Conservation laws, gravitational waves, and mass losses in the Dicke-Brans-Jordan theory of gravity

In the Dicke-Brans-Jordan theory of gravity, far away from a bounded system, orbiting test particles measure the total, active gravitational mass $M$ while orbiting test black holes measure the "tensor" mass $M_T$. Their difference ($M-M_T$) is the scalar mass $M_S$. In this paper, conservation laws for $M_S$, $M_T$, and $M$ are delineated and are used to show the following: (i) A spin-2 gravitational plane wave carries tensor mass, but does not carry scalar mass; the flux of tensor mass is proportional to the square of the time-integrated amplitude of the Riemann tensor ${\left|\int {\Psi }_4\mathrm{dt}\right|}^2$. (ii) A spin-0 gravitational plane wave carries both tensor mass (flux proportional to the square of the time-integrated Riemann amplitude ${\left|\int {\Phi }_{22}\mathrm{dt}\right|}^2$) and scalar mass (flux proportional to the Riemann amplitude ${\Phi }_{22}$—or, equivalently, proportional to the second time derivatives of the amplitude of the scalar field $\frac{{\partial }^2\phi }{\partial t^2}$). (iii) The tensor mass in a gravitational wave curves up the background spacetime through which the wave propagates; the scalar mass does not. (iv) The tensor mass in a wave is positive-definite; the scalar mass is not. (v) If a dynamical spherical system emits gravitational waves that change its scalar mass by $\Delta M_S$ in time $\tau$ ($\Delta M_S$ may be positive or negative), then these waves will also reduce its tensor mass by an amount $\ge \frac{{\left(\Delta M_S\right)}^2}{\tau }$. The response of gravitational-wave antennas to scalar waves is discussed. It is shown that, whereas antennas of negligible self-gravity respond only to the tidal forces of the wave (${\Phi }_{22}$), antennas with significant self-gravity respond about equally to the tidal forces ${\Phi }_{22}$ and the oscillating Cavendish gravitation constant $\phi$. Because of the unique phase and amplitude relations of ${\Phi }_{22}$ and $\phi$, the two responses are coherent—and can even cancel each other perfectly for a "carefully designed" detector.