# Interpretation of the Weyl tensor

My question arises from the interpretation of Ricci curvature. If I am not mistaken :

• The Ricci scalar measures the change of volume of small geodesic balls, compared to a euclidean space, it is a generalization of the notion of Gauss curvature.

• The Ricci tensor measures the change of volume of small spherical geodesic cones. So it accounts for a "directional" or "partial" change of volume of small geodesic balls in a way.

As a consequence, if the Ricci tensor is not zero, a null Ricci scalar means that geodesic balls do not change volume, but are still deformed. Like a ball becomes an ellipsoid of same volume, say.

Now if we take a Ricci-flat space with the Schwarzschild metric for example, the space outside the Schwarzschild radius is empty of matter-energy and Ricci-flat. So no change in volume nor form. But, there is still a curvature because the Riemann tensor is not zero, and the spherical mass has to curve space anyway.

Since the only non-vanishing part left of the Riemann tensor must be the Weyl tensor, most books and articles I've read attribute the space curvature to the tidal forces of gravitational waves. BUT, tidal forces would deform the geodesic balls, wouldn't they ? So there couldn't be any !

Evidently, I am missing something in the interpretation of the Weyl tensor. Is it geodesic deviation ? Can there be gravitational waves without tidal forces ?

[EDIT : My mistake was to think the deformation of geodesic balls was measured only by the Ricci tensor, when in fact both it and the Weyl tensor account for deformation, though for different reasons.]

• Weyl tensor biological analogy is astigmatism, I am sure there is a dupe of this here somewhere... – user139561 Dec 21 '16 at 16:23
• – Qmechanic Dec 21 '16 at 16:38
• Thanks Qmechanic, I saw these posts but I somehow missed a reference of the first : Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes. I'm reading it and it seems to answer part of my question : – Johann Dec 21 '16 at 19:52
• - Yes, what is left in Vacuum is the Weyl tensor - There seem to be tidal effects in vacuum, though it only mentions "tidal acceleration" and not "tidal deformation" so far. Until I read more, I'm assuming the latter needs a non vanishing Ricci tensor, so there wouldn't be any contradiction with my above statement - Another interpretation of the Weyl tensor : frame-dragging – Johann Dec 21 '16 at 19:57
• Aaand I'm back where I started : "stretching" and "squeezing" tidal effects are mentionned in the paper, it counts as "deforming". So again : how can we have "stretching" and "squeezing" with a null Ricci tensor ? I'm at a loss. – Johann Dec 21 '16 at 20:23

Actually there are some equations and results that make the Weyl tensor more intuitively clear, and that also makes the intuitive description that it denotes the deformation of a sphere, whereas the trace of the Riemann tensor is more closely related to its volume growth.

That deformation of the shapes, is due to the tidal forces, and more technically to the shear in the spacetime. The Weyl tensor describes it.

First is the fact that it is also called the conformal tensor, or the conformal Weyl tensor. That is, for two metrics that are conformally related, i.e., $g' = fg$, with $f$ a conformal transformation function of the spacetime coordinates, then, $C' = C$, with $g$ the metric and $C$ the Weyl tensor.

That means that a conformal (or simplistically) 'volume' change leaves the Weyl tensor invariant. The transformations or variations which affect the Weyl tensor are more the deformations than the 'volume' changes. This is explained a little better in Wikipedia at https://en.m.wikipedia.org/wiki/Weyl_tensor.

But one can do better, and understand what it denotes in spacetime.

In essence, the Weyl tensor describes the shear of null geodesics. Thus, how geodesic balls deform. From a nice description in https://www.physicsforums.com/threads/geometrical-meaning-of-weyl-tensor.708383/ and with mostly a copy/paste below while leaving out unnecessary parts,

'Let $k^a$ represent the tangent vector field to a congruence of null geodesics. We want to find a way to describe the behavior of a neighboring collection of null geodesics in the congruence relative to each other and for this we need a spatial deviation vector connecting a given null geodesic in the congruence to infinitesimally nearby ones in the congruence.' The equation turns out to be (also from the same reference)

$$k^{c}\nabla_{c}\sigma_{ab} = -\theta {\sigma}_{ab}+ > C_{cbad}k^{c}k^{d}$$

where $\sigma_{ab}$ is the shear, and describes the deformation of a ball into an ellipse for instance, and $\theta$ is the expansion factor of the volume. C is the Weyl tensor. Those are in the subspace orthogonal to $k_a$. If $\theta$ = 0, then the C term in the equation above is the rate of change of the shear along the null geodesic. This is described for instance in Wald.

It has some very useful properties to describe and analyze how a shape in spacetime changes along null geodesics, I.e., along gravitational waves.

It is useful in analyzing gravitational waves. One is the classification of Petrov types, with a general Weyl tensor asymptotically (as we go towrds infinity, ie, further from the course) able to be described as a sum of successively faster decaying Weyl Tensors, as higher inverse powers of u, with u a parameter along the null geodesics towards infinity. This is called the Peeling Theorem, and allows for the classification of Ricci flat spacetimes, and analysis of gravitational waves in vacuum. See a simple description at https://en.m.wikipedia.org/wiki/Peeling_theorem

Another useful tool derived from the Weyl tensor, the Newman-Penrose formulation, uses the Weyl scalars, 5 complex scalars which are formed from the 10 independent components of the Weyl tensor. One of those complex scalars is the outgoing gravitational wave, another is incoming, one is a static term (like the static Schwarzschild field), and two define gauges, at large distances (i.e., as u gets large). See https://en.m.wikipedia.org/wiki/Weyl_scalar

• All right ! That does it. Apparently my mistake was to think the deformation of geodesic balls was measured only by the Ricci tensor, when in fact both it and the Weyl tensor account for deformation, though for different reasons. Much clearer now, thanks ! – Johann Dec 22 '16 at 8:52