The optical geometry definition of the total deflection angle of a light ray in curved spacetime. (arXiv:2006.13435v3 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Arakida_H/0/1/0/all/0/1">Hideyoshi Arakida</a>
Assuming a static and spherically symmetric spacetime, we propose a novel
concept of the total deflection angle of a light ray. The concept is defined by
the difference between the sum of internal angles of two triangles; one of the
triangles lies on curved spacetime distorted by a gravitating body and the
other on its background. The triangle required to define the total deflection
angle can be realized by setting three laser-beam baselines as in planned space
missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total
deflection angle is, in principle, measurable by gauging the internal angles of
the triangles. The new definition of the total deflection angle can provide a
geometrically and intuitively clear interpretation. Two formulas are proposed
to calculate the total deflection angle on the basis of the Gauss–Bonnet
theorem. It is shown that in the case of the Schwarzschild spacetime, the
expression for the total deflection angle $alpha_{rm Sch}$ reduces to
Epstein–Shapiro’s formula when the source of a light ray and the observer are
located in an asymptotically flat region. Additionally, in the case of the
Schwarzschild–de Sitter spacetime, the expression for the total deflection
angle $alpha_{rm SdS}$ comprises the Schwarzschild-like parts and coupling
terms of the central mass $m$ and the cosmological constant $Lambda$ in the
form of ${cal O}(Lambda m)$ instead of ${cal O}(Lambda/m)$. Furthermore,
$alpha_{rm SdS}$ does not include the terms characterized only by the
cosmological constant $Lambda$.
Assuming a static and spherically symmetric spacetime, we propose a novel
concept of the total deflection angle of a light ray. The concept is defined by
the difference between the sum of internal angles of two triangles; one of the
triangles lies on curved spacetime distorted by a gravitating body and the
other on its background. The triangle required to define the total deflection
angle can be realized by setting three laser-beam baselines as in planned space
missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total
deflection angle is, in principle, measurable by gauging the internal angles of
the triangles. The new definition of the total deflection angle can provide a
geometrically and intuitively clear interpretation. Two formulas are proposed
to calculate the total deflection angle on the basis of the Gauss–Bonnet
theorem. It is shown that in the case of the Schwarzschild spacetime, the
expression for the total deflection angle $alpha_{rm Sch}$ reduces to
Epstein–Shapiro’s formula when the source of a light ray and the observer are
located in an asymptotically flat region. Additionally, in the case of the
Schwarzschild–de Sitter spacetime, the expression for the total deflection
angle $alpha_{rm SdS}$ comprises the Schwarzschild-like parts and coupling
terms of the central mass $m$ and the cosmological constant $Lambda$ in the
form of ${cal O}(Lambda m)$ instead of ${cal O}(Lambda/m)$. Furthermore,
$alpha_{rm SdS}$ does not include the terms characterized only by the
cosmological constant $Lambda$.
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