Spatial gradient examples

Difficulty:   ★★★☆☆   undergraduate

Last time we discussed the “spatial gradient” or “3-gradient”, and here we follow up with two examples. Recall from before that a scalar field \Phi has gradient d\Phi, and the part of this which is orthogonal to an observer 4-velocity \mathbf u is, as a vector:

    \[^{(3)}(d\Phi)^\sharp := (d\Phi)^\sharp + \langle d\Phi,\mathbf u\rangle \mathbf u.\]

This direction has the greatest increase of \Phi, for any vector in \mathbf u’s 3-space (that is, orthogonal to \mathbf u), per length of the vector.

As an example, suppose the 4-gradient vector (d\Phi)^\sharp is a null, future-pointing vector. It can be decomposed E(\mathbf u+\boldsymbol\xi), where E := -\langle d\Phi,\mathbf u\rangle, and \boldsymbol\xi is a unit spatial vector orthogonal to \mathbf u. Physically, this gradient may be interpreted as a null wave or photon, which the observer determines to have energy (or related quantity, such as frequency) E, and to move in the spatial direction \boldsymbol\xi. The 3-gradient vector is E\boldsymbol\xi, hence the direction of relative velocity also has the steepest increase of \Phi, within the observer’s 3-space.

Suppose now (d\Phi)^\sharp is a unit, timelike, future-pointing vector, so that we may interpret it as the 4-velocity \mathbf v of a second observer. Then ^{(3)}(d\Phi)^\sharp = \mathbf v - \gamma\mathbf u, where \gamma = - \langle\mathbf u,\mathbf v\rangle is the Lorentz factor between the pair. But we also have the “relative velocity” decomposition \mathbf v = \gamma(\mathbf u + \mathbf V), where \mathbf V is the relative velocity of \mathbf v as determined in \mathbf u’s frame, as I discussed previously. Combining these, ^{(3)}(d\Phi)^\sharp = \gamma\mathbf V. Hence within the observer’s 3-space, \Phi again increases most sharply in the direction of the relative velocity.

timelike 1-form
Spacetime diagram, from the observer \mathbf u‘s perspective. The timelike 1-form d\Phi \equiv \mathbf v^\flat is suggested by dotted blue lines, given at intervals of 1/4 for more resolution. These are orthogonal to the vector \mathbf v, in the Lorentzian sense.

The figure shows the single tangent space — think of this as the linearisation of what is happening locally over the manifold itself. The hyperplanes are numbered by \Phi, where only the differences between them are relevant, as an overall constant was not specified. Observe \mathbf v crosses four of them, spanning an interval \Delta\Phi = \langle d\Phi,\mathbf v\rangle = -1, so \Phi is the negative of \mathbf v’s proper time; see a previous post for more background. In both our examples, the scalar decreases towards the future (or can vanish in the null case), even though the gradient vectors are future-pointing. That is, the gradient vectors actually point “down” the slope! This quirk is due to our −+++ metric signature, and would apply to spacelike gradients if +−−− were used instead. This really hurt my brain, until I drew the diagram. 🙁

To construct it, consider the action of d\Phi on the axes. The horizontal axis is the relative velocity direction, with unit vector \hat{\mathbf V} := \mathbf V/\beta. One can show \langle d\Phi,\hat{\mathbf V}\rangle = \beta\gamma. Also \langle d\Phi,\mathbf u\rangle = -\gamma, but I find it easier to think of: \langle d\Phi,-\mathbf u\rangle = \gamma. These give the number of hyperplanes crossed by the unit axes vectors, then you can literally “connect the dots” since the 1-form is linear. In the figure \beta = 1/2, so \gamma \approx 1.15. (As for the 3-gradient, it vanishes in the \mathbf u direction, hence \mathbf u must cross no contours of ^{(3)}d\Phi. It would be drawn as vertical lines, with corresponding vector pointing to the right.)

Most of our discussion applies to arbitrary 1-forms, not just gradients which are termed exact 1-forms. I derived the work here independently, but the literature contains some similar material. It turns out Jantzen, Carini & Bini 1992  §2 explicitly define the “spatial gradient”, as they most appropriately call it. A few textbooks discuss scalar waves, for which the 3-gradient vector is the wave 3-vector, which is orthogonal to the wavefronts within a given frame, as discussed shortly.

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