I Galaxy recession and Universe expansion

[cianfa72]

TL;DR Summary

About the Galaxy recession and universe expansion that cannot be understood as an ordinary velocity

Hi,
I'm aware of the measured recession of the galaxies in our universe and the universe expansion itself cannot be understood as an "ordinary" velocity/speed (for instance in the FRW solutions of Einstein's equations).

Can you kindly help me clarify this topic ?

Thank you.

 

[Hill]

I think that "ordinary" velocity is relative velocity in flat (Minkowski) spacetime. The galaxies in question are not.

 

[jbriggs444]

We can certainly consider a relative velocity between two things that are right next to one another. Either the velocity of A in the rest frame of B. Or the velocity of B in the rest frame of A. Both will yield the same number. There is no ambiguity.

In the flat space time of special relativity, there is no problem doing the same comparison at a distance. The inertial rest frame of A extends to cover B and the velocity of B relative to A is still well defined. Similarly for the rest frame of B and the velocity of A relative to B. There is still no ambiguity.

In the curved space times of general relativity, inertial rest frames are local. They do not extend globally. If A and B are at a distance from one another, there is no unambiguous way to extend a locally inertial coordinate system at A so that it covers B. Accordingly, the velocity of B relative to A is ambiguous. It depends on a choice of coordinate system.

You can do a trick known as "parallel transport" to incrementally carry a coordinate system (or at least a direction) from A to B. But it turns out that in curved space times, the result depends on the path you use. So your velocity measurement is still ambiguous.

In the flat space time of special relativity, parallel transport is independent of path. It works fine.

 

[cianfa72]

You can do a trick known as "parallel transport" to incrementally carry a coordinate system (or at least a direction) from A to B. But it turns out that in curved space times, the result depends on the path you use. So your velocity measurement is still ambiguous.

Yes, I'm aware of it. The point I made, however, is different: we can try to define, taking into account all the aforementioned ambiguities that arise from spacetime curvature, a relative velocity for bodies following a timelike path "inside" the spacetime.

For the entire universe, however, is there an analogous concept of expansion velocity/rate ?

 

[jbriggs444]

we can try to define, taking into account all the aforementioned ambiguities that arise from spacetime curvature, a relative velocity for bodies following a timelike path "inside" the spacetime.

We can try to define a velocity for a single body. But unless there is a symmetry that picks out a global coordinate system, can we succeed in doing so unambiguously?

For the entire universe, however, is there an analogous concept of expansion velocity/rate ?

In our universe, there is an identifiable foliation called out by a symmetry of the space time. Co-moving coordinates. So yes, we can define an expansion rate.

 

[A.T.]

I'm aware of the measured recession of the galaxies in our universe and the universe expansion itself cannot be understood as an "ordinary" velocity/speed (for instance in the FRW solutions of Einstein's equations).

Can you kindly help me clarify this topic ?

A good illustration of the distinction between "ordinary" relative motion and recession due to metric expansion is the ant on a rubber rope:
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope

 

[cianfa72]

In our universe, there is an identifiable foliation called out by a symmetry of the space time. Co-moving coordinates. So yes, we can define an expansion rate.

As far as I understand, the symmetry of spacetime does mean it admits a timelike KVF. Its (timelike) integral curves define a 1-dimensional foliation of spacetime. Therefore the timelike coordinate $t$ of Co-moving coordinates actually represents in that global chart such KVF integral curves.

Hypersurfaces of constant coordinate time $t$ are spacelike and define a 3-dimensional foliation of spacetime.

Edit: reading on PF's threads about this, are the Co-moving coordinates the coordinates w.r.t. the CMB is "at rest" ?

 

[PeterDonis]

As far as I understand, the symmetry of spacetime

Which spacetime? There is no such thing as "the symmetry of spacetime"; there is only symmetry of particular spacetimes, i.e., particular solutions of the Einstein Field Equation. Some solutions have symmetries, some don't.

It appears that in this particular case you mean FRW spacetime, i.e., the solution we use to describe our expanding universe. In that case, your statements are wrong. See below.

the timelike coordinate $t$ of Co-moving coordinates actually represents in that global chart such KVF integral curves.

No, it doesn't. There is no timelike KVF in FRW spacetime.

Hypersurfaces of constant coordinate time $t$ are spacelike and define a 3-dimensional foliation of spacetime.

This is true, but it does not mean there is a timelike KVF. It means the congruence of comoving worldlines is hypersurface orthogonal, but that congruence is not associated with a timelike KVF.

are the Co-moving coordinates the coordinates w.r.t. the CMB is "at rest" ?

They are coordinates in which the CMB, like the universe in general, is isotropic--the same in all directions--and homogeneous--the same at every point in a given surface of constant coordinate time $t$.

 

[PeterDonis]

I'm aware of the measured recession of the galaxies in our universe

This "measured recession" is not a speed. We measure redshift, brightness, and angular size. Any "recession speed" is a calculation, not a measurement.

the universe expansion itself cannot be understood as an "ordinary" velocity/speed (for instance in the FRW solutions of Einstein's equations).

That is correct.

When cosmologists talk about "recession speed", what they mean is this: from the observed redshift, brightness, and angular size of a particular galaxy, using our best current model of the spacetime geometry of the universe, calculate which comoving worldline the galaxy is on. Then calculate the coordinate speed [Edit: rate of change of proper distance with respect to time], in FRW coordinates, of that comoving worldline in the surface of constant coordinate time $t$ that corresponds to our "now" here on Earth. That coordinate speed [Edit: rate of change] is then given as the "recession speed" of the galaxy.

Note that at least two things are being done here that cosmologists don't usually talk about when they give recession speeds. First, we are assuming that the galaxy is on a comoving worldline, but most galaxies are not; they have some nonzero "peculiar velocity" relative to comoving worldlines in their vicinity. (This is true of our own galaxy and of us here on Earth.) Second, we are taking observations of the galaxy as it was some time in the past--because the light we see now took time to get to us--and extrapolating them forward to our "now".

 

[cianfa72]

They are coordinates in which the CMB, like the universe in general, is isotropic--the same in all directions--and homogeneous--the same at every point in a given surface of constant coordinate time $t$.

Ok, so isotropic and homogeneous are actually properties on each spacelike hypersurface of constant comoving coordinate time $t$.

So, in general, comoving coordinate time $t$ is not the same as the FRW coordinate time (i.e. the coordinate time of the FRW solution in standard FRW coordinates).

 

[Ibix]

So, in general, comoving coordinate time is not the same as the FRW coordinate time (i.e. the coordinate time of the FRW solution in standard FRW coordinates).

What do you mean by "comoving coordinate time" if you don't mean FLRW coordinate time?

 

[cianfa72]

What do you mean by "comoving coordinate time" if you don't mean FLRW coordinate time?

I don't know exactly. @PeterDonis talked in post#9 about comoving worldlines of galaxies.

 

[PeterDonis]

isotropic and homogeneous are actually properties on each spacelike hypersurface of constant comoving coordinate time $t$.

More precisely, that particular set of spacelike hypersurfaces is defined as the set which is everywhere orthogonal to comoving worldlines, and comoving worldlines are defined as the worldlines of observers that always see the universe as homogeneous and isotropic.

So, in general, comoving coordinate time $t$ is not the same as the FRW coordinate time (i.e. the coordinate time of the FRW solution in standard FRW coordinates).

Wrong. They are the same. I have no idea why you would think they are not.

 

[PeterDonis]

I don't know exactly. @PeterDonis talked in post#9 about comoving worldlines of galaxies.

Post #9 says nothing whatever about any coordinate time. As you note, it talks about comoving worldlines.

 

[cianfa72]

and comoving worldlines are defined as the worldlines of observers that always see the universe as homogeneous and isotropic.

What do you mean with worldlines of observers that always see the universe as homogeneous and isotropic ?

 

[PeterDonis]

What do you mean with worldlines of observers that always see the universe as homogeneous and isotropic ?

I'm not sure what is unclear about the statement. Do you know what "worldlines" are?

 

[cianfa72]

Do you know what "worldlines" are?

Yes, I know. I don't fully understand the usage of the term see in relation with those worldlines of observers.

 

[PeterDonis]

I don't fully understand the usage of the word see in relatton with those worldlines.

It means every observer following such a worldline observes/measures what I described.

More precisely, as an example, every such observer would measure the CMB to have the same temperature in all directions. That is an indication that the universe is seen to be isotropic. And if we pick out one particular spacelike hypersurface, at the event on each such worldline that is in that hypersurface, each observer would measure the CMB temperature to have the same numerical value. That is an indication that the universe is seen to be homogeneous.

 

[cianfa72]

More precisely, as an example, every such observer would measure the CMB to have the same temperature in all directions. That is an indication that the universe is seen to be isotropic.

Ah ok, however I believe the measurement of CMB temperature done from comoving observers in all spatial directions is actually a local measure.

Wrong. They are the same. I have no idea why you would think they are not.

Just to be sure: are comoving worldlines the same as the timelike curves of constant spacelike coordinates and varying comoving coordinate time $t$ in FRWL standard coordinate global chart ?

 

[PeterDonis]

I believe the measurement of CMB temperature done from comoving observers in all spatial directions is actually a local measure.

It's local for each observer, taking just that observer's measurements in isolation. But the measurements of different comoving observers at different locations are not local to each other.

are comoving worldlines the same as the timelike curves of constant spacelike coordinates and varying comoving coordinate time in FRWL standard coordinate global chart ?

Yes. That's how the FRW coordinate chart is defined.

 

[cianfa72]

When cosmologists talk about "recession speed", what they mean is this: from the observed redshift, brightness, and angular size of a particular galaxy, using our best current model of the spacetime geometry of the universe, calculate which comoving worldline the galaxy is on. Then calculate the coordinate speed, in FRW coordinates, of that comoving worldline in the surface of constant coordinate time $t$ that corresponds to our "now" here on Earth. That coordinate speed is then given as the "recession speed" of the galaxy.

Maybe I misinterpreted this point. Assume that a galaxy is on a comoving worldline (i.e. it has fixed spacelike coordinates in FRWL global coordinate chart). How can we calculate its coordinate speed in the hypersurface of constant comoving coordinate time $t$ that corresponds to our "now" on the Earth ?

 

[Ibix]

How can we calculate its coordinate speed in the hypersurface of constant comoving coordinate time that corresponds to our "now" on the Earth ?

Depends on your coordinate chart.

In the usual FLRW coordinates the coordinate speed of comoving observers is zero.

 

[PeterDonis]

How can we calculate its coordinate speed in the hypersurface of constant comoving coordinate time that corresponds to our "now" on the Earth ?

The term I actually used in post #9 was not "coordinate speed", it was "recession speed". They're not the same, as can be seen from this correct observation by @Ibix:

In the usual FLRW coordinates the coordinate speed of comoving observers is zero.

The recession speed of a comoving observer relative to another one depends on the specific FRW model we are using. Our best current model of the universe is a spatially flat FRW model with a dark energy density "now" of about $0.7$ times the critical density, and a total matter density "now" of about $0.3$ times the critical density. Davis & Lineweaver 2003 [1], Figure 2, shows a graph of recession speed vs. redshift $z$ for galaxies in this model, as well as a few other idealized models.

[1] https://arxiv.org/abs/astro-ph/0310808

 

[cianfa72]

The recession speed of a comoving observer relative to another one depends on the specific FRW model we are using.

Surely I misinterpreted your claim in post #9. So the recession speed is a relative speed between two FRWL comoving observers. I'm not sure to grasp how it is actually calculated.

 

[PeterDonis]

the recession speed is a relative speed between two FRWL comoving observers.

That depends on what you mean by "relative speed". It is most certainly not a "relative speed" in the sense of Special Relativity.

 

[PeterDonis]

I'm not sure to grasp how it is actually calculated.

I gave you a reference. Go read it.

 

[cianfa72]

I gave you a reference. Go read it.

From that reference, the recession speed of a galaxy w.r.t. a comoving observer at the origin of FRWL spatial coordinates is defined as the derivative of the proper distance $D$ between them (derivative w.r.t. the cosmological coordinate time $t$) evaluated on an spacelike hypersurface of constant cosmological coordinate time $t$ (i.e. the timelike coordinate of FRWL solution in FRWL global chart -- it should be actually the proper time of comoving observers)
$$D(t)=R(t)\chi$$
In the following derivative (Eq 17 in that reference) does not apper also the derivative of $\chi(z)$ w.r.t. the coordinate time $t$.
$$v_{rec}(t,z)=\dot R(t)\chi(z)$$
Maybe the point is that the galaxy redshift $z$ changes w.r.t. coordinate time $t$ however the value of the composite map $\chi(z(t))$ for it does not (i.e. a galaxy on a comoving worldline has fixed spatial comoving coordinates in FRWL global chart).

 

[Ibix]

$\chi$ is the comoving coordinate of the galaxy, which doesn't change with time.

Print a square grid on a rubber sheet. Choose one of the intersections as the origin and at every other one note the distance in units of grid square sides from the origin. Stretch the sheet so you have a grid twice the size. The distance notations remain correct and unchanging; these are the $\chi$ values. The $R$ value measures the grid square size, and it will have doubled.

Note that this is an idealised model where every galaxy is an ideal FLRW observer at rest in those coordinates.

 

[cianfa72]

Print a square grid on a rubber sheet. Choose one of the intersections as the origin and at every other one note the distance in units of grid square sides from the origin. Stretch the sheet so you have a grid twice the size. The distance notations remain correct and unchanging; these are the $\chi$ values. The $R$ value measures the grid square size, and it will have doubled.

Ok, so basically $R(t)$ is the scaling factor that allows to get the proper distance between points of fixed coordinates (i.e. between events with fixed $\chi$ values on a spacelike hypersurface of constant cosmological coordinate time).

In that reference there is another expression for $v_{rec}$ namely (Eq 18)
$$v_{rec}(t,z) = H(t)D(t)$$ The function $H(t)$ should be related to Hubble's law...

Btw: Robertson-Walker (RW) is just a shorthand for Friedmann-Lemaitre-Robertson-Walker (FLRW) ?

 

[Ibix]

$H(t)$ is Hubble's constant (which is a constant across FLRW spatial slices, but not in time) at cosmological time $t$. The value now is usually denoted $H_0$.

 

[cianfa72]

The value now is usually denoted $H_0$.

You mean it is its value at "now" cosmological time (as you said it does not depends on any other FRWL coordinates except that $t$).

 

[Jaime Rudas]

In the following derivative (Eq 17 in that reference) does not apper also the derivative of $\chi(z)$ w.r.t. the coordiante time $t$.
$$v_{rec}(t,z)=\dot R(t)\chi(z)$$

The total velocity is equal to the recession velocity plus the peculiar velocity, which is what includes the derivative of $\chi$, as shown in equations 19 and 20:
$$ \dot D= \dot R \chi +R \dot \chi $$ $$ v_{tot} = v_{rec} + v_{pec} $$

 

[Jaime Rudas]

In that reference there is another expression for $v_{rec}$ namely (Eq 18)
$$v_{rec}(t,z) = H(t)D(t)$$ The function $H(t)$ should be related to Hubble's law...

Equation 18 is Hubble's law (or, more precisely, the Hubble-Lemaître law).

Btw: Robertson-Walker (RW) is just a shorthand for Friedmann-Lemaitre-Robertson-Walker (FLRW) ?

Yes

 

[PeterDonis]

From that reference, the recession speed of a galaxy w.r.t. a comoving observer at the origin of FRWL spatial coordinates is defined as the derivative of the proper distance $D$ between them (derivative w.r.t. the cosmological coordinate time $t$) evaluated on an spacelike hypersurface of constant cosmological coordinate time $t$

Yes.

(i.e. the timelike coordinate of FRWL solution in FRWL global chart -- it should be actually the proper time of comoving observers)

They're the same.

In the following derivative (Eq 17 in that reference) does not apper also the derivative of $\chi(z)$ w.r.t. the coordiante time $t$.
$$v_{rec}(t,z)=\dot R(t)\chi(z)$$

That's because $\chi$ is constant for a comoving observer; $\chi$ is the spatial coordinate in the notation the paper is using.

a galaxy on a comoving worldline has fixed spatial comoving coordinates in FRWL global chart

Yes.

 

[PeterDonis]

The term I actually used in post #9 was not "coordinate speed", it was "recession speed".

I see that I did incorrectly use "coordinate speed" later on in that post. I have edited it to correct those statements.

 

[cianfa72]

I see that I did incorrectly use "coordinate speed" later on in that post. I have edited it to correct those statements.

Ok thanks, now it makes sense.

 

[cianfa72]

More precisely, as an example, every such observer would measure the CMB to have the same temperature in all directions. That is an indication that the universe is seen to be isotropic. And if we pick out one particular spacelike hypersurface, at the event on each such worldline that is in that hypersurface, each observer would measure the CMB temperature to have the same numerical value. That is an indication that the universe is seen to be homogeneous.

I think it is mathematically described from the FLRW metric that on spacelike hypersurfaces of constant cosmological time $t$ is homogeneous and isotropic.

 

[PeterDonis]

I think it is mathematically described from the FLRW metric that on spacelike hypersurfaces of constant cosmological time $t$ is homogeneous and isotropic.

No, it isn't; that doesn't even make sense.

 

[cianfa72]

No, it isn't; that doesn't even make sense.

It doesn't make any sense to say that a metric (restricted on spacelike hypersurfaces) is homogeneous and isotropic ?

 

[PeterDonis]

It doesn't make sense to say that a metric (restricted on spacelike hypersurfaces) is homogeneous and isotropic ?

I thought your post #37 was saying that $t$ (the coordinate) is homogeneous and isotropic. That doesn't make sense.

It makes sense to say that the geometry of a spacelike hypersurface of constant $t$ is homogeneous and isotropic, so if by "metric" you mean "geometry" then it would also make sense to say the metric is homogeneous and isotropic.

It also makes sense to say that the distribution of stress-energy on such a spacelike hypersurface is homogeneous and isotropic; that's the original physical statement that is used to derive the FRW spacetime geometry.

 

[cianfa72]

It makes sense to say that the geometry of a spacelike hypersurface of constant $t$ is homogeneous and isotropic, so if by "metric" you mean "geometry" then it would also make sense to say the metric is homogeneous and isotropic.

From a mathematical point of view, the homogeneous property of a Riemannian manifold (that is any spacelike hypersurface of constant cosmological time) boils down to the existence of a transitive isometry map on it.

What about the mathematical definition of isotropic for that manifold ?

 

[PeterDonis]

From a mathematical point of view, the homogeneous property of a Riemannian manifold (that is any spacelike hypersurface of constant cosmological time) boils down to the existence of a transitive isometry map on it.

Yes.

What about the mathematical definition of isotropic for that manifold ?

It means that, if we pick a point in the manifold, there is an SO(3) isometry centered on that point.

Note that it is possible to have a manifold that is isotropic about one point, but not homogeneous. However, as should be evident, if a manifold is homogeneous, and we can show that it is isotropic about one point, then it must be isotropic about every point.

 

[cianfa72]

It means that, if we pick a point in the manifold, there is an SO(3) isometry centered on that point.

SO(3) is the special continuous group of linear transformations that preserves orthogonality. In this context the orthogonality is w.r.t. the (positive definite) metric tensor defined on each point on spacelike hypersurfaces.

An element of SO(3) group acts on vectors of tangent vector space defined at each point. Therefore, I think, the definition of isotropy at a point does require SO(3) group elements act on tangent space's vectors as isometries.

 

[PeterDonis]

SO(3) is the special continuous group of linear transformations that preserves orthogonality.

SO(3) is the group of rotations in 3 dimensional space. These are linear transformations and they do preserve the orthogonality, but they are hardly the only such group of transformations. So I don't see where you are getting "the" (bolded in the quote above) from.

An element of SO(3) group acts on vectors of tangent vector space defined at each point.

Yes.

Therefore, I think, the definition of isotropy at a point does require SO(3) group elements act on tangent space's vectors as isometries.

That's what I said.

 

[PeterDonis]

An element of SO(3) group acts on vectors of tangent vector space defined at each point.

Note that, while this is correct, it is not the only possible way to interpret the SO(3) group. It can also be interpreted as a three-parameter group of Killing vector fields on the spacetime. A spacetime with such a group of KVFs is called "spherically symmetric", and will be isotropic about at least one point, but possibly only one. For example, a spacetime containing a single non-rotating body surrounded by vacuum will be spherically symmetric, and will be isotropic about the center of mass of the body, but not about any other point.

In the case of FRW spacetime, however, the spacetime is isotropic about every point. That means it is spherically symmetric no matter which point we choose as our center of symmetry. It also means that there is a different SO(3) group of KVFs centered on each such point. In other words, a much stronger set of properties than just an SO(3) isometry on tangent spaces.

 

[cianfa72]

SO(3) is the group of rotations in 3 dimensional space. These are linear transformations and they do preserve the orthogonality, but they are hardly the only such group of transformations. So I don't see where you are getting "the" (bolded in the quote above) from.

Ok, so you are saying that there could be other groups of possibly non-linear transformations that preserve orthogonality and length as well.

Note that, while this is correct, it is not the only possible way to interpret the SO(3) group. It can also be interpreted as a three-parameter group of Killing vector fields on the spacetime. A spacetime with such a group of KVFs is called "spherically symmetric", and will be isotropic about at least one point, but possibly only one.

Ok, so integral curves of such KVFs lie on spacelike hypersurfaces, I believe.

Sorry, when you say "point" you actually mean an event in spacetime.

 

[DrGreg]

there could be other group of possibly non-linear transformations that preserve orthogonality and length as well

... or groups of linear transformations that do preserve orthogonality but don't preserve length

 

[PeterDonis]

integral curves of such KVFs lie on spacelike hypersurfaces, I believe.

They lie within each spacelike hypersurface of constant FRW coordinate time.

when you say "point" you are actually mean an event in spacetime.

Yes. But we can also consider points in a spacelike hypersurface of constant FRW coordinate time, since the integral curves of the KVFs in question like entirely within such surfaces.

 

[cianfa72]

In the case of FRW spacetime, however, the spacetime is isotropic about every point. That means it is spherically symmetric no matter which point we choose as our center of symmetry. It also means that there is a different SO(3) group of KVFs centered on each such point. In other words, a much stronger set of properties than just an SO(3) isometry on tangent spaces.

That means the three-parameters family of KVFs (that form a Lie algebra) are the infinitesimal generators of a group of isometries centered on any point on each spacelike hypersurface of constant cosmological time. Such groups are different instances each isomorphic to SO(3).

 

[PeterDonis]

the three-parameters family of KVFs (that form a Lie algebra) are the infinitesimal generators of a group of isometries centered on any point on each spacelike hypersurface of constant cosmological time.

Of SO(3) KVFs, yes. Note that the isometries associated with homogeneity, the spatial translations in a spacelike hypersurface of constant time, are also generated by a three-parameter group of KVFs, but the group is not SO(3).

Such groups are different instances each isomorphic to SO(3).

I guess you could think of them that way, although I don't think the term "instances" appears in this connection in the literature.