What is covariant metric tensor?

The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor.

What are covariant and contravariant tensors?

In differential geometry, the components of a vector relative to a basis of the tangent bundle are covariant if they change with the same linear transformation as a change of basis. They are contravariant if they change by the inverse transformation.

What is meant by metric tensor?

Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean theorem: (1)

What is metric tensor in special relativity?

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation.

What is the difference between covariance and Contravariance?

Covariance and Contravariance in Delegates. Covariance permits a method to have a return type that is a subtype of the one defined in the delegate. Contravariance permits a method to have a parameter type that is a base type of the one defined in the delegate type.

What is transformation of tensor?

Tensors are defined by their transformation properties under coordinate change. One distinguishes covariant and contravariant indexes. Number of indexes is tensor’s rank, scalar and vector quantities are particular case of tensors of rank zero and one. In general, the position of the indexes matters.

What is coordinate transformation in tensor?

The coordinate transform of a vector in matrix and tensor notation is. v′=Q⋅vandv′i=λijvj. The coordinate transform of a tensor in matrix and tensor notation is. σ′=Q⋅σ⋅QTandσ′mn=λmiλnjσij. Note that Q and λij λ i j are the same transformation matrix.