Black Holes - the two points of view.

[atyy]

You can if you want. You can be the exception that proves the rule. It is not a mathematical issue, nor is it in doubt that GR predicts them. The issue is what you think this says about GR as a physical theory. To me, and every physicist I know of who has written significantly on this, singularity is taken (by assumption, belief, not mathematical inconsistency) to represent the breakdown of the theory as a model of our universe, and the need for an alternative.

Yes, that makes sense to me (actually, from the physics point of view, QM would kick in first, so this point should be mathematical, not physical:) BTW, is it clear that GR is mathematically consistent in the presence of singularities? I've been assuming that it is because we can make sense of eg. FRW or Schwarzschild solutions. But would a mathematician see it the same way?

 

[PAllen]

Yes, that makes sense to me (actually, from the physics point of view, QM would kick in first, so this point should be mathematical, not physical:) BTW, is it clear that GR is mathematically consistent in the presence of singularities? I've been assuming that it is because we can make sense of eg. FRW or Schwarzschild solutions. But would a mathematician see it the same way?

Sure, why not? Not that I'm a mathematical expert, but mathematicians study and classify singularities all the time. The proof of the Poincare Conjecture didn't involve banning singularities - just showing that they weren't of an unmanageable type.

 

[TrickyDicky]

BTW, is it clear that GR is mathematically consistent in the presence of singularities? ...would a mathematician see it the same way?

I started a thread that is now 10 pages long with those questions in mind.
I don't think a mathematician would object singularities per se, but I'd imagine they might object calling smooth manifold a space with singularities.

 

[atyy]

Sure, why not? Not that I'm a mathematical expert, but mathematicians study and classify singularities all the time. The proof of the Poincare Conjecture didn't involve banning singularities - just showing that they weren't of an unmanageable type.

I don't see why not, but I have zero understanding of anything from a rigourous point of view, so I wish to be handed the definitive answer from "on high"

 

[atyy]

I started a thread that is now 10 pages long with those questions in mind.
I don't think a mathematician would object singularities per se, but I'd imagine they might object calling smooth manifold a space with singularities.

I have no idea of rigour, but I've sometimes tried to read http://relativity.livingreviews.org/Articles/lrr-2005-6/ . There also seems to be something called the BKL conjecture, which is apparently well-defined enough to be a mathematical conjecture even though it's about singularities, eg. http://arxiv.org/abs/1102.3474 .

 

[Austin0]

Redshift is always calculated the same way: take two timelike worldlines, on one pick two nearby events, calculate the proper time between them, find a null geodesic from each event to an event on the other worldline, calculate the proper time between those, the redshift is the ratio.

This procedure works in flat spacetime, curved spacetime, with or without motion.

Just to clarify. Here you seem to be talking about relative dilation which requires two events in each worldline

I'm just saying that there is only one process for calculating redshift. You don't have to do two calculations and add or multiply them together.

If we are considering an EM emission from a free falling frame at a particular potential altitude to a receiver at infinity does it still hold? Or does it require calculation of the g dilation and the relativistic Doppler due to velocity?
I guess in a way I am just asking if there are two separate effects or only one, and if not two ; why not? Thanks

 

[PAllen]

Just to clarify. Here you seem to be talking about relative dilation which requires two events in each worldline
If we are considering an EM emission from a free falling frame at a particular potential altitude to a receiver at infinity does it still hold? Or does it require calculation of the g dilation and the relativistic Doppler due to velocity?
I guess in a way I am just asking if there are two separate effects or only one, and if not two ; why not? Thanks

It still holds exactly as Dalespam described it, in all cases he listed (SR pure doppler, GR any case, any space time, any world lines). One calculation, one core phenomenon. Any possibility of separation into 'gravitational redshift' versus doppler in GR depends on the special case of a sufficiently static metric. This is often done as a matter of convenience, but is undefinable if the geometry is far from static.

There is a less intuitive version of the same concept, due to J.L. Synge (1960) [in the cosmological framework, this approach was pushed in a well known paper by Bunn and Hogg (2008), but it was demonstrated in greater generality by Synge in 1960]. This approach also is true for every SR and GR case , one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case. Synge took the view that there was really no such thing as gravitational or cosmological red shift per se, only relative velocity doppler corrected for the case of dynamic metric of GR (using the specified parallel transport). He especially railed against gravitational redshift because it is definable only for the special case of (sufficiently) static metric.

[Admittedly, Synge had somewhat icon iconoclastic views on GR. For example, he felt the principle of equivalence was mathematically false, and therefore of no value except historically.]

 

[Austin0]

It still holds exactly as Dalespam described it, in all cases he listed (SR pure doppler, GR any case, any space time, any world lines). One calculation, one core phenomenon. Any possibility of separation into 'gravitational redshift' versus doppler in GR depends on the special case of a sufficiently static metric. This is often done as a matter of convenience, but is undefinable if the geometry is far from static.

There is a less intuitive version of the same concept, due to J.L. Synge (1960) [in the cosmological framework, this approach was pushed in a well known paper by Bunn and Hogg (2008), but it was demonstrated in greater generality by Synge in 1960]. This approach also is true for every SR and GR case , one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case. Synge took the view that there was really no such thing as gravitational or cosmological red shift per se, only relative velocity doppler corrected for the case of dynamic metric of GR (using the specified parallel transport). He especially railed against gravitational redshift because it is definable only for the special case of (sufficiently) static metric.

[Admittedly, Synge had somewhat icon iconoclastic views on GR. For example, he felt the principle of equivalence was mathematically false, and therefore of no value except historically.]

If I am understanding you correctly then in the case I outlined there is only one effect.
It sounds like the resulting shift is purely dependent on the relative velocity and in this case the received frequency would be the same whether the gravitating mass was there or not. Is this right??
If this is so then I don't understand what you mean when you say it can only be separated into gravitational and velocity components in a static metric. it sounds like there is no gravitational component to be separated in any metric.
If Synge doubted the g redshift to what did he attribute the observed relative dilation related to potential that occurs with static sources and receivers and remote clocks??
In what way is the EP not mathematically supported according to Synge??
Thanks

 

[PAllen]

If I am understanding you correctly then in the case I outlined there is only one effect.
It sounds like the resulting shift is purely dependent on the relative velocity and in this case the received frequency would be the same whether the gravitating mass was there or not. Is this right??
If this is so then I don't understand what you mean when you say it can only be separated into gravitational and velocity components in a static metric. it sounds like there is no gravitational component to be separated in any metric.
If Synge doubted the g redshift to what did he attribute the observed relative dilation related to potential that occurs with static sources and receivers and remote clocks??
In what way is the EP not mathematically supported according to Synge??
Thanks

You can factor it for a static metric because there is an identifiable class of static observers. Then you define redshift relations between these observers (computed e.g. with either Dalespam's approach or Synge's) as 'gravitational'. Then for, other observers, you figure total redshift, compare to instantly co-located static observers and call the difference kinematic. But for non-static metric, there is no class of static observers to perform this separation.

It is not true that mass makes no difference under this scheme. Stress-energy and geometry are interlinked, and parallel transport is affected by geometry (as are the way null paths connect world lines in Dalespam's approach). It is just that there is no need to factor it into separate effects, and in the general case, you can't.

Synge's position was that the spacetime was either curved or not, period. And that the difference is detectable mathematically in an arbitrarily small region; in the limit at a single point. Therefore he felt it was simply false to say gravity and acceleration in flat spacetime were locally equivalent. I don't agree this makes the principle useless - it just defines the bounds of its accuracy.

 

[Austin0]

You can factor it for a static metric because there is an identifiable class of static observers. Then you define redshift relations between these observers (computed e.g. with either Dalespam's approach or Synge's) as 'gravitational'. Then for, other observers, you figure total redshift, compare to instantly co-located static observers and call the difference kinematic. But for non-static metric, there is no class of static observers to perform this separation.

Forgive me if I am a little slow tonight. But let me see if I've got it right:
Given static observers S₁, S₂ with S₂at infinity and free falling observer FF with rel velocity v wrt S₂
S₁ and FF emit identical signals at the moment of co-location.
As received at S₂ the signal from FF will be equivalent to the signal from S₁ with the addition of a purely classical Doppler shift for relative velocity v.
Is this right?
If we consider another inertial frame I ,in flat spacetime with the same v relative to S₂ would there be any difference in received signals at S₂, between those from FF and I ?

It is not true that mass makes no difference under this scheme. Stress-energy and geometry are interlinked, and parallel transport is affected by geometry (as are the way null paths connect world lines in Dalespam's approach). It is just that there is no need to factor it into separate effects, and in the general case, you can't.

I thought that parallel transport of a vector along a geodesic left the vector unchanged. is this incorrect?
Thanks for your patience

 

[pervect]

Forgive me if I am a little slow tonight. But let me see if I've got it right:
Given static observers S₁, S₂ with S₂at infinity and free falling observer FF with rel velocity v wrt S₂
S₁ and FF emit identical signals at the moment of co-location.
As received at S₂ the signal from FF will be equivalent to the signal from S₁ with the addition of a purely classical Doppler shift for relative velocity v.
Is this right?

Imagine that S₁ rebroadcasts the signal received from FF. The signal as received by S₁ will be redshifted by the relative velocity between FF and S₁. What will be received by S₂ will be the rebroadcasted signal redshifted by an additional gravitational redshift factor, the one between S₁ and S₂.

So the answer is yes, though I'd reverse the order from your original phrasing, because the velocity between FF and S₁ is well defined as they are at the same spot, and that way you don't have to worry about multiplication being commutative (though it is).

 

[Austin0]

Imagine that S₁ rebroadcasts the signal received from FF. The signal as received by S₁ will be redshifted by the relative velocity between FF and S₁. What will be received by S₂ will be the rebroadcasted signal redshifted by an additional gravitational redshift factor, the one between S₁ and S₂.

So the answer is yes, though I'd reverse the order from your original phrasing, because the velocity between FF and S₁ is well defined as they are at the same spot, and that way you don't have to worry about multiplication being commutative (though it is).

It appears to me that the signal received at S₂ in your relayed adaptation would not be equivalent to a signal sent directly from FF to S₂ as I outlined.
In your case there would only be transverse Doppler between FF and S ₁
so there would not be any classical kinematic component ,only a simple gamma dilation factor.
But from your input it appears I was wrong in my first conclusion. It seems you are saying there are two effects in operation. So a direct signal from FF to S₂ would be red shifted by the full relativistic Doppler factor (which includes a gamma dilation component) and the additional gravitational dilation factor due to potential location. Do I have it yet??
Thanks

 

[pervect]

It appears to me that the signal received at S₂ in your relayed adaptation would not be equivalent to a signal sent directly from FF to S₂ as I outlined.

Why not? FF and S1 are at the point in space-time, and the light cones for signals emitted by FF and S1 will be identical.

I am imagining that FF is falling into the black hole, so that FF, S1, and S2 will always be in a straight line. I don't think it necessarily matters if they aren't, but I'll agree it's not as obvious if S1 doesn't automatically "intercept" the signal from FF "en-route" to S2.

In your case there would only be transverse Doppler between FF and S ₁
so there would not be any classical kinematic component ,only a simple gamma dilation factor.

No, you want to use the relativistic doppler shift formula, See http://en.wikipedia.org/w/index.php?title=Relativistic_Doppler_effect&oldid=509495441. FF can't help but move away from S1, so there would be a doppler shift factor of sqrt[ (1 - beta) / (1 +beta) ], as per the wiki article. This doppler shift factor incorporates relatiavistic "time dilation" into the formula.

Do I have it yet??
Thanks

I'm not following you 100%, so there's probably still some confusion somewhere.

 

[harrylin]

[..]
I'm not following you 100%, so there's probably still some confusion somewhere.

And I'm not following either of you for 100%. Austin, a little clarification of S1, S2 and FF together with a sketch (even in ASCII) would be helpful to clear up what you are talking about.

 

[Dale]

There is a less intuitive version of the same concept, due to J.L. Synge (1960) [in the cosmological framework, this approach was pushed in a well known paper by Bunn and Hogg (2008), but it was demonstrated in greater generality by Synge in 1960]. This approach also is true for every SR and GR case , one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case.

Thanks PAllen, that is great to know. That is a much simpler calculation than the one that I was proposing. It is easy to see that this is a completely equivalent way of doing it. As you take two events which are separated by an infinitesimal amount of proper time you get the tangent vector.

 

[Austin0]

It appears to me that the signal received at S2 in your relayed adaptation would not be equivalent to a signal sent directly from FF to S2 as I outlined.

Why not? FF and S1 are at the point in space-time, and the light cones for signals emitted by FF and S1 will be identical.

See below

I am imagining that FF is falling into the black hole, so that FF, S1, and S2 will always be in a straight line. I don't think it necessarily matters if they aren't, but I'll agree it's not as obvious if S1 doesn't automatically "intercept" the signal from FF "en-route" to S2.

Yes I was assuming a proximate parallel path with a single short transmission , even a single photon in principle.(assuming ideal detection)
So FF is falling by S₁ and emits the photon in passing.

In your case there would only be transverse Doppler between FF and S 1
so there would not be any classical kinematic component ,only a simple gamma dilation factor.

No, you want to use the relativistic doppler shift formula, See http://en.wikipedia.org/w/index.php?...ldid=509495441. FF can't help but move away from S1, so there would be a doppler shift factor of sqrt[ (1 - beta) / (1 +beta) ], as per the wiki article. This doppler shift factor incorporates relatiavistic "time dilation" into the formula.

I was using the relativistic Doppler. which as far as I know simply reduces to the gamma factor for transverse reception just as I stated.
And no moving away from S₁,,, a single transmission,

So a direct signal from FF to S₂ would be red shifted by the full relativistic Doppler factor (which includes a gamma dilation component)

Do I have it yet??

I'm not following you 100%, so there's probably still some confusion somewhere.

Yep still some confusion but hopefully this might clear up some of it.
___SO if you might reread my previous post and see if it now tracks. Thanks_______________

 

[Dale]

If we are considering an EM emission from a free falling frame at a particular potential altitude to a receiver at infinity does it still hold? Or does it require calculation of the g dilation and the relativistic Doppler due to velocity?
I guess in a way I am just asking if there are two separate effects or only one, and if not two ; why not?

As mentioned earlier, the process I was describing includes both effects automatically. When you calculate the null geodesics you intrinsically include the effect of the curvature of spacetime between the emitter and the receiver, and the remaining description is simply how you calculate the Doppler effect in flat spacetime.

 

[TrickyDicky]

one operation: parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame. This will give the correct shift for every case.

How do you ascertain the 4-velocity of the emitter in the GR case and compare it to the vector in the local frame of the receiver?, for a emitter sufficiently distant, isn't comparing vectors not well defined due to path-dependence of parallel transport on a curved manifold?

Basically in practice you have to assume validity of the Hubble parameter to calculate a coordinate velocity of the emitter and obtain the redshift.

 

[PAllen]

How do you ascertain the 4-velocity of the emitter in the GR case and compare it to the vector in the local frame of the receiver?, for a emitter sufficiently distant, isn't comparing vectors not well defined due to path-dependence of parallel transport on a curved manifold?

Basically in practice you have to assume validity of the Hubble parameter to calculate a coordinate velocity of the emitter and obtain the redshift.

Just read what I wrote, all your points are answered. Path dependence is removed by specifying parallel transport along the null path followed by light from emitter to receiver. This does not remove the general ambiguity of distant comparison of 4-velocities; however, for this purpose, a unique transport path is specified, with a unique result. Nothing assumed about Hubble or any cosmology feature, nor any feature of specific geometry. Reread what you quoted, it already answered all of your questions:

"parallel transport the 4 velocity of the emitter at moment of emission, along the null path the light follows to the receiver, then apply SR doppler formula using the transported emitter 4-velocity expressed in the local frame of the receiver, and the null path tangent also expressed in this local receiver frame"

It is a complete, unambiguous prescription, which Synge showed to always yield the correct result.

 

[pervect]

The null geodesic approach is a good one, I don't mean to imply by discussing other approaches that its not.

Do note that if you have multiple images, you can in general have a different doppler shift for each image - so it doesn't necessarily solve the path dependence problems.

 

[PAllen]

The null geodesic approach is a good one, I don't mean to imply by discussing other approaches that its not.

Do note that if you have multiple images, you can in general have a different doppler shift for each image - so it doesn't necessarily solve the path dependence problems.

Good point, but this is certainly true of any approach to red/blue shift. For each image, you must analyze null path corresponding to that image. Clearly, no model based on gravitational potential will work - source and target are unique, only thing that differs are null paths.

Thus, Synge's (and Dalespam's equivalent) approach handle this case naturally: For each image, you use the the corresponding null path. I don't know what other approach you can use for this case.

In any case, the following wording was misleading:

"a unique transport path is specified, with a unique result"

You don't have to worry about all paths, but you do have to worry about all null paths light actually follows, and compute a separate redshift for each.

 

[Mike Holland]

I haven't posted here for a few days, as I have had a lot to think about, but I have found your conversatiion interesting, and I am learning all the time.

I realize now that I was wrong about redshifts. There are what I think of as two processes, gravitational redshift and Doppler redshift, though some of you say it is all the same process. But I had thought there was an additionaln redshift because successive light signals from the falling body take longer and longer to escape the gravitation well. I see now that as I am thinking about separate signals at fixed intervals, this does not appear as a redshift, although it will add an added apparent time dilation as seen by a stationary distant observer.

Have been reading up on naked singularities, and am about to email Saul Teukolsky to get more info from the horse's mouth. We got our B.Sc.s at the University of the Witwatersrand, but different generations.

Mike

 

[TrickyDicky]

Just read what I wrote, all your points are answered. Path dependence is removed by specifying parallel transport along the null path followed by light from emitter to receiver. This does not remove the general ambiguity of distant comparison of 4-velocities; however, for this purpose, a unique transport path is specified, with a unique result. Nothing assumed about Hubble or any cosmology feature, nor any feature of specific geometry. Reread what you quoted, it already answered all of your questions

No, my specific question is not answered, I asked how you exactly calculate the relative velocity of the distant emitter.
As you admit below path dependence is not removed, so I won't enter into that issue.

It is a complete, unambiguous prescription, which Synge showed to always yield the correct result.

I don't think is complete, you still need to calculate the distant relative velocity. And to do it you must use some additional information like the observed Hubble parameter that not only uses observed redshifts but luminosities, to pick the unique null path that leads to a unique relative velocity for the distant emitter, in the end the formula must equal the scale factors ratio of the cosmological redshift, Synge's is basically an algebraic reordering of this.

I'm not saying Synge (and Hogg and Bunn) "prescription" is wrong, I think it is purely an interpretational(almost just about terminology) issue, no matter how you call it (Doppler or cosmological redshift) the result must be the same, and yes, it can be computed in one step.

Do note that if you have multiple images, you can in general have a different doppler shift for each image - so it doesn't necessarily solve the path dependence problems.

In any case, the following wording was misleading:

"a unique transport path is specified, with a unique result"

You don't have to worry about all paths, but you do have to worry about all null paths light actually follows, and compute a separate redshift for each.

 

[PAllen]

No, my specific question is not answered, I asked how you exactly calculate the relative velocity of the distant emitter.
As you admit below path dependence is not removed, so I won't enter into that issue.

Path dependence issue remains only to the extent of multiple light paths from emitter to receiver. Then, a separate calculation is needed for each light path. Other possible paths are not a concern.

I don't think is complete, you still need to calculate the distant relative velocity. And to do it you must use some additional information like the observed Hubble parameter that not only uses observed redshifts but luminosities, to pick the unique null path that leads to a unique relative velocity for the distant emitter, in the end the formula must equal the scale factors ratio of the cosmological redshift, Synge's is basically an algebraic reordering of this.

You need only the metric to compute the null path(s) light follows. All you need is emitter world line, receiver world, and metric to apply the method. You don't need to care about what is cosmological or gravitational. Effectively, all geometric influences come into play by determining the null geodesics and also how the parallel transport acts.

I'm not saying Synge (and Hogg and Bunn) "prescription" is wrong, I think it is purely an interpretational(almost just about terminology) issue, no matter how you call it (Doppler or cosmological redshift) the result must be the same, and yes, it can be computed in one step.

 

[TrickyDicky]

You need only the metric to compute the null path(s) light follows.

When you say the metric I hope you realize that except the static case(where you can clearly separate the gravitational and doppler parts), this metric must include a scale factor that is a proper distances ratio, to decide the proper distance from the emitter we need to use information independent of the Hubble law, like luminosities, cosmological distance ladder, etc to try to come up with the most accurate Hubble parameter. All that must be included in the metric's scale factor and in a reliable calculation of the distant emitter relative velocity because that additional info is what allows us to pick the "unique" path.
For some reason you seem to avoid admitting that.

You don't need to care about what is cosmological or gravitational.

Certainly, I never said you needed to.

 

[PAllen]

When you say the metric I hope you realize that except the static case(where you can clearly separate the gravitational and doppler parts), this metric must include a scale factor that is a proper distances ratio, to decide the proper distance from the emitter we need to use information independent of the Hubble law, like luminosities, cosmological distance ladder, etc to try to come up with the most accurate Hubble parameter. All that must be included in the metric's scale factor and in a reliable calculation of the distant emitter relative velocity because that additional info is what allows us to pick the "unique" path.
For some reason you seem to avoid admitting that.

I'm just saying you don't need to worry about it in any explicit way. Given a spacetime with metric, and two world lines, you have a recipe to follow. How you arrive at a metric for the real world is an independent questions.

 

[TrickyDicky]

How you arrive at a metric for the real world is an independent questions.

Yes, but it needs to be answered for any real computation. I just wanted to point out ( for those that might have taken wrong "Synge's prescription") that using the relativistic doppler formula with relative velocity in GR is equivalent to the cosmological redshift formula normally used in cosmology. It adds nothing new.

It all really comes down to the fact there is only one redshift observed (and I'm restricting here to the remote or cosmological case), whatever one wants to call it, and that redshift can be arbitrarily decomposed in a doppler and a gravitational part in any possible way depending on the context and the interpretational bias, from purely doppler as in Synge's (also Hogg&Bunn) way, in a gravitational part plus a doppler part with all the range of different proportions, and in a purely gravitational redshift way although this last interpretation is less frequent.

 

[Dale]

that redshift can be arbitrarily decomposed in a doppler and a gravitational part in any possible way depending on the context and the interpretational bias, from purely doppler as in Synge's (also Hogg&Bunn) way, in a gravitational part plus a doppler part with all the range of different proportions, and in a purely gravitational redshift way although this last interpretation is less frequent.

I agree. Because of that, I would rather not decompose it at all and simply state what the redshift is without attributing one arbitrary part of it to gravity and the rest to motion. Only the total redshift is measurable.

 

[PAllen]

Yes, but it needs to be answered for any real computation. I just wanted to point out ( for those that might have taken wrong "Synge's prescription") that using the relativistic doppler formula with relative velocity in GR is equivalent to the cosmological redshift formula normally used in cosmology. It adds nothing new.

It all really comes down to the fact there is only one redshift observed (and I'm restricting here to the remote or cosmological case), whatever one wants to call it, and that redshift can be arbitrarily decomposed in a doppler and a gravitational part in any possible way depending on the context and the interpretational bias, from purely doppler as in Synge's (also Hogg&Bunn) way, in a gravitational part plus a doppler part with all the range of different proportions, and in a purely gravitational redshift way although this last interpretation is less frequent.

What it adds is a uniform algorithm for computing the most general cases. This algorithm is intuitive and has a reasonable physical interpretation: the light carries information about emitter motion as it travels to the receiver.

As to factoring, I agree: my point has always been that there is no need to factor, and factoring is only possible if there is well defined family of static observers (or a well defined family of comoving observers for the cosmological case). Even in such cases, factoring is optional - it is just a computational shortcut.

Also, I don't interpret Synge's procedure as saying 'all is relative motion'. I would describe it as: red shift is caused by relative motion mediated by intervening curvature in a specific way. The curvature determines the light path, and also determines how the emitter motion is 'carried' to the receiver (that is, it affects how the parallel transport carries the emitter 4-velocity).

 

[TrickyDicky]

I agree. Because of that, I would rather not decompose it at all and simply state what the redshift is without attributing one arbitrary part of it to gravity and the rest to motion. Only the total redshift is measurable.

Yeah, that would haved saved quite a few disputes around here and in the cosmology subforum.