Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions

In static, spherically symmetric spacetimes, the deflection angle of photons in the strong deflection limit exhibits a logarithmic divergence. We introduce an analytical framework that clarifies the physical origin of this divergence by employing local, coordinate-invariant geometric quantities alongside the properties of the matter distribution. In contrast to conventional formulations—where the divergence rate <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mover accent="true"> <a:mi>a</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> </a:math> is expressed via coordinate-dependent metric functions—our approach relates <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mover accent="true"> <e:mi>a</e:mi> <e:mo stretchy="false">¯</e:mo> </e:mover> </e:math> to the components of the Einstein tensor in an orthonormal basis adapted to the spacetime symmetry. By applying the Einstein equations, we derive the expression <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mover accent="true"> <i:mrow> <i:mi>a</i:mi> </i:mrow> <i:mrow> <i:mo stretchy="false">¯</i:mo> </i:mrow> </i:mover> <i:mo>=</i:mo> <i:mfrac> <i:mrow> <i:mn>1</i:mn> </i:mrow> <i:mrow> <i:msqrt> <i:mrow> <i:mn>1</i:mn> <i:mo stretchy="false">−</i:mo> <i:mn>8</i:mn> <i:mi>π</i:mi> <i:msubsup> <i:mrow> <i:mi>R</i:mi> </i:mrow> <i:mrow> <i:mi mathvariant="normal">m</i:mi> </i:mrow> <i:mrow> <i:mn>2</i:mn> </i:mrow> </i:msubsup> <i:mrow> <i:mo stretchy="false">(</i:mo> <i:msub> <i:mrow> <i:mi>ρ</i:mi> </i:mrow> <i:mrow> <i:mi mathvariant="normal">m</i:mi> </i:mrow> </i:msub> <i:mo stretchy="false">+</i:mo> <i:msub> <i:mrow> <i:mi mathvariant="normal">Π</i:mi> </i:mrow> <i:mrow> <i:mi mathvariant="normal">m</i:mi> </i:mrow> </i:msub> <i:mo stretchy="false">)</i:mo> </i:mrow> </i:mrow> </i:msqrt> </i:mrow> </i:mfrac> </i:mrow> </i:math> , where <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msub> <u:mi>ρ</u:mi> <u:mi mathvariant="normal">m</u:mi> </u:msub> </u:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:msub> <x:mi mathvariant="normal">Π</x:mi> <x:mi mathvariant="normal">m</x:mi> </x:msub> </x:math> denote the local energy density and tangential pressure evaluated at the photon sphere of areal radius <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi>R</bb:mi> <bb:mi mathvariant="normal">m</bb:mi> </bb:msub> </bb:math> . This result reveals that <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mover accent="true"> <eb:mi>a</eb:mi> <eb:mo stretchy="false">¯</eb:mo> </eb:mover> </eb:math> is intrinsically governed by the local matter distribution, with the universal value <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mover accent="true"> <ib:mi>a</ib:mi> <ib:mo stretchy="false">¯</ib:mo> </ib:mover> <ib:mo>=</ib:mo> <ib:mn>1</ib:mn> </ib:math> emerging when <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:msub> <mb:mi>ρ</mb:mi> <mb:mi mathvariant="normal">m</mb:mi> </mb:msub> <mb:mo>+</mb:mo> <mb:msub> <mb:mi mathvariant="normal">Π</mb:mi> <mb:mi mathvariant="normal">m</mb:mi> </mb:msub> <mb:mo>=</mb:mo> <mb:mn>0</mb:mn> </mb:math> . Notably, this finding resolves the long-standing puzzle of obtaining <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:mover accent="true"> <rb:mi>a</rb:mi> <rb:mo stretchy="false">¯</rb:mo> </rb:mover> <rb:mo>=</rb:mo> <rb:mn>1</rb:mn> </rb:math> in a class of spacetimes supported by a massless scalar field. Furthermore, these local properties are reflected in the frequencies of quasinormal modes, suggesting a profound connection between strong gravitational lensing and the dynamical response of gravitational wave signals.