Sign and physical meaning of the cosmological constant

Question

I've heard that a cosmological constant can be used to model dark energy (e.g. $\Lambda$-CDM model), and that the constant $\Lambda$ should be positive. But my (quite small) understanding of dark energy is that it acts to expand things, which should correspond to $R_{00} < 0$ rather than $R_{00} > 0$, where here $R_{\mu\nu}$ is the Ricci tensor and $0$ is some fixed timelike direction with, say, $g_{00} = -1$.

But in Einstein's equations with a cosmological constant, $$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = T_{\mu\nu}, $$ it seems to have the opposite effect, as follows. If $\Lambda > 0$, then putting it on the other side and letting $\mu = \nu = 0$ gives $R_{00} = \cdots + \Lambda$, so $\Lambda$ has the effect of increasing Ricci curvature, i.e. causing the focusing of timelike geodesics, rather than the supposed "spreading out" of things.

Edit 1: This question addresses the positivity of the scalar curvature, but I am interested in how it affects the Ricci curvature, in particular the value of $\text{Ric}(T,T)$ for a timelike vector $T$.

Any help is appreciated!

Answer 1

The natural equation to understand whether the role of a given matter type in general relativity is attractive or repulsive is the Raychaudhuri equation, which describes precisely how a congruence of geodesics expands or contracts in the presence of some specific curvature. For this particular analysis I will consider a twist-free, shear-free congruence. Hence, we will only see if the expansion $\theta$ grows or diminishes with time, which will let us know whether the effect is attractive or repulsive.

You can find a detailed discussion of the Raychaudhuri equation in the books by Wald (General Relativity) or Poisson (A Relativist's Toolkit), for example. I'll lay out the basic ideas in this answer. Consider a timelike geodesic congruence with unit tangent vector $u^a$. I define $\theta = \nabla^a u_a$ and $h_{ab} = g_{ab} + u_a u_b$. My previous assumptions state that $\nabla_{[a}u_{b]} = 0$ and $\nabla_{(a}u_{b)} - \frac{1}{3} \theta h_{ab} = 0$, but these two conditions are meant just to simplify the analysis. $\theta$ is known as the expansion of the congruence of geodesics and it $\theta > 0$ the congruences are diverging (and hence expanding), while $\theta < 0$ mean they are converging (and hence contracting).

The Raychaudhuri equation for timelike geodesics states, under the previous assumptions, that $$\frac{\mathrm{d}\theta}{\mathrm{d}\tau} = - \frac{1}{3}\theta^2 - R_{ab}u^a u^b.$$

Let us compute the latter term on the right-hand side. We assume the only matter content on the spacetime is the cosmological constant. Notice that $$R_{ab} - \frac{1}{2} R g_{ab} = - \Lambda g_{ab}$$ implies, upon contracting both sides with the metric in four-dimensions, that $$R - 2 R = - 4\Lambda,$$ and thus $$R = 4 \Lambda.$$

Hence, \begin{align*} R_{ab} &= \frac{1}{2} R g_{ab} - \Lambda g_{ab}, \\ &= 2 \Lambda g_{ab} - \Lambda g_{ab}, \\ &= \Lambda g_{ab}. \end{align*}

Using this, we find that $$R_{ab}u^a u^b = - \Lambda,$$ and the Raychaudhuri equation leads to $$\frac{\mathrm{d}\theta}{\mathrm{d}\tau} = - \frac{1}{3}\theta^2 + \Lambda. \tag{1}$$

Notice then that positive $\Lambda$ means $\frac{\mathrm{d}\theta}{\mathrm{d}\tau}$ gets larger (more positive), and hence this makes the geodesics expand more (in the sense that it makes the expansion increase). Negative $\Lambda$ has the opposite effect, and makes the geodesics contract (i.e., the expansion $\theta$ tends to decrease under the influence of $\Lambda$). Therefore, the cosmological constant has exactly the intuitive role we would expect, in the sense that a positive cosmological constant tends to make the geodesics expand more.

You can repeat the analysis with a more complete stress-energy tensor, or with the other relevant terms in the Raychaudhuri equation as well. I didn't do this because they wouldn't add anything to the discussion, but they don't change the results: the role of $\Lambda$ is obviously still the same.

I should mention Eq. (1) can be easily solved with Mathematica, for example. You can then plot the solution for $\theta(0) = 0$ and see that the sign of $\Lambda$ does force $\theta$ to assume the expected values immediately after the geodesic starts evolving. For $\Lambda < 0$ you can also see the appearance of caustics, which are points in which the expansion diverges to $-\infty$ (meaning the geodesics all cross each other).