Stars closely approaching the massive black hole in the center of the Galaxy provide a unique opportunity to probe post-Newtonian physics in a yet unexplored regime of celestial mechanics. Recent advances in infrared stellar spectroscopy allow the precise measurement of stellar Doppler shift curves and thereby the detection of beta 2 post-Newtonian effects (gravitational redshift in the black hole's potential and the transverse Doppler shift). We formulate a detection procedure in terms of a simplified post-Newtonian parametrization. We then use simulations to show that these effects can be decisively detected with existing instruments after about a decade of observations. We find that neglecting these effects can lead to statistically significant systematic errors in the derived black hole mass and distance.