From what I understand, in an expanding universe given by $$ds^2=-dt^2+e^{2Ht}dx_{com}^2,$$ where I believe that $dx_{com}$ denotes a comoving spatial line element, the distance between a comoving observer and a point located at $x_{com}=x_c$ is constant, although the physical distance $x_{phys}=x_{com}e^{Ht}$, is growing. However, this contrasts with my understanding of a comoving observer, which is just for example a person standing on a point in space. This person, as far as I understand, perceives that galaxies, stars, and so on, is getting further from it, and therefore the distance is not constant. What is the source of my confusion?
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The coordinate distance $\int \mathrm{d} x_\mathrm{com}$ is constant in time between a pair of comoving observers. This is just a feature of the coordinate system and says little about the physical evolution.
The proper distance $\int \mathrm{d}s$, integrated in a straight line along a conventional spatial surface (so $\mathrm{d}t=0$), grows in time proportionally with $\mathrm{e}^{Ht}$. This is a more physical quantity, which you could measure for example using a sequence of comoving rulers. There is still some influence from the coordinate choice, because the surfaces defined by $\mathrm{d}t=0$ depend on the coordinate system.
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$\begingroup$ Thank you for your answer! However, there is something I do not understand: when expressing the line element as $ds^2=-dt^2+a^2(t)dx^2$, $x$ is here a comoving coordinate, right?. Then, why is it used to calculate the proper distance? $\endgroup$ Commented Sep 17 at 10:40
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$\begingroup$ @TopoLynch Proper distance is $\int ds=a\int dx$. $\endgroup$– StenCommented Sep 17 at 13:52