Newton–Cartan theory
Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of
Classical spacetimes
In Newton–Cartan theory, one starts with a smooth four-dimensional manifold and defines two (degenerate) metrics. A temporal metric with signature , used to assign temporal lengths to vectors on and a spatial metric with signature . One also requires that these two metrics satisfy a transversality (or "orthogonality") condition, . Thus, one defines a classical spacetime as an ordered quadruple , where and are as described, is a metrics-compatible covariant derivative operator; and the metrics satisfy the orthogonality condition. One might say that a classical spacetime is the analog of a relativistic spacetime , where is a smooth
Geometric formulation of Poisson's equation
In Newton's theory of gravitation, Poisson's equation reads
where is the gravitational potential, is the gravitational constant and is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential
where is the inertial mass and the gravitational mass. Since, according to the weak equivalence principle , the corresponding equation of motion
no longer contains 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
represents the equation of motion of a point particle in the potential . The resulting connection is
with and (). 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 and under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by
where the brackets mean the antisymmetric combination of the tensor . The
which leads to following geometric formulation of Poisson's equation
More explicitly, if the roman indices i and j range over the spatial coordinates 1, 2, 3, then the connection is given by
the Riemann curvature tensor by
and the Ricci tensor and Ricci scalar by
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
References
Bibliography
- Cartan, Élie (1923), "Sur les variétés à connexion affine et la théorie de la relativité généralisée (Première partie)" (PDF), Annales Scientifiques de l'École Normale Supérieure, 40: 325,
- Cartan, Élie (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (Première partie) (Suite)" (PDF), Annales Scientifiques de l'École Normale Supérieure, 41: 1,
- Cartan, Élie (1955), Œuvres complètes, vol. III/1, Gauthier-Villars, pp. 659, 799
- Renn, Jürgen; Schemmel, Matthias, eds. (2007), The Genesis of General Relativity, vol. 4, Springer, pp. 1107–1129 (English translation of Ann. Sci. Éc. Norm. Supér. #40 paper)
- Chapter 1 of ISBN 90-277-0369-8