Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of
Newtonian gravity
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
first introduced 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 geometr ...
and
Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski and Horneffer, Ehlers, Havas, Künzle, Lottermoser,
Trautman, and others. In this re-formulation, the structural similarities between Newton's theory and
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's
general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
are readily seen, and it has been used by Cartan and 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
Jürgen Ehlers (; 29 December 1929 – 20 May 2008) was a German physicist who contributed to the understanding of Albert Einstein's theory of general relativity. From graduate and postgraduate work in Pascual Jordan's relativity research gro ...
to extend this correspondence to specific
solutions
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
of general relativity.
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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
, where
is a smooth
Lorentzian metric on the manifold
.
Geometric formulation of Poisson's equation
In Newton's theory of gravitation,
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
reads
:
where
is the gravitational potential,
is the gravitational constant and
is the mass density. The weak
equivalence principle
In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (su ...
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 according equation of motion
:
does not 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
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
:
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
Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
is given by
:
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
Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction. This lifting is considered to be useful for non-relativistic
holographic
Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, i ...
models.
References
Bibliography
*
*
*
* (English translation of Ann. Sci. Éc. Norm. Supér. #40 paper)
*Chapter 1 of
Theories of gravity
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