# The cosmological redshift is not the result of a Lorentz transformation

#### marcus

Gold Member
Dearly Missed
We have a theoretical issue here.
There is a misconception floating around PF about the relation of the cosmological redshift to present and past recession velocity.

If a redshift is Doppler in origin then in the context of Special Relativity one has Einstein's correction of the Doppler formula
1+z = sqrt[(1+beta)/(1-beta)] where beta=v/c

But the relation of redshift to speed is different in cosmology and depends on how the expansion of the universe is modeled. One gets comparably simple formulas, but different ones, in some simple cases, but none of the usual models give the SR formula. Here are two formulas which Ned Wright mentions:

v = c ln(1+z) ------empty universe case

v = 2c[1-(1+z)-0.5] --------critical density, zero cosmological constant

These are obviously not the result of a Lorentz change of coordinates as in Special Relativity! They simply result from space stretching out and in the process lengthening wavelength, the usual explanation of cosmological, as opposed to Doppler, redshift. But for a side-by-side comparison with the Doppler formula we can do a little algebra on the Doppler formula and solve for v.

beta = [(1+z)2 - 1]/[(1+z)2 +1]

v = c [(1+z)2 - 1]/[(1+z)2 +1]

Ned Wright discusses the cosmological redshift in

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

Since it's a short passage containing the two formulas I first mentioned, I will quote the whole thing:

Can objects move away from us faster than the speed of light?

Again, this is a question that depends on which of the many distance definitions one uses. However, if we assume that the distance of an object at time t is the distance from our position at time t to the object's position at time t measured by a set of observers moving with the expansion of the Universe, and all making their observations when they see the Universe as having age t, then the velocity (change in D per change in t) can definitely be larger than the speed of light. This is not a contradiction of special relativity because this distance is not the same as the spatial distance used in SR, and the age of the Universe is not the same as the time used in SR. In the special case of the empty Universe, where one can show the model in both special relativistic and cosmological coordinates, the velocity defined by change in cosmological distance per unit cosmic time is given by v = c ln(1+z) which clearly goes to infinity as the redshift goes to infinity, and is larger than c for z > 1.718. For the critical density Universe, this velocity is given by v = 2c[1-(1+z)^-0.5] which is larger than c for z > 3 .
http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

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#### jeff

Ned's handling of the topic is not entirely correct because he failed to point out that gravitational fields also cause redshifting. Thus his empty universe formula is valid because there are no gravitating masses, but in all other cases such formulae are misleading because we don't know exactly how much of the redshifting is being caused gravitationally. If we did, we could factor it out and obtain a more precise recessional velocity. Unfortunately, it's still very common to hear about some high redshift quasar receeding at some percentage of the speed of light.

#### marcus

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Originally posted by steinitz
....Unfortunately, it's still very common to hear about some high redshift quasar receeding at some percentage of the speed of light.
Dont know if its unfortunate tho its common enough. If the speed of light is one's measuring stick for speed one hasnt much choice.
For example a recent quasar had redshift 6.4 corresponding to comoving distance (observed at rest w/rt Hubble flow) 28 billion LY and therefore Hubble law recession speed currently 2c.

Would you prefer if this were expressed as 6E8 meters per second? Or 6E5 km/s? Dont see point of "unfortunately"
It is a good yardstick.

#### jeff

Originally posted by marcus
Dont know if its unfortunate tho its common enough. If the speed of light is one's measuring stick for speed one hasnt much choice.
For example a recent quasar had redshift 6.4 corresponding to comoving distance (observed at rest w/rt Hubble flow) 28 billion LY and therefore Hubble law recession speed currently 2c.

Would you prefer if this were expressed as 6E8 meters per second? Or 6E5 km/s? Dont see point of "unfortunately"
It is a good yardstick.
By unfortunate, I'm referring to the way people like you who are just learning about cosmological redshifts are mislead by the way that professionals - even though they know better - report without qualification redshifts in terms of recessional velocities. It's better just to report the redshift itself and leave it at that.

#### marcus

Gold Member
Dearly Missed
Originally posted by steinitz
By unfortunate, I'm referring to the way people like you who are just learning about cosmological redshifts are mislead by the way that professionals - even though they know better - report without qualification redshifts in terms of recessional velocities. It's better just to report the redshift itself and leave it at that.
I am embarrassed for you---by you I mean steinitz

I do not know to whom you are referring in saying
"you who are just learning about cosmological redshifts"
"you who are misled by ....professionals"

When astronomers report their findings in journals or
at conferences they often do simply give redshift--but
not always---sometimes km/s speeds of recession are
given.

What you-steinitz-may be reading in the popular press, which
I gather must be a great source of information for you, does
not concern me. If you disapprove of what astronomers say to

In a professional journal, which is what I am talking about, if someone uses natural units c=G=hbar=1 and converts
velocity results into fractions of the speed of light that is
fine with me, and may actually be helpful to the people reading
the results.

I dont have time for this. Sorry. Bye

#### jeff

Originally posted by marcus
I am embarrassed for you---by you I mean steinitz

I do not know to whom you are referring in saying
"you who are just learning about cosmological redshifts"
"you who are misled by ....professionals"

When astronomers report their findings in journals or
at conferences they often do simply give redshift--but
not always---sometimes km/s speeds of recession are
given.

What you-steinitz-may be reading in the popular press, which
I gather must be a great source of information for you, does
not concern me. If you disapprove of what astronomers say to

In a professional journal, which is what I am talking about, if someone uses natural units c=G=hbar=1 and converts
velocity results into fractions of the speed of light that is
fine with me, and may actually be helpful to the people reading
the results.

I dont have time for this. Sorry. Bye
I post on sci.physics.research which is the newsgroup for professional researchers, but you can post there to. What you'll find is that smolin's book has been widely read by other high energy theorists. In fact it's quite common for scientists to read popular treatments by other scientists, not so much to learn anything that will fundamentally change their views, but to see how cleverly or elegantly the author was able to explain (or finesse where necessary) the difficult issues. In fact there are a number of sizeable threads on smolin's book because of it's controversiality, being scorned as it is by string theorists who regard it as excessively biased and self-serving. Go see for yourself.

As for you're extreme reaction to my honest responses to your posts, well that's really got nothing to do with me, now has it marcus? If you think I didn't realize immediately why you were so careful to claim that you'd never heard of one of the most well known recent works in popular science writing, you, my tempestuous friend, are mistaken. I answered you straightforwardly about the book out of politeness because I didn't want to embarrass you.

So, why am I here? Like most young researchers, I find it helpful to teach. But sci.phys is so heavily and inefficiently moderated that it takes days for posts to appear. So I'm giving this forum a spin. If you want to learn, I'm happy to help. You'll also find that when I discover I'm wrong - even when helping someone who I think knows less than I do - I have few qualms about admitting it. So don't take things so personally because you're really not doing yourself any favours.

Anyway, I'm not going anywhere and will continue to respond to you're posts.

Last edited:

#### marcus

Gold Member
Dearly Missed
Originally posted by steinitz
I post on sci.physics.research which is the newsgroup for professional researchers, but you can post there to. What you'll find is that smolin's book has been widely read by other high energy theorists. In fact it's quite common for scientists to read popular treatments by other scientists, not so much to learn anything that will fundamentally change their views, but to see how cleverly or elegantly the author was able to explain (or finesse where necessary) the difficult issues. In fact there are a number of sizeable threads on smolin's book because of it's controversiality, being scorned as it is by string theorists who regard it as excessively biased and self-serving. Go see for yourself.

As for you're extreme reaction to my honest responses to your posts, well that's really got nothing to do with me, now has it marcus? If you think I didn't realize immediately why you were so careful to claim that you'd never heard of one of the most well known recent works in popular science writing, you, my tempestuous friend, are mistaken. I answered you straightforwardly about the book out of politeness because I didn't want to embarrass you.

So, why am I here? Like most young researchers, I find it helpful to teach. But sci.phys is so heavily and inefficiently moderated that it takes days for posts to appear. So I'm giving this forum a spin. If you want to learn, I'm happy to help. You'll also find that when I discover I'm wrong - even when helping someone who I think knows less than I do - I have few qualms about admitting it. So don't take things so personally because you're really not doing yourself any favours.

Anyway, I'm not going anywhere and will continue to respond to you're posts.

#### marcus

Gold Member
Dearly Missed
Jeff ("steinitz" the name of a world chess champion) says he is a young researcher who has come to PF because he finds it helpful to teach.

The field mentioned here is "high energy theory"

Jeff says he posts at a newsgroup for professional researchers---suggesting that he is a professional researcher----tho not quite saying so outright.

and I honestly did not know what book he was referring to.
It turns out that he was referring to a book called " 3 roads to
q. gravity". Anyway Jeff seems to be accusing me of lying when I said I didnt know what book of Smolin's he meant.

I'm reserving comment but feel there is more here than meets the eye.

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