Thread by @mpoessel

Here's a non-expert version of my earlier tweet threads about cosmology. The key question behind those threads was and is: what is a good way to think about cosmic expansion? Should you think about space itself expanding? Or about galaxies moving (through space)? #GenRelEdu 1/

I've been arguing (as have a number of others) that the latter viewpoint has more pedagogical merit. (But a judgement like that has of course a subjective component. Others may disagree.) And I admit that the viewpoint introduces a complication. There are at least two 2/

ways of defining the "speed at which a distant galaxy moves away from us as the universe expands". The first way follows directly from the definition of cosmic expansion, which is that distances between far-away galaxies change in a very specific way: they all change by the 3/

same factor. Crank up that factor from 1 to 1.1 over time, and all distances between galaxies will be 10% larger than they were initially. As long as you don't worry about what the words "time" and "distances" even mean in that context, you can introduce what cosmologists 4/

call the recession speed of a distant galaxy: the way the galaxy's distance from us changes over time ("first time derivative of the [time-dependent] distance"). That recession speed v is what you need in the famous Hubble-Lemaître relation v=Hd, with H the Hubble constant and 5/

d the galaxy's distant from us. Plotting v vs. d for distant galaxies and finding something like a straight line is what made astronomers sit up and take cosmic expansion seriously in the first place! 6/

But Einstein's combination of space and time are tricky, and so is the notion of speed in an Einsteinian universe. For more distant galaxies, there are good reasons to ascribe to a galaxy another "speed at which they are moving away from us", which we might 7/

call "relativistic radial speed". (I am not distinguishing conscientiously between speeds and velocities here, admittedly.) That second kind of speed has several advantages: It's closer to usual physical notions of speed than the recession speed (but you need to dive into 8/

general relativity a bit deeper to see that for yourself). One aspect of this is that, for all the galaxies we see around us, the relativistic radial speed always stays below the speed of light. (The recession speed becomes faster than light speed for galaxies beyond a certain 9/

distance, which has people wondering how that can possibly be compatible with Einstein's statement that the speed of light is the highest speed there is.) Also, for the relativistic radial speed, the so-called cosmological redshift is a so-called Doppler shift, something that 10/

many people are familiar with from basic physics. (The acoustic version is something you are probably familiar with from everyday situations when an ambulance passes you by, the pitch of it's siren higher when it is moving towards you, lower when it's moving away.) 11/

For not-all-that-distant galaxies, the way that the galaxy's light is shifted to longer wavelengths (=redshift) can be interpreted as a Doppler shift, either of the recession speed or of the relativistic radial speed – at those low speed values, the difference between the two 12/

doesn't matter. At higher distances, you can still write the cosmological redshift as a Doppler shift in terms of the relativistic radial speed – but there is no such interpretation in terms of the recession speed. In my view, a clear advantage of the former. It lets you 13/

use a helpful interpretation of a fundamental phenomenon (the cosmological redshift) not just as an approximation, but up to speeds arbitrarily close to the speed of light. And that Doppler interpretation is helpful for another reason. If you follow a light particle from a 14/

distant galaxy to our own, the cosmological redshift will mean that the light particle has a lower energy when we measure its energy in our own galaxy, when the light particle arrives, compared to what an observer on a distant galaxy measured at the time the light particle 15/

began its journey. If you don't know that there is relative motion between the galaxies, that might make you wonder where that extra energy went. Isn't energy supposed to be conserved? In fact, energy conservation *is* somewhat tricky in general relativity, but you don't need 16/

to go there in this particular case: Even in classical physics, energy values change if you transition from one reference frame to a another frame in relative motion. Think about a gnat and a truck approaching each other. In the system where the gnat is at rest, the total 17/

energy of that situation is huge - it's the kinetic energy, 1/2 times the relative speed of gnat and truck times the mass of the truck! In the frame where the truck is at rest, the energy is small: It's again the kinetic energy, which this time is 1/2 times the relative 18/

speed of gnat and truck, but in this frame times the mass of the gnat – a factor of more than a billion, given that a truck weighs several tons, and a gnat several milligrams. But nobody expects energy values to be the same if you change from one frame to another! There's 19/

no law demanding that energy values remain the same when you change your frame of reference. So that is where viewing galaxies in an expanding universe as being in relative motion again helps to understand what is happening – and once you talk about relative motion and 20/

the cosmological Doppler effect, you'll need to specify what the relative speed is. It is, as I mentioned, the relativistic radial speed. Another advantage of that relativistic radial speed is apparent when you think about which regions of the cosmos can communicate with 21/

us. There is a simple picture that goes like this: as we consider ever more distant galaxies, the speed at which they fly away from us gets faster and faster. (This statement holds true for both types of speed - recession and relativistic radial.) 22/

What would you expect when we get to regions where the speed approaches (for the relativistic radial speed) or even exceeds (for the recession speed) the speed of light? Naively, you might expect that this defines a kind of limit – for slower, less distant galaxies you would 23/

expect their light to reach us; for more distant galaxies, you might expect that we couldn't even see those galaxies – they are flying away so fast that their light, which an after all only fly at light-speed, can never catch up with us! 24/

For recession speed, that intuition fails. The "Hubble sphere", defined as the location of all galaxies so distant that their recession speed equals the speed of light, has no direct connection with the "cosmic event horizon", that is, with the boundary that separates 25/

those regions of the cosmos whose light we will be able to receive, at some distant future time, from those regions whose light can never reach us, however long we wait. (Those latter regions are "behind the horizon".) 26/

This situation, too, is much more convenient for the relativistic radial velocity. In that case, the boundary that is the cosmic horizon coincides with the boundary where the relativistic radial velocity approaches the speed of light. So instead of having to explain away the 27/

simple intuition "galaxies behind the cosmic horizon are moving away so fast their light will never reach us," you can make it work for you. There are some reservations (the relativistic radial velocity is simply not defined for galaxies behind the cosmic horizon – no simple 28/

"they are faster than light!" there) but the basic intuition leads to the right idea.

Of course, the relativistic radial speed has a fundamental disadvantage as well. When talking about cosmic expansion, you will need to introduce not only one speed, but two: 29/

You cannot get around the recession speed, that is needed for the Hubble-Lemaître relation v=Hd, and it has a simple relation to the basis of cosmic expansion (that thing with the scale factor) that is too good to miss. So you cannot just leave that out. 30/

But that means that, if you want to take advantage of the relativistic radial speed, you will need to introduce another speed concept later on. And the mathematics behind that is not simple; when talking to the general public or high-school students, you will need to 31/

just state "here's the way it is!" without being able to present a derivation your audience can follow. That *is* a drawback, but I think it is not a very big one. The reason is that, even if you assiduously avoid introducing the relativistic radial speed, you cannot 32/

get around the little "and this is where it gets more complicated, I will just tell you the result" dance. That is because, where it counts, namely when it comes to a relationship that astronomers can actually *observe*, that relationship is simple for less-distant 33/

galaxies – use the classical formula for the Doppler effect to get from the recession speed to a Doppler redshift; relate that redshift to the distance via the Hubble-Lemaître relation v=Hd. But after that, for more distant galaxies, it gets complicated. If you want to 34/

mention more modern observations, e.g. of distant supernovae as distance markers, the point will inevitably come where you will need to say "and this is where it gets more complicated, I will just tell you the result" anyway. And since you need to do this anyway, I 35/

don't think that adding "also, the speed at which galaxies move away from us becomes more complicated than v=Hd" makes things fundamentally worse or more difficult. That is why I think talking about the usual recession speed for less-distant galaxies and switching to 36/

relativistic-radial speeds for more distant galaxies is a good way to think about cosmic expansion. And remember, at low speed values, the two speeds are basically the same. You can keep it simple by talking just about "the speed", meaning the relativistic 37/

radial speed but, as you talk about less-distant galaxies, exploiting the fact that in that regime at least there's no difference to the usual recession speed, and doing the cool derivations of the Hubble-Lemaître relation. That way, it's somewhat the best of both worlds. 38/

Anyway, the connection to my earlier threads. In the previous thread, I explained the same as here, but for people who are already somewhat familiar with the cosmology behind it. I've gone into more basic detail here, but this is the short version:

https://twitter.com/mpoessel/status/1551163616031903744

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The thread before that was about the physical and mathematical background, for people with basic knowledge of general relativity. Both this and the other thread I mentioned have references for those who want to dig deeper:

https://twitter.com/mpoessel/status/1551114774926204929

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The first thread was heavier on the sociology of it all. The point of view I'm presenting here is not new/original, but it hasn't gained all that much traction. Most physicists/astronomers do not use the relativistic radial velocity when they talk about cosmology to a lay 41/

audience. In the first thread, I went into the history, and why I think that the literature on the subject hasn't caught on as much as it should have:

https://twitter.com/mpoessel/status/1551109240995155969

- so, now 4 threads all in all. Part of the reason: I was sorting my own thoughts along the way! 42/42