Measurement of the space–time interval in modified gravity theories in Palatini formalism

Abstract

There are strong motivations that lead cosmologists to consider alternatives to Einstein’s theory of gravity in Palatini formalism. In addition, there are two distinguishable local frames in this formalism in which one of them is local inertial frame and the equivalence principle is satisfied. Different features of speed of light such as the causal structure constant, electromagnetic and gravitational wave velocities and Einstein velocity will not coincide in this local inertial frame for extended gravity theories in Palatini formalism. On the other hand, both the measurement of time and exchange of a signal between the distant points are required to determine spatial distances. In a particular situation where these aspects of the speed of light do not coincide, the distance determination will become more demanding because light will follow a time-like geodesic of the metric. In modified gravity theories in Palatini approach, theories involve a varying speed of photon. Therefore these kinds of theories must be based on some other technique of measuring spatial distances than radar or some terms should be corrected in the line element in the proposed model. We found out we should consider a coefficient which is proportional to energy density in each era, in the line element in order to be able to use radar for measuring distance in modified gravity theories in Palatini formalism. Analysis of some observational data will be affected by considering this coefficient in the line element.