Equivalence Principle for Massive Bodies Including Rotational Energy and Radiation Pressure

The Einstein gravitational theory and the Brans-Dicke gravitational theory are investigated for the gravitational-to-inertial mass ratio ($\frac{m_g}{m_i}$) they yield for massive systems. We examine models for stars in equilibrium but undergoing rotation or containing electromagnetic radiation. General conditions on the space-time metric of many mass sources are obtained which lead to $\frac{m_g}{m_i}=1\mathrm{}$ for the massive systems. It is found that $\frac{m_g}{m_i}\ne 1$ for a nonrotating star in the Brans-Dicke theory. The violation is of order ${10}^{-6\mathrm{}}$ for the Sun. In the Brans-Dicke theory there are also contributions to $\frac{m_g}{m_i}\ne 1$ due to the radiation content of a star. This violation is of order (radiation energy)/(stellar mass), yielding a correction of about ${10}^{-8\mathrm{}}$ for the Sun.