Can someone derive Einstein's general relativity field equation for one spatial dimension and one 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.
I am expecting an explanation starting from equivalence principle or anything related. Something like this https://www.youtube.com/watch?v=pES_tNZJm3Q but simplified to a single spatial dimension for a beginner to be able to mess with and get a feel of the model.
For example, I want to be able to describe things like,
If this is the space time, what would happen if I add a big mass at (0,8). What would happen if I add 2 big masses at (0,6) and (0,10). How would the space time curve. What are the world lines of these objects. How do they change with the mass and their position.
Best Answer
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.
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.