Einstein & Relativity: Metric, Tensor Fields, Scalar Product

[kent davidge]

I was reading a document from the time of Einstein where the metric is referred to as "the fundamental tensor". That made me wonder if it's possible to derive all concepts of tensor field, scalar product, connectivity etc. starting from the requirement that the metric is invariant under change of coordinate patches?

Also, did the mathematics of the time of Einstein and other physicists/mathematicians differ from the current mathematics for Relativity that we encounter currently in Textbooks/etc?

 

[sweet springs]

For TOR Einstein applied Riemann geometry that had been established at that time.

 

[Daverz]

The metric is not more fundamental than any other tensor field, but it does allow us to define a scalar product and thus to do geometry (e.g. construct geodesics). Connections are a more general concept not dependent on defining a metric, but you can use the metric to construct a unique connection, the metric connection.

The math Einstein used was due mainly due to Italian geometers like Ricci, Levi-Cevita, and Bianchi. Einstein had some trouble finding his field equations because he did not know of the Bianchi identities, which were not widely known outside this Italian school of differential geometers.

https://en.wikipedia.org/wiki/Contracted_Bianchi_identities

Some discussion of the history here.

 

[Orodruin]

but you can use the metric to construct a unique connection, the metric connection.

Just to add that metric compatibility is not sufficient to uniquely define a connection. You also need to impose that the connection is torsion free.