In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics. In the language of mechanics it is measures the rate of relative acceleration of two particles moving toward one another on neighbouring paths. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ... A particle is Look up Particle in Wiktionary, the free dictionary In particle physics, a basic unit of matter or energy. ...

In textbooks it is usually derived in a handwaving manner. It can however be derived from the second covariant variation of the point particle Lagrangian, or from the first variation of a combined Lagrangian. The lagrangian approach has at other advantages: 1) it allows various formal approaches of quantization to be applied to the geodesic deviation system, 2) 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). The term handwaving is used in mathematics and physics to describe arguments that are not mathematically rigorous. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... World line of the orbit of the Earth depicted as a circle in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, Z, making the circle appear as a helix. ...

References

General relativity - an introduction to the theory of the gravitation field. Hans Stephani, Cambridge University Press 1982, 1990. ISBN 0-521-37066-3. - ISBN 0-521-37941-5 (pbk.)

External links

• gr-qc/0404094 arXiv.org