I'm sorry, I'm not sure what your question is. Could you please clarify?

[exmarine]

In an inertial reference frame - and zero gravity field - we believe that any passing photons go in Euclidean straight lines. If I have some constant velocity towards the path of a passing photon, it still goes in a straight line, just at a different angle. But if I am accelerating towards that passing photon’s path, then it appears to curve toward me. And GRT says that is indistinguishable from the case of a gravitational field with source behind me.

So consider photons raining down toward our sun from the North Pole region. Since the Earth is constantly accelerating radially inward toward the sun, then those photon paths should appear to curve radially outward from the sun. But the sun’s gravity field should be causing them to curve radially inward. I haven’t worked out the math yet, but I am guessing that the photon path in a gravity field appears - to a free-falling observer like us here on the Earth - to be a straight line again?

The only GRT effects I see for stellar aberration seem to be in the radial direction with respect to the sun, i.e., no tangential effects with respect to our orbit - to be consistent with what we observe. Is this correct? So does GRT offer no help with understanding the observed changes in stellar aberration angles?

 

[PeterDonis]

GRT says that is indistinguishable from the case of a gravitational field with source behind me.

More precisely, GR says that *proper* acceleration is indistinguishable from *being at rest* in a gravitational field. This makes a difference; see below.

So consider photons raining down toward our sun from the North Pole region. Since the Earth is constantly accelerating radially inward toward the sun, then those photon paths should appear to curve radially outward from the sun.

No, because the Earth is not undergoing *proper* acceleration; it is in free fall. The equivalence principle says that free fall is indistinguishable from free fall (), which means that, as far as the equivalence principle can tell, the photons will travel in straight lines as seen from the Earth.

the sun’s gravity field should be causing them to curve radially inward.

With respect to an object that is sitting at rest in the sun's field, under proper acceleration, yes. But, as above, the Earth is not such an object.

I haven’t worked out the math yet, but I am guessing that the photon path in a gravity field appears - to a free-falling observer like us here on the Earth - to be a straight line again?

It does, but not because of two effects "canceling out". See above.

The only GRT effects I see for stellar aberration seem to be in the radial direction with respect to the sun

The bending of light by the sun is experimentally confirmed. I'm not sure whether you would call that "radial" or "tangential" with respect to the Earth's orbit; you need to re-think this scenario in the light of the above.

So does GRT offer no help with understanding the observed changes in stellar aberration angles?

If you mean, do you need to take the curvature of spacetime into account in understanding the observed changes in stellar aberration angles due to the Earth changing direction in its orbit about the sun, no, you don't. But you certainly *do* need to take spacetime curvature into account to explain how the Earth can be in a circular orbit about the Sun even though it's in free fall. You also need to take spacetime curvature into account to explain the bending of light by the sun; I'm not sure if you consider that an example of "aberration" or not (I think most physicists don't).

 

[pervect]

In an inertial reference frame - and zero gravity field - we believe that any passing photons go in Euclidean straight lines. If I have some constant velocity towards the path of a passing photon, it still goes in a straight line, just at a different angle. But if I am accelerating towards that passing photon’s path, then it appears to curve toward me. And GRT says that is indistinguishable from the case of a gravitational field with source behind me.

So consider photons raining down toward our sun from the North Pole region. Since the Earth is constantly accelerating radially inward toward the sun, then those photon paths should appear to curve radially outward from the sun. But the sun’s gravity field should be causing them to curve radially inward. I haven’t worked out the math yet, but I am guessing that the photon path in a gravity field appears - to a free-falling observer like us here on the Earth - to be a straight line again?

The only GRT effects I see for stellar aberration seem to be in the radial direction with respect to the sun, i.e., no tangential effects with respect to our orbit - to be consistent with what we observe. Is this correct? So does GRT offer no help with understanding the observed changes in stellar aberration angles?

If you do all the math, you should find that your coordinate-based explanation is equivalent to the idea that there is some gravitational lensing of light. I'm interpreting the analysis you propose as calculating the geodesic paths in some specific sun-centered (or perhaps barycentric) coordinate system, then comparing your results to what you'd get if you did a 1:1 mapping from your coordinates to some Euclidean model, traced the straight line in the Euclidean model, then did the inverse mapping of this straight line back to the actual (and in generally curved) space-time manifold.

You'd then have both the Earth and the light undergoing coordinate acceleration, which would be of roughly the same order.

I think you'd get some insights into what GR would predict by using an "effective refraction index model", then noting that the refractive index due to the suns field varies radially but not tangentially. I've seen statements made that the effects of GR can be calculated using this simpler model (for instance http://www.slac.stanford.edu/econf/C041213/papers/0305.PDF) but I haven't seen a detailed justification for the model. I assume the result would still be approximate, giving the effects of light bending to the same sort of accuracy as PPN does (which is good enough for solar system effects).

According to the theory of General Relativity (GRT), a light ray propagating in the Sun’s gravi-tational field is effectively refracted, with the vacuum index of refraction n
augmented by a refractivity inversely proportional to the distance r from the Sun’s center of mass.

Unfortunately, this paper does not provide a more detailed reference on this point. Perhaps someone else can supply one.

There would be some similar effects to to the gravitational field of the Earth (which would have tangential variations in the effective index of refraction) as well. I can't say whether they would be significant or not.

 

[exmarine]

Thanks for making statements more precise, etc. I will download the paper and study. There is a rather long recent thread on here, also about stellar aberration, trying to understand it in terms of special relativity. It seemed to generate more heat and confusion than light, and I was wondering if general relativity might be needed to understand it. After all, is it not the changes in our velocity, i.e., our acceleration, that makes it an observable phenomenon?

 

[exmarine]

Well I don't see any way to get a tangential component out of the Schwarzschild metric for the annual change in aberration angles - what Bradley observed in 1725. Of course, I don’t know a whole lot about what I am trying to do either. How about the Kerr metric? Does the sun have much rotation?

 

[PeterDonis]

I don't see any way to get a tangential component out of the Schwarzschild metric for the annual change in aberration angles - what Bradley observed in 1725.

I'm not sure what's leading you to that conclusion. To me it seems like the other way around: the Schwarzschild metric tells you that light paths that are tangential to the gravitating body are bent (like light passing the Sun), but light paths that are radial to the gravitating body are not bent--the light is just redshifted (like light rising radially from the Sun), or blueshifted if it is falling instead of rising.

At the distance of the Earth's orbit about the Sun, though, any such effect is extremely small; the order of magnitude will be ratio of the Sun's "gravitational radius" ($2 G M_{sun} / R_{sun}$), about 3 km, to the radius of the Earth's orbit, about 150 million km, i.e., about $2 \times 10^{-8}$. I don't think we can even measure stellar aberration with that precision. (Tests of light deflection by the Sun at the distance of the Earth's orbit, like the one described in the paper pervect linked to, have been done using radio waves with spacecraft and specialized detectors.)

How about the Kerr metric? Does the sun have much rotation?

The Kerr metric is harder to work with than the Schwarzschild metric, so I don't think trying it is likely to be helpful. AFAIK the Sun's rotation is far too small to have a significant effect on the metric in the solar system.

 

[exmarine]

By "tangential" I mean tangent to our orbit - what Bradley observed in 1725 when the stars appear slightly "ahead" of our radial position... I thought that was the common understanding of "stellar aberration" - though I think of that as actually the annual CHANGES in the aberration angle. Schwarzschild doesn’t seem to offer any such component in the geodesics. I thought Kerr’s metric might?

 

[PeterDonis]

By "tangential" I mean tangent to our orbit - what Bradley observed in 1725 when the stars appear slightly "ahead" of our radial position...

In other words, by "tangential" you mean "the star's light is coming in purely radially, relative to the Sun; but because of the Earth's motion in its orbit, we on Earth see a small tangential component in the direction of the star's light".

I thought that was the common understanding of "stellar aberration"

Aberration is not restricted to starlight coming in purely radially, relative to the Sun; that's just the simplest case for many people to visualize.

though I think of that as actually the annual CHANGES in the aberration angle.

The changes are how we know (at least if we restrict ourselves to observations made on Earth) that the aberration is present at all. If we could only make observations in one state of motion, we would never be able to tell that the aberration was present.

Schwarzschild doesn’t seem to offer any such component in the geodesics.

Yes, for purely radial motion, there is no bending of light by the Sun; just redshift or blueshift.

I thought Kerr’s metric might?

Yes, it does--for example, starlight falling radially onto, say, a pulsar (a rapidly rotating neutron star) would acquire a tangential component of motion due to the pulsar's gravity. But the Sun's rotation, as I said before, is way too small for such an effect to be detectable.

 

[pervect]

I'm not sure what's leading you to that conclusion. To me it seems like the other way around: the Schwarzschild metric tells you that light paths that are tangential to the gravitating body are bent (like light passing the Sun), but light paths that are radial to the gravitating body are not bent--the light is just redshifted (like light rising radially from the Sun), or blueshifted if it is falling instead of rising.

My ooops here. The index-of-refreaction model predicts what Peter describes and not what I said it did earlier.

For instance, if you consider refreaction through an atomsphere of varying density, horiziontal beams are refracted even though the index of refreaction is contant vs height. SImilarly, vertical beams aren not refracted.