# Calculating distance from redshift

1. Jul 3, 2012

### chris1969

Hello everyone

I've starting to learn about Hubble's law and I have a very simple question. How are the velocities to distant objects calculated from a redshift? I understand the basic principle, that faster objects have longer wavelengths, but I'm not sure about the formula which links the two.

The wikipedia page has some formula for redshift, but the cosmological formula doesn't seem to have a term in velocity.

Chris

2. Jul 4, 2012

### Chalnoth

Right, that's because it's usually not considered in cosmological contexts, because the velocity of far-away objects is not well-defined. Usually it's just taken to be Hd, where H is the current Hubble parameter and d is the comoving distance.

3. Jul 4, 2012

### marcus

Re: Calculating [velocities] from redshift

For small local velocities a handy rule of thumb is simply that a Doppler shift of 1/1000 corresponds to radial (towards or away) speed of c/1000.
That is, about 300 km per second.

We distinguish between small DOPPLER shifts caused by small local motions, and COSMOLOGICAL REDSHIFTS caused by the expansion of the universe's geometry---the distance expansion rates---the rates we see large scale distances increasing without anybody actually getting anywhere.

Hubble law distance expansion rates are a different story from Doppler. You should probably get familiar with the convenient online calculators. For example, google "wright calculator" and put in a redshift, like 3, and press calculate.
It will give you a distance. Unfortunately it does not give a distance expansion rate, but you can calculate that yourself using Hubble law, if you want.

If you are interested in expansion rates, a handy shortcut is to use an online calculator with more features like where it says "...ocalc.2010.htm" in my signature. That one gives you the distance expansion rate too, as well as the distance itself.
Put in 3 for the redshift and it will tell you that the current recession rate is some multiple of the speed of light.
I just checked. The rate it gives is 1.53 c. About 53% faster than the speed of light.

Hubble law distance expansion rates should really not be called "velocities". It confuses people because it makes them think that geometry expansion is like ordinary motion (where you get somewhere).
In geometry expansion nobody gets anywhere---distances between everybody just get larger. Typically at rates faster than the speed of light. (The recession rate is proportional to distance and most objects we observe are far away enough that the distances to them are expanding faster than c.)

Last edited: Jul 4, 2012
4. Jul 5, 2012

### chris1969

Many thanks both. I think I'm getting there, but I have a follow-up question if that is ok. Ned Wright's tutorial re-produces Hubble's 1929 plot of velocity against distance : does this refer to velocity as a layperson would conventionally understand it, or the distance expansion rate.

When we say the velocity of far away objects are not defined, does this mean that they could be moving quickly or slowly with respect to us? To put it another way, geometric expension increases with distance, but velocity needn't?

Many thanks once again

Chris

5. Jul 5, 2012

### marcus

For small distances it doesn't make any difference---you can think of it either way. And use either formula.
cosmologists' distance expansion redshift formula: 1+z = R(now)/R(then)
or
Doppler shift formula: z = v/c

Everything you do in cosmology involves fitting mathematical maps or models to nature and the fit is always approximate (even when amazingly good.)
So it's a bit like pasting flat maps over the surface of the earth. The fit is good if the chunk is small. VELOCITY as we usually think of it is defined in terms of some straight foursquare framework (not curved not expanding). The locally adequate "flat map" idea.

If you take too big a chunk of spacetime a simple framework won't accurately encompass/represent the whole thing.
Two things with the same velocity (same speed and direction) might eventually crash, or veer widely apart. Curvature is bad enough but expanding geometry (a kind of curvature in 4D) makes it even worse.

Simple flat rectilinear coordinate maps don't fit when you try to cover large distances and timespans. So good local concepts like VELOCITY become problematical.

So one rule of thumb might be that for redshifts z > 0.01 try to think in terms of "distance expansion rate" and use the formula 1+z = R(now)/R(then).

But for redshifts z = 0.01 that would only correspond, if it were a Doppler effect of ordinary motion, to one percent of the speed of light: 3000 kilometers per second. So if you like, think of it as a Doppler shift! And the distances and times involved would be comparatively small. The Hubble expansion rate changes over time, but only slowly. So over short intervals of time (cosmologically speaking) it is a constant "the Hubble constant".

It's just an arbitrary rule of thumb but it might help resolve things for you.

Last edited: Jul 5, 2012
6. Jul 9, 2012

### Lino

Marcus, This is the first time that I have heard this described in this fashion. It does make alot of sense, but can I ask it is intended to be taken literally or is it more of a reflection on margins of error with variables / measurements?

Regards,

Noel.

7. Jul 9, 2012

### andrewkirk

I found Bunn & Hogg's paper very helpful in coming to a sort of understanding of this issue.

In section III they describe how the recessional velocity can be meaningfully defined by parallel transporting the four-velocity vector of the distant galaxy, at the time it emits the light we see now, along a geodesic to the observer's current spacetime event. We then measure the recessional velocity as vrel = sqrt(1-1/g(vob,vem)2), where vob and vem are the observer's current four-velocity and the emitter's parallel transported four-velocity respectively.

Bunn and Hogg claim that if the geodesic along which we parallel transport is the lightlike geodesic that the light we see follows then the formula gives us a recessional velocity that gives the observed redshift when plugged into the SR Doppler redshift formula. I wonder whether this is the recessional velocity usually intended when physicists refer to recessional velocities.

I have been wondering what would happen if we instead parallel transported the emitter's four-velocity at the current cosmic time, along a spacelike geodesic, to our current spacetime event. Would that be the best way to define a "current" recessional velocity? And would the value be uniquely defined, ie would there be only one such spacelike geodesic in a LDCM universe?

Last edited: Jul 9, 2012