# How does the Hubble Redshift work?

I am a little confused about the workings of the Hubble Redshift. I do understand the classical Doppler-effect, however in special relativity the velocity of light c is a natural velocity limit. So how does Hubble Redshift come about?

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I'm going to assume you're just starting with your studies in relativity, in which case this paper will be way over your head. However, it can be fun to look through Appendix B, which lists many cases where very smart, well-known physicists got confused and mixed up Hubble expansion with the Doppler effect, when their physical mechanisms aren't related at all. –  Chris White Sep 15 '12 at 16:56
@ChrisWhite that would probably be better posted as part of an answer. –  David Z Sep 15 '12 at 21:55
A useful related question: physics.stackexchange.com/q/7522 –  David Z Sep 15 '12 at 21:56
@ChrisWhite: but, all of that is a matter of interpretation. All interpretations will require general relativity to analyze, but it is possible to have explosions of matter where you have horizons form relative to the comoving observers, but it looks like a perfectly normal explosion to the outside observers. In that case, is it a doppler effect due to relative motion or do to space expansion? It's both and neither--the interpretation is coordinate dependent. –  Jerry Schirmer Sep 15 '12 at 23:52
@Chris sure, no problem - it's never too late to post a good answer ;-) –  David Z Sep 16 '12 at 5:10

This is a great question, as it is both centrally important to modern astrophysics and cosmology, and it is misunderstood by very many people, including scientists themselves. Now the full, rigorous treatment requires general relativity, which I won't discuss in detail here. However, this is a topic that can be explained somewhat intuitively, so I'll give that a shot.

Coordinates: First, we're going to need a consistent coordinate system to work in. Let's say everyone measures time by the temperature of the cosmic microwave background (CMB). If you're not familiar with this, you can think of it as the background temperature of the universe, which we know to be monotonically related to time for any observer - it decreases as the universe cools off from the Big Bang. Now, I should be able to get the same physical results from any reference frame, inertial or not, but the interpretation of those results depends on the frame, and this is an intuitive one to work in.

Okay, that covers time. What about distance? Well, this is where cosmology gets complicated right away, and a proper explanation of distance goes a long way toward explaining many cosmological issues, including the cosmological (Hubble) redshift.

Suppose we had a network of intergalactic beacons stretching all the way from a distant galaxy (A) to us (B). Suppose they're so close together you could be at one of them and reach out your arm and touch the next. At a single moment in time (remember, all the beacons are synced to the CMB), every beacon measures the distance to its neighbor (whether with lasers or a meter stick, it doesn't matter on such small scales), and all these distances are summed up. That sum is called the proper distance between A and B.

Note on distances: Now you may think the distance between two things is a pretty uniquely defined concept, but in fact there are other useful distances. Things like comoving distance, luminosity distance, and angular diameter distance are all defined differently. All these distances would agree but for two caveats. First, GR allows spacetime to be "curved" (though we've found experimentally that this happens to be negligible in our universe). Second, and most relevant to our discussion, GR allows for space itself to be "expanding" or "contracting" as time goes on, and indeed this is happening in our universe.

Of course, lots of people say things like "space is expanding" without giving any clue as to what they mean. What I mean by that is that the proper distance between A and B will grow over time, even excluding changes brought on by any relative velocity per se. How could we exclude the possibility of relative motion? Let's say A and B are both at rest with respect to the CMB. This is doable, since if you were moving (in the special relativity sense) with respect to this background, photons would be (Doppler) blueshifted in front of you and redshifted behind you. In fact, we see a slight dipole in the CMB here on Earth, so we're not quite at rest with respect to it. So even with observers at A and B both at rest, the separation between them will be growing.

Emergence of redshift: If you can wrap your head around that (and you shouldn't be able to on first exposure - this stuff takes time to get used to), the rest is pretty straightforward. Suppose someone standing at A sends a beam of light toward B, and that light is fixed to have a wavelength of 1 meter at A. For concreteness, I'll be using numbers which reflect the actual state of things in our universe. Let's also pin down a time $t_\text{then}$ at 8 billion years ago, and say that the proper distance between A and B was at the time $d_\text{then} = 5\times10^{25}\ \mathrm{m}$ (that's around 5.5 billion light years).

If the universe were static, this is what would happen: After 5.5 billion years, B would receive the first photons from A. The wavelength would still be 1 meter, and if our intergalactic beacons had wavefront sensors they would tell us that there are indeed $5\times10^{25}$ peaks in the electromagnetic field evenly spaced between A and B.

However, the universe isn't static. That proper distance is growing, and so the photons have to travel a bit longer to reach B. In our universe, as it turns out, they have to travel for a total of 8 billion years. So what do we see here at B, at time $t_\text{now}$? Well, first let's consider what has been done at A. Suppose the beam was held steady for all this time, making a new EM peak every $(1\ \mathrm{m})/c$ unit of time (about 3 nanoseconds). There are enough peaks in this wave that they would extend 8 billion light years if they were all evenly spaced at 1 meter. Just for a consistency check, though, we query the intergalactic beacon network, and ask how far away A is (of course, we might have to wait billions of years for the information to get to us, but still, we can measure that number in principle).

To our great surprise, A is quite far away. Certainly more than 5.5 billion light years, and in fact more than 8 billion light years. As it turns out (because of the numbers I chose) A is exactly twice as far away as when we started this whole experiment! That is, $d_\text{now} = 2d_\text{then}$. Now picture the wave peaks. There are enough to cover 8 billion light years, but they're stretched over 11 billion light years. Clearly they're not all evenly spaced at 1 meter anymore. Actually, if we once more make use of our intergalactic beacons and have them examine the series of peaks and troughs, we would learn that the light close to A has a wavelength of 1 meter, and that close to B has a wavelength of a whole 2 meters.

In fact this shouldn't surprise us given that the total distance was multiplied by 2. The first wavelength covered 1 part in $5\times10^{25}$ of $d_\text{then}$ when emitted, and whatever "stretching" occurred should have occurred uniformly everywhere, just as much in the space between those first two consecutive peaks in the wave as in all the rest of intervening space. Today, as we receive the photon, it's wavelength is 1 part in $5\times10^{25}$ of $d_\text{now} = 2d_\text{then}$. In fact, the universe is twice the size (linear size, volumes have increased eightfold) now as compared to 8 billion years ago.

Comparison to Doppler: The big difference between what I've described and the Doppler shift is that cosmological redshift happens over time. In a typical textbook question about SR, you have two rockets with some relative velocity, and it doesn't matter how far apart they are, since the shift in wavelength just occurs whenever you switch to the other frame. Here the wavelength had a definite, continuously growing value the whole time in this CMB frame.

Final note on accelerated expansion: Okay, if you've made it this far, you're more of an expert than the vast majority of the world. But let's go just a little further, since this is the material of the most recent Nobel Prize in physics (as of this writing, September 2012). We can do a lot with these sorts of measurements, measuring simultaneously distance and redshift. Since we don't want to wait billions of years, we use the more accessible luminosity distance (that link has a definition but not much else in the way of content, alas). In any event, we can make redshift measurements at a bunch of different distances, and then we can put all this together to get an expansion history of the universe. Astronomers have been very interested in doing this ever since Edwin Hubble found that the universe wasn't static by fitting a line to the plot of redshift as a function of distance. (Side note: his data was awful, limited to only nearby galaxies where other effects dominate, but he was audacious if nothing else, so he gets props for that.) In particular, is the rate of expansion changing? We know that today the rate is $H_0 = 70\ (\mathrm{km}/\mathrm{s})/\mathrm{Mpc}$. That is, objects at a proper distance of 1 million parsecs will have that proper distance increase at a rate of 70 kilometers every second. In the late '90s, the people who eventually got that Nobel Prize did make more precise measurements, and they found that in fact the expansion was accelerating, much to everyone's surprise. The name we give to the as-yet-little-understood cause of this is "dark energy," (not to be confused with "dark matter") but that's a topic for another post.

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+1 fantastic treatment. I learned a lot. –  Brandon Enright Apr 27 '13 at 5:54
+1 It's the clearest explanation I've come across –  hdhondt Apr 13 '14 at 23:48

In addition to the special-relativistic version of the Doppler effect, there are other sources of redshift due to general relativity, in particular gravitational redshift (a consequence of gravitational time dilation which can also manifest as a blueshift) and cosmological redshift (a consequence of an expanding spacetime).

Hubble's law is due to the cosmological redshift: While light is travelling from source to observer, the wave is stretched because the distance between wavefronts increases.

Observed shifts will of course include contributions from all these effects.

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Technically, it's space that's expanding rather than spacetime, i.e., it's the metric expansion of space. –  Alfred Centauri Sep 15 '12 at 21:38
@AlfredCentauri: depends on choice of coordinates - use of conformal time leads to an overall scale factor –  Christoph Sep 15 '12 at 22:14
Very well then, I'll be more precise - it's the proper expansion of space. –  Alfred Centauri Sep 16 '12 at 15:34

I always like simplistic but consistent facts to understand something a bit better. Let's neglect cosmological expansion first and rather focus on the immediate vicinity. A rocket A is fired and a rocket B is considered to be a point at rest with respect to A's initial position. The visible spectrum of light is sent form A to B and B can see A's spectrum through a telescopic as a flat as monochromatic red light. As A reaches higher velocities the picture projected from B will have a color change as observed from A. The amount of shift or color change is directly proportional to the relative difference in velocity between the two rockets. If A was moving towards B at exactly the same velocity, but in the opposite direction, the color will move away from a red towards a blue color(whatever he transition between red and blue requires over the visible light spectrum . In other words, the amount of color change is directly proportional to the relative velocity between A AND B. Please tell me if I am right about this.

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Redshift happens due to the speed of the source. The speed of the medium is always treated as a constant. The size of the redshift is governed by the ratio of the relative speed of the source and observer and the speed of light (assumed to be less than one)

If you are interested in waves generated by an object moving faster than the speed of the waves in a medium, look into Cherenkov radiation

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Cosmological redshift really has nothing to do with actual velocities. Two objects both at rest with respect to the CMB will appear redshifted to one another due simply to the expansion of space between them. –  Chris White Sep 15 '12 at 18:57
@ChrisWhite: relative velocities are all there are. The only reason that you can say that sentence is that the CMB gives a preferred reference frame. An expanding universe is indistinguishible from a "hubble bubble" scenario, which WOULD be everything moving away from everything else. –  Jerry Schirmer Sep 15 '12 at 23:36
@ChrisWhite: note the Oppenheimer-Snyder solution to Einstein's equation, which models a collapsing pressureless matter distribution as a RW-spacetime in the interior of the distribution, and as Schwarzschild spacetime in the exterior. Now, imagine that the distribution is larger than the visible universe... –  Jerry Schirmer Sep 15 '12 at 23:41
What I mean is that no motions in, for instance, a global Minkowski space can duplicate what we observe. Recession velocities greater than $c$ can only come about if spacetime is not that of SR. –  Chris White Sep 16 '12 at 4:58
@ChrisWhite: fine, that doesn't contradict what I said. Why the downvote? –  Jerry Schirmer Sep 16 '12 at 5:30

I think it all has to do with wavelengths and frequency. As the object goes further, the wavelength is increased and the frequency in which you can interpret the wave is decreased. As the wavelength is increased and the frequency decreased, the natural light emissions encompass more red whereas alternatively, they encompass more blue [blueshift] i.e. in the visible spectrum. This does not mean that the light emission is actually being converted to red or blue at the source, it just means that saturation occurs [colour loss and gain respectively]. This gives the effect of added red. For instance, I think that if we were to watch the sun [giving the natural emissions being interpreted as yellow] from mercury [without aether], it will be a much more brilliant yellow or maybe even white as compared to watching it from jupiter, where it would look orange. The same concept can be applied to how we interpret sunlight passing through aether. The wavelength and frequency of the sunlight is slimmer at midday as compared to during sunset/sunrise, where the sun's light must travel further to reach your eyes. As a result, the light is saturated and the spectrum is more/less endorsed according to the distance variable.

I've provided an updated image to try to visually display how I see colour saturation, composition and redshift every day. I think that most people associate redshift only with astrophysics, when these phenomena are all available to us everyday right here on earth, only on a miniscule level. It's cool if it's ridiculed, but hey, it's physics and I'm no physicist, yeah?

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This is not correct at all. Light does not change color because it get tired from traveling long distances. There is certainly no cosmological redshift within our solar system, or even within our galaxy. Moreover, the aether theory was debunked about 100 years ago. –  Chris White Sep 15 '12 at 18:59
-1: This answer is completely wrong. Everyone, please downvote so no one gets confused by this wrong answer. –  FrankH Sep 15 '12 at 19:06
It doesn't get tired, only saturated –  Dorian Lyder Sep 15 '12 at 19:30
@Chris White my suggestion is based on my interpretation of light and my vision impairment. Due to my short sightedness, I see wavelengths much larger than they actually are, hence, normal streetlights at night display large light echos and colours become more saturated to me e.g. white lights gain colour. I could very much account for what takes place during redshift on a small scale (everyday) and I'm not speaking out of a BOOK... I don't propose that light gets tired, only that by increasing wavelengths, SATURATION occurs. –  Dorian Lyder Sep 15 '12 at 19:54
@FrankH, only those that understand why the answer is wrong should down-vote this answer. Folks, down-voting just because someone else asks you to is just plain wrong no matter how incorrect the answer may be. –  Alfred Centauri Sep 15 '12 at 21:32