Originally Posted by

**geordief**
Do you think that you could perhaps provide a short definition of the "**connection" **and the "**metric" **

I hope I am not being patronising if I say I suspect you want an intuitive definition, and would not welcome the full mathematics.

So. Given a space (or spacetime), and a single point therein, relativistic theories insist that one is free to use any coordinates one chooses

*provided only* that one can specify a coordinate transformation formula that leaves everything invariant.

Suppose that we can do this for any

*single* spacetime point. Problems arise, however, when you try to do a coordinate transformation from one spacetime point to another such point. In particular, for a non-scalar object (vector, tensor) defined at one point, there is no coordinate transformation that leaves it invariant at another.

The resolution of this dilemma is to introduce a gadget called a

**connection** - basically it allows you to do calculus by treating different points as the same.

The

**metric** is easier to describe - it is a means of measuring things. Basically the length of vectors and the angle between them.

Relativistic theories state that vector length (spacetime interval) depend upon the choice of coordinates, and since different coordinates very likely apply at different spacetime points, one talks (or should talk) about the metric

*field*.

This field is the principal object in the General Theory.