## Saturday, 18 July 2020

### Coordinates and basis vectors

I was still not really sure what is meant by the statement that partials form a coordinate basis. Then if they do, is their character (null, timelike or spacelike) related to the metric? At last I asked on Physics forums and PeterDonis had the final word. So now I do know what is meant by partials forming a coordinate basis and there is some relationship to the metric.

Carroll writes$$\frac{d}{d\lambda}=\frac{dx^\mu}{d\lambda}\partial_\mu$$"Thus the partials $\left\{\partial_\mu\right\}$ do indeed represent a good basis for the vector space of the directional derivatives, which we can therefore safely identify with the tangent space."

We have a parabola parameterized by $\lambda$ given by$$t=\frac{\lambda^2}{10}\ ,\ \ x=\lambda$$so$$\frac{dt}{d\lambda}=\frac{\lambda}{5}\ \ ,\ \ \ \frac{dx}{d\lambda}=1$$and the tangent vector $V=d/d\lambda$ has components $\left(dt / d\lambda , dx/ d \lambda\right)$ at $\left(t,x\right)$.

In plane polar coordinates $\left(r,\theta\right)$ the $\partial_\theta$ basis vector is along lines with parameters $\left(k_r,\lambda\right)$. So the $\theta$ basis vectors lie on concentric circles around the origin. Similarly $r$ basis vectors are radial.
All you need for plane polar coordinates is a line segment to measure $\theta$ from and one end to serve as the origin. The line is normally drawn horizontally. That is not essential.

See how it all hangs together: Commentary 2.3 Coordinates and basis vectors.pdf
And the thread on physics forums here.