Einstein's field equations in (1+1) spacetime?

Question

Can someone derive Einstein's general relativity field equations for 1 spatial dimension and 1 time dimension "from the beginning"? I think this will help many beginners to get a feel and understand details and effects of spacetime curvature easily.

If possible, I would like an explanation starting from equivalence principle (i.e., something like this video but simplified to a single spatial dimension). This would allow a beginner to get a feel of the model.

For example, I want to be able to describe things like the world lines of two point-like masses on this (1+1) spacetime.

Answer 1

The Einstein field equations may be derived using the action principle from the action,

$$S = \frac{1}{2\kappa^2}\int d^Dx \, \sqrt{|g|} \, \mathcal R,$$

potentially supplemented by a cosmological constant term, or a matter Lagrangian with other fields if coupling gravity to another theory. The Einstein field equations follow from the variation with respect to $g^{\mu\nu}$ and at no point does one assume $D=4$, so the derivation for $D= 2$ is the same.

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The Atiyah-Singer index theorem applied to the de Rham complex for a manifold $\mathcal M$ reads,

$$\chi(\mathcal M) = \int_{\mathcal M} e(T\mathcal M)$$

where $\chi$ is the Euler characteristic, a topological invariant and $e(T\mathcal M)$ is the Euler class of the tangent bundle of $\mathcal M$. In $D= 2$, this integral reduces to the Einstein-Hilbert action, up to constants and thus gravity in $D=2$ is classically purely topological.

Since $S$ becomes topological, $\frac{\delta S}{\delta g^{\mu\nu} }=0$ which implies stress-energy $T_{\mu\nu} = 0$ vanishes. Solutions are manifolds, of varying genus, otherwise they are seen as the same system by the action, due to the homeomorphism invariance.

Answer 2

Due to various topological constraints, the Einstein field equation in 2 dimensions are almost trivial. That is, any matter content will go with any metric, except if you decide to add a cosmological term, in which case the equations become

$$\Lambda g_{\mu\nu} = T_{\mu\nu}$$

This is due to the Hilbert action being a constant term due to the Gauss Bonnet theorem. $$S_H = \int dt dx R(x,t) = C$$

Its variation will therefore always be 0 no matter the metric. The addition of a cosmological term will give you the above equation.

Answer 3

As explained in previous answers the Einstein--Hilbert (EH) action is topological in two dimensions $$ \int d^2 x \sqrt{g}\, R = 4\pi\chi $$ where $\chi$ is the Euler characteristics. Equivalently this can be seen by the fact that the symmetries of the Ricci tensor $R_{\mu\nu}$ imply that $$ R_{\mu\nu} = \frac{R}{2}\, g_{\mu\nu} $$ and thus that the Einstein tensor is identically zero $$ G_{\mu\nu} = R_{\mu\nu} - \frac{R}{2}\, g_{\mu\nu} \equiv 0. $$ Another explanation is that the action is invariant under both diffeomorphisms (fixing two components of the metric) and Weyl symmetry (fixing the last component). So all the usual arguments used to infer the Einstein equations in $D = 4$ dimensions (or any $D > 2$, for what matters) cannot be applied in two dimensions.

There has been some attempts to recover the Einstein equation in vacuum but this involves some baroque constructions (see for example Teitelboim '83). Other researchers have preferred to generalize the action by introducing a dilaton (see for example hep-th/9204002 or gr-qc/9309018) or by searching what is the natural generalization of the EH action in 2d (it has been shown that a properly defined limit $\epsilon \to 0$ of $D = 2 + \epsilon$ yields the Liouville action, for example look at hep-th/9303123).

Finally note that $2d$ gravity presents a lot of pathologies. First if one considers that there is just a cosmological constant $$ S_\mu = \mu \int d^2 x \sqrt{g} $$ besides the EH term then the equation of motion reduces to $$ \mu = 0 $$ which has no solution. The same result is found if the Lagrangian contains matter that is invariant under the Weyl symmetry. Finally if one considers matter not invariant under the Weyl symmetry one can still show that many models have no dynamics or even don't exist (this is the topic of a recent paper I have written).

You asked for a simple and intuitive derivations and thus the above comments may look complicated, but I think it is necessary to explain why considering gravity in two dimensions is not a good idea.

Answer 4

Note: present answers are already very good, here I complement them with some references. See also the question General relativity (gravitation) in time and one spatial dimension and Einstein tensor in 2d.

It is not obvious how to construct General Relativity in (1+1) dimensions. In fact, taking the (3+1) Einstein field equations and interpreting the indexes as if they were in (1+1) dimensions do not work: the gist is that in (1+1) space-time there can be curvature but no matter, as shown in the paper General relativity in 2 and 3 dimensional space–times. Moreover, the theory is conformally invariant.

The point is that there can be no matter in this (1+1) spacetime unless you also allow for a cosmological constant (in the linked paper you can find the derivation of the "Einstein equations": $T_{\mu\nu}=\Lambda g_{\mu\nu}$, see also this answer or the notes Aspects of General Relativity in 1+1 Dimensions that contain a simple and complete derivation.

Less "trivial" 1+1 models: There are at least two ways of building a more interesting (1+1) theory of gravity. Both are discussed in the paper General relativity in (1+1) dimensions. Note also that the "paradoxes" of Newtonian gravity in (1+1) space dimensions (see this example related to the non-well-posedness of the Laplace equation) can be traced back to the peculiar nature of (1+1) spacetime (namely, there should be no matter!).

Spacetimes of different dimensionality: it is interesting to make a "tour" of the properties of spacetime as the space and time dimensions are varied. A good way to start is from On the dimensionality of spacetime by M. Tegmark, which collects many known properties of spacetimes of different dimensionalities, see also the answers in Is 3+1 spacetime as privileged as is claimed?.