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I'm currently in 11th grade working on a science project about measuring distances using redshifts. I understand how the wavelength of a light wave increases when an object moves away from the point of observation. Now suppose a star, say 20 light years away, emits a light wave at this moment. Then that specific light wave must reach us after 20 years. If we use the redshift of this light wave to calculate the distance, then aren't we basically getting the position of the star as it was 20 years ago? Cause by this time the star has travelled much further away. Can someone please explain how the redshift-distance relation works and if I've gone wrong somewhere?

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    $\begingroup$ I am not an expert in this subject, but I think the key is thinking about why does the distant galaxy move away. Under the considerations of the cosmological principle (homogeneity and isotropy of the universe at large scales), the galaxy is not moving from its coordinates, but the space between its coordinates and ours is expanding. Therefore, the reason for the redshift is not that the galaxy has a velocity with respect to space coordinates, but the expansion of the universe itself (cosmological redshift). Therefore, it tells you how much the space expanded during the travel of the light $\endgroup$
    – SrJaimito
    May 10, 2022 at 8:06
  • $\begingroup$ Redshift is only a significant indicator at large distances of milllions of lightyears. $\endgroup$
    – my2cts
    May 10, 2022 at 9:27
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    $\begingroup$ I don't see a point here. With astronomy we learn about past, that's it. Lightspeed is finite so that's true about any observations, but with astronomy is most prominent. $\endgroup$
    – Mithoron
    May 10, 2022 at 12:50
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    $\begingroup$ You may also want to change the title, as it does not really reflect the content of the question. $\endgroup$
    – Toffomat
    May 11, 2022 at 15:37
  • $\begingroup$ This may not be detailed enough for a real answer, but I wanted to point out that the cosmological redshift is not caused by the original speed of the emitting galaxy, it is caused by what happens to the emitted light between being emitted and observed (or maybe more accurately, what happens to the Earth in that time). There's a great video on this youtu.be/9DrBQg_n2Uo $\endgroup$
    – RC_23
    May 12, 2022 at 21:15

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I have found the perfect graph which I believe can explain the concepts.

enter image description here

Let’s suppose we have an object located at a comoving distance $r$. This objects emits light at $t=t_1$ (or $z$) and reaches us around $t=t_0$ (or $z=0$). By looking at this graph, we can define some distances.

  1. Comoving distance, $r$

The comoving distance $r$, can be written as

$$r(z) = \frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$

As you can see from the graph, the comoving distance does not change with time.

  1. Proper distance, $d(z)$

The proper distance of an object can be written as,

$$d(z) = \frac{r}{1+z} \equiv \frac{1}{1+z}\frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$

In this case, from the graph, we can define 2 proper distances.

$$d_0(z) = r \equiv \frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$ $$d_1(z) = \frac{r}{1+z} \equiv \frac{1}{1+z}\frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$

By setting $a(t)=1$, we see that the comoving distance at current instance is equal to the proper distance of the object.

  1. Conformal Time, $\eta(z)$

The conformal time can be defined as

$$\eta(z) = \frac{1}{H(z)}\int_0^z\frac{dz'}{E(z')}$$

which describes the distance traveled by the photon from $t_1$ to $t_0$.

If we use the redshift of this light wave to calculate the distance, then aren't we basically getting the position of the star as it was 20 years ago?

The redshift of the object and given cosmological model can help us to determine the distance of an object. In general there are 5-6 different distance definitions (such as angular diameter distance, luminosity distance and transverse comoving distance, lookback time). However, the basics are like this. So when you observe a galaxy and measure it's redshift you can obtain these distances.

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