# Interpretation of the Raychaudhuri equation and the attractive nature of gravity

In many books it is stated that the Raychaudhuri equation is a sort of "proof" that in general relativity gravity is attractive. This is done by considering a bundle of geodesics, and proving that under certain reasonable conditions its expansion $\theta$ has a negative time derivative. This implies that a bundle of geodesics tends to converge as time goes on, and hence it would imply that the curvature of spacetime produces an attractive force.

I have a problem with this interpretation, though. A bundle of geodesics implies a bundle of test particles: objects whose gravitational influence can be neglected. The spacetime curvature is supposed to be sourced by some other matter distribution, not by our geodesics. We seem to be proving that test particles are attracted to each other! But that's not what gravity does: test particles should be attracted by matter, not by each other. Am I misunderstanding the interpretation of the Raychaudhuri equation?

In fact, I can think of a situation where gravity could have a diverging effect (please let me know if this part should be a separate question). Consider two neighboring test particles heading in the general direction of a planet, with impact parameters $b$ and $b + \delta b$: one of them will pass closer to the planet than the other. Then the closer particle will be deflected more than the farther particle, so that their trajectories will diverge as they pass near the planet. Does this not contradict the Raychaudhuri equation? Or, if it doesn't, doesn't this example show that gravity can be attractive and yet have a diverging effect?

There is a nice introduction to the Raychaudhuri equation in the paper On the Raychaudhuri equation by George Ellis, Pramana Vol. 69, No. 1, July 2007 ${}^1$. He gives the following intuitive explanation of how it shows the attractive nature of gravity:

Ellis gives the equation in the form:

$$\dot\Theta +\tfrac{1}{3}\Theta^2 + 2(\sigma^2 - \omega^2) - \dot{u}^a{}_{;a} + \tfrac{1}{2}\kappa\left(\mu + \frac{3p}{c^2}\right) - \Lambda = 0$$

He then makes the point that if we choose some representative length scale $\ell$ this is related to $\Theta$ by:

$$\Theta = \frac{3\dot\ell}{\ell}$$

And substituting this gives us:

$$\frac{3\ddot\ell}{\ell} = -2(\sigma^2 - \omega^2) + \dot{u}^a{}_{;a} - \tfrac{1}{2}\kappa\left(\mu + \frac{3p}{c^2}\right) + \Lambda$$

The left side of the equation tells us whether the rate of change of volume of space is increasing or decreasing. A positive value means the expansion is accelerating and a negative value means the expansion is decelerating. So we need only look at the signs of the parameters on the right hand side to see what effect they have.

So in this form it is immediately obvious that a positive cosmological constant drives expansion, since it appears on the right hand side with a positive sign. Likewise the matter term $\mu+3p/c^2$ drives contraction since it appears with a negative sign.

But this assumes some distribution of matter $\mu(\mathbf x)$. You give an example of two particles on a hyperbolic orbit round a planet but the matter density is zero outside the planet so assuming the CC is negligible too the equation becomes:

$$\frac{3\ddot\ell}{\ell} = \dot{u}^a{}_{;a}$$

which is true but not very interesting. In this case the Raychaudhuri equation does not tell us anything useful.

${}^1$ I found a copy online but that link has disappeared now. I'm afraid it looks as if you'll have to pay for the paper.

• I think I'm getting it now, but there's still a thing that bothers me a little. Ellis says: "This equation (..) shows that shear, energy density and pressure tend to make matter collapse (...)". Intuitively I would expect that geodesics would be attracted to regions of higher density, independently of whether they converge or diverge. The equation seems to say nothing about the former, and instead it states that in the presence of matter geodesics always converge. Is there really no situation in which geodesics can diverge due to gravity (ignoring vorticity)? – Javier Apr 20 '18 at 13:22
• @Javier the Raychaudhuri equation is not telling you about the rate of convergence/divergence of geodesics, because that is just set by your initial conditions i.e. the initial state of your system. The equation is telling you how the rate of convergence/divergence changes. That is, anything that appears with a minus on the right hand side of my second equation will make the rate of divergence decrease/the rate of convergence increase with time. It's basically like the second Friedman equation. – John Rennie Apr 20 '18 at 14:45
• Yes, I misspoke. Replace "geodesics converge" by "the convergence of geodesics is decreasing" in my comment, and I think my question still stands. – Javier Apr 20 '18 at 15:05
• @Javier I've extended my answer to address what I think you're asking ... – John Rennie Apr 21 '18 at 9:18

In many books it is stated that the Raychaudhuri equation is a sort of "proof" that in general relativity gravity is attractive.

I would be interested in seeing a quote from a book that actually says this. It just sounds wrong to me. A more reasonable statement would be the following. Many forms of matter that we know of (but not dark energy) obey certain energy conditions. One of these energy conditions is the strong energy condition (SEC), which basically says that gravity is attractive. General relativity does not claim that gravity is always attractive, and in fact we now know from cosmological observations that gravity is not always attractive.

The Raychaudhuri equation describes the result of the SEC. For example, if the SEC holds, then the Raychaudhuri equation says that cosmological expansion must always decelerate, never accelerate. And because the SEC is false for dark energy, cosmological expansion is actually currently accelerating.

Or, if it doesn't, doesn't this example show that gravity can be attractive and yet have a diverging effect?

Both the strong energy condition and the Raychaudhuri equation describe a parcel of test particles in three dimensions. You can't probe that with only two test particles.

• First of all, did you mean dark energy instead of dark matter? AFAIK dark matter obeys the same energy conditions as normal matter. But anyway, quoting Hawking and Ellis: The importance of the weak energy condition is that it implies that matter always has a converging (or more strictly non-diverging) effect on congruences of null geodesics. Or as Poisson (2004) says: This is the statement of the focusing theorem. Its physical interpretation is that gravitation is an attractive force when the strong energy condition holds (...). – Javier Apr 8 '18 at 1:34
• Still, I don't feel this answers my question. The part about the needing three dimensions is a good point, but it seems like you just restate the question when you say "the SEC says that gravity is attractive". That is the statement I doubt. – Javier Apr 8 '18 at 1:35
• Yes, I meant dark energy, not dark matter. Thanks for the correction. At this point I'm not really clear on what you're asking. Maybe someone else could zero in better on what it is that is bugging you and write an answer that would help you more. – user4552 Apr 8 '18 at 21:14

In many books it is stated that the Raychaudhuri equation is a sort of "proof" that in general relativity gravity is attractive. T

An equation doesn't really prove anything. And since it's "important as a fundamental lemma for the Penrose-Hawking singularity theorems" I would say you are right to be questioning it.

This is done by considering a bundle of geodesics, and proving that under certain reasonable conditions its expansion $\theta$ has a negative time derivative. This implies that a bundle of geodesics tends to converge as time goes on, and hence it would imply that the curvature of spacetime produces an attractive force.

The curvature of spacetime is associated with the tidal force, which relates to the second derivative of potential. Not the force of gravity, which relates to the first derivative of potential - the gradient in the potential. There's no detectable tidal force in the room you're in, but your pencil still falls down. Light curves downward too. People tend to say "light follows a geodesic", where a geodesic is the shortest possible line between two points on a curved surface. But that light isn't following the curvature of spacetime. For an analogy imagine a stiff board. Pick up one side then, roll a marble across it. The path of the marble curves because the board has a gradient, not because it's curved. Of course, in the rubber sheet analogy you need some curvature to have a gradient, but it's important to appreciate that light curves most where the gradient in gravitational potential is greater, not where the curvature is greater.

I have a problem with this interpretation, though. A bundle of geodesics implies a bundle of test particles: objects whose gravitational influence can be neglected. The spacetime curvature is supposed to be sourced by some other matter distribution, not by our geodesics. We seem to be proving that test particles are attracted to each other! But that's not what gravity does: test particles should be attracted by matter, not by each other. Am I misunderstanding the interpretation of the Raychaudhuri equation?

I think so. It concerns the tidal force, where er, the soccer ball gets turned into a rugby ball. Or an American "football" if you prefer.

In fact, I can think of a situation where gravity could have a diverging effect (please let me know if this part should be a separate question). Consider two neighboring test particles heading in the general direction of a planet, with impact parameters $b$ and $b + \delta b$: one of them will pass closer to the planet than the other. Then the closer particle will be deflected more than the farther particle, so that their trajectories will diverge as they pass near the planet. Does this not contradict the Raychaudhuri equation? Or, if it doesn't, doesn't this example show that gravity can be attractive and yet have a diverging effect?

The tidal force is a "diverging" effect too. If you're falling into a black hole people say you get spaghettified. But you don't get spaghettified widthways. Instead you get "focussed" widthways. However this is only because the downward force of gravity at your feet is greater than the downward force of gravity at your head. It isn't because the force of gravity is always downward, even though it is. It's always towards the concentration of energy that "conditions" the surrounding space because this conditioning is akin to setting up an energy-density gradient in space. It's non-linear, hence spacetime curvature. See Einstein's 1920 Leyden Address: "According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration".