Confusion about the math of the Comoving Coordinates

[Arman777]

From the FLRW metric Proper distance can be derived like this,

$$ds^2=-c^2dt^2+a^2(t)[dr^2+S_k(r)^2d\Omega^2]$$

Let us fixed the time at $t=t_0$ for the measurement and assume that the object has only radial component, then the metric equation turns out to be,

$$ds^2=a^2(t_0)dr^2$$
$$\int_0^{s_p}ds=a(t_0)\int_0^{r}dr$$

or

$$s_p=a(t_0)r~~(Eqn.1)$$ where $s_p$ is the proper distance and $r$ is the comoving distance.

To derive the comoving coordinate we can consider the path of the light

Now for the derivation of the comoving coordinate, which it writes in this paper https://arxiv.org/abs/astro-ph/0310808, the comoving distance can be defined as the path that taken by the light from emission time to observation time.

$$r=c\int_{t_e}^{t_o}\frac {dt} {a(t)}~~(Eqn.2)$$

Now let us suppose, that r to be found at 100Mpc. Is this means that $r=100Mpc$ at $a(t_0)=1$ ?
So can we say the proper distance at $a(t)=1/2$ will be $s_p=50Mpc$ ?

Another problem that I noticed is that when we derive the Hubble Law from the proper distance
$$s_p=a(t)r$$
we take as $dr/dt=0$ so that we claim
$$V=H(t)r$$ but as above its clear that $\frac {dr} {dt}=\frac {c} {a(t)}≠0~~(Eqn.3)$

So if r means comoving distance in both cases then what is the problem here?

 

[PeterDonis]

To derive the comoving coordinate

What do you mean by this? You don't have to "derive" any comoving coordinate: the coordinates $t$, $r$, $\theta$, $\phi$ are comoving coordinates. However, you are using non-standard labeling for $r$; see below.

which it writes in this paper https://arxiv.org/abs/astro-ph/0310808, the comoving distance can be defined as the path that taken by the light from emission time to observation time

In this paper, they are calling the radial coordinate $\chi$ instead of $r$. This appears to have confused you. The metric given in equation (15) of the paper is the same as yours, just with variable labels changed.

However, the equation for $r$ that you give, where now you are using $r$ to mean the comoving distance (which is not the same as the coordinate $\chi$, which you mistakenly label $r$), appears nowhere in this paper that I can find. Where are you getting it from?

 

[PeterDonis]

Another problem that I noticed is that...

Is this something you saw in the same paper, or somewhere else?

 

[Orodruin]

Another problem that I noticed is that when we derive the Hubble Law from the proper distance
$$s_p=a(t)r$$
we take as $dr/dt=0$ so that we claim
$$V=H(t)r$$ but as above its clear that $\frac {dr} {dt}=\frac {c} {a(t)}≠0$

Huh? It is certainly not "clear from above" that $dr/dt \neq 0$. In the equation you are referring to for a light signal, there is no reference whatsoever to $t$ on the right-hand side, it is just an integration variable and you cannot differentiate with respect to it. Since you are dealing with a light signal, you certainly cannot find the proper distance at a particular time from that expression, nor any sort of time dependence of $r$. The expression in itself is derived with an assumption of particular $r$ coordinates for the end points of the light-like geodesic. The expression appears in certain derivations of the cosmological redshift and the entire point of that argument is that $r$ is constant for comoving observers so that the integral on the right-hand side must be the same regardless of what $t_e$ is (what $t_o$ is will follow from this).

 

[Arman777]

which is not the same as the coordinate χχ\chi, which you mistakenly label rrr)

So what is the difference then ?

Where are you getting it from?

İf you are meaning where I get the $$r=c\int_{t_e}^{t_o}\frac {dt} {a(t)}$$ its in the equation 22. Well yes there is no $r$ but $χ$. They are not the same ? Or the comoving coordinate $r$ is not the $χ$.

I want to suggest that, we can forget about the $χ$ and think like this;

When we are deriving the proper distance we are saying that $r$ is the comoving distance. In that sense. $r$ that we used to derive the proper distance and (Eqn.2) are the same.

Is this something you saw in the same paper, or somewhere else?

I derive that conclusion myself.

Huh? It is certainly not "clear from above" that $dr/dt \neq 0$. In the equation you are referring to for a light signal, there is no reference whatsoever to $t$ on the right-hand side, it is just an integration variable and you cannot differentiate with respect to it. Since you are dealing with a light signal, you certainly cannot find the proper distance at a particular time from that expression, nor any sort of time dependence of $r$. The expression in itself is derived with an assumption of particular $r$ coordinates for the end points of the light-like geodesic. The expression appears in certain derivations of the cosmological redshift and the entire point of that argument is that $r$ is constant for comoving observers so that the integral on the right-hand side must be the same regardless of what $t_e$ is (what $t_o$ is will follow from this).

You are saying equations but I cannot understand which equations you are referring to. I ll put numbers so maybe you can refer to them ? Are you talking about Eqn.1 for the entire time ?

Yes I agree that comoving coordianates must be time-independent. Eqn.2 "Finds" some distance. Then what that distance represents if its not a comoving distance ?

I guess I understand your point.Since we calculated the path taken by light, its meaningless to take the time derivate of it.

 

[PeterDonis]

So what is the difference then ?

See my comments below.

When we are deriving the proper distance we are saying that $r$ is the comoving coordinate.

How you label coordinates is irrelevant. However, since your choice of labeling appears to be confusing you, I would strongly suggest forgetting whatever you think $r$ means and focusing on equation (15) in the paper, which gives the metric in the coordinates they are using, where the radial coordinate is $\chi$. Then a "proper distance" is just a spacelike interval $ds^2$ for which $dt = 0$ and $t = t_0$. To make things simple, we usually orient the coordinates so that the only separation is radial, in which case we have $ds = R(t_0) d\chi$, which, if we assume that the observer (us) is at $\chi = 0$, integrates easily to $s = R(t_0) \chi$, where $\chi$ is the radial coordinate of some other object. This is basically what you wrote as your equation 1 in the OP to this thread, which is fine as far as it goes (but again, I would strongly recommend using the standard labeling in the paper to avoid confusion).

But you seem to be missing an important aspect of the above: it has nothing to do with the integral given in equation (22) of the paper (the one you write, using your own choice of labeling, as equation 2 in your OP). The integral in equation (22) of the paper is not an integral over a spacelike interval; it's an integral over a null (lightlike) interval, the one traversed by a light ray traveling from some distant object that emitted light at coordinate time $t_{\text{em}}$, to us, here and now, seeing the light at coordinate time $t_0$. Furthermore, as noted in the paper, equation (22) itself is not much use practically since the quantities in it are not directly observable (we have no way of knowing at what coordinate time the light from a particular object that we are seeing was emitted); equation (24) gives an integral in terms of observable quantities (the redshift and the Hubble parameter as a function of the redshift).

when we derive the Hubble Law from the proper distance

We don't. At least, not as far as actually trying to derive it from observations is concerned. We have no way of observing the proper distance to distant objects. Nor, as I noted above, do we have any way of directly observing coordinate time.

I derive that conclusion myself.

Then I think you need to go back and start again. It does not appear that you have a sound understanding of this topic yet.

 

[Arman777]

One last thing that I want to ask, Is there a mathematical definition or derivation of the comoving distance (like proper distance for example,) I never have seen it. Maybe we can define as its $χ=a(t_0)s_p$ or it's just some kind of a coordinate system that we choose and it has only a definition but not derivation.

 

[Orodruin]

The comoving distance is just the distance along the surface of simultaneity when the scale factor is equal to one. You cannot "derive" it since it is a definition.

 

[Arman777]

Thanks for your helps. I understand it now