Research Theme: Brans-Dicke theory in Bondi-Sachs form
This post reviews two papers that treat asymptotically flat spacetimes in Brans-Dicke theory. In the first paper, we gave the conditions for asymptotic flatness in this theory, computed the asymptotic solutions of the field equations, determined the asymptotic symmetries, obtained the conserved charges, and understood how the memory effects could be computed through flux laws. In the second paper, we applied the results of the first to compute gravitational-wave memory effects from compact binaries in the post-Newtonian approximation.
Papers Highlighted
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S. Tahura, D. A. Nichols, and K. Yagi, “Gravitational-wave memory effects in Brans-Dicke theory: Waveforms and effects in the post-Newtonian approximation.” Phys. Rev. D 104, 104010, (2021), arXiv:2107.02208.
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S. Tahura, D. A. Nichols, A. Saffer, L. C. Stein, and K. Yagi, “Brans-Dicke theory in Bondi-Sachs form: Asymptotically flat solutions, asymptotic symmetries and gravitationalwave memory effects.” Phys. Rev. D 103, 104026, (2021), arXiv:2007.13799.
Summary of the Papers
The set of symmetries of asymptotically flat spacetimes in general relativity is the Bondi-Metzner-Sachs (BMS) group. This group consists of the rotations, boosts, and the infinite-dimensional group of supertranslations (“angle-dependent” translations around an isolated source). These symmetries have corresponding conserved quantites, and changes in the conserved quantities and their fluxes are responible for producing gravitational-wave memory effects. It was not known what the symmetries of asymptotically flat solutions are in modified theories of gravity, or if there is the same relationship between memory effects and conserved quantities in these theories. To investigate this topic in more detail, we focused on a well known and extensively studied theory called Brans-Dicke theory, which has a third (scalar-type) polarization of gravitational waves.
In work lead by UVA Physics graduate student Shammi Tahura, we constructed asymptotically flat solutions in Brans-Dicke theory, found the symmetries that preserve the asymptotic properties of these solutions, computed the corresponding conserved charges, and understood their relationship to gravitational-wave memory effects. Specifically, we found that the tensor polarizations of gravitational waves are related to the changes in the conserved quantities related to the asymptotic spacetime symmetries. The scalar polarizations are not (though it was shown elsewhere that they are related to large asymptotic gauge symmetries of a dual theory).
The fact that the tensor-polarized gravitational-wave memory effect is related to flux-balance laws allowed us to compute the effect in the post-Newtonian approximation for compact binaries. We found a few differences between the memory effects in general relativity and in Brans-Dicke theory for these sources. First, the scalar-polarized radiation produces a small effect of a negative post-Newtonian order. Second, this radiation also changes the sky pattern of the memory effect around the source. Because the size of these effects is small, it will be challenging to detect with current (and even future) gravitational-wave detectors. There is the possibility that these slight differences could be found in a population of events (if indeed Brans-Dicke theory describes the gravitational interaction in nature).