# Is there a cosmological constant $\Lambda$ in 2D?

It is well known that the Einstein tensor is trivially 0 in any two dimensions spacetime: $$$$\tag{1} G_{\mu \nu}^{(2)} \equiv 0.$$$$ Thus, it is not possible to formulate the Einstein equation when the spacetime dimension is $$D = 2$$ and introduce the cosmological constant in the usual way: $$$$\tag{2} G_{\mu \nu}^{(D)} + \Lambda_D \, g_{\mu \nu}^{(D)} = -\, 8 \pi G_D \, T_{\mu \nu}^{(D)}.$$$$ This equation imposes $$\Lambda_2 = 0$$ and $$G_2 = 0$$ when $$D = 2$$.

So is there a way to introduce a non-trivial cosmological constant in 2 dimensional spacetimes? Is $$\Lambda_2 \equiv 0$$ really an unique necessity?

One thing to note first is that pure gravity (i.e. $$\Lambda = 0$$) is a topological theory. What this means is that on any manifold, the Einstein action $$S_E = \int d^2 x \sqrt{|g|} R$$ only depends on quantities on the boundary. In fact, due to the Gauss-Bonnet theorem, if your manifold is compact and has no boundary, then

$$$$\tag{1} \int d^2 x \sqrt{|g|} R = n \in \mathbb{Z}$$$$

where $$n = 2 - 2 g$$ is the Euler characteristic of a genus g surface, and is an integer which is importantly independent of the metric's specific values. This manifests itself in the fact that in the equations motion for the $$R$$ term vanish identically, i.e. that (exercise)

$$$$\tag{2} \frac{\delta \sqrt{|g|} R}{\delta g^{\mu\nu}} = 0$$$$

So to compute the equations of motion for the cosmological constant, you get

$$$$\tag{3}$$$$ $$\begin{split} & \frac{\delta \Lambda \sqrt{|g|}}{\delta g^{\mu\nu}} \\ &= \Lambda \frac{\delta \sqrt{|g|}}{\delta g^{\mu\nu}} \\ &= -\Lambda \frac{1}{2} g_{\mu \nu} \sqrt{-g} = 0 \end{split}$$

so the equations of motion are $$g_{\mu\nu} = 0$$ identically everywhere.

It is worth mentioning that most places where people talk about gravity in 2D is 2D dilaton gravity, where the gravitational field is coupled to an additional scalar $$\phi$$ with an action looking like

$$$$\tag{4} S_E = \int d^2x e^{\phi}(R + \Lambda + (\partial \phi)^2 )$$$$

These types of actions have relevance to string theory, and in more recent developments, they are thought to arise as low energy theories of a quantum mechanical model known as the SYK model.

• I think there's a missing $\Lambda$ in your equation, in the middle, which should imply $\Lambda = 0$ and not $g_{\mu \nu} = 0$. Also, please, consider adding a number to your main equations, using the command \tag{number} – Cham Aug 14 at 2:45
• Thanks, fixed it. But Lambda is a constant that you stick in by hand, not a dynamical field. So it would mean that $g_{\mu\nu}=0$ (assuming that $\sqrt{g} \neq 0$) – Joe Aug 14 at 2:47