Gradient one-form

Question

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's association with the "gradient vector" is a one-to-one map through the metric tensor (Lorentz metric in the book). I am having trouble understanding what the components of the gradient one-form $ \tilde d \phi$ (of some scalar field $\phi$) $\{\frac {\partial \phi}{\partial x^\alpha}\}$ stand for. What meaning do these component values just by themselves hold at a point {$x^\alpha$}, given that the rate of change of $\phi$ depends on the direction we take. Is it the case that, only when $ \tilde d \phi$ is supplied with a "unit" vector it gets a physical meaning, "the rate of change of $\phi$ in that direction"?

Answer 1

I think your last sentence shows you're on the right track. A one form is a linear functional that maps vectors to real numbers. You give it a vector as an input, and, as you say, it returns the rate of change in the implied direction.

Let's say we have a path whose tangent at a point is defined by the vector $v^j\,\partial_j$ - the differential operator that operates on a smooth scalar field $\psi$ and returns the total derivative $\mathrm{D}_v\psi = v^j\,\partial_j\,\psi$. A one form $\mathrm{d}u$, being a linear, homogeneous functional of vectors, is defined wholly by its values at the basis vectors - i.e. by its values $u_1,\,u_2,\,\cdots$ at the basis tangent vectors $\partial_1,\,\partial_2,\,\cdots$ (in components, the latter are $(1,0,0,\cdots),\, (0,1,0,\cdots),\,\cdots$)

Well the gradient $\nabla\phi$ is just such a beast. Its value when we input the tangent vector $\partial_j$ with components $(0,0,,\cdots,1,\,\cdots)$ (with one "1" in the $j^{th}$ position), the value of the functional is $\partial\phi/\partial x^j$.

We simply sum up these basis values by superposition: take the inner product with the vector $(v^1,\,v^2,\,\cdots)$ to find the functional's value and your value is $v^j\,\partial\phi/\partial x^j$ - the total rate of change of $\phi$ in the direction implied by $\vec{v}$.

I also like Schutz's visualization of a one-form as a system of hyperplanes: the value being the rate at which a vector pierces the hyperplanes. The classic, prototypical one form is the wavenumber , where you imagine phase fronts. The rate at which a vector $\vec{r}$ pierces the phasefronts is $k(\vec{r})$; in optics we often loose sight of this and treat $k(\cdot)$ as a vector. If we want to reconcile this with the above, we understand that the wavenumber has meaning as an inner product, so the wavevector in optics, if we were calculating in curved space, is the wavenumber one-form converted to a vector by raising its index with the metric: the $\vec{k}$ we ken and love in optics is in fact the "sharpened" $\vec{k} = k()^\sharp$ (a shorthand for raising an index) so that $\vec{k}^\mu = g^{\mu\,\nu}\,k(\cdot)_\nu$. I think it would be enlightening for you to read and think about the wavenumber in Minkowski spacetime.