Derivative as contraction of a 1-form and vector

Difficulty level:   ★ ★ ★

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  §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 , 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.

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