Normal coordinates

Riemann normal coordinates ( after Bernhard Riemann, and normal coordinates or Exponentialkoordinaten ) forming a special coordinate system, which is considered in the differential geometry. Here the tangent space to be used as a local map of the manifold in a neighborhood of. Such coordinates are easy to handle and therefore are also used in the general theory of relativity.

Definition

Let be a differentiable manifold with a connection and be an arbitrary curve that satisfies the geodesic equation. The tangent space is denoted at the point and going for with

The exponential referred. By choosing an orthonormal basis of obtained an isomorphism

Which is given by. Let further an open neighborhood of on which the exponential map is an isomorphism and, applies to some. Then we obtain a mapping

Here and on the corresponding domains of definition isomorphisms ( diffeomorphisms ) are also also diffeomorphic and can thus be regarded as a map image. The local coordinates, which are obtained by these cards are called Riemann normal coordinates.

Properties

Let be a Riemannian manifold and be centered around Riemannian normal coordinates, then:

• For all, the geodesic which starts with the velocity vector in Riemannian normal coordinates the representation

• The coordinates of are.
• The components of the Riemannian metric on are.
• The Christoffel symbols are zero.
• If the Levi- Civita connection (or other metric connection ), then

Physical view

Physically describe normal coordinates in the space-time point, the rest frame of a freely falling observer in point. This point is defined as the origin of the coordinate system. Normal coordinates are suitable for the description of the equivalence principle of general relativity. In normal coordinates all geodesics are straight lines through the origin in the four-dimensional space-time. This is understandable, which means the equivalence of freely falling observers, including observers in inertial frames. Since only the geodesics by a single space-time point line, the equivalence principle is exactly valid only in a single space-time point. The curved geodesics that do not run through the origin are explained by the observer by tidal forces.

In normal coordinates can be the metric tensor at a point as a series expansion specify the coordinates of this point:

Here are the components in normal coordinates, the components of the Minkowski metric and the components of the Riemannian curvature tensor. It is used the Einstein summation convention. With increasing distance of the point from the origin of the metric tensor deviates more and more from the flat Minkowski metric. This leads to tidal forces that directly depend on the curvature tensor.