Metric Tensor

A metric tensor defines the way length is measured.

Metric tensors make it possible to measure angles.

Metric tensors generalize some things you learned when studying dot products of vectors in Euclidean spaces.

A metric tensor will take two tangent vectors of a point on a smooth manifold and output a scalar.

There may be an issue here that metric tensor is a definition and anything that matches the definition gets to call itself a metric tensor and if we show you something to say this is a metric tensor the problem is, we are only showing you one example.

In other work with “vector”, we started with a definition that said almost nothing and we were promised that later it would be given more structure to make it the vector that we want to use.

Telling you what a metric tensor does might be the simplest way to define it.

In tensor notation we will see \delta_{mn} if we are working in a flat space.

The metric tensor is \eta when we are doing spacetime stuff in a minkowski space.

If we want to talk about a metric and we haven’t chosen a space, then we use g.

We can show you some examples below:

Below is a template filled with placeholders.

\begin{bmatrix} g_{11} & g_{12} & g_{13} \\ g_{21} & g_{22} & g_{23} \\ g_{31} & g_{32} & g_{33} \end{bmatrix}

Below is an actual example:

\begin{bmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 sin (\theta) \end{bmatrix}

\begin{bmatrix} content \end{bmatrix}

Knowing how to build a metric tensor might be part of what you have to do before they give you a certificate.

Brandon Ross article about metric tensors to answer a Quora question.