Not yet reviewed by Pith; the record is open.

The next generation of galaxy surveys will allow us to test one of the most fundamental assumptions of the standard cosmology, i.e., that gravity is governed by the general theory of relativity (GR). In this paper we investigate the ability of the Javalambre Physics of the Accelerating Universe Astrophysical Survey (J-PAS) to constrain GR and its extensions. Based on the J-PAS information on clustering and gravitational lensing, we perform a Fisher matrix forecast on the effective Newton constant, $\mu$, and the gravitational slip parameter, $\eta$, whose deviations from unity would indicate a breakdown of GR. Similar analysis is also performed for the DESI and Euclid surveys and compared to J-PAS with two configurations providing different areas, namely an initial expectation with 4000 $\mathrm{deg}^2$ and the future best case scenario with 8500 $\mathrm{deg}^2$. We show that J-PAS will be able to measure the parameters $\mu$ and $\eta$ at a sensitivity of $2\% - 7\%$, and will provide the best constraints in the interval $z = 0.3 - 0.6$, thanks to the large number of ELGs detectable in that redshift range. We also discuss the constraining power of J-PAS for dark energy models with a time-dependent equation-of-state parameter of the type $w(a)=w_0+w_a(1-a)$, obtaining $\Delta w_0=0.058$ and $\Delta w_a=0.24$ for the absolute errors of the dark energy parameters.