Kundt geometries and memory effects in the Brans–Dicke theory of gravity
Siddhant Siddhant, Indranil Chakraborty, Sayan Kar
Abstract
Abstract Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To this end, we introduce, in BD theory, novel Kundt spacetimes (without and with gyratonic terms), which serve as backgrounds for the ensuing analysis on memory. The BD parameter $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> and the scalar field ( $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> ) profile, expectedly, distinguishes between different solutions. Choosing specific localised forms for the free metric functions $$H'(u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> (related to the wave profile) and J ( u ) (the gyraton) we obtain displacement memory effects using both geodesics and geodesic deviation. An interesting and easy-to-understand exactly solvable case arises when $$\omega =-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> (with J ( u ) absent) which we discuss in detail. For other $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> (in the presence of J or without), numerically obtained geodesics lead to results on displacement memory which appear to match qualitatively with those found from a deviation analysis. Thus, the issue of how memory effects in BD theory may arise and also differ from their GR counterparts, is now partially addressed, at least theoretically, within the context of this new class of Kundt geometries.