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It is well known that the Einstein tensor is trivially 0 in any two dimensions spacetime: \begin{equation}\tag{1} G_{\mu \nu}^{(2)} \equiv 0. \end{equation} 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: \begin{equation}\tag{2} G_{\mu \nu}^{(D)} + \Lambda_D \, g_{\mu \nu}^{(D)} = -\, 8 \pi G_D \, T_{\mu \nu}^{(D)}. \end{equation} 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?

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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

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

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)

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

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

\begin{equation}\tag{3} \end{equation} \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

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

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.

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  • $\begingroup$ 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} $\endgroup$ – Cham Aug 14 at 2:45
  • $\begingroup$ 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 $\sqrt{g} \neq 0$) $\endgroup$ – Joe Aug 14 at 2:47

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