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 has gradient , and the part of this which is orthogonal to an observer 4-velocity is, as a vector:

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

As an example, suppose the 4-gradient vector is a null, future-pointing vector. It can be decomposed , where , and is a unit spatial vector orthogonal to . 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) , and to move in the spatial direction . The 3-gradient vector is , hence the direction of relative velocity also has the steepest increase of , within the observer’s 3-space.

Suppose now is a unit, timelike, future-pointing vector, so that we may interpret it as the 4-velocity of a second observer. Then , where is the Lorentz factor between the pair. But we also have the “relative velocity” decomposition , where is the relative velocity of as determined in ’s frame, as I discussed previously. Combining these, . Hence within the observer’s 3-space, again increases most sharply in the direction of the relative velocity.

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 , where only the differences between them are relevant, as an overall constant was not specified. Observe crosses four of them, spanning an interval , so is the negative of ’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 on the axes. The horizontal axis is the relative velocity direction, with unit vector . One can show . Also , but I find it easier to think of: . 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 , so . (As for the 3-gradient, it vanishes in the direction, hence must cross no contours of . 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.