# How the speed of light is constant with the particle horizon moving toward us?

I asked a question earlier with the same subject in mind. However, I was rather upset at the time and it didn't make much sense. I'm taking an astronomy course and seem to be misunderstanding something about the speed of light. I understand how light behaves locally. However, I'm encountering a problem understanding it on the larger scale.

If the expansion of the universe is accelerating$^1$ and as a consequence, the cosmological horizon$^2$ is slowly getting closer to us, then in relation to our position there are distant galaxies moving away from us at or faster than the speed of light. Thus I came to the conclusion that the speed of these galaxies is purely in reference to where you are measuring from, that the perceived time distortion is an illusion caused by our relative speed, time is not really grinding to a halt for distant galaxies, and if you were in one of them, light would still move the same speed in relation to that location.

I've been informed in class that I was mistaken, and no object can go the speed of light, but when I asked for clarification, I was told to do the readings again. On here it was confirmed that I'm confused, but my previous question was not well written, so it was difficult to say how. I hope this time my misconception is more clear, and apologize if this isn't much of a physics question.

Update: I've been informed by Thriveth that Recession velocity is the correct term for increasing distance over time due to the expansion of spacetime. and a big thanks to Heather informing me that speed is an inappropriate term and that using it this way was leading to misunderstandings and unintended implications that contradict the speed of light as a constant.

• Can you clarify what you mean by 'partical'? – Emilio Pisanty Jul 14 '16 at 21:33
• The particle horizon (also called the cosmological horizon, the comoving horizon (in Dodelson's text), or the cosmic light horizon) is the maximum distance from which particles could have traveled to the observer in the age of the universe. en.wikipedia.org/wiki/Particle_horizon – Harold Saxon Jul 14 '16 at 21:34
• My two cents, until an expert writes an answer, is that the speed of light is a constant relative to space, but in general relativity space itself can be stretched, so you can have something that "looks like" it is moving faster than light, but it is not really moving faster, it is the illusion created when you put a lot of "new" space between you and that galaxy. – Wolphram jonny Jul 14 '16 at 21:46
• Just going to same thing as @Wolphramjonny, but have a read of this link: curious.astro.cornell.edu/about-us/104-the-universe/… – user108787 Jul 14 '16 at 21:48
• Thanks Wolphram jonny & count_to_10, but i still can't see what i'm getting wrong. – Harold Saxon Jul 14 '16 at 22:05

The other answers seems to answer most of your questions, but I think one confusion remains: The speed of light as a maximum speed in the Universe (which is not the case).

First off, redshift doesn't go to infinity for objects receding at $v = c$. We easily see galaxies recede at superluminal velocities. In fact, this is the case for all galaxies with a cosmological redshift of $\gtrsim\!1.5$.

To understand how this is possible, imagine a photon emitted from the galaxy GN-z11 in our direction when the Universe was 400 Myr (million years) old, and the distance to that galaxy was $d_\mathrm{then}=800\,\mathrm{Mpc}$ (roughly 2.7 billion lightyears). At this time, the scale factor (the size of the Universe relative to today) was $a=0.08$, and the Hubble parameter was $H_\mathrm{then} \sim 1600\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, so the recession velocity of GN-z11 was $$v_\mathrm{then} = H_\mathrm{then} d_\mathrm{then} = 1\,300\,000\,\mathrm{km}\,\mathrm{s}^{-1} \simeq 4.3c.$$

In the beginning, although the photon departed from GN-z11 at $v=c$ as it should, and even though it traveled in our direction, it was carried away from us by the expansion of the Universe faster than this, and thus receded from us at $v=4.3c-c=3.3c$.

However, expansion is ubiquitous, and thus also helped carrying the photon away at an increasing speed from GN-z11. At some point (when the Universe was 3.7 Gyr old), expansion had helped it reach the midpoint between GN-z11 and the Milky Way, and although its local velocity was always $c$, for a brief moment it stood still wrt. both MW and GN-z11. Then it began to increase its velocity, until it hit the Hubble Space Telescope last year with a speed of $c$.

Today, the distance of GN-z11 has increased to $d_\mathrm{now} = 9.9\,\mathrm{Gpc}$, and it thus recedes from us at $$v_\mathrm{now} = H_\mathrm{now} d_\mathrm{now} = 670\,000\,\mathrm{km}\,\mathrm{s}^{-1} \simeq 2.2c,$$ which is still superluminal. However, due to the accelerated expansion of the Universe, photons emitted from GN-z11 today will never reach us.

Edit: Clearing up a few misunderstandings

the cosmological horizon is slowly getting closer to us

In fact, the opposite is the case: The particle horizon, which is what you link to, is defined as the farthest we can see, which is given by the distance that light has had to travel since the Big Bang. Since time increases, this distance always increases.

there are distant galaxies moving away from us at or faster than the speed of light.

This is true, as described above.

the speed of these galaxies is purely in reference to where you are measuring from

Again true. While we measure the recession speed of GN-z11 to be $2.2c$, an observer midway between us and them would measure half this speed.

the perceived time distortion is an illusion caused by our relative speed, time is not really grinding to a halt for distant galaxies

Time does indeed run slower in distant galaxies, as seen from us. But it does not come to a halt at $v=c$. In fact, time is simply dilated by a factor $(1+z) = 1/a$. That is, time in GN-z11, which lies at redshift $z=11.1$, runs slower by a factor $12.1$.

However, this is not an illusion, but a real effect. This is the very essence of relativity; you must accept that time (and space and simultaneity, etc.) is relative for different observers in order for physics to make sense. An example of when this time dilation is important is the time it takes for the brightness of distant supernovae (SNe) to decrease. All type 1a SNe have the same explosion mechanism, and increase and decrease in brightness in the same way (modulo a known effect but forget about that for now). But distant SNe are seen to decrease more slowly by a factor that exactly corresponds to the redshift they're at (plus one).

if you were in one of them, light would still move the same speed in relation to that location.

Indeed true.

I've been informed in class that I was mistaken, and no object can go the speed of light

Again, no object or information can travel through space faster than light. But space itself can expand at any rate.

• This is helpful, and has helped me understand more about the subject, however, everyone seems to be saying that I'm saying the speed of light is not a constant, but i fail to see how I'm saying that. I know the speed of light is a constant, and am missing what I'm saying wrong. – Harold Saxon Jul 15 '16 at 16:45
• @HaroldSaxon: Okay, a few misunderstandings remain in your question: I'll update my answer to address those. – pela Jul 15 '16 at 22:57

The problem with the assumptions in your second paragraph is that space is moving, not the galaxies. Space itself can travel faster than the speed of light - that is not forbidden by general relativity. The speed of light as a constant therefore still holds, removing the implications you bring up.

As an analogy, imagine you have a coordinate grid, and you mark a couple of points on there to represent galaxies. Now imagine you transfer this coordinate grid to some stretchy mesh. Now stretch the mesh. The galaxies aren't moving, but the space between them is expanding.

Here's another picture to keep in mind. Let's say you have a great-uncle twice removed in another galaxy that you think is moving away from you. Now, let's imagine that you are magically able to communicate faster than the speed of light, so you are able to talk to each other. Here's what the conversation might go like: "Hey, uncle, you're moving away from us!" "What are you talking about?" "The galaxy you're in, it's moving away!" "Right, uh, no. Your galaxy is moving away." Neither galaxy is truly moving. It is space that is moving. From each galaxy, it looks like every other galaxy is moving away from you.

It should be noted that the speed of light is constant and time is not, so some of your points do not make much sense (i.e., "light would still move the same speed", "time is not really grinding to a halt", etc).

Finally, I answered your previous question (I believe, though I can't double check as it is now deleted), and the explanation in my answer of the speed of light in a vacuum's constancy still applies. To summarize, the speed of light is the speed limit of the universe, no exceptions.

I hope this helps!

• Ah, no offense, but isn't that the same as my assumptions? That they appear to be moving faster than the speed of light to us, as red shift becomes infinite, but they locally don't have that motion? so the effects of reaching the speed of light do not actually take place? I really don't see what i have wrong. I'm sorry if it's plain to see and I'm missing it. – Harold Saxon Jul 15 '16 at 0:36
• Okay, so let me see if I've got your question straight. You're saying that in relation to our position of the universe, galaxies are moving away from us at or above the speed of light, and that therefore the speed of light is not a constant, right? Well, that's what I'm talking about. If you are in one of those galaxies, you would think that your galaxy wasn't moving at all. Space is moving, not the galaxies. And it's okay for space to move faster than the speed of light. – heather Jul 15 '16 at 12:46
• Oh, no, I'm not saying that at all! My teacher says the Hubbles constant requires correction for specal relitivity as distant objects propetually aproch the speed of light. However, the handout clearly says that this effect scales with distance, so if you mesure from the half way point you get half the speed. So I figure this must not be 'real' motion, and dose not fall under specal relitivity. The handouts back this up with the observible universe getting smaller. If they were only aproaching the speed of light, we'd be able to see the whole universe. – Harold Saxon Jul 15 '16 at 13:39
• @HaroldSaxon, first, that's totally different from what your question said. Second, it's still not very clear. It doesn't help that there are a lot of spelling errors making it difficult to tell what words you mean. What exactly do you not understand? – heather Jul 15 '16 at 14:22
• Well, Everyone agrees that I'm claiming the speed of light is not a constant, However I don't see how I'm saying that. " Thus I came to the conclusion that the speed of these galaxies is purely in reference to where you are measuring from" says nothing about the speed of light. What am i saying that goes against the speed of light? I know the speed of light is a constant, and am missing what I'm saying wrong. – Harold Saxon Jul 15 '16 at 16:51

Heather is right and it is not much more complex than that. Except you might need to follow the math to understand it. Dodelson certainly has the math.

Light goes at c. Period. If you want to find the geodesics of light you set the metric ds^2 = 0. But space itself expands, and it can expand at any spee, it is not a particle or wave or object, it is just the metric of the spatial part. For FLRW universes, the standard cosmology, the spatIal part has a scale factor a(t) which sets the scale of the spatial sections. The Hubble parameter H is a dot/a, and defines the expansion factor per unit distance. The Hubble constant is its value now, typically expressed in km/sec/Mpsec. It is about 71. You can calculate that means the speed c, 300,000 Kms/sec, ocurs for objects about 4200 Mpsecs away. That is their distance now. You can calculate (but search in Google or Dodelson for how) with general relativity (careful not a simple multIplication) that objects at those distances (which inevitably , if we can see them are galaxies or their central cores) were 4.6 billion years old. The light they emitted was emitted 4.6 billion years ago, and it is now reaching us, and now that galaxy is receding from us at a speed of about c. Any light it emits now will be the last from it that reaches us, any light emitted after that will not reach us as space expands faster than c.

We see a lot of galaxies now, and will see for a very long time, their light was emitted in the past when space was expanding slower, and we will see the light emitted from galaxies further away, up to the cosmological horizon.

This may not answer your question, you just need to understand that space expanding faster than c does not mean the objects there are locally going faster than c. They travel slower than c, emit their light, and space then expands between them and us. Follow the math and it'll be easier to believe it. I don't know a more intuitive way. MTW's (Misner, Thorne and Wheeler, you can Google it) textbook explains it well also.

• Do you mean $MTW$ in your last line? Might be good to spell it out, even though it sounds as though the OP would know of this book, – WetSavannaAnimal Jul 24 '16 at 11:20