Black Holes - the two points of view.

PAllen

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

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 ???

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

pervect

Imagine that S₁ rebroadcasts the signal recieved from FF. The signal as received by S₁ will be redshifted by the relative velocity between FF and S₁. What will be recieved 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

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

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.

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

harrylin

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.

DaleSpam

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

See below

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.

[QUOTE=Austin0;4050761] 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,


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_______________

DaleSpam

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

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

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

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 realise 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

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.

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.

PAllen

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.
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.

TrickyDicky

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.

Certainly, I never said you needed to.