# Distance in Hubble law vs special relativity

Came across this very informative website https://astro.ucla.edu/~wright/cosmo_02.htm , and it has the following two diagrams. I'm having trouble figuring out why "But the Hubble law distance $$D_{now}$$, which is measured now, of these most distant galaxies is infinity (in this model)." (highlighted in the screenshot). Which particular point or line in which of the two diagram indicates that the "now" distance is $$\bf {infinite}$$? It'd be great if someone could point out what I'm missing. Thank you.

The time and distance used in the Hubble law are not the same as the $$x$$ and $$t$$ used in special relativity, and this often leads to confusion. In particular, galaxies that are far enough away from us necessarily have velocities greater than the speed of light:

The light cones for distant galaxies in the diagram above are tipped over past the vertical, indicating $$v > c$$. The space-time diagram below shows a "zero" (really very low) density cosmological model plotted using the $$D_\text{now}$$ and $$t$$ of the Hubble law.

Worldlines of comoving observers are plotted and decorated with small, schematic lightcones. The red pear-shaped object is our past light cone. Notice that the red curve always has the same slope as the little light cones. In these variables, velocities greater than $$c$$ are certainly possible, and since the open Universes are spatially infinite, they are actually required. But there is no contradiction with the special relativistic principle that objects do not travel faster than the speed of light, because if we plot exactly the same space-time in the special relativistic $$x$$ and $$t$$ coordinates we get:

The grey hyperbolae show the surfaces of constant proper time since the Big Bang. When we flatten these out to make the previous space-time diagram, the worldlines of the galaxies get flatter and giving velocities $$v = dD_\text{now}/dt$$ that are greater than $$c$$. But in special relativistic coordinates the velocities are less than $$c$$. We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance $$x = c\cdot t_o/2$$. But the Hubble law distance $$D_\text{now}$$, which is measured now, of these most distant galaxies is infinity (in this model). Furthermore, this galaxy with infinite Hubble law distance and hence infinite Hubble law velocity is visible to us, since in this model the observable Universe is the entire Universe. The relationships between the Hubble law distance and velocity ($$D_\text{now}$$ & $$v$$) and the redshift $$z$$ for the zero density model are given below:

• You might want to look at the 2003 version of "Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe", by Lineweaver & Davis, which provides several diagrams of the relation between spatial expansion generally & the expansion of the Hubble Sphere: As the UCLA piece straddles 2003, it may not take account of an earlier misinterpretation of Special Relativity that's acknowledged at the end of L&D's 2003 paper (which contains several diagrammatic representations of "now"). Oct 15, 2022 at 4:43
• Sorry for the need to copy & paste L&D's title, but their paper's web address is even longer. Oct 15, 2022 at 4:45

Horizontal lines in the upper figure, gray hyperbolas in the lower figure, are "spatial" surfaces in this model. Focusing on the lower figure, the uppermost hyperbola is the "present time" spatial surface. Note that its total length is infinite ("space is infinite").

The black lines are worldlines of comoving objects like galaxies. Since space is infinite, there are of course infinitely many of them. In the lower figure they appear to get bunched together at large distances, but that's just an artifact of drawing Minkowski geometry on a Euclidean computer screen. All of these lines are in fact equally separated along the aforementioned spatial surfaces. If you were to draw the same diagram from the point of view of any other galaxy, it would look the same.

Now, if you trace a light ray (red line) backward, it crosses the worldline of every galaxy, even galaxies that are arbitrarily far "at the present time" (along the uppermost hyperbola). That's what "$$D_\mathrm{now}$$ is infinite" means.

• Thank you very much for this insight! Could the same be said using the upper diagram? I mean since the horizon lines in the upper diagram are the "spatial" surfaces, tracking the red line back can we say it also crosses all the world lines of all galaxies, even though at t=0 it only intersects at the origin (zero distance)? Does that mean the light emitted at t=0 that we see today was from an object at zero distance to us at the time, but now is infinitely faraway?
– ABC
Oct 13, 2022 at 19:40
• Thanks to you I understand the math a lot more now, but I desperately need an intuitive explanation of the meaning of this. How does it make sense intuitively that we can see things that are now infinitely far away? Light takes time to travel, galaxies take time to move away, within the $\bf{finite}$ amount of time as of "now", how could it be possible to see things infinitely far away in this particular model universe?
– ABC
Oct 13, 2022 at 19:45
• The basic thing to realize is that "now" has no intrinsic meaning over long distances. In cosmology we define "now" to mean that the proper time elapsed for the other object is the same as the proper time elapsed for us. But as objects approach the speed of light, time dilation means that their elapsed time gets arbitrarily short, in relation to ours. That means they have been able to travel an arbitrarily long distance away from us by the time that we take to be "now" in this context.
– Sten
Oct 14, 2022 at 12:04
• I can't see how "now" ever has any intrinsic meaning, so, on the strength of its 1st sentence, I've upvoted Sten's last comment. In my own comment on the OP's question, I've mentioned a line marked "now" that's printed across L&D's diagrams, but the actual now is in the center of that line's (not any entire diagram's) width. Oct 15, 2022 at 16:03
• @Edouard Indeed we do not (and may never) know that space is infinite. I only note that it is infinite in the model at hand -- which, it should be noted, isn't a viable model for our universe anyway.
– Sten
Oct 15, 2022 at 17:19