Fundamentals of Time and Relativity

Light Aberration

  • Aberration of light is the apparent angular displacement of the observed position of a light source (e.g., a star) as seen by a moving receiver.
  • The effect is explained by the Lorentz transformation from a static inertial frame to the frame of the receiver and, hence, is independent of the distance between receiver and source.

In the case of the annual aberration of starlight, the direction of incoming starlight as seen in the Earth's moving frame is tilted relative to the angle observed in the Sun's frame. Since the direction of motion of the Earth changes during its orbit, this tilting changes during the course of the year, and causes the apparent position of the star to differ from its true position as measured in the inertial frame of the Sun.

To obtain the basic aberration formula, let S 1 and S 2 be two distant stellar sources from which light is emitted with frequency four-vectors k=(κ,k) and l=(λ,l) , respectively, as seen by some inertial receiver O . Furthermore, suppose that this receiver measures the angle subtended by the two stars to be θ , which implies k.l=κλ(1−cosθ ) .

Then a second receiver O ′ , moving relative to O with four-velocity v=γ(v)(1,vk/κ) directly away from the star S 1 , measures the angle θ ′ defined by the relation

6.3

k⋅l:= κ ′ λ ′ (1−cos θ ′ )=(κ⋅v)(λ⋅v)(1−cos θ ′ )

The two observed angles are related by Einstein's relativistic aberration formula

6.4

cos θ ′ = cosθ−v 1−vcosθ

To show this one may use the identities k⋅v=κγ(v)(1−v) and l⋅v=λγ(v)(1−vcosθ) . Then from (6.3) the result (6.4) follows, or equivalently, after some trigonometric manipulations

6.5

tan 1 2 θ ′ = 1+v 1−v tan 1 2 θ

The velocity-dependent factor at the right-hand side is called the Bondi k-factor. It is another name for the Doppler factor, when source and observer are moving directly away from or towards each other.

Since the tangent-functions in (6.5) are monotonic for θ, θ ′ =[0,π] , it follows that θ ′ is greater or less than θ according as v is positive or negative, respectively. If θ≠0 , so that the stars are separated in the sky, then θ ′ →π as v→1 .

  • As the second observer looks in the direction of his motion relative to the first and accelerates toward the velocity of light, stars ahead of him seem brighter and more closely clustered together than the stars behind.
  • The effect of aberration is about 50 times larger than that of parallax.