Geodesic deviation equation
Encyclopedia
In general relativity
, the geodesic deviation equation is an equation involving the Riemann curvature tensor
, which measures the change in separation of neighbouring geodesic
s or, equivalently, the tidal force
experienced by a rigid body moving along a geodesic. In the language of mechanics it measures the rate of relative acceleration
of two particles moving forward on neighbouring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation
.
Let T a be the tangent vector to a given geodesic γ, and X a a vector field along γ connecting it to an infinitesimally near geodesic (the deviation vector). The relative acceleration of the infinitesimally near geodesic is defined by
The geodesic deviation equation asserts that
To more rigorously formulate the equation, let γs(t) be a 1-parameter variation through geodesics: i.e., for each fixed s, the curve swept out by γs(t) as t varies is a geodesic with affine parameter. The tangent vector and deviation vector are respectively defined by
In order that γs be a variation through geodesics, a necessary condition is that the geodesic equation holds:
The geodesic deviation equation can be derived from the second variation of the point particle Lagrangian
along geodesics, or from the first variation of a combined Lagrangian. The Lagrangian approach has two advantages. First it allows various formal approaches of quantization
to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any dynamical system
which has a one spacetime
indexed momentum appears to have a corresponding generalization of geodesic deviation).
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the geodesic deviation equation is an equation involving the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
, which measures the change in separation of neighbouring geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s or, equivalently, the tidal force
Tidal force
The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...
experienced by a rigid body moving along a geodesic. In the language of mechanics it measures the rate of relative acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
of two particles moving forward on neighbouring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation
Jacobi field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space...
.
Let T a be the tangent vector to a given geodesic γ, and X a a vector field along γ connecting it to an infinitesimally near geodesic (the deviation vector). The relative acceleration of the infinitesimally near geodesic is defined by
The geodesic deviation equation asserts that
To more rigorously formulate the equation, let γs(t) be a 1-parameter variation through geodesics: i.e., for each fixed s, the curve swept out by γs(t) as t varies is a geodesic with affine parameter. The tangent vector and deviation vector are respectively defined by
In order that γs be a variation through geodesics, a necessary condition is that the geodesic equation holds:
The geodesic deviation equation can be derived from the second variation of the point particle Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
along geodesics, or from the first variation of a combined Lagrangian. The Lagrangian approach has two advantages. First it allows various formal approaches of quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
which has a one spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
indexed momentum appears to have a corresponding generalization of geodesic deviation).