# Metric $f(R)$ Instability

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I was reading this manuscript about Metric $f(R)$ instability, could anyone explain why the value of $\mu^4$ creates strong instability in Eq. (4)?

One piece I don't understand is where it says "Thus, the $T^{3}/6\mu^{4}$ dominates the coefficient in front of $R_1$ term in Eq. (6) and leads to strong instability" just below Eq.(7). Why is the value of the coefficient of $R_1$ may create instabilities.

Consider the field equation:$D^{2}R-3\frac{D_{a}R D_{a}R}{R}+\frac{R^{4}}{6{\mu}^{4}}-\frac{R^{2}}{2}=-\frac{T R^{3}}{6{\mu}^{4}}$, why the value of $\mu^4$, whether positive or negative, may create instabilities?

This question and the first comment below it was deleted from Physics Stack Exchange and has been restored from an archive.

edited Apr 30, 2014

Consider the field equation: $D^{2}R-3\frac{D_{a}R D_{a}R}{R}+\frac{R^{4}}{6{\mu}^{4}}-\frac{R^{2}}{2}=-\frac{T R^{3}}{6{\mu}^{4}}$, why the value of $\mu^4$, whether positive or negative, may create instabilities?

A comment by David Zaslavasky has been omitted from repost, since the linked manuscript is merely 4 pages long.

I am extremely sorry @physicsnewbie, it seems you are correct. I found from a meta.SE discussion that SE "redistributes [deleted content] to 10k+rep users and moderators", and whether it redistributes content at all is a choice it makes, but it still owns its content.

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I actually dealt this model some in my thesis work.  (Shameless plug: Stability of spherically symmetric solutions in modified theories of gravity.)  The basic picture in the Dolgov-Kawasaki paper is that they start with a background solution where the Ricci scalar is equal to the trace of the stress-energy tensor, $R_0 = T$ (inside a star, say). This solution should match up to some exterior solution with $R_0 \approx \mu^4$ (one of the cosmological models that were motivating Carroll, Duvvuri, Trodden, & Turner);  however, if $\mu^4 < 0$, this turns out to be impossible.  (Basically, the Ricci scalar tends to diverge at large distances from the star.)

Having dispensed with the $\mu^4 < 0$ case, they look at at the $\mu^4 > 0$ case.  In this case, one can obtain a well-behaved solution in which $R_0 \approx T$ inside the star and $R_0 \approx \mu^4$ asymptotically.  The next step is to look at the behavior of perturbations about this solution;  the linearized perturbation equation is eq. (5) in their paper.  The equation of motion for the first-order perturbations grows exponentially with time, since the last term on the left-hand side of eq. (5) is so large in magnitude and negative.

I should mention that while Dolgov & Kawasaki's proof isn't iron-clad (see my comments at the start of Section III.B in the paper I linked to above), it does persist in a more rigorous analysis.  You might also take a look at The Large Scale Structure of f(R) Gravity, which obtains a similar result.

answered May 2, 2014 by (70 points)

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