Gravitational tensor and scalar modes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> nonmetric gravity

Abstract

We investigate gravitational waves generated in $f(Q,B)$ nonmetric gravity, i.e., a theory of gravity described by a nonmetric compatible connection, free of torsion and curvature. It is an extension of symmetric teleparallel gravity, equipped with a boundary term $B$. This theory exhibits gravitational waves regardless of the gauge adopted: they are the standard massless tensors plus a massive scalar gravitational wave like in the case of $f(R)$ gravity. It is precisely the boundary term $B$ that generates the massive scalar mode with an effective mass ${m}_{B}$ associated to a Klein-Gordon equation in the linearized boundary term. As in $f(Q)$ gravity also in $f(Q,B)$ nonmetric gravity, a free test particle follows a geodesic motion due to the covariant conservation with respect to the Levi-Civita connection of the energy and momentum densities on-shell. Therefore, in $f(Q,B)$ gravity, the proper acceleration between two neighboring worldlines traveled by two free pointlike particles is governed by a first-order geodesic deviation equation in the metric perturbation ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$. Thanks to this approximate linear equation, $f(Q,B)$ nonmetric gravity shows three polarization modes: two massless transverse tensor radiation modes, with helicity equal to 2, reproducing the standard plus and cross modes, exactly as in general relativity, and an additional massive scalar wave mode with transverse polarization of zero helicity. We obtain the same result both by considering the coincident gauge and by leaving the gauge free. This happens because, at first order, the boundary term $B$ remains unchanged as well as the linearized field equations of metric tensor and connection in vacuum. Furthermore, we derive the energy and momentum balance equations and the equations of motion of the deviation, both in a generic theory of gravitation, where the matter stress-energy tensor ${{T}^{\ensuremath{\alpha}}}_{\ensuremath{\beta}}$ is Levi-Civita covariantly not conserved, i.e., ${\mathcal{D}}_{\ensuremath{\alpha}}{{T}^{\ensuremath{\alpha}}}_{\ensuremath{\beta}}\ensuremath{\ne}0$. In summary, three degrees of freedom propagate in the $f(Q,B)$ linearized theory with amplitudes ${\stackrel{\texttildelow{}}{h}}^{(+)}$ and ${\stackrel{\texttildelow{}}{h}}^{(\ifmmode\times\else\texttimes\fi{})}$ for tensor modes and amplitude ${\stackrel{\texttildelow{}}{h}}^{(s)}$ for the scalar mode. Specifically, both $f(Q,B)$ and $f(R)$ gravity involve the same massive transverse scalar perturbation.

Topics & Concepts