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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation, one of several alternatives to general relativity. The theory was first proposed by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...
in 1922.


Overview

Einstein–Cartan theory differs from general relativity in two ways: : (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not; : (2) an additional set of equations are posed that relate torsion to spin. This difference can be factored into :: general relativity (Einstein–Hilbert) → general relativity (Palatini) → Einstein–Cartan by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, which results in the additional set of equations for spin and torsion, as well as the addition of extra spin-related terms in the Einstein field equations themselves. The theory of general relativity was originally formulated in the setting of
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
by the Einstein–Hilbert action, out of which arise the
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
. At the time of its original formulation, there was no concept of Riemann–Cartan geometry. Nor was there a sufficient awareness of the concept of gauge symmetry to understand that Riemannian geometries do not possess the requisite structure to embody a locally gauged Lorentz symmetry, such as would be required to be able to express continuity equations and conservation laws for rotational and boost symmetries, or to describe
spinors In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
in curved spacetime geometries. The result of adding this infrastructure is a Riemann–Cartan geometry. In particular, to be able to describe spinors requires the inclusion of a spin structure, which suffices to produce such a geometry. The chief difference between a Riemann–Cartan geometry and Riemannian geometry is that in the former, the affine connection is independent of the metric, while in the latter it is derived from the metric as the
Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
, the difference between the two being referred to as the contorsion. In particular, the antisymmetric part of the connection (referred to as the torsion) is zero for Levi-Civita connections, as one of the defining conditions for such connections. Because the contorsion can be expressed linearly in terms of the torsion, it is also possible to directly translate the Einstein–Hilbert action into a Riemann–Cartan geometry, the result being the Palatini action (see also Palatini variation). It is derived by rewriting the Einstein–Hilbert action in terms of the affine connection and then separately posing a constraint that forces both the torsion and contorsion to be zero, which thus forces the affine connection to be equal to the Levi-Civita connection. Because it is a direct translation of the action and field equations of general relativity, expressed in terms of the Levi-Civita connection, this may be regarded as the theory of general relativity, itself, transposed into the framework of Riemann–Cartan geometry. Einstein–Cartan theory relaxes this condition and, correspondingly, relaxes general relativity's assumption that the affine connection have a vanishing antisymmetric part (
torsion tensor In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X,Y, that produces an output vector T(X,Y) representing the displacement within a t ...
). The action used is the same as the Palatini action, except that the constraint on the torsion is removed. This results in two differences from general relativity: : (1) the field equations are now expressed in terms of affine connection, rather than the Levi-Civita connection, and so have additional terms in Einstein's field equations involving the contorsion that are not present in the field equations derived from the Palatini formulation; : (2) an additional set of equations are now present which couple the torsion to the intrinsic angular momentum ( spin) of matter, much in the same way in which the affine connection is coupled to the energy and momentum of matter. In Einstein–Cartan theory, the torsion is now a variable in the principle of stationary action that is coupled to a curved spacetime formulation of spin (the spin tensor). These extra equations express the torsion linearly in terms of the spin tensor associated with the matter source, which entails that the torsion generally be non-zero inside matter. A consequence of the linearity is that outside of matter there is zero torsion, so that the exterior geometry remains the same as what would be described in general relativity. The differences between Einstein–Cartan theory and general relativity (formulated either in terms of the Einstein–Hilbert action on Riemannian geometry or the Palatini action on Riemann–Cartan geometry) rest solely on what happens to the geometry inside matter sources. That is: "torsion does not propagate". Generalizations of the Einstein–Cartan action have been considered which allow for propagating torsion. Because Riemann–Cartan geometries have Lorentz symmetry as a local gauge symmetry, it is possible to formulate the associated conservation laws. In particular, regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field.


History

The theory was first proposed by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...
, who was inspired by Cosserat elasticity theory, in 1922 and expounded in the following few years.
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the
electromagnetic field tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
as part of a unified field theory. This line of thought led him to the related but different theory of teleparallelism. Dennis Sciama and Tom Kibble independently revisited the theory in the 1960s. Einstein–Cartan theory has been historically overshadowed by its torsion-free counterpart and other alternatives like Brans–Dicke theory because torsion seemed to add little predictive benefit at the expense of the tractability of its equations. Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
. In the Einstein–Cartan theory, the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
becomes nonlinear when it is expressed in terms of the Levi-Civita connection, though it remains linear when expressed in terms of the connection native to the geometry. Because the torsion does not 'propagate', its relation to the spin tensor of the matter source is algebraic and it is possible to solve in terms of the spin tensor. In turn, the difference between the connection and Levi-Civita connection (the contorsion) can be solved in terms of the torsion. When the contorsion is back-substituted for in the Dirac equation, to reduce the connection to the Levi-Civita connection (e.g. in passing from equation (4.1) to equation (4.2) in ), this results in non-linear contributions arising, ultimately, from the Dirac field itself. If two or more Dirac fields are present, or other fields that carry spin, the non-linear additions to the Dirac equation of each field would include contributions from all of the other fields, as well. Even though renowned physicists such as Steven Weinberg "never understood what is so important physically about the possibility of torsion in differential geometry", other physicists claim that theories with torsion are valuable.


Field equations

The Einstein field equations of general relativity can be derived by postulating the Einstein–Hilbert action to be the true action of spacetime and then varying that action with respect to the metric tensor. The field equations of Einstein–Cartan theory come from exactly the same approach, except that a general asymmetric affine connection is assumed rather than the symmetric
Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
(i.e., spacetime is assumed to have torsion in addition to
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
), and then the metric and torsion are varied independently. Let \mathcal_\mathrm represent the Lagrangian density of matter and \mathcal_\mathrm represent the Lagrangian density of the gravitational field. The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the Ricci scalar: :\mathcal_\mathrm=\fracR \sqrt :S=\int \left( \mathcal_\mathrm + \mathcal_\mathrm \right) \, d^4x , where g is the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the metric tensor, and \kappa is a physical constant 8\pi G/c^4 involving the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
and the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. By Hamilton's principle, the variation of the total action S for the gravitational field and matter vanishes: :\delta S = 0. The variation with respect to the metric tensor g^ yields the Einstein equations: : \frac -\fracP_=0 : where R_ is the Ricci tensor and P_ is the ''canonical'' stress–energy–momentum tensor. The Ricci tensor is no longer symmetric because the connection contains a nonzero torsion tensor; therefore, the right-hand side of the equation cannot be symmetric either, implying that P_ must include an asymmetric contribution that can be shown to be related to the spin tensor. This canonical energy–momentum tensor is related to the more familiar ''symmetric'' energy–momentum tensor by the Belinfante–Rosenfeld procedure. The variation with respect to the torsion tensor _c yields the Cartan
spin connection In differential geometry and mathematical physics, a spin connection is a connection (vector bundle), connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field gene ...
equations :\frac -\frac^c =0 : where ^c is the spin tensor. Because the torsion equation is an algebraic constraint rather than a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
, the torsion field does not propagate as a
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
, and vanishes outside of matter. Therefore, in principle the torsion can be algebraically eliminated from the theory in favor of the spin tensor, which generates an effective "spin–spin" nonlinear self-interaction inside matter. Torsion is equal to its source term and can be replaced by a boundary or a topological structure with a throat such as a "wormhole".


Avoidance of singularities

Recently, interest in Einstein–Cartan theory has been driven toward nonsingular black hole models and
cosmological Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
implications, most importantly, the avoidance of a gravitational singularity at the beginning of the universe, such as in the black hole cosmology, quantum cosmology, static universe, and
cyclic model A cyclic model (or oscillating model) is any of several cosmological models in which the universe follows infinite, or indefinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 ...
. Singularity theorems which are premised on and formulated within the setting of Riemannian geometry (e.g. Penrose–Hawking singularity theorems) need not hold in Riemann–Cartan geometry. Consequently, Einstein–Cartan theory is able to avoid the general-relativistic problem of the singularity at the
Big Bang The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
. The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin–spin self-interaction, which becomes significant inside
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
ic matter at extremely high densities. Such an interaction is conjectured to replace the singular Big Bang with a cusp-like Big Bounce at a minimum but finite scale factor, before which the
observable universe The observable universe is a Ball (mathematics), spherical region of the universe consisting of all matter that can be observation, observed from Earth; the electromagnetic radiation from these astronomical object, objects has had time to reach t ...
was contracting. This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic, providing a physical alternative to cosmic
inflation In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...
. Torsion allows fermions to be spatially extended instead of "pointlike", which helps to avoid the formation of singularities such as
black holes A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
, removes the ultraviolet divergence in quantum field theory, and leads to the toroidal ring model of electrons. According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole. In the Einstein–Cartan theory, instead, the collapse reaches a bounce and forms a regular Einstein–Rosen bridge (
wormhole A wormhole is a hypothetical structure that connects disparate points in spacetime. It can be visualized as a tunnel with two ends at separate points in spacetime (i.e., different locations, different points in time, or both). Wormholes are base ...
) to a new, growing universe on the other side of the
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive c ...
; pair production by the gravitational field after the bounce, when torsion is still strong, generates a finite period of inflation.


Other

Einstein–Cartan theory seems to allow gravitational shielding and the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of massless neutrinos without violating the
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
. In addition, the Einstein–Cartan theory is also related to geometrodynamics and the vortex theory of the atom.


See also

* Alternatives to general relativity * Metric-affine gravitation theory * Gauge theory gravity * Loop quantum gravity


References


Further reading

* * * * * * * * Lord, E. A. (1976). "Tensors, Relativity and Cosmology" (McGraw-Hill). * * * * * de Sabbata, V. and Gasperini, M. (1985). "Introduction to Gravitation" (World Scientific). * de Sabbata, V. and Sivaram, C. (1994). "Spin and Torsion in Gravitation" (World Scientific). * * * {{DEFAULTSORT:Einstein-Cartan theory Theories of gravity Albert Einstein