In this paper, the author formulates a $(1+1)$-dimensional theory of gravity by taking the trace of the Einstein equations $$\left(1 - \frac{D}{2}\right)R = 8\pi G_D T,\tag{2}$$ (where $G_D$ is the gravitational constant in $D$ dimensions) and defining $$G_2 := \lim_{D \to 2}\frac{G_D}{1 - D/2},\tag{4}$$ so that the above equation becomes $$R = 8\pi G_2 T.\tag{5}$$ However, I find this definition of $G_2$ to be suspect, as the limit does not exist. From this post, $G_D$ is proportional to the usual constant $G_4$, so the definition of $G_D$ does not fix the problem. My question is then whether this is actually allowed, and if so, why?
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2$\begingroup$ I mean, it's a definition, so it's allowed if he says it's allowed. Whether it's useful or not is an entirely different question. $\endgroup$– PraharCommented Jun 8 at 7:18
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The $R = 8\pi G_2 T$ equation Mann writes down is nothing but Nordstrom's theory which predates general relativity. The big observational problem with it was that $T = 0$ for electromagnetic fields. Since the geodesic equation can be derived from the field equation for the metric (unlike the Lorentz force law which cannot be derived from Maxwell's equations), Nordstrom incorrectly predicted that there would be no gravitational lensing.
So why is Mann playing this game of imagining a set of universes with non-integer dimension $D$ whose Newton constants are such that $G_D/(1 - D/2)$ has a finite limit as $D \to 2$? Beats me... it would have been better to just say he's reconsidering Nordstrom's theory in two dimensions as a toy model.
There are at least two things I can answer though. Prescribing the properties of $G_D$ as a function of $D$ would only contradict the post you mentioned if Mann had also specified that $G_{D_1}$ was obtained from a model with Newton constant $G_{D_2}$ by compactifying $D_2 - D_1$ dimensions. He is not doing that. The other point worth keeping in mind is that using something other than general relativity in 2d is often essential because the 2d Einstein-Hilbert action is the topological invariant \begin{equation} \int_M d^2x \sqrt{g} R = 4\pi \chi(M) \end{equation} which is the Gauss-Bonnet theorem. Most recent works on 2d gravity have evaded the Gauss-Bonnet theorem by considering a different modification of general relativity. This would be JT gravity.
