General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr–de Sitter Spacetime

• Main Author: Hideyoshi Arakida
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Universe
• Subjects: bendin of light ray; cosmological constant; aberration equation; effect of motion of observer
• Online Access: https://www.mdpi.com/2218-1997/7/6/173

As an extension of our previous paper, instead of the total deflection angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, we will mainly focus on the discussion of measurable angle of the light ray <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>P</mi></msub></semantics></math></inline-formula> at the position of observer <i>P</i> in Kerr–de Sitter spacetime, which includes the cosmological constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>. We will investigate the contribution of the radial and transverse motion of the observer which are connected with radial velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>v</mi><mi>r</mi></msup></semantics></math></inline-formula> and transverse velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><msup><mi>v</mi><mi>ϕ</mi></msup></mrow></semantics></math></inline-formula> (<i>b</i> is the impact parameter) as well as the spin parameter <i>a</i> of the central object which induces the gravito-magnetic field or frame dragging and the cosmological constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>. The general relativistic aberration equation is employed to take into account the influence of motion of the observer on the measurable angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>P</mi></msub></semantics></math></inline-formula>. The measurable angle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>P</mi></msub></semantics></math></inline-formula> derived in this paper can be applicable to the observer placed within the curved and finite-distance region in the spacetime. The equation of light trajectory will be obtained in such a sense that the background is de Sitter spacetime instead of Minkowski one. As an example, supposing the cosmological gravitational lensing effect, we assume that the lens object is the typical galaxy and the observer is in motion with respect to the lensing object at a recession velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>v</mi><mi>r</mi></msup><mo>=</mo><mi>b</mi><msup><mi>v</mi><mi>ϕ</mi></msup><mo>=</mo><msub><mi>v</mi><mi>H</mi></msub><mo>=</mo><msub><mi>H</mi><mn>0</mn></msub><mi>D</mi></mrow></semantics></math></inline-formula> (where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>0</mn></msub></semantics></math></inline-formula> is a Hubble constant and <i>D</i> means the distance between the observer and the lens object). The static terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">Λ</mi><mi>b</mi><mi>m</mi><mo>,</mo><mi mathvariant="sans-serif">Λ</mi><mi>b</mi><mi>a</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> are basically comparable with the second order deflection term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, and they are almost one order smaller that the Kerr deflection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>4</mn><mi>m</mi><mi>a</mi><mo>/</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>. The velocity-dependent terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">Λ</mi><mi>b</mi><mi>m</mi><msup><mi>v</mi><mi>r</mi></msup><mo>,</mo><mi mathvariant="sans-serif">Λ</mi><mi>b</mi><mi>a</mi><msup><mi>v</mi><mi>r</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for radial motion and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">Λ</mi><msup><mi>b</mi><mn>2</mn></msup><mi>m</mi><msup><mi>v</mi><mi>ϕ</mi></msup><mo>,</mo><mi mathvariant="sans-serif">Λ</mi><msup><mi>b</mi><mn>2</mn></msup><mi>a</mi><msup><mi>v</mi><mi>ϕ</mi></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for transverse motion are at most two orders of magnitude smaller than the second order deflection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We also find that even when the radial and transverse velocity have the same sign, asymptotic behavior as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> approaches 0 is different from each other, and each diverges to opposite infinity.