# Is this a coincidence?

The smallest distance probed in any experiment so far conducted is $$10^{-18}$$m and the largest distance we can have is the radius of the universe, nearly $$10^{26}$$m. The ratio is $$10^{44}$$.

In another case, if we think about time, the shortest time studied is about $$10^{-26}$$ sec and the longest time is "the lifetime of the universe", which is estimated to be $$13.7$$ billion years, or nearly $$10^{18}$$ sec. The ratio is again $$10^{44}$$!

Is this(the ratio belongs is of same order) a coincidence or there is some deeper explanation(probably space and time are related etc.)? I am an undergrad student, don't know much about astronomy. I wondering about this question while studying quantum mechanics from the book The Quantum World, which claims that it is not a coincidence.

The book explains it in the following way -

"This is not a coincidence. The outermost regions of the universe are moving at a speed near the speed of light, and the particles flying about in the subatomic world are also moving at such speed. On both subatomic and cosmological frontiers, the speed of light is the natural link between distance and time measurements."

I haven't posted exactly what the book is saying because I wanted to hear what other people are saying about this and also because I haven't understood what the book is saying.

• BTW, the radius of the observable universe is roughly 46.6 billion light-years, or $4.4\times 10^{26}$ metres. – PM 2Ring Jan 20 '19 at 16:00
• The only thing which makes sense in this whole SE page is that time and distance are directly proportional for anything moving at constant speed. Hope we don't waste time on here anymore. Starting myself, of course. – Alchimista Jan 21 '19 at 13:56

Giving it some thought, I think this is a question worth some examination, so I voted your question up.

The largest distance we can see and the amount of time we can see by looking as far in the past as we can are directly related. We can see $$13.8$$ billion light years away, though we can infer that those most distant objects are currently 44 billion light years away (a variation of 3.2), and we can also infer that the diameter of the observable universe is twice that, but let's ignore the expansion and just use the farthest distant object we can observe $$- 13.8$$ billion light years and $$13.8$$ billion years. Using those numbers doesn't significantly change your question and those two numbers are clearly related.

Translating to seconds and meters, we get $$4.35 \times 10^{17}$$ seconds (let's not call that nearly $$10^{18}$$, it's less than half).

And translating light years to meters, we get $$1.31 \times 10^{26}$$ meters. About 300 million times the other, which makes sense cause that's how many meters light travels in 1 second.

OK, let's look at small and see if we get the same.

The Large Hadron Collider gives a pretty precise measurement of the smallest distance we can measure, see this question for details. That number is what you said, about $$10^{-18}$$ meters.

The shortest time we can measure isn't quite as clear cut. This article gives a time of 250 atto seconds ($$250 \times 10^{-18}$$ seconds):

Most recently, the techniques of pulse generation and measurement have been extended to the ultraviolet and soft x-ray region of the spectrum where pulses on the order of 250 attoseconds have been created and used to study the motion of electrons around the nucleus of a neon atom.

But it mentions that quantum models can infer time periods much shorter. The shortest is the estimated average lifespan of the top quark at about 0.4 yoctoseconds or about $$4 \times 10^{-25}$$ seconds (40 times your estimate of the 1 to the power of -26).

Events as short as about $$10^{-25}$$ second have been indirectly inferred in extremely energetic collisions in the largest particle accelerators. For example, the mean lifetime of the top quark, the most massive elementary particle so far observed, has been inferred to be about 0.4 yoctosecond.

So, when I run the numbers, I get the shortest estimated time and the age of the universe have a ratio of about $$1.1 \times 10^{42}$$ and the shortest measured distance and the longest observable distance have a ratio of about $$1.3 \times 10^{44}$$. A variation of more than 100. That's pretty close, but nothing remarkable.

Also, I can't for the life of me see what the lifespan of the top quark and the energy they can generate per collision in the LHC would have to do with each other, so I would say that there is no relation at all, and at risk of taking on your author, I would disagree with his claim.

• On discussing what we can see you cuts the legs of the chair you are sitting on. I didn't read the rest but I wonder the question didn't get close and further got voted and accepted. Going to up vote the one brlow that basically say "forget this question". – Alchimista Jan 21 '19 at 10:26
• I just see the very last part now and I feel better ;) – Alchimista Jan 21 '19 at 10:27
• @Alchimista I'll see what I can do about tidying up my writing style. – userLTK Jan 21 '19 at 17:25
• Dear userLTK my comment was mostly on the fact that the Q doesn't deserve a deep discussion. At least if no reason are given for the assertion "is not a coincidence". There is no link between the minimum measurable distance and the maximum one. Who say the opposite should explain why, else. I an afraid the book linked space and time by speed. So what? Your question is ok as you point out expansion but neglect it according to the Q. – Alchimista Jan 22 '19 at 8:27

It's indeed a coincidence. The human limits on observables cannot have some deeper explanation.

In fact, we can claim many things like this, without having some sort of deep meaning.

The other thing is they are not "exactly" equal. Their ratios just happens to be in the same order. There are also uncertainties in all these measurements and the ratios are never equal to each other. If they were exactly equal in some sense then we may thought there is some deeper meaning.

• Thanks for your answer. I am getting that they are not exactly equal and the ratio is of same order. But the book also claims that it is not a coincidence. – tarit goswami Jan 20 '19 at 18:34
• What the book claims ? – Layla Jan 20 '19 at 18:44
• It says - "this(what I have explained in question) is not a coincidence".. after reading it I also thinking in same way you are saying now.. that is, human limits on observables can't have any deeper explanation.. – tarit goswami Jan 20 '19 at 18:46
• Indeed, thats what I am thinking – Layla Jan 20 '19 at 19:10
• The only thing that I can consider not coincidence is the fine-structure constant – Layla Jan 20 '19 at 19:15

Actually, I favor the idea that it is not a coincidence, at least it's not a coincidence that those two ratios bear some strong resemblance. The reason is because at the large scale, as seen already, the largest time is we can detect is the age of the universe T, and the largest distances L we can measure are going to be directly related to that age by L~cT, at least in rough order (the expansion of the universe makes distance L a rather nonunique issue). On the smallest scales, the energies E we require to probe the shortest times t obey Et ~ h by the uncertainty relation, but if we assume those energies will be manifested in relativistic particles, the best we can localize those particles in a length scale l obeys E/c ~ h/l, also by the uncertainty principle. Eliminating E gives l ~ ct, and so forming the ratio of the largest and smallest scales gives L/l ~ (cT)/(ct) ~ T/t. So whatever that ratio comes out, the two should not be all that much different.

So that those ratios are similar is not nearly as surprising as that they come out so incredibly huge. Why does our universe exhibit such incredible contrasts of times and distances? One might look to the anthropic principle there, in the sense that life requires both long timescales to appear and short timescales to support the required complexity, but then there's still the issue of whether or not one regards anthropic explanations as explanations at all. Also, even if we take anthropic thinking, it seems likely the universe could have produced intelligent life even if it was much smaller, and could have supported such life using much slower processes than those we can detect in our supercolliders. It appears our universe is built with a significant degree of "overkill" in regard to both the volume available for events to happen, and the smallest possible distance scales on which they do happen.