Derivative as contraction of a 1-form and vector

Suppose you seek the derivative of a quantity along a curve, such as the rate of change of a scalar by proper time along a worldline: $d\Phi/d\tau,$ or perhaps the rate of change of pressure by proper distance along a given spatial direction: $dp/ds$ . These derivatives are conveniently expressed as a contraction between a 1-form (the gradient of the scalar) and a tangent vector to the curve. For the first example,

$\frac{d\Phi}{d\tau} = \frac{\partial\Phi}{\partial x^\mu} \frac{dx^\mu}{d\tau} = (d\Phi)_\mu u^\mu = d\Phi(\mathbf u),$

where $(x^\mu)$ is some coordinate system, $d\Phi$ is the 1-form with components $(d\Phi)_\mu = \Phi_{,\mu} = \partial\Phi/\partial x^\mu$ , and $u^\mu = dx^\mu/d\tau$ is the 4-velocity. $d\Phi(\mathbf u)$ is the contraction of the vector and 1-form, yielding a scalar. Schutz 2009 book §3.3 gives this derivation.

A spacelike path can be parametrised by proper distance. Then $d\Phi/ds = d\Phi(\boldsymbol\xi)$ , where $\xi^\mu := dx^\mu/ds$ is the unit tangent vector. An example of a paper which uses this is Gibbons 1972 research paper (astronomy) , for the change in stress along a rigid cable, see the line after his Equation 4.

For a null path there is no natural parameter, at least not without additional context. But for any chosen parameter $\lambda$ , we have $d\Phi/d\lambda = d\Phi(\boldsymbol\xi)$ as before, where $\xi^\mu := dx^\mu/d\lambda$ is the tangent vector. Of course this applies to the other cases as well. Note all these calculations occur within a single tangent space.

2 thoughts on “Derivative as contraction of a 1-form and vector”

2 thoughts on “Derivative as contraction of a 1-form and vector”

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