The Lyra–Schwarzschild Spacetime

In this paper, we provide a complete analysis of the most general spherical solution of the Lyra scalar-tensor (LyST) gravitational theory based on the proper definition of a Lyra manifold. Lyra’s geometry features the metric tensor and a scale function as fundamental fields, resulting in generalizations of geometrical quantities such as the affine connection, curvature, torsion, and non-metricity. A proper action is defined considering the correct invariant volume element and the scalar curvature, obeying the symmetry of Lyra’s reference frame transformations and resulting in a generalization of the Einstein–Hilbert action. The LyST gravity assumes zero torsion in a four-dimensional metric-compatible spacetime. In this work, geometrical quantities are presented and solved via Cartan’s technique for a spherically symmetric line element. Birkhoff’s theorem is demonstrated so that the solution is proven to be static, resulting in the Lyra–Schwarzschild metric, which depends on both the geometrical mass (through a modified version of the Schwarzschild radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>S</mi></msub></semantics></math></inline-formula>) and an integration constant dubbed the Lyra radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>L</mi></msub></semantics></math></inline-formula>. We study particle and light motion in Lyra–Schwarzschild spacetime using the Hamilton–Jacobi method. The motion of massive particles includes the determination of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>ISCO</mi></msub></semantics></math></inline-formula> and the periastron shift. The study of massless particle motion shows the last photon’s unstable orbit. Gravitational redshift in Lyra–Schwarzschild spacetime is also reviewed. We find a coordinate transformation that casts Lyra–Schwarzschild spacetime in the form of the standard Schwarzschild metric; the physical consequences of this fact are discussed.