Newton–Cartan theory

Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity developed by Élie Cartan. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Kurt Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity.

Geometric formulation of Poisson's equation

In Newton's theory of gravitation, Poisson's equation reads

\Delta U=4\pi G\rho \,

where $U$ is the gravitational potential, $G$ is the gravitational constant and $\rho$ is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential $U$

m_{t}{\ddot {\vec {x}}}=-m_{g}\nabla U

where $m_{t}$ is the inertial mass and $m_{g}$ the gravitational mass. Since, according to the weak equivalence principle $m_{t}=m_{g}$ , the according equation of motion

{\ddot {\vec {x}}}=-\nabla U

doesn't contain anymore a reference to the mass of the particle. Following the idea that the solution of the equation then is a property of the curvature of space, a connection is constructed so that the geodesic equation

{\frac {d^{2}x^{\lambda }}{ds^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{ds}}{\frac {dx^{\nu }}{ds}}=0

represents the equation of motion of a point particle in the potential $U$ . The resulting connection is

\Gamma _{\mu \nu }^{\lambda }=\gamma ^{\lambda \rho }U_{,\rho }\Psi _{\mu }\Psi _{\nu }

with $\Psi _{\mu }=\delta _{\mu }^{0}$ and $\gamma ^{\mu \nu }=\delta _{A}^{\mu }\delta _{B}^{\nu }\delta ^{AB}$ ( $A,B=1,2,3$ ). The connection has been constructed in one inertial system but can be shown to be valid in any inertial system by showing the invariance of $\Psi _{\mu }$ and $\gamma ^{\mu \nu }$ under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by

R_{\kappa \mu \nu }^{\lambda }=2\gamma ^{\lambda \sigma }U_{,\sigma [\mu }\Psi _{\nu ]}\Psi _{\kappa }

where the brackets $A_{[\mu \nu ]}={\frac {1}{2!}}[A_{\mu \nu }-A_{\nu \mu }]$ mean the antisymmetric combination of the tensor $A_{\mu \nu }$ . The Ricci tensor is given by

R_{\kappa \nu }=\Delta U\Psi _{\kappa }\Psi _{\nu }\,

which leads to following geometric formulation of Poisson's equation

R_{\mu \nu }=4\pi G\rho \Psi _{\mu }\Psi _{\nu }

More explicitly, if the roman indices i and j range over the spatial coordinates 1, 2, 3, then the connection is given by

\Gamma _{00}^{i}=U_{,i}

the Riemann curvature tensor by

R_{0j0}^{i}=-R_{00j}^{i}=U_{,ij}

and the Ricci tensor and Ricci scalar by

R=R_{00}=\Delta U

where all components not listed equal zero.

Note that this formulation does not require introducing the concept of a metric: the connection alone gives all the physical information.

Bargmann lift

It was shown that four-dimensional Newton–Cartan theory of gravitation can be reformulated as Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction.^[1] This lifting is considered to be useful for non-relativistic holographic models.^[2]

References

↑ C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)
↑ Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069

Bibliography

Cartan, Elie (1923), "Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)" (PDF), Ann. Ecole Norm., 40: 325
Cartan, Elie (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)" (PDF), Ann. Ecole Norm., 41: 1
Cartan, Elie (1955), OEuvres Complétes, III/1, Gauthier-Villars, pp. 659, 799
Renn, Jürgen; Schemmel, Matthias, eds. (2007), The Genesis of General Relativity (English translation of Ann. Ecole Norm. #40 paper), 4, Springer, pp. 1107–1129
Chapter 1 of Ehlers, Jürgen (1973), "Survey of general relativity theory", in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, ISBN 90-277-0369-8

Isaac Newton

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	"Hypotheses non fingo" "Standing on the shoulders of giants"

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Newton (unit) Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose

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