidean and higher-dimensional geometry. This prompted mathematicians such as Gauss and Riemann to investigate that area of mathematics. The mathematics that Riemann developed was used by Einstein in his theory ofgeneral relativity, but that was not as extensive a description as Boscovich's theory, for which the mathematics had been only incompletely developed. Lancelot Law Whyte's ideas were adopted for experimental work by Leo Baranski, who planned a series of books based upon this theory. Only Baranski's first book was published before his death, upon which this line of investigation based upon classical physics was abandoned by academia.
[edit]Eddington's affine geometry
Sir Arthur Stanley Eddington was a noted astronomer who became an enthusiastic and influential promoter of Einstein's general theory of relativity. He was among the first to propose an extension of the gravitational theory based on the affine connection as the fundamental structure field rather than the metric tensor which was the original focus of general relativity. Affine connection is the basis for parallel transport of vectors from one space-time point to another; Eddington assumed the affine connection to be symmetric in its covariant indices, because it seemed plausible that the result of parallel-transporting one infinitesimal vector along another should produce the same result as transporting the second along the first. (Later workers revisited this assumption.)
Eddington emphasized what he considered to be epistemological considerations; for example, he thought that the cosmological constant version of the general-relativistic field equation expressed the property that the universe was "self-gauging". Since the simplest cosmological model (the De Sitter universe) that solves that equation is a spherically symmetric, stationary, closed universe (exhibiting a cosmological red shift, which is more conventionally interpreted as due to expansion), it seemed to explain the overall form of the universe.
Like many other classical unified field theorists, Eddington considered that in the Einstein field equations for general relativity the stress-energy tensor 
The Dirac equation for the relativistic quantum electron caused Eddington to rethink his previous conviction that fundamental physical theory had to be based on tensors. He subsequently devoted his efforts into development of a "Fundamental Theory" based largely on algebraic notions (which he called "E-frames"). Unfortunately his descriptions of this theory were sketchy and difficult to understand, so very few physicists followed up on his work.[5]
[edit]Einstein's geometric approaches
When the equivalent of Maxwell's equations for electromagnetism is formulated within the framework of Einstein's theory of general relativity, the electromagnetic field energy (being equivalent to mass as one would expect from Einstein's famous equation E=mc2) contributes to the stress tensor and thus to the curvature of space-time, which is the general-relativistic representation of the gravitational field; or putting it another way, certain configurations of curved space-time incorporate effects of an electromagnetic field. This suggests that a purely geometric theory ought to treat these two fields as different aspects of the same basic phenomenon. However, ordinary Riemannian geometry is unable to describe the properties of the electromagnetic field as a purely geometric phenomenon.
Einstein tried to form a generalized theory of gravitation that would unify the gravitational and electromagnetic forces (and perhaps others), guided by a belief in a single origin for the entire set of physical laws. These attempts initially concentrated on additional geometric notions such asvierbeins and "distant parallelism", but eventually centered around treating both the metric tensor and the affine connection as fundamental fields. (Because they are not independent, the metric-affine theory was somewhat complicated.) In general relativity, these fields are symmetric (in the matrix sense), but since antisymmetry seemed essential for electromagnetism, the symmetry requirement was relaxed for one or both fields. Einstein's proposed unified-field equations (fundamental laws of physics) were generally derived from a variational principle expressed in terms of the Riemann curvature tensor for the presumed space-time manifold.[6]
In field theories of this kind, particles appear as limited regions in space-time in which the field strength or the energy density are particularly high. Einstein and coworker Leopold Infeld managed to demonstrate that, in Einstein's final theory of the unified field, true singularities of the field did have trajectories resembling point particles. However, singularities are places where the equations break down, and Einstein believed that in an ultimate theory the laws should apply everywhere, with particles being soliton-like solutions to the (highly nonlinear) field equations. Further, the large-scale topology of the universe should impose restrictions on the solutions, such as quantization or discrete symmetries.
The degree of abstraction, combined with a relative lack of good mathematical tools for analyzing nonlinear equation systems, make it hard to connect such theories with the physical phenomena that they might describe. For example, it has been suggested that the torsion (antisymmetric part of the affine connection) might be related to isospin rather than electromagnetism; this is related to a discrete (or "internal") symmetry known to Einstein as "displacement field duality".
Einstein became increasingly isolated in his research on a generalized theory of gravitation, and most physicists consider his attempts ultimately unsuccessful. In particular, his pursuit of a unification of the fundamental forces ignored developments in quantum physics (and vice versa), most notably the discovery of the strong nuclear force and weak nuclear force.[7] On the other hand, by 1930 Einstein had already considered the Einstein-Maxwell-Dirac System [Dongen]. This system is (heuristically) the super-classical [Varadarajan] limit of (the not mathematically well-defined) Quantum Electrodynamics. One can easily extend this system to include the weak and strong nuclear forces to get the Einstein-Yang-Mills-Dirac System. This system has complete solutions without singularities, solitons, and a Cyclic Universe solution. (The system has negative energy density; hence doesn't satisfy the positivity conditions in the Penrose-Hawking Singularity Theorems.) The E-Y-M-D equations provide an alternative approach to a Cyclic Universe which Penrose [Penrose] has recently been advocating. They also imply that the massive compact objects now classified as Black Holes are actually Quark Stars, possibly with event horizons, but without singularities.6 A Super version [Varadarajan] of the above-including super-neutrinos-might be needed to explain Dark Matter. The E-Y-M-D is also a totally geometricized theory as a non-commutative geometry [Connes] [Connes-Marcolli]; the charge e and the mass m of the electron are geometric invariants of the non-commutative geometry analogous to pi. Unfortunately, there are quantum phenomena, such as EPR, for which this beautiful theory doesn't make adequate predictions. (One can still have a Block Universe: see [Goldstein], [Nottale].)
[edit]Schrödinger's pure-affine theory
Inspired by Einstein's approach to a unified field theory and Eddington's idea of the affine connectionas the sole basis for differential geometric structure for space-time, Erwin Schrödinger from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational theory. Although he initially assumed a symmetric affine connection, like Einstein he later considered the nonsymmetric field.
Schrödinger's most striking discovery during this work was that the metric tensor was induced upon the manifold via a simple construction from the Riemann curvature tensor, which was in turn formed entirely from the affine connection. Further, taking this approach with the simplest feasible basis for the variational principle resulted in a field equation having the form of Einstein's general-relativistic field equation with a cosmological term arising automatically.[8]
Skepticism from Einstein and published criticisms from other physicists discouraged Schrödinger, and his work in this area has been largely ignored.
[edit]Later work
After the 1930s, progressively fewer scientists worked on classical unification, due to the continual development of quantum theory and the difficulties encountered in developing a quantum theory of gravity. Einstein continued to work on unified field theories of gravity and electromagnetism, but he became increasingly isolated in this research, which he pursued until his death. Despite the publicity of this work due to Einstein's celebrity status, it never resulted in a resounding success.
Most scientists, though not Einstein, eventually abandoned classical theories. Current mainstream research on unified field theories focuses on the problem of creating quantum gravity and unifying such a theory with the other fundamental theories in physics, which are quantum theories. (Some programs, most notably string theory, attempt to solve both of these problems at once.) With four fundamental forces now identified, gravity remains the one force whose unification proves problematic.
Although new "classical" unified field theories continue to be proposed from time to time, often involving non-traditional elements such as spinors, none has been generally accepted by physicists.
[edit]References
- ^ Weyl, H. (1918). "Gravitation und Elektrizität". Sitz. Preuss. Akad. Wiss.: 465.
- ^ Eddington, A. S. (1924). The Mathematical Theory of Relativity, 2nd ed.. Cambridge Univ. Press.
- ^ Mie, G. (1912). "Grundlagen einer Theorie der Materie". Ann. Phys. 37 (3): 511–534. Bibcode1912AnP...342..511M. doi:10.1002/andp.19123420306.
- ^ Reichenbächer, E. (1917). "Grundzüge zu einer Theorie der Elektrizität und der Gravitation". Ann. Phys. 52 (2): 134–173. Bibcode 1917AnP...357..134R. doi:10.1002/andp.19173570203.
- ^ Kilmister, C. W. (1994). Eddington's search for a fundamental theory. Cambridge Univ. Press.
- ^ Einstein, A. (1956). The Meaning of Relativity. 5th ed.. Princeton Univ. Press.
- ^ Gönner, Hubert F. M.. "On the History of Unified Field Theories". Living Reviews in Relativity. Retrieved August 10, 2005.
- ^ Schrödinger, E. (1950). Space-Time Structure. Cambridge Univ. Press.
[Connes] Noncommutative Geometry, http://www.amazon.com/Noncommutative-Geometry-Alain-Connes/dp/012185860X/ref=sr_1_1?s=books&ie=UTF8&qid=1321029105&sr=1-1
[Connes-Marcolli] Noncommutative Geometry, Quantum Fields and Motives (Colloquium Publications),http://www.amazon.com/Noncommutative-Geometry-Quantum-Colloquium-Publications/dp/0821842102/ref=sr_1_1?s=books&ie=UTF8&qid=1321029602&sr=1-1
[Dongen] Einstein's Unification,http://www.amazon.com/Einsteins-Unification-Jeroen-van-Dongen/dp/0521883466/ref=sr_1_1?ie=UTF8&qid=1321025679&sr=8-1
[Goldstein] Bohmian Mechanics, http://plato.stanford.edu/entries/qm-bohm/
[Nottale] Scale Relativity And Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics,http://www.amazon.com/Scale-Relativity-Fractal-Space-Time-Mechanics/dp/1848166508/ref=sr_1_1?ie=UTF8&qid=1323694470&sr=8-1
[Penrose] BEFORE THE BIG BANG: AN OUTRAGEOUS NEW PERSPECTIVE AND ITS IMPLICATIONS FOR PARTICLE PHYSICS,http://accelconf.web.cern.ch/AccelConf/e06/PAPERS/THESPA01.PDF
[Varadarajan] Supersymmetry for Mathematicians: An Introduction (Courant Lecture Notes),http://www.amazon.com/Supersymmetry-Mathematicians-Introduction-Courant-Lecture/dp/0821835742/ref=sr_1_1?ie=UTF8&qid=1321023252&sr=8-1
