# Expansion of space

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• Martyn Arthur

#### Martyn Arthur

TL;DR Summary
The expansion of space between the Milky Way and galaxies beyondf visible space.

Say we have two galaxies, the Milky Way (MW) and galaxy A separated along a straight line 20 x the radius of the Universe visible to us.

If I understand correctly then the rate of the physical expansion of space between MW and A could be at the rate of say, arbitrarily, 10 x c.
They aren't moving through space, the entirety of space between them is physically expanding.

If we then have galaxy D positioned along the same line, just at the edge of our visible Universe which we observe being redshifted at the rate of 1 x c.
Yet the entirety of space between MW and A is expanding at the [arbitrary[ rate of 10 x c.

So that the elastic band comprising the space between MW and A must presumably be carrying everything that is within it at the same rate.
So then why then is it that galaxy D is not being transported within that elastic band towards MW at 1/2 x 10 x c?

Thanks
Martyn

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Delta2

Say we have two galaxies, the Milky Way (MW) and galaxy A separated along a straight line 20 x the radius of the Universe visible to us.
If I understand correctly then the rate of the physical expansion of space between MW and A could be at the rate of say, arbitrarily, 10 x c.
I haven't done the math but I believe it would be considerably larger than 10c
They aren't moving through space, the entirety of space between them is physically expanding.
That's probably not a helpful way to look at it. Spacetime is geometry. Things are just getting farther apart.
If we then have galaxy D positioned along the same line, just at the edge of our visible Universe which we observe being redshifted at the rate of 1 x c.
Too low. Things at the outer reaches of the OU are receding at about 3c.
Yet the entirety of space between MW and A is expanding at the [arbitrary[ rate of 10 x c.
A is receding from us at 10c (again, a wrong number but yes according to your scenario). BUT ... 10c is a SPEED but recession is a rate, so "rate of 10c" is nonsensical.
So that the elastic band comprising the space between MW and A must presumably be carrying everything that is within it at the same rate.
So then why then is it that galaxy D is not being transported within that elastic band towards MW at 1/2 x 10 x c?
I'm not clear on how you did your math here but as nearly as I can tell it is inconsistent and nonsensical even within your own misunderstanding between speed and rate.

That's not how the expansion works.

The elastic band model has severe limitations, but it works reasonably well here - you just seem to be misapplying it. Imagine three dots on a rubber band in a straight line, each one 10cm from the next. Now stretch the rubber band so that the end dots, which were 20cm apart, are now 40cm apart. Where is the middle dot? It's still half way between the end dots, so 20cm from each end dot. So, from the perspective of the end dot the next one moved 10cm and the far end one moved 20cm in the same time. So the speed of galaxies grows with distance.

That's why the expansion rate is characterised as a rate not a speed. There isn't a single speed that things move at. Usually we use the Hubble constant, which has dimensions of inverse time, not distance per unit time.

malawi_glenn, PeroK and Melbourne Guy
Expansion of space is fully explained by General Relativity or do we need some additional theory like the one we use to partially explain dark energy?

Expansion of space is fully explained by General Relativity or do we need some additional theory like the one we use to partially explain dark energy?
It's fully explained by GR.
When he called the cosmological constant his "greatest blunder" it really was a blunder; if he had instead listened to what the equations told him, he could have predicted the expanding Universe!

Instead, others figured it out using Einstein's equations for GR.

https://www.forbes.com/sites/starts...lly-was-a-tremendous-mistake/?sh=7be2f58e50f4

Expansion of space is fully explained by General Relativity or do we need some additional theory like the one we use to partially explain dark energy?
I wouldn't say dark energy is an additional theory. You can treat it as an additional term in the field equations if you want, but you can also lump it in with the rest of the stress-energy tensor. I'd say it's closer to adding a then-unknown 8th planet to solar system models to explain discrepancies in the motions of the known seven than a new theory. Admittedly we're positing a new kind of stuff rather than more of the same (Neptune is just another planet) but even then I wouldn't really regard it as a change to GR. YMMV, I guess.

PeroK and PeterDonis
even then I wouldn't really regard it as a change to GR. YMMV, I guess.
I don't think it's even a matter of mileage varying. The simplest way to derive the Einstein Field Equation is from the Einstein-Hilbert Lagrangian, and the way that Lagrangian is defined makes it clear that the cosmological constant term belongs in the Lagrangian and hence in the field equation, since it meets the same constraints as the Ricci scalar term (a Lorentz scalar involving no higher than second derivatives of the metric--the constant is a zeroth derivative, so to speak, but that counts as no higher than a second derivative). Whether or not the numerical value of the term is significant is something the theory alone can't tell you; you have to look at observations, and we now know from observations that its numerical value is indeed significant for cosmology. But even if it had turned out not to be, the term would still be valid theoretically as a part of the EFE and it would not be correct to look at it as a "change" to GR.

Ehm guys I don't have the background to fully grasp your replies but are you saying that GR even explains dark energy?

Thanks for these replies.
Excuse me for my lack of detailed knowledge; I seek just to understand not to challenge!
I am just a first-year undergraduate and haven't got to the equations yet.
I note what was said about the rate of change.
Moving back then to the visible universe.
Space seems to be expanding circa c at the edge of that 'visibility'
Is there a calculation/formula that dictates the rate of expansion from our stationery perspective to a galaxy distance of 20 x the radius of the visible universe?
Is there a general formula/equation that defines/quantifies the expansion of space between two galaxies at a separation of 20 x the radius of our visible universe, without there being any other factors?
Thanks
Martyn

Ehm guys I don't have the background to fully grasp your replies but are you saying that GR even explains dark energy?
Perhaps better to say that dark energy requires no modification or rethinking of any part of GR, or even that applying GR to the observed behavior of distant objects leads us to predict the existence of dark energy.

Delta2
are you saying that GR even explains dark energy?
It explains how dark energy affects the expansion of the universe, yes.

It doesn't "explain" what exactly dark energy is, but GR in general does not "explain" what any components of the stress-energy tensor "are" in terms of what they are made of. That's not what GR is for. GR is for explaning why the spacetime geometry is what it is, given whatever particular stress-energy tensor is present.

Delta2
Ehm guys I don't have the background to fully grasp your replies but are you saying that GR even explains dark energy?
Dark energy is just one more kind of stress-energy as far as GR is concerned. You compare observation to models not including it and you back out an equation of state to describe how its density and pressure evolve and there you are. It doesn't tell you anything about what it is, just a little about how it behaves.
Whether or not the numerical value of the term is significant is something the theory alone can't tell you; you have to look at observations, and we now know from observations that its numerical value is indeed significant for cosmology. But even if it had turned out not to be, the term would still be valid theoretically as a part of the EFE and it would not be correct to look at it as a "change" to GR.
Sure. I was just thinking that you sometimes see it as a cosmological constant with ##\Lambda g_{ab}## written out as a separate term on the left of the Einstein field equations. That suggests to me that some people do think of it as a change to the model rather than just another stress-energy tensor term, whatever I think.

Is there a calculation/formula that dictates the rate of expansion from our stationery perspective to a galaxy distance of 20 x the radius of the visible universe?
That's just the Hubble's law: ##V_{rec}=H_0d## where ##H_0## is the value of the Hubble parameter at present epoch (aka the Hubble's constant), V is the current recession velocity, and d current distance (of the proper kind). As you can see, it's a simple proportionality. So twice the distance means twice the recession velocity. 20x the distance means 20x the velocity. And so on.

Space seems to be expanding circa c at the edge of that 'visibility'
This is not correct. It's closer to 3c.

Is there a general formula/equation that defines/quantifies the expansion of space between two galaxies at a separation of 20 x the radius of our visible universe, without there being any other factors?
Yes and no. It's all governed by the Friedmann equations, which can be derived from the Einstein Field Equations for the special case of a homogeneous and isotropic universe. However, those equations have different solutions depending on the amounts and mix of matter, radiation and dark energy you put in. You can plug in the numbers as far as we know them, but you'll need a numerical integrator to get an answer.

Space seems to be expanding circa c at the edge of that 'visibility'
So, I take it that for some reason, you didn't believe me when I told you in post #2 that it's 3C, not c. A factor of 3 might be trivial in some things but this isn't one of them.

One reason it matters is that when you quote "c" for that, some people will see that as being based on the fact that nothing travels faster than c, and this makes it harder for them to understand recession, which is not "traveling" in the sense of proper motion.

Ehm guys I don't have the background to fully grasp your replies but are you saying that GR even explains dark energy?
In a sense. As others noted, GR doesn't care about the contents of the universe.

But there's also a deeper sense in which GR (sort of) predicted dark energy: the cosmological constant. This constant is a necessary feature of the theory, based upon how the equations are derived. But because the value needs to be a very tiny number for us to exist at all, it was assumed for a long time to simply be zero.

Lo and behold, we measure a tiny value for "dark energy", which the cosmological constant could explain. It could be something else too, which is why we still call it dark energy instead. But there really is no theory which is nearly as clean and reasonable as the cosmological constant.

Though many theorists still feel that the value of the cosmological constant is too small to be believed, so they continue to look for some other explanation.

Delta2
many theorists still feel that the value of the cosmological constant is too small to be believed
Can you be more specific? As I understand it, there are no particular theoretical reasons from within GR itself to think that the value of the cosmological constant we actually measure is too small to be believed; and anthropic arguments (which are admittedly much more speculative) indicate that the value we observe is just small enough to allow us to exist.

The main line of argument I'm aware of that would indicate that the value of the cosmological constant is too small to be believed is that based on quantum field theory, which gives an answer (at least on a certain set of more or less reasonable assumptions) about 120 orders of magnitude larger than the value we observe. But the QFT line of argument itself is based on viewing the "cosmological constant" term in the EFE as not being due to any intrinsic property of spacetime itself (which is what the term "cosmological constant" normally implies), but due to the quantum vacuum having nonzero stress-energy (which would be more aptly described as a theoretical explanation for dark energy).

Thank you all.
The technical understanding of the debate into GR et al is yet beyond my first-year BSc Physics level but that hopefully will come to me.
Just one last layperson question then if I may please, for a first-year physics student's understanding, specifically defined by redshift.
Just basic redshift measurements have identified that galaxies at the perimeter of our visual universe are receding from the Milky Way at circa the speed of light.
Leaving aside the various debates for reasons, could we expect that a galaxy 20 x the perimeter of our visual universe would show a commensurate increase in redshift?
Thanks
Martyn

Delta2
Just basic redshift measurements have identified that galaxies at the perimeter of our visual universe are receding from the Milky Way at circa the speed of light.
Three times the speed of light - as has been stated several times in this thread.
could we expect that a galaxy 20 x the perimeter of our visual universe would show a commensurate increase in redshift?
Well, it's beyond our observable universe, so it doesn't make sense to try to work out what redshift we would observe. No light ever reaches us from it, not even in principle, so there's no way to calculate what frequency it would have when it reached us.

I am not seeking to be difficult and apologize if this is abstract.
Should I take from this that, from what is "known";
Given the MW as a stationery point then the separation between them, the distance, however, defined would be increasing as a factor of c.?
Thanks
Martyn

Given the MW as a stationery point then the separation between them, the distance, however, defined would be increasing as a factor of c.?
The rate at which the distance to the distant galaxy grows is well defined (in our models - obviously we can't directly conform it). It's Hubble's Law, as @Bandersnatch noted, so something 20× further away goes 20× as fast - so around 60c in this case. However, this is not associated with a redshift because you can't model light traveling from such a distance to us in order to calculate the red shift.

Leaving aside the various debates for reasons, could we expect that a galaxy 20 x the perimeter of our visual universe would show a commensurate increase in redshift?
What can be said about this, though, is that redshift doesn't scale linearly with distance. It's not the case that 2x any distance corresponds to 2x the redshift. In particular, redshift goes to infinity (as per the model) at a finite distance, called the cosmic event horizon. This horizon is approx 1.5x further out than the current extent of the observable universe. So you will never be able to even observe a galaxy currently at 20x the distance. I.e. the concept of redshift for such a galaxy doesn't apply.

Delta2 and phinds
Thank you again for your patience with this old fella (age 70 in his first year of a Physics BSc.).
I am trying to get a basic, basic "interpretation".
Can I please leave aside the absolute equations etc and revert to trying to get a concept of the overall concept of the the situation of just then two items.
The Milky Way and a galaxy a distance 20 x that of the radius of the visible Universe.
What can be deduced/predicted/suggested as a rate of the expansion of space between those two items?
What are the potentially anticipated consequences if this is extrapolated to an infinite rate of expansion?
Thanks
Martyn

Delta2
The Milky Way and a galaxy a distance 20 x that of the radius of the visible Universe.
What can be deduced/predicted/suggested as a rate of the expansion of space between those two items?
I'm struggling to understand what you want to know that we haven't already said. We can calculate the speed of the galaxy. We cannot translate this to a redshift because light never reaches us. We cannot actually test our claims about its speed because we cannot observe it, but the observations underlying our model seem to hold good so we think we're right.

What are the potentially anticipated consequences if this is extrapolated to an infinite rate of expansion?
I don't understand what you are asking.

What are the potentially anticipated consequences if this is extrapolated to an infinite rate of expansion?
I think they (@Bandersnatch ) said something about the doppler shift being infinite beyond a certain distance, not the rate of expansion being infinite.

Sorry, it's me that's simple not you guys; trying to phrase a suggestion in layperson terms to those who function in the mathematical et al environment.
Please put this question down to a nonmathematical person.
The rate, my understanding, the measured rate, of expansion of our known visible Universe is I understand to be x.
That is that say, galaxies are moving apart (space is expanding) at a defined rate of x within our visible universe.
Without knowing anything at all about circumstances beyond our visual Universe.
Can we / should we extrapolate that the rate of expansion of space, of a universe that is / may be infinite, is potentially infinite?
Martyn

Delta2
We have no reason to believe that the rate of expansion is anything different outside the observable universe than it is inside the observable universe. The rate is approximately 73.3 kilometers per second per megaparsec.

Delta2
73.3 kilometers per second per megaparsec.
what does it mean the per megaparsec , that in distance 10 megaparsec from here, the rate of expansion is 733 kilometers per second?

what does it mean the per megaparsec , that in distance 10 megaparsec from here, the rate of expansion is 733 kilometers per second?
That is not a rate, that is a speed. The distance between two objects 10 Mpc apart grows by 733 km every second.

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Can we / should we extrapolate that the rate of expansion of space, of a universe that is / may be infinite, is potentially infinite?
When you say 'rate', it has a specific meaning in cosmology. It's the Hubble parameter (whose present value is called Hubble's constant). Its meaning is similar to the rates banks give you when you deposit your savings - all <quantity> grow by certain percentage per unit time.
The <quantity> in case of banks is the deposited money. I.e. every month or year, or whatever time period, whatever anyone has put in there will grow by this percentage. Regardless of who has how much money in the bank, the savings will grow at the same rate.

In cosmology, the quantity is distance. Every unit of time (continuously, unlike in a bank) all distances, regardless how large, or between what two points, grow by this percentage.

This rate is (or, so the model says) constant across the entire universe. It doesn't go up the farther you go, same as rates in a bank are the same, and don't go up the more money you have there.

But, if you deposit more money in a bank than some other person, the same rate as everyone else's will nett you more money every month. This is analogous to how larger distances in the universe grow faster (faster recession velocity). You can get the recession velocity however high you want, as long as you pick two sufficiently distant points. But at each moment in time the rate stays the same across the entire universe.

Or at least there are good reasons to think it ought to, since it can't be measured everywhere.

Ibix
That is not a rate, that is a speed. The distance between two objects 10 Mpc apart grows by 733 km every second.

That is not a rate, that is a speed. The distance between two objects 10 Mpc apart grows by 733 km every second.
And after approximately 30,000,000,000 years (twice the current age of the universe) they will be 20mpc apart and receding at 1466 km / sec

The velocity due to expansion of space we call it recession velocity. How do we call the "real" velocity due to real motion?

The velocity due to expansion of space we call it recession velocity. How do we call the "real" velocity due to real motion?
Proper motion.

Delta2
Proper motion.
What is the cause of expansion of space and what is the cause of proper motion?