What comprises a SRT relative velocity?

1. Dec 19, 2011

Tracer

What exactly comprises a relative velocity in special relativity? I find it confusing when distant galaxies can be moving away from the earth at speeds approaching the speed of light and yet that separation speed contributes nothing to a relative velocity. The usual answer provided is that both the earth and the distant galaxies are motionless in their local space. Therefore their relative velocity is zero. However, this means that to correctly determine a relative velocity the differential motion of space between the two bodies also must be considered.

2. Dec 19, 2011

Mentz114

In SRT relative velocity can be operationally defined as a function of the measured red/blue-shift of light between the frames. Applying this to the expanding cosmological model we can attribute the red-shift to relative velocity. But actually the attribution of red-shift to velocity or some other cause is coordinate and model dependent.

3. Dec 20, 2011

Tracer

There are still questions.
How is the observed red shift converted to a relative velocity?
Method #1.
Is the value of Fobserved divided by Fsource just plugged into the relativistic Doppler effect equation (RDE) and then solved for RV? Using this method, if Fo/Fs is 0.5773503 then RV will be determined to be 0.5c.

However the Relativistic Doppler effect equation treats RV as being applied to both a classical non-relativistic Doppler effect and a relativistic Doppler effect. This is wrong in this scenario since there should be no relativistic effects between the earth and the distant galaxy since both objects are at rest in their own local space.(except for very small values of peculiar motions of the bodies involved).

Method #2.
The way to determine an RV based on the observed red shift when only separation velocities due to Hubble expansion are involved, should be to use just the classical Doppler effect. Using this approach RV = (1-Fo/Fs)/(Fo/Fs)
which is V = 0.7320507c for Fo/Fs = 0.5773503.

Last edited: Dec 20, 2011
4. Dec 20, 2011

ghwellsjr

If R is the Ratio of the two frequencies, then:

β = |(1-R2)/(1+R2)|

This works for both red and blue shifts and for ratios of time intervals or periods as well as for frequencies, assuming that the motion is in line with the observer.

5. Dec 21, 2011

Tracer

Thank you George for your reply. However my question is whether or not β caused by hubble expansion causes relativistic effects. My previous post shows that the value of β will be considerably different if there are relativistic effects and when there are not. Which method is correct?

6. Dec 21, 2011

Staff: Mentor

Note that although the title and the first sentence specify special relativity, the remainder of the thread discusses cosmology where special relativity is not valid and general relativity must be used.

7. Dec 21, 2011

Tracer

That is curious. At what RV or distance does SRT become invalid?

8. Dec 21, 2011

Tracer

If R is the Ratio of the two frequencies, then:

β = |(1-R2)/(1+R2)|

This works for both red and blue shifts and for ratios of time intervals or periods as well as for frequencies, assuming that the motion is in line with the observer.

I take that as a vote for "hubble expansion causes time dilation". Correct me if that is not what you meant.

9. Dec 21, 2011

Staff: Mentor

SR is invalid whenever curvature is significant.

10. Dec 22, 2011

ghwellsjr

No, that's not what I meant. I simply was showing the equation for the sake of others that you used to do your calculation for method 1 which applies only to Special Relativity. I wasn't answering your question.

11. Dec 22, 2011

Tracer

Does GR consider hubble expansion to be curvature?

12. Dec 22, 2011

Naty1

Hi Tracer: first, it's good when you realize you are confused..... This whole cosmology thing IS very confusing....and still confounds me. [If I get something wrong here, somebody please correct it.] After reading this, you may be sorry you asked about velocity in GR!!(LOL)

As Dalespam notes, your title and question apply to two different regimes, SR and GR, respectively.

Distant galaxies CAN be moving away from the earth at speeds GREATER than the speed of light....and so much, but not all, and probably most, of the universe is way beyond our ability to detect via light.

Wikipedia explains further this way:

http://en.wikipedia.org/wiki/Metric_expansion_of_space#Measuring_distance_in_a_metric_space

and for a simple illustration of the "curve" :
from the above source.

Forgetting increasing spatial separation,expansion of the universe, for a few moments, and expanding on the Wikipedia comments:

In SR, flat spacetime, relative velocities at distant points can be compared. Distances are well defined; In GR, curved spacetime, velocity comparisons can only be made at some common measurement point. Distances are NOT well defined because of curvature; There is no global frame of reference in GR.
[This is what Dalespam explained:
In general relativity, an inertial reference frame is only an approximation that applies in a region that is small enough for the curvature of space to be negligible. So space is curved over larger distances but we keep our observations to only a small region for comparisons. Another way to say this: In curved spacetime there's no globally valid transformation between frames as there is in flat spacetime.

To define a frame (or observer, who has clocks and rulers ) in curved spacetime we must know something about the worldline, the curve (in space and time) along which the frame is carried. This is called parallel transport and different worldlines give different results!!!

I'll stop here because how all this exactly relates to redshift interpretations has been argued in these forums and I am not confident I understand it.

Expansion: I saved this explanation from Chalnoth and Marcus (whom I trust) :

I am aware the Hubble constant apparently varies over time....how that affects the above is another issue that confuses me!

If anyone can simplify this, BRAVO!!!

Last edited: Dec 22, 2011
13. Dec 22, 2011

Tracer

Thank you Naty1. Your post helped a lot. I now understand why others have said SRT does not apply.

14. Dec 22, 2011

Staff: Mentor

Yes. Roughly speaking it is curvature in the time dimension.