Effort to get us all on the same page (balloon analogy)

Hello mint,
sorry, I didn't see your post until just now.
There are two Einstein Relativities. The 1905 one was SR (special) and the later extension to include curved and dynamically changing geometry is the 1915 (general) one.

Every man-made mathematical theory has limits to its applicability and cannot be pushed too far. As a student when you learn the math model of something you are also told the limits---how far you can trust its predictions, where it blows up and fails to compute meaningful numbers, or starts to diverge from reality.

A major fault of pop-sci journalism is it often fails to make this clear.

The 1905 special only applies locally in situations that are approximately uncurved and approximately non-expanding. It is only perfectly right in situations that dont exist, namely where space is perfectly uncurved, or flat, triangles always add up to 180 degrees, parallel lines...etc. etc. and that simply is never true. And locally means temporarily too, since we are talking spacetime.

Basically, 1915 general trumps 1905 special. But as a rule nature deviates from static flat geometry so slightly and slowly that 1905 works admirably well! It is only over very long distances and time periods that deviation is pronouced enough to be intrusive.

Be sure you have watched Wright's balloon model, the animated film. Better watch several times.
A galaxy (the white spiral whirls) is at rest with respect to the microwave background if it stays at the same longitude latitude on the balloon. All the galaxies are at rest. The distances between them increase.

We have this rule that information cannot travel faster than c.

Think about two galaxies, both sitting still relative to background, and the distance between them increasing. Neither is going anywhere, no information is being transmitted from one place to another, by the mere fact that the distance between is increasing.

Neither of the galaxies could overtake and pass a photon!

The speed law only applies to motion within a given approximately flat frame of reference, where SR applies.

So two things can sit still, and the distance between them can be increasing at 4 or 5 times the speed of light, and nobody is breaking any rules.

You can see that in the balloon animation. The wiggles that change color and have lengtheningwavelengths represent photons. They move at a standard speed like one millimeter per second. But if you watch alertly you will see that after a photon has left the neighborhood of some galaxy the separation between it and the galaxy will begin to grow faster than c, and after a while will be several times c. It's remoteness is increasing several times the rate it can travel on its own.

http://www.astro.ucla.edu/~wright/Balloon2.html

I'll have to go and have a study up on this, it's totally counter-intuitive. I don't understand how red shift is possible within this model. It seems that within this model that light speeds greater than c are directly proportional to the distance from source and my understanding of red shift as a measure of age and distance (probably wrong, I'm just a lay-nut) is that wavelengths stretch over distance because of increasing recession with distance, not despite it.
Anyway I'll go look at the animation and see if it clicks with me..

One of the first things you learn in an introductory cosmo class is not to think of the redshift as a doppler effect. It is not the result of some particular speed.
The formula involves the entire factor by which distances have been expanded during the whole time the light has been traveling.


roughly speaking

1+z = size(now)/size(then)

Technically the index of size used is called the "scalefactor" usually written a(t) as a function of universal time. Intuitively it is just a handle on the size of the universe or (if that is too vague and undefined) the average distance between galaxies. The exact definition involves a differential equation modeling the growth of a(t), the expansion history of the universe.

So you get taught that
1+z = a(now)/a(then)
the factor by which distances have increased from the moment the light was emitted until the moment it reached our telescope.

Since z is the fractional increase in wavelength, that is the amount added to it, it must be that 1+z is the ratio by which wavelength(now) is bigger than wavelength(then).

So wavelengths have increased by the same factor that large astronomical distances have, during the same time interval.

==================
Pop-sci journalism often misleads readers by presenting the redshift z as a Doppler effect.
Presumably the Doppler shift due to some particular recession speed at one moment in history. But it is not. It is the integrated stretch due to the whole history of expansion during the light's transit.

Just another instance of pop-sci media damage that we are constantly having to recover from.

One of the first things you learn in an introductory cosmo class is not to think of the redshift as a doppler effect. It is not the result of some particular speed.
The formula involves the entire factor by which distances have been expanded during the whole time the light has been traveling.


roughly speaking

1+z = size(now)/size(then)

Technically the index of size used is called the "scalefactor" usually written a(t) as a function of universal time. Intuitively it is just a handle on the size of the universe or (if that is too vague and undefined) the average distance between galaxies. The exact definition involves a differential equation modeling the growth of a(t), the expansion history of the universe.

So you get taught that
1+z = a(now)/a(then)
the factor by which distances have increased from the moment the light was emitted until the moment it reached our telescope.

Since z is the fractional increase in wavelength, that is the amount added to it, it must be that 1+z is the ratio by which wavelength(now) is bigger than wavelength(then).

So wavelengths have increased by the same factor that large astronomical distances have, during the same time interval.

==================
Pop-sci journalism often misleads readers by presenting the redshift z as a Doppler effect.
Presumably the Doppler shift due to some particular recession speed at one moment in history. But it is not. It is the integrated stretch due to the whole history of expansion during the light's transit.

Just another instance of pop-sci media damage that we are constantly having to recover from.

So there are 2 kinds of doppler effect, one is from motion the other from space expansion.

I feel better in understanding universe and physics from this dialogue.
I have a new question:
Suppose that we got a new spectrum picture from a star, and that picture shows several dark lines with shift, and we speculate the object might be moving very fast but do not know the distance. Now from that shift pattern can we tell if it comes from space expansion or local motion ?

Regarding the distance advanced by light in expanding universe , I did some calculation to get the result:

[tex]
D(T)\ = {c \over r} (\left( 1 + {r*dt} \right)^{T \over dt} -1)
[/tex]

Taking the limit dt going to 0,
[tex]
D(T)\ = {c \over r} (e^{rT} -1)
[/tex]
where
D(T): distance advanced by light during period T.
c: speed of light
T: time from emission to present.
r : space expansion rate 1/140 % per million.
dt: the arbitrary small time intervals in T.
** In case r goes to 0, D(T) goes to c*T as expected.

I got this formula by adding each light path segment advanced for each dt, that is after the last dt, the D0 (distance advanced of the last dt) is D1=c*dt*(1+r*dt), and the 2nd last one D2=c*(1+r*dt)^2, and so on .. Dn=c*(1+r*dt)^n. From summing D1 D2 ..Dn, I got above formula.
I do not want to use the word speed to avoid confusion, but it is just the distance of light advanced after a period T.

wavelength(now)/wavelength(then) = distances(now)/distances(then)
or more formally:
1+z = a(t_rec)/a(t_em)

I can't see anything in this that I'm not understanding

the same basic equation can be used to calculate Doppler for sound waves.

Oh I see what you mean. What a(t) means here is the universe scale factor. You are drawing an analogy where a(t) is the distance between emitter and receiver, and the emitter is moving in still air.

Ich! I am glad to see you. My memory is unreliable but I have the notion (perhaps wrong) that you live somewhere in south Germany and know a fair bit of mathematics. I am glad that you sometimes glance at this thread. Thanks for any and all help!

Mint, if we were in a situation where Doppler applied, we would use

[tex]1+z = \sqrt{\frac{1+\beta}{1-\beta}}[/tex]

The correct Doppler formula for light in special relativity. We would not use the formula appropriate for sound from moving source in still air, which by coincidence looks like the correct one for redshift if you interpret the scalefactor a(t) as distance between source and receiver.
When I think Doppler, I think the formula I wrote for you there.
It goes crazy when recession rates equal or exceed the speed of light. The Doppler formula (which is correct for actual motion) is completely different and completely wrong for redshift. (Only works as approx for nearby slow receding things.)

Ich! I am glad to see you. My memory is unreliable but I have the notion (perhaps wrong) that you live somewhere in south Germany and know a fair bit of mathematics. I am glad that you sometimes glance at this thread. Thanks for any and all help!