The metric tensor (also metric tensor or Maßtensor ) serves mathematical spaces, in particular differentiable manifolds, equipped with a measure of distances and angles.
This measure does not necessarily meet all the conditions to be met by a metric in the definition of a metric space: in Minkowski space of special relativity theory, these conditions are only valid for distances that are either uniformly space-like or time-like uniform.
For the differential geometry and general relativity is significant that the metric tensor, unlike a metric defined on inner product and norm, may depend on the place.
- 5.1 Euclidean space
- 5.2 Minkowski space ( special relativity )
Definition and meaning
The metric tensor g on an affine point space A with a real shift vector space is a mapping of A into the space of scalar products on VDH is, for each point
A positive definite, symmetric bilinear form.
Based on the distinction between metric and pseudo- metric and the case is sometimes considered that g ( P) for some or all points P is positive semidefinite, ie the requirement of definiteness
Is attenuated to
One of such tensor is then called pseudometric tensor.
A metric tensor defines a ( dependent from the point P ) length on the vector space V:
Similar to the standard scalar product of the angle at point P is between two vectors is defined by:
When a local coordinate system is chosen on the basis of V with V to write the components of a. Using the Einstein summation convention is then for the vectors and
In terms of category theory the metric tensor is contravariant, since under linear ( affine ) injective mappings from a metric tensor on (B, W) be a metric tensor on (A, V) can be constructed naturally,
In physics, the metric tensor, or better called its coordinate representation covariant, since its components transform under a change of coordinates in each index as the base. Is a change of coordinates as
Given, then the basis vectors transform as
And it applies to the metric tensor
Length of curves
If a differentiable curve given in affine point space, so this has in any time t a tangent vector
The entire curve or a segment of it, you can now with the help of the metric tensor a length
Again using the summation convention, called line element. Is substituted according to the chain rule
Therefore, the integrand of the above integral for determining a curve length.
Induced metric tensor
Has a one -dimensional manifold of a Riemannian space with the metric which is given by the parameter representation () (, also called induced coordinates), and considered a curve
On this submanifold, we obtain for the arc length according to the chain rule
Is called the induced metric tensor. With this, the curve length ultimately results as
In a Euclidean space with Cartesian coordinates is the metric tensor by the unit matrix
Given. In Euclidean space, the scalar product is given and, by assumption, namely to the metric tensor corresponding to this scalar product. So apply for this in local coordinates where the vectors of the standard basis are. For arbitrary vectors and the Euclidean space
Here the Einstein summation convention.
For the curve length
And the angle
We obtain the usual formulas of vector analysis.
If a manifold into a Euclidean space with Cartesian coordinates is embedded, then her metric tensor of the Jacobian matrix gives the embedding as
In some other coordinate systems is the metric tensor of the Euclidean space as follows:
- In polar coordinates:
- In cylindrical coordinates:
- In spherical coordinates:
Minkowski space ( special relativity )
The flat Minkowski space of special relativity describes a four-dimensional space-time without gravity. Spatial distances and periods of time hang in this room by choosing from an inertial frame; when describing a physical process in two different uniformly moving against each other inertial frames, they can assume different values.
Invariant under Lorentz transformations, however, is the so-called four- distance spatial and temporal distances summarizing. Using the speed of light c, this four distance calculated from the distance and a time period as
In Minkowski space of contravariant local four-vector is defined by.
The metric (more precisely: pseudo- metric ) tensor is in a convention that is mainly used in quantum field theory ( signature -2, ie , -, -, -)
In a convention, which is mainly used in the general theory of relativity ( signature 2, ie, -, , , ), one writes
In general relativity, the metric tensor is location-dependent, and therefore, forms a tensor, since the curvature of space-time at various points is usually different.
- Riemannian geometry