Vacuum Brans–Dicke theory in the Jordan and Einstein frames: Can they be distinguished by lensing?
Abstract
Vacuum Brans-Dicke (BD) theory continues to receive widespread attention since it is consistent with solar and cosmological experiments. The theory can be self-consistently described in two frames, the Jordan frame (JF) and the conformally rescaled Einstein frame (EF), the transformations providing an easy passage from one frame to the other at the level of actions and solutions. While coordinate transformations do not change curvature properties, conformal transformations do change them leading to corresponding changes in the numerical values of observables. A previous article by Bhadra et al.10 did exemplify this change between JF and EF using the diagnostic of second-order light deflection. This important work leaves room for further improvements on two points, which we do here. First, the measurement of second-order effect faced technically unsurmountable difficulties even around the Sun, hence actually abandoned. Second, the comparison of quantitative values between JF and EF should be based on a common value ofω connecting the two frames. Keeping these in mind, we investigate a technically easier diagnostic, viz., the weak field lensing (WFL) and compare the quantitative changes at common ω to show that the two frames can indeed be distinguished by lensing experiments. Specifically, the predictions of light deflection, image position, total magnification and magnification factor are computed in the EF and compared with those recently obtained (by Gao et al.22) directly in the JF BD class I solution. The use of the value of BD coupling constant |ω|=50,000, suggested by the Cassini spacecraft solar experiment, reveals that an exceptionally high degree of accuracy is needed to experimentally rule out one or the other frame by means of WFL measurements.
