Difference between vielbein and the Jacobian matrix

In a nutshell, vielbeins $e^a_{mu}$ work more generally for abstract manifolds (up to topological obstructions), and generalize the Jacobian $J^a_{mu}=partial y^a/partial x^{mu}$, which only works for affine spaces. Unlike the vielbeins, the Jacobian always satisfies an integrability condition $partial J^a_{mu}/partial x^{nu}=(muleftrightarrow nu)$.

A choice of coordinates $x^mu$ for some patch of spacetime automatically defines a corresponding basis for the tangent space at each point, with basis vectors $frac{partial}{partial x^mu}$. This is referred to as a coordinate basis, or sometimes as a holonomic basis.

Of course, a choice of basis is in principle independent of a choice of coordinates. The fact that there is a natural coordinate-induced basis available doesn’t mean we have to use it.

This might lead one to wonder if there are choices of basis which cannot be induced by a coordinate chart, and the answer is a resounding yes. As an example, one can show that the familiar orthonormal polar unit vectors $hat r$ and $hat theta $ are such a choice.

When we go from one coordinates chart to another, the Jacobian matrix provides the corresponding transformation between coordinate-induced bases. However, if a non-holonomic basis is involved then there’s obviously no corresponding Jacobian because the non-holonomic basis doesn’t correspond to a choice of coordinates. Therefore, the change of basis needs to be described by a more general object. This is the vielbein matrix $e_mu^{ nu}$.

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Consider the following example for the standard Euclidean plane with Cartesian coordinates $(x,y)$. This choice of coordinates corresponds to the (holonomic) basis $left{frac{partial}{partial x},frac{partial}{partial y}right}$.

If we shift to polar coordinates $(r,theta)$, we can find a corresponding polar basis $left{frac{partial}{partial r},frac{partial}{partial theta}right}$. Since we have

$$x = rcos(theta) qquad y = rsin(theta)$$ it follows that

$$frac{partial}{partial r} = frac{partial x}{partial r} frac{partial}{partial x} + frac{partial y}{partial r} frac{partial}{partial y} = cos(theta)frac{partial}{partial x}+sin(theta)frac{partial}{partial y}$$ and similarly for $frac{partial}{partial theta}$. Letting $yequiv (r,theta)$, this can be written compactly as

$$frac{partial}{partial y^mu} = frac{partial x^nu}{partial y^mu} frac{partial}{partial x^nu} equiv J^nu_{ mu} frac{partial}{partial x^nu}$$

with $J$ the Jacobian. In this basis, the metric takes the form

$$g = pmatrix{1& 0 \ 0 & r^2}$$

which means that this polar basis is orthogonal but not orthonormal. In contrast, consider the basis

$$hat r equiv cos(theta)frac{partial}{partial x} + sin(theta)frac{partial}{partial y}$$ $$hat theta equiv -sin(theta)frac{partial}{partial x} + cos(theta)frac{partial}{partial y}$$

One can show without much effort that these basis vectors are orthonormal. They are not holonomic, however; one can see this by noting that for a smooth function $f$, $hat r(hat theta f) neq hat theta(hat r f)$, which means that they cannot be expressed as

$$hat r = frac{partial}{partial u} qquad hat theta = frac{partial}{partial v}$$ for some coordinates $(u,v)$. Therefore, we cannot write a Jacobian for this coordinate transformation. Instead, writing $(hat r,hattheta) equiv (hat e_r, hat e_theta)$, the change of basis is provided by

$$e_mu^{ nu} = pmatrix{cos(theta) & sin(theta) \ -sin(theta) & cos(theta)}$$ $$hat e_mu = e_{mu}^{ nu} frac{partial}{partial x^nu}$$