Research Theme: Scalar-tensor gravity in Bondi-Sachs form

This post reviews three papers that treat asymptotically flat spacetimes in scalar-tensor theories of gravity. The first two papers dealt one theory called Brans-Dicke (BD) theory, whereas the third paper dealt with an extension of the theory by Damour and Esposito-Farese (DEF). 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. In the third paper, we revisited the calculations in the first two papers in the more general DEF theory.

Papers Highlighted

• S. Tahura, D. A. Nichols, and K. Yagi, “Gravitational-wave memory effects in the Damour-Esposito-Farese extension of Brans-Dicke theory,” (2025). arXiv:2501.07488.

• 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.

• 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 gravitational-wave 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).

The most recent paper extended the results of the previous two works to DEF gravity. The most significant difference in DEF theory is that the scalar field has a nonlinear interaction with itself, which can give rise to enhancements in the magnitude of the scalar field, particularly in strong-gravity regions of the spacetime. In the asymptotically flat part of the spacetime, these nonlinearities give rise to small differences in the memory effects. Because the effects are nonlinear, they are of second-order in a small coupling parameter, which would make these modifications small and challenging to distinguish observationally from the effects in BD theory.