# What is Minkowski metric？

1. Jun 22, 2010

### xlzhang0910

If a Minkowski metric is just a constant Riemann metric? I am confused with the concept in the Finsler geometry,there Minkowski metric is defined as g=g_{ij}(y)dx^idx^j.

2. Jun 22, 2010

### Fredrik

Staff Emeritus
Riemannian metric: An inner product on each tangent space

Lorentzian metric: A symmetric non-degenerate bilinear form on each tangent space. (Same thing as an inner product except that <x,x> can be negative).

Minkowski metric: A specific Lorentzian metric on $\mathbb R^4$ with the standard manifold structure. The metric can be defined using the identity map I as a coordinate system. Define $\langle x,y\rangle=x^T\eta y$, where $\eta$ is the diagonal 4×4 matrix with -1,1,1,1 (or 1,-1,-1,-1) on the diagonal, the x and y on the left are tangent vectors, and the x and the y on the right are 4×1 matrices with components equal to the components of those tangent vectors in the "coordinate system" I.

Finsler metric: A norm on each tangent space.

Last edited: Jun 22, 2010
3. Jun 22, 2010

thanks