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The following are facts of the prevailing cosmological model.

  1. The age of the universe is about 13.772 billion years.

  2. Nothing with mass can exceed the speed of light.

  3. The diameter of the observable universe is 28.5 gigaparsecs, i.e. 93 billion light-years.

How could two particles never exceeding 186,000 miles per second ever end up farther than 27.544 ± light years apart? Is it possible to explain this seeming contradiction in a way that appeals to reason and doesn't require a pencil and paper?

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    $\begingroup$ Possible duplicate of Why is the observable universe so big? $\endgroup$
    – knzhou
    Commented Aug 7, 2016 at 7:01
  • $\begingroup$ "but since those light-years that you measured have expanded since the photon passed through" - this is the premise from the answer you incorrectly assert answers my question. $\endgroup$
    – Jim McMillan
    Commented Aug 7, 2016 at 7:07
  • $\begingroup$ Tell your friend to assume that space is like a cloth which can stretch. Tell him that light always moves along the space fabric (say that this is a fact). There is some godly force which keeps expanding the space, i.e: be the god and hold your piece of cloth and stretch it from two ends. The light will continue to move along the new elongated space fabric as usual but the distance between two points increases just like the distance between your hands increased. $\endgroup$
    – Yashas
    Commented Aug 7, 2016 at 7:10
  • $\begingroup$ @JimMcMillan I still think it is a duplicate, but the answers in that question dance around the issue you raised. So I made a separate answer to address it. $\endgroup$
    – knzhou
    Commented Aug 7, 2016 at 7:24

3 Answers 3

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Is it possible to explain this seeming contradiction in a way that appeals to reason and doesn't require a pencil and paper?

I think so, but it does require some imagination/visualization (and perhaps a better explanation than I can provide).

How could two particles never exceeding 186,000 miles per second ever end up farther than 27.544 light years apart?

A short answer would be "space is expanding, which increases the distance between the particles, without them ever having to travel faster than the speed of light".

Think of two particles in a small universe (a.k.a. the early universe) moving away from each other at the speed of light. If the universe didn't expand, some time ($t$) later they would just be $2ct$ apart from each other. In an expanding universe, they will now be separated by a distance $$ 2d = 2ct + 2v_{space}(t)t $$ where $2v_{space}(t)$ is the speed at which the distance between them expanded. If you then try and calculate the average speed of a single particle $(d/t)$ you get $$ v_{avg} = \frac{d}{t} = c + v_{space}(t) $$ which is greater than the speed of light. Crucially, relativity hasn't broken - the particles always traveled at $c$, it was the space that expanded.

A realistic/intuitive analogy is difficult because these kinds of effects simply don't happen on our every-day scale. A thought experiment you could think about is: Two cars travelling at maximum speed, in opposite directions along a very long and straight road. What happens to the distance between them if the road stretches?

or is it just the kind of thing one doesn't understand unless they are a physicist?

Absolutely not, although having the time to think about these kinds of things and seeing the maths helps a lot.

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  • $\begingroup$ In the early universe, wouldn't they be only c*t apart, because velocity addition is different when moving with relativistic sppeds? $\endgroup$
    – Graipher
    Commented Aug 7, 2016 at 10:23
  • $\begingroup$ I don't think so, but perhaps you know more than I do? If you fire one photon north, and another photon south. Both photons individually move at the speed of light, but because they're moving in opposite directions the distance between them is increasing at twice the speed of light. In other words, I'm not adding velocities I'm adding distances here. $\endgroup$
    – Judge
    Commented Aug 7, 2016 at 10:33
  • $\begingroup$ Yeah, I think you are right, in the centre-of-momentum frame of the two photons they should both be receding with c. However from the rest frame of one of the photons, the other one would have to recede only with c. But I think that will fail because you can't boost into the restframe of a photon (?), because there is no frame where the photon is at rest, it is always moving with c. $\endgroup$
    – Graipher
    Commented Aug 7, 2016 at 12:18
  • $\begingroup$ Yes, or more simply: it's impossible for one of the photons to ever know about the other, because the messenger particle would have to catch up to something moving away at $c$, thus requiring a velocity greater than $c$. So in the rest frame of one photon, the other photon is not visible. Yes you're right, because if you try to Lorentz boost into the photon's rest frame ($v=c)$, the Lorentz factor $\gamma(v=\pm c)=\infty$ will make it very difficult to make sense of things (e.g. with infinite time dilation and infinite length contraction). $\endgroup$
    – Judge
    Commented Aug 7, 2016 at 14:33
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Lets assume you have to explain the truth without any complications to your friend.

Firstly, give your friend some introduction and draw some analogy between physics terms and some nicer terms.

"Space is like a piece of cloth, a fabric which can stretch. There are some godly powers, forces which keep stretching the space fabric (the cloth) continuously. Light always moves along the fabric of space like a super fast worm."

Now ask your friend to be the god by holding two diagonally opposite corners of a cloth and ask him to stretch it quickly (faster than the worm). After stretching, tell him that light still continues to move along the fabric at the speed of light being oblivious to all the stretching magic which happened.

Ask him if the distance between his two hands has increased. Also, ask how far the worm has travelled during the process...

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  • $\begingroup$ Ok, that seems bite sized enough for a non physicists to handle and after careful consideration I believe I understand. I do have one question though. It was implied that the first light years have stretched a considerable amount a when compared in distance to light years today. Do we know if light travelled faster than 186,000 miles per second? or if those miles have just stretched in length since early universe? If it is the latter then shouldn't that stretching be quantified and updated like leap seconds? or maybe that's the same thing?.... mind blown... ok marked as answer. Thank you $\endgroup$
    – Jim McMillan
    Commented Aug 12, 2016 at 2:27
  • $\begingroup$ Initially, the space was just a point which expanded to, say a light year, the expansion continued and it is now 92 billion light years in diameter. If I got your question right, then what we today is a fully stretched extremely small point. Light cannot travel faster than light and this is a fact. However, the theory of relativity and quantum mechanics do not work at the very beginning of the universe. Stretching of the universe is already quantified but involves high level math. $\endgroup$
    – Yashas
    Commented Aug 12, 2016 at 9:17
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As Yashas said, the standard "math-free" explanation is that the space in between the particles stretches, so that the distance between the particles is growing faster than the speed of light. Now we have to explain why this doesn't contradict your statement, "the particles never exceed 186,000 miles per second".

The problem is we need to specify what this speed is relative to. In special relativity, you can compute the speed of any object relative to any other object (e.g., the speeds of the two particles relative to each other). However, this is only possible since spacetime in special relativity is uniform. Every point is the same, so anything at any two points can be compared.

In general relativity, spacetime is no longer homogeneous, and as a result it is impossible to compare vectors, such as velocities, at two different points. You can say that a particle has just moved past you at some speed, but it's meaningless to calculate the relative velocity of two distant particles.

As an analogy, suppose we stand at two different spots on the equator, both pointing North. If we wanted to compare the directions we're pointing, we have to meet in the same spot. If we meet on the equator, our directions will agree. If we both march up to the North pole and compare, they'll disagree! This happens because the Earth's surface is curved. For identical reasons, it is impossible to compare distant velocities in curved spacetime. There's no unique way to do it.

As such, the fact that the distance between them grows faster than $c$ doesn't contradict relativity. Nothing is ever traveling faster than $c$ according to a local observer.

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