Understanding Stellar Aberration and Its Relationship to Transverse Velocity

[kmarinas86]

...tellar aberration is independent of the distance of a celestial object from the observer, and depends only on the observer's instantaneous transverse velocity with respect to the incoming light beam, at the moment of observation...Another way to state this is that the emitting object may have a transverse velocity with respect to the observer, but any light beam emitted from it which reaches the observer, cannot, for it must have been previously emitted in such a direction that its transverse component has been "corrected" for. Such a beam must come "straight" to the observer along a line which connects the observer with the position of the object when it emitted the light.

Is this transverse velocity one that exists at right angles to the space-time geodesic on which that light travels?

 

[Bill_K]

This is one of the poorest 'explanations' of stellar aberration that I have ever seen. Everything in what you quoted is either irrelevant or incorrect, and I'm trying to be diplomatic when I say that!

Aberration is an effect resulting from a change in the observer's velocity. The velocity does not have to be transverse to anything. And it's rather simple to calculate. If the beam of light has a propagation 3-vector k in the observer's rest frame, you ask yourself what will be the propagation vector k' in some other frame moving with velocity v. Both k and k' are unit vectors, and the aberration is the angle between them.

 

[kmarinas86]

This is one of the poorest 'explanations' of stellar aberration that I have ever seen. Everything in what you quoted is either irrelevant or incorrect, and I'm trying to be diplomatic when I say that!

Aberration is an effect resulting from a change in the observer's velocity. The velocity does not have to be transverse to anything. And it's rather simple to calculate. If the beam of light has a propagation 3-vector k in the observer's rest frame, you ask yourself what will be the propagation vector k' in some other frame moving with velocity v. Both k and k' are unit vectors, and the aberration is the angle between them.

Oh, so in other words, the observer has to be accelerated (non-inertial), correct?

 

[bcrowell]

Aberration is an effect resulting from a change in the observer's velocity.

I don't think this is right -- or maybe I'm misinterpreting what you wrote. You seem to be saying that the observer has to be accelerating in order to get aberration, but that's not true.

The velocity does not have to be transverse to anything.

Again, I think either you're wrong or I'm misinterpreting you. If an observer is moving directly toward the object being observed, then by symmetry there is no aberration. WP is correct when it states that only the transverse component of the observer's velocity creates aberration.

Is this transverse velocity one that exists at right angles to the space-time geodesic on which that light travels?

What we care about is the *component* of the vector that's perpendicular, $v_\perp$. There is no need to talk about anything as fancy as geodesics, since the effect can be explained locally, and locally, you only need SR, not GR. When it comes to the question of what line the perpendicularity is defined relative to, I think there is probably more than one such line you could define, and that would lead to differing definitions of $v_\perp$. The result for the aberration would then have different forms depending on the definition of $v_\perp$. For applications like aberration of starlight due to the orbital motion of the earth, I don't think it matters much what you pick, because the effect is small. E.g., you could define $v_\perp$ as the component perpendicular to the direction in which the star appears to lie, or perpendicular to the direction in which the star actually lies, but the two definitions would give almost the same number.

 

[kmarinas86]

I don't think this is right -- or maybe I'm misinterpreting what you wrote. You seem to be saying that the observer has to be accelerating in order to get aberration, but that's not true.


Again, I think either you're wrong or I'm misinterpreting you. If an observer is moving directly toward the object being observed, then by symmetry there is no aberration. WP is correct when it states that only the transverse component of the observer's velocity creates aberration.


What we care about is the *component* of the vector that's perpendicular, $v_\perp$. There is no need to talk about anything as fancy as geodesics, since the effect can be explained locally, and locally, you only need SR, not GR. When it comes to the question of what line the perpendicularity is defined relative to, I think there is probably more than one such line you could define, and that would lead to differing definitions of $v_\perp$. The result for the aberration would then have different forms depending on the definition of $v_\perp$. For applications like aberration of starlight due to the orbital motion of the earth, I don't think it matters much what you pick, because the effect is small. E.g., you could define $v_\perp$ as the component perpendicular to the direction in which the star appears to lie, or perpendicular to the direction in which the star actually lies, but the two definitions would give almost the same number.

Does GR give one answer for $v_\perp$?

When you say, "There is no need to talk about anything as fancy as geodesics, since the effect can be explained locally..." are you saying that GR doesn't give a better answer? Or by saying this do you imply that GR gives an answer that is not so different than what you can get in SR?

 

[bcrowell]

When you say, "There is no need to talk about anything as fancy as geodesics, since the effect can be explained locally..." are you saying that GR doesn't give a better answer?

Right.

 

[kmarinas86]

The result for the aberration would then have different forms depending on the definition of $v_\perp$. For applications like aberration of starlight due to the orbital motion of the earth, I don't think it matters much what you pick, because the effect is small. E.g., you could define $v_\perp$ as the component perpendicular to the direction in which the star appears to lie, or perpendicular to the direction in which the star actually lies, but the two definitions would give almost the same number.

I think determining the direction of $v_\perp$ is straight-forward, but 1) what determines the magnitude of $v_\perp$? It would seem that light would have to travel in a system of coordinates in which anything stationary (relative to these coordinates) could not share the same inertial frame as the observer provided that the magnitude of $v_\perp$ is not zero. 2) Is there any basis in SR for determining such coordinates?

 

[kmarinas86]

This is one of the poorest 'explanations' of stellar aberration that I have ever seen. Everything in what you quoted is either irrelevant or incorrect, and I'm trying to be diplomatic when I say that!

Aberration is an effect resulting from a change in the observer's velocity. The velocity does not have to be transverse to anything. And it's rather simple to calculate. If the beam of light has a propagation 3-vector k in the observer's rest frame, you ask yourself what will be the propagation vector k' in some other frame moving with velocity v. Both k and k' are unit vectors, and the aberration is the angle between them.

Oh I see. Aberration can be seen as the accumulative effect that is found when taking multiple measurements at multiple parts of an orbit, in which Earth takes on changes in velocity. In other words, what we are observing through the data sets are differences in the aberration which can be attributed the different velocities that the observer takes over the course of an orbit. Aberrations measured over the course of one year does not account for the total aberration attributed to long-term velocity changes through the galaxy, which may last longer than the most of the very light which is being picked up in these experiments.

I guess the only problem here I am still left wondering about is this:

If I were to be at exactly zero perpendicular speed relative to the light path, what inertial frame would I be in? This would give a true account as to where in the sky the light source was (assuming simple Euclidean geometry for the propagation of light waves), though in practice we cannot "know" where that light source in sky was at the time of emission, so unfortunately we cannot derive the inertial frame from that. However, suppose we run an experiment on the table top. It should be then possible to prove what that inertial frame would be (co-(transverse)-moving with the source at emission?).

 

[bcrowell]

Oh I see. Aberration can be seen as the accumulative effect that is found when taking multiple measurements at multiple parts of an orbit, in which Earth takes on changes in velocity.

No, this is incorrect. It's an instantaneous effect.

 

[kmarinas86]

No, this is incorrect. It's an instantaneous effect.

It seems that some people sometimes use the word aberration when they really mean changes in aberration. When a person means it like that, it would seem that Bill_K's definition would apply. Of course, it would be more rigorous to use the correct definition and say change in aberration when one is referring to the change of aberration.

I guess the reason some cut it short is follows: If one only had one snapshot in the sky and did not know the velocity, then it would be hard to tell what the actual positions of the stars were. Of course, if one had two snapshots and knew the velocity of one of them, then the other velocity could be deduced. Similarly, if one had one picture and two velocities, with one of them corresponding to that picture, one could predict the image seen at the other velocity. So it would seem to make no sense to speak of measuring aberration without having taken at least two pictures with different aberration, although one could still speak of aberration even with one picture if it meant deducing aberration in a second picture.

 

[harrylin]

It seems that some people sometimes use the word aberration when they really mean changes in aberration. When a person means it like that, it would seem that Bill_K's definition would apply. Of course, it would be more rigorous to use the correct definition and say change in aberration when one is referring to the change of aberration.

I guess the reason some cut it short is follows: If one only had one snapshot in the sky and did not know the velocity, then it would be hard to tell what the actual positions of the stars were. Of course, if one had two snapshots and knew the velocity of one of them, then the other velocity could be deduced. Similarly, if one had one picture and two velocities, with one of them corresponding to that picture, one could predict the image seen at the other velocity. So it would seem to make no sense to speak of measuring aberration without having taken at least two pictures with different aberration, although one could still speak of aberration even with one picture if it meant deducing aberration in a second picture.

Yes indeed: We even cannot tell "the velocity" of ourselves in an absolute sense (relativity principle).
"Stellar aberration" necessarily refers to observed changes in the apparent positions of stars due to changes of the Earth's velocity as it orbits the Sun.

PS. Obviously a transverse component is required to see an effect: a change in speed along the line that connects the Earth with a star will not result in aberration.

Cheers,
Harald

 

[bcrowell]

Yes indeed: We even cannot tell "the velocity" of ourselves in an absolute sense (relativity principle).
"Stellar aberration" necessarily refers to observed changes in the apparent positions of stars due to changes of the Earth's velocity as it orbits the Sun.

I see. OK, in that sense I can agree that what is relevant is changes in velocity.

 

[harrylin]

I see. OK, in that sense I can agree that what is relevant is changes in velocity.

Good to see that we now all agree.
And I take the opportunity to thank you for the link to the really clarifying paper by Davis and Lineweaver on interpretation of cosmological expansion and superluminal velocities in the other thread (which suddenly closed!).