It is usually pointed out that FRW metric is invariant under time reparametrization. Consider the flat case for simplicity


The choice of function $N(t)$ is arbitrary because one can make a change of variable $d\tau=N(t)dt$.

Now I'm trying to see how this freedom is manifested on the action level. The Lagrangian is found to be


If the choice of $N(t)$ is arbitrary, I expect it to not alter the equation of motion (be non dynamical). However the equation of motion for $N(t)$ turns out not to be trivial

$$\frac{\partial\mathscr{L}}{\partial N}-\frac{\partial}{\partial t}(\frac{\partial\mathscr{L}}{\partial \dot{N}})=0$$


$$\Rightarrow \ddot{a}=-3N^{-1}\dot{N}\dot{a}$$

How do we claim invariance under this parametrization if it clearly influences the dynamics?

  • $\begingroup$ Pointed out where? $\endgroup$ – Qmechanic Sep 11 '19 at 21:49
  • $\begingroup$ For example c.f. Gravitation and cosmology by Weinberg. 13.5 "spaces with maximally symmetric subspaces". See the result highlighted in 13.5.32 $\endgroup$ – Ammar Qasim Sep 12 '19 at 8:54
  • $\begingroup$ If you rewrite your equation of motion in terms of $\tau$, can you get rid of $N$? $\endgroup$ – Avantgarde Sep 15 '19 at 14:14

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