# Copernican Principle and Metric Expansion

• FireBones
In summary: However, I don't think it follows that the shift must be cosmological in origin simply because the Copernican Principle tells us that we are not at the center of the universe. In fact, I think your argument might be evidence that the shift is not cosmological in origin.

#### FireBones

I have seen over and over a description of how the Copernican Principle led people to believe Hubble's redshifts were cosmological (due to metric expansion) rather than typical Doppler shifts. I'm trying to understanding the logic of this, but it appears like just a revision of history. From what I can tell, the Copernican Principle (by itself) gives us no reason to suspect Hubble's red shifts are due to cosmological effects rather than plain, vanilla velocity-induced Doppler Shifts.

Hubble's observations showed all galaxies are red-shifted (receding from Earth) and the magnitudes of these redshifts (the speed) was proportional to their distance away from the Earth. The argument I've seen (implicitly) goes something like this.

1. Either the redshift seen by Hubble was

i)due to the Earth honestly being at the center of the universe and everything was receding from it due to ordinary velocity, in which case the redshift is from the Doppler Effect.

OR

ii) The redshift is not from the Doppler effect but is due to metric expansion of the universe itself (the stretching of space...in which case the balloon analogy is used to explain how this is different from mundane velocity that would show up as the Doppler effect)

2. The Copernican Principle rules out option "i)" above because we are not at the center of the universe.

3. Hence, "ii)" is correct and the red shift is due to metric expansion.

QED

But this just looks like shoddy logic. From what I can tell, there is absolutely nothing in option "i" that requires Earth to be at the center of the universe.

Indeed, simple vector algebra should show that if everything is doppler-shifted away from any point in space in a way consistent with Hubble's law, then the same applies to all points in the space without resorting to metric expansion.

If Alex if flying away from me at 45 miles per hour, then (in Alex's rest frame) I am flying away from Alex at the same speed.

If Bob is halfway between me and Alex, and I see Bob running away from me toward Alex, but only at half the speed, then Alex (in his rest frame) see's Bob running away from him toward me, but only at half the speed.

This reminds me of one way of seeing part of the Coriolis effect. We normally think that the North Pole is "special' in that someone sitting on a frictionless dais would see the Earth rotate underneath him. But it turns out that the same is true for ANY point on the Earth (other than the Equator). If you took a frictionless seat mounted on a pole anywhere at all and sat down, you would see the Earth rotate under you (less rotational speed the closer you are to the equator). Having a net rotation under your seat does not mean you are at the north pole, and seeing everything moving away from you (even if it is just due to standard, everyday velocity-induced redshift) does not mean you are the center of the universe...in fact, it means that you see the same thing that everyone else does.

Nothing here is meant to suggest that Hubble's redshifts are not cosmological. Rather, I am replying to the common argument mentioned above that presumes to show that the Copernican Principle dictates which explanation is used. Since neither Doppler nor Cosmological red shifts would require the Earth to be near the center of the universe, the Copernican principle has no relevance whatsoever.

Of course, if I'm wrong about the above, I'd be interested in seeing what I am missing...but at the very least it looks like people who make this argument (which I see everywhere) are trying to sweep things under the rug, hoping their audience will just blindly accept their argument without thinking it through.

it looks like people who make this argument (which I see everywhere) are trying to sweep things under the rug, hoping their audience will just blindly accept their argument without thinking it through.
Well, most of the time, they get away with it.

i) doesn't work, because the expansion is basically uniform. All you need to do is show that if you move over to the side by some amount, the expansion looks exactly the same.

The second point is that in order to have "pure doppler shifts", you would need an empty universe, one with no mass whatsoever. There's a lot of mass out there, so that's clearly wrong.

I don't think you understand the point of my post.
I'm not debating whether the shifts are Doppler, gravitational, or cosmological. I'm referring to the use of the Copernican Principle. In fact, your first point seems to be reinforcing my post since you ended up saying precisely what I did later.

I think this is a very interesting post FireBones. I agree with your argument that a Doppler redshift is equally consistent with the Copernican Principle as the cosmological redshift. However, the Copernican Principle has been supplanted by Cosmological Principle: "NOBODY is at the center of the universe". In this case, Doppler redshifts fail because there must be some point of marking the center of the universe. Am I understanding you correctly?

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FireBones said:
I don't think you understand the point of my post.
I'm not debating whether the shifts are Doppler, gravitational, or cosmological. I'm referring to the use of the Copernican Principle.

If the isotropic Hubble red shift were doppler in nature then almost every other galaxy would be rushing away from us alone, which would place us in a very special place, at the 'centre of expansion'. This is against the Copernican Principle which says there is nothing special about our location, i.e. not the Earth, nor the Sun, nor the Milky Way are in the centre of the universe.

If our galaxy is not uniquely in the centre of the expansion then observers in every galaxy must observe a similar effect that is, the other galaxies are rushing away from them. If this is so then the whole universe must be expanding and all the galaxies are in fact rushing away from each other.

In your thought experiment about Bob and Alex, if that were true for observers in every galaxy then the expansion would be truly isotropic and Hubble red shifts are due to cosmological expansion.

The question is what is this 'cosmological expansion'? Is it a metric effect or due to actual velocities relative to a background non-spatially expanding space-time?

The answer depends on what gravitational theory you use to analyse the Hubble observation. If that theory is GR, and solar system experiments lead us to accept GR as the most accurate gravitational theory we have, then the cosmological solution of that theory predicts such a cosmological expansion as an expansion of space. That is, test particles move apart in successive foliations (slices) of cosmological space-time.

I hope this helps,
Garth

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bapowell said:
I think this is a very interesting post FireBones. I agree with your argument that a Doppler redshift is equally consistent with the Copernican Principle as the cosmological redshift. However, the Copernican Principle has been supplanted by Cosmological Principle: "NOBODY is at the center of the universe". In this case, Doppler redshifts fail because there must be some point of marking the center of the universe. Am I understanding you correctly?

Actually, the "Cosmological Principle" does not state that "Nobody is at a special location," it simply says that the universe looks the same locally at every point. It is a statement of homogeneity. You can violate this principle while keeping the Copernican principle and vice-versa (almost).

What you refer to is better labeled the "Strong Copernican Principle," that not only we, but also no one else, is in a special location. However, that is a bit of misnomer, since the sun was obviously at the center of his cosmology.

But getting back to the meat of your post: Doppler redshifts do not fail any of these principles (so long as we allow an Open Euclidean space). If everything is moving away from Earth at a rate linearly dependent to its distance from Earth, then the same is true everywhere else. Earth would not be at the center of this expansion, it would only appear to be the center of it.

Here is a simple mathematical representation. Pick any point in the universe you want and label it as (0,0,0) [note, this is the center of our coordinate system, but NOT the center of the universe...it is just for mathematical convenience.] Now, let r(p,0) be the position of point p in this coordinate system at time 0. Now, if you let r(p,t) = (At+1)*r(p,0) [in other words, Hubble's law with respect to the point (0,0,0) assuming Doppler shifts], then you will find that EVERY OTHER POINT in your system also obeys Hubble's law (assuming this motion).

This is the basic vector math I mentioned in my post. Even when the red shift is plain, old, ordinary Doppler shift, if any place in the universe obey's Hubble's Law, then all points in the universe do so, and hence a Doppler shift interpretation of Hubble's Law contravenes neither the strong nor weak versions of the Copernican principle.

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Right. But your assumed motion needs to be symmetric about some point, no? If I have a universe with randomly moving particles in it, then nobody will derive Hubbles Law from their location. In the case that expansion causes the redshift, then as long as peculiar velocities are small, then observers at all points in the universe will derive Hubble's Law. In the case of the Doppler effect causing the redshift, then all points must have the prescribed motion you've mentioned, which is necessarily symmetric about some point. This is the difference that I see between these two.

Garth said:
If the isotropic Hubble red shift were doppler in nature then almost every other galaxy would be rushing away from us alone, which would place us in a very special place, at the 'centre of expansion'. This is against the Copernican Principle which says there is nothing special about our location, i.e. not the Earth, nor the Sun, nor the Milky Way are in the centre of the universe.

But this is exactly what I am claiming is not true. If the Hubble red shifts were doppler in nature, then it WOULD NOT put us at the center of expansion, as the vector math I gave earlier shows. Yes, we would see everything moving away from us isotropically, but an observer at any other point would see the same thing, simply due to basic vector math, hence my Bob and Alex example.

Let the Earth be at point (0,0,0). To be moving away [not cosmologically, just plain old, everyday velocity] isotropically means that the chunk of matter that is at point (x,y,z) at time 0 must obey the motion law r(t) = (At+1)*<x,y,z>

Now, consider the point (1,1,1), this motion law can be rewritten as:

r(t) = (At+1)*<x-1,y-1,z-1> + (At + 1) * <1,1,1>

So the distance between this chunk of matter and the chunk of matter that had been at point (1,1,1) at time 0 is (at any other time):

q(t) = r(t) - (At + 1) * <1,1,1> = (At +1) * <x-1, y-1, z-1>

But that is exactly Hubble's law with regard to the point (1,1,1).

So, as I said earlier, once Hubble's law is true (using doppler interpretation of red shift) at any point in the universe, it becomes true FOR ANY OTHER POINT as well (using doppler interpretation)...and hence the Copernican Principle has nothing to do with it because no point because the single "center of expansion." Each point simply thinks it is.

bapowell said:
Right. But your assumed motion needs to be symmetric about some point, no? If I have a universe with randomly moving particles in it, then nobody will derive Hubbles Law from their location. In the case that expansion causes the redshift, then as long as peculiar velocities are small, then observers at all points in the universe will derive Hubble's Law. In the case of the Doppler effect causing the redshift, then all points must have the prescribed motion you've mentioned, which is necessarily symmetric about some point. This is the difference that I see between these two.

Hubble's law presume's symmetric motion, and I'm assuming Hubble's law.

Yes, the motion is symmetric about some point, but it is also symmetric about every point. That is my point. Hubble's law (regardless of whether you take the redshift as being doppler or cosmological) does not merely call for symmetric expansion from a single point. It calls for symmetric expansion from every point...which is not a problem w.r.t. the Copernican principle.

Just imagine an infinite string [1-dimension] with a person at each unit distance. Now, imagine the person at x=1 starts walking to the right at 1 mph, the person at x=2 starts walking to the right at 2 mph, etc. The person at x=-1 starts walking to the LEFT at 1 mph , the person at x=-2 starts walking to the left at 2 mph.

Clearly, the person at x=0 sees himself as the center of expansion. People to his right are moving to the right, and people to his left are moving to his left.

But what about the person at x=1? To him, the person at x=0 is moving to his left at 1 mph, the person at x=-1 is moving to his left at 2 mph, etc. But the person at x=2 is moving to his right at 1 mph, etc. In other words, the person at x=1 ALSO sees himself as the center of expansion.

So, instead of creating a single "center of expansion," hubble's law being observed from the Earth simply means every point in the universe becomes a center for expansion, regardless of whether that expansion is mechanical, everyday, ordinary velocity or whether it is cosmological expansion.

Firebones You are correct, this was Milne's point in his original model of 'Kinematic Relativity'.

The Copernican Principle simply says the Earth/Milky Way is not at a special place in the universe.

The fact remains that having GR as a theory of gravitation verified by means of the precession of perihelion of Mercury and deflection of light, means its cosmological model is the one to use to interpret the Hubble red shift observations.

Garth

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Thanks, Garth, for verifying my point regarding the Copernican Principle.

However, isn't it possible for some solution to the GR equations to allow for a doppler interpretation? Not FLRW, obviously, but my understanding is that there is a wide variety of solutions to GR equations, especially if other assumptions are weakened. Several inflation models, after all, violate the cosmological principle at the greatest scale.

Furthermore, there is no need for all the redshift to be of one type. Even in a cosmologically expanding universe, the red shift could be a combination of both doppler and cosmological redshift.

FireBones said:
Several inflation models, after all, violate the cosmological principle at the greatest scale.
Could you elaborate on this claim?

bapowell said:
Could you elaborate on this claim?

Any inflation model where false vacuum regions persist beyond the initial moments would qualify, and Guth's original version (where the vast, vast, vast majority of the Universe is false vacuum) certainly is included in that.

The cosmological principle claims that the universe "looks the same at large scales wherever you go," but a big blotch of false vacuum certainly does not "look the same" as a big blotch of "normal" matter. It does not even obey the same laws of physics...at least in some (and I would assume all) articulations.

[I'm using "false vacuum" since that was Guth's original articulation. Feel free to use your own name for inflationary regions.]

Garth said:
If that theory is GR, ... then the cosmological solution of that theory predicts such a cosmological expansion as an expansion of space. That is, test particles move apart in successive foliations (slices) of cosmological space-time.
Garth, I hesitate to respond because I don't want to get into a debate about whether the 'expanding space' or 'kinematic' paradigm is more correct. I do not believe there is any principled basis for distinguishing between the two paradigms.

But I do not see any basis for claiming that the 'expanding space' paradigm is required or preferred by GR.

Comoving coordinate charts portray receding galaxies as if they were stationary on an expanding background, but that is merely a useful convention, and is only one of numerous alternative ways to chart the expansion. So perhaps the 'expanding space' paradigm is just the 'kinematic' paradigm charted in a different coordinate system.

If GR applies in all respects to a closed system comprised of two galaxies moving away from each other 'through' space, then it should be equally applicable to an open system comprised of an infinite number of such galaxies receding 'through' space. How would the latter necessarily violate GR?

Where is it written in the EFE that motion of recession caused by Inflation has an inherently different physical character than the peculiar motion caused by, say, gravity? A massive particle can be boosted to a peculiar velocity and then that peculiar motion will decay spontaneously at the rate of 1/a until only the Hubble motion remains. The two kinds of motion seem completely interchangeable, as if the Hubble motion were in fact kinematic motion.

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nutgeb said:
Garth, I hesitate to respond because I don't want to get into a debate about whether the 'expanding space' or 'kinematic' paradigm is more correct. I do not believe there is any principled basis for distinguishing between the two paradigms.
Milne's theory of Kinematic Relativity was set in flat space-time. (Note: this is not the same as the Milne FWR model of GR, which is simply the limiting case of the empty universe - though both had a linearly expanding scale factor a(t) = t and in the empty universe case space-time is also flat in GR.) KR uses Special Relativity alone, which is why the experimental evidence of the deflection of light and precession of perihelia and the geodetic precession falsify it.
But I do not see any basis for claiming that the 'expanding space' paradigm is required or preferred by GR.

Comoving coordinate charts portray receding galaxies as if they were stationary on an expanding background, but that is merely a useful convention, and is only one of numerous alternative ways to chart the expansion. So perhaps the 'expanding space' paradigm is just the 'kinematic' paradigm charted in a different coordinate system.

If GR applies in all respects to a closed system comprised of two galaxies moving away from each other 'through' space, then it should be equally applicable to an open system comprised of an infinite number of such galaxies receding 'through' space. How would the latter necessarily violate GR?
It doesn't, unless you are saying the galaxies recede at a constant velocirty
Where is it written in the EFE that motion of recession caused by Inflation has an inherently different physical character than the peculiar motion caused by, say, gravity?
(Note: Cosmic expansion is not caused by Inflation, that hypothesis simply enormously enhances the expansion while the Inflation scalar field is rolling down its potential energy slope.)

Cosmic expansion is simply an effect of the cosmological gravitational field as described by GR.
A massive particle can be boosted to a peculiar velocity and then that peculiar motion will decay spontaneously at the rate of 1/a until only the Hubble motion remains. The two kinds of motion seem completely interchangeable, as if the Hubble motion were in fact kinematic motion.
If GR accurately describes the phenomena of gravity then the motion is under the influence of gravitational forces, i.e. space-time curvature. It is not kinematic. Certainly you can interpret the Hublle red shift observations by Milne Kinematic Relativity, however doing so you would be ignoring GR while holding onto SR.

Garth

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Garth said:
Milne's theory of Kinematic Relativity was set in flat space-time. (Note: this is not the same as the Milne FWR model of GR, which is simply the limiting case of the empty universe - though both had a linearly expanding scale factor a(t) = t and in the empty universe case space-time is also flat in GR.) KR uses Special Relativity alone, which is why the experimental evidence of the deflection of light and precession of perihelia and the geodetic precession falsify it.

It doesn't, unless you are saying the galaxies recede at a constant velocirty
I guess I wasn't clear about terminology. When I use the term 'kinematic', I do not intend to invoke Milne's particular idiosynchratic model. On the contrary, I intend it to mean that the expansion rate is affected by gravity and the cosmological constant (if there is one), so it behaves completely as required by GR. The only difference is that I intend 'kinematic' to mean that for analytical purposes galaxies are treated as if they are moving 'through' space, not being carried along by 'expanding space.' GR certainly recognizes the concept that objects can move 'through' space.
Jorrie said:
Cosmic expansion is simply an effect of the cosmological gravitational field as described by GR.
I don't follow what you mean by that statement. I start from the assumption that the cosmological gravitational field definitely causes the Hubble rate to decelerate progressively over time, as a function of density. (And the cosmological constant supposedly causes it to re-accelerate.) But I don't see any basis for saying that the 'cosmological gravitation field' literally was the original 'cause' of the cosmic expansion.

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nutgeb said:
Cosmic expansion is simply an effect of the cosmological gravitational field as described by GR.
I don't follow what you mean by that statement. I start from the assumption that the cosmological gravitational field definitely causes the Hubble rate to decelerate progressively over time, as a function of density. (And the cosmological constant supposedly causes it to re-accelerate.) But I don't see any basis for saying that the 'cosmological gravitation field' literally was the original 'cause' of the cosmic expansion.

(Actually I think it was me not Jorrie!)

I'm not saying anything here about the 'original cause', the theory breaks down at t=0, although we can get pretty close to it. However after t=0 the cosmological solution of the GR field equation requires the universe to be expanding or contracting, not static, we just happen to be in an expanding stage.

Garth

Don't know how Jorrie crept in!

Anyway, yes I agree that GR requires expansion and sometimes contraction. All I'm saying is that expansion 'through space' seems to be as consistent with GR as expansion 'of space.'

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## What is the Copernican Principle?

The Copernican Principle states that the Earth is not in a special or central position in the universe, and that the laws of physics are the same everywhere in the universe.

## How does the Copernican Principle relate to Metric Expansion?

The Copernican Principle is closely related to Metric Expansion, as it implies that the universe is homogeneous and isotropic, meaning that it looks the same in all directions and at all locations. Metric Expansion is the idea that the space between objects in the universe is constantly increasing, causing the universe to expand.

## What evidence supports the Copernican Principle?

There is a vast amount of evidence that supports the Copernican Principle, including observations of the cosmic microwave background radiation, the large-scale structure of the universe, and the redshift of distant galaxies. All of these observations align with the idea that the universe is homogeneous and isotropic.

## Does the Copernican Principle apply to other planets and solar systems?

Yes, the Copernican Principle applies not only to the Earth but to all planets and solar systems in the universe. This principle is not limited to our own solar system, but extends to all objects in the universe and their relationship to one another.

## How does the Copernican Principle impact our understanding of the universe?

The Copernican Principle has greatly impacted our understanding of the universe, as it has led to the development of the Big Bang theory and the concept of Metric Expansion. It has also challenged our perceptions of the Earth's place in the universe and has allowed us to more accurately model and study the universe as a whole.