My question is concerning wikipedia article on Oh-my-God particle, to be precise, this paragraph:

This particle had so much kinetic energy it was travelling at 99.99999999999999999999951% the speed of light. This is so near the speed of light that if a photon were travelling with the particle, it would take 220,000 years for the photon to gain a 1 centimeter lead. Also, due to special relativity effects, from the proton's reference frame it would have only taken it around 10 seconds to travel the 100,000 light years across the Milky way galaxy. [1]

I would like to see demonstration how the special relativity effect allows the particle to travel the distance in 10 seconds.


Thanks for all responses, I have one more question: you all explain the situation from the "Proton reference frame". What about from "Observer reference frame"? We can imagine the observer (and also the whole universe around) moving at 99.99999999999999999999951% the speed of light comparing to the "stationary" proton. How will the proton look from this reference frame?


This was not a homework, just a weekend curiosity while reading wikipedia :-)

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    $\begingroup$ Note the all-important "from the proton's reference frame". $\endgroup$ Mar 28 '16 at 11:42
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    $\begingroup$ What would it feel like if it hit your skin? $\endgroup$
    – Chloe
    Mar 28 '16 at 21:32
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    $\begingroup$ It has as much energy as a baseball flying at nearly 100 km/h. It is also charged. But I am actually curious, if it is traveling so fast, will it be able to interact much with the tissues of the human. Will it get absorbed, reemitted or scattered? Will just ionize a single nucleus or could it cause a chain reaction. That is actually a great question, now that I think of it. Would love to see thoughts on that matter. $\endgroup$
    – Ilya Lapan
    Mar 28 '16 at 21:47
  • $\begingroup$ @IlyaLapan: the Wikipedia article starts to address that, saying that it can't convert all that kinetic energy to other forms in a single interaction (I believe this is on account of the need to conserve momentum). So if it interacts at all then I suppose there must be a chain reaction in the sense of the produced particles having further interactions, producing a particle shower as it does in air. Given that it interacted with air (and this flash was spotted by the detector) there must be some chance of it interacting with your body, but I've no idea what chance... $\endgroup$ Mar 29 '16 at 13:57
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    $\begingroup$ @IlyaLapan: You have found a satisfying answer to a rather serious para-phenomenon that has been hunting us for centuries and that not even Dana Scully could explain. :-) $\endgroup$
    – Damon
    Mar 30 '16 at 8:52

Key point in your quote is: "from protons reference frame". In the reference frame, travelling at a relativistic speed, length contraction is experienced. All the lengths in the direction of travel of the particle are contracted by Lorentz factor: $$ l'=\frac{l}{\gamma}$$ $$ \gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$$ So $ \gamma = \frac{1}{\sqrt{1-(0.9999999999999999999999951})^2}=3.19*10^{11}. $ In the reference frame of the particle, Milkyway is contracted by this factor. So the proton sees it only $2.96 * 10^9 m$ long. Now you can do the usual calculation to find time using the new contracted length and see that it would take only $2.96 * 10^9/ 3*10^8 = 9$ seconds to cross the Milkyway.

Length contraction is kind of a consequence of 4D space-time we live in. If you look at time dilation (which also can be used to derive this result but is less intuitive in my opinion), the length contraction naturally arises from it. If you want to know more about length contraction you can easily more information on it. It is a topic which is usually well explained in any special relativity book,and I bet there are a lot of question on the topic on this website, search tags and .

  • $\begingroup$ Note that you can type [tag:special-relativity] to get a link to all questions with that tag (i.e., special-relativity). $\endgroup$
    – Kyle Kanos
    Mar 28 '16 at 11:25
  • $\begingroup$ Thanks for advice, Kyle, I made the tags into a link now! $\endgroup$
    – Ilya Lapan
    Mar 28 '16 at 11:31
  • $\begingroup$ The two tags you suggest are actually linked by the tag synonym system: length-contraction maps to special-relativity. $\endgroup$ Mar 28 '16 at 13:59
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    $\begingroup$ Maybe worth pointing out that from the reference frame of a photon, nothing takes any time. As an objects speed becomes more like that of a photon, the time it takes to move any distance in its own reference frame becomes closer to zero. $\endgroup$
    – bdsl
    Mar 28 '16 at 16:05
  • $\begingroup$ @DanHenderson I know, I just meant that as the speed of the proton is 'near' that of a photon it makes sense that the proper time of the proton crossing the milky way is 'near' zero. $\endgroup$
    – bdsl
    Mar 28 '16 at 18:15

Relativity makes time relative (what a surprise! :)). It makes a difference, whether we look at the particle from "outside", or if we travel along with the same speed.

So viewed from outside the particle is a normal fast particle, and need some hundreds of thousands of years for the milky way. The "oh-my-god"-ity does nothing here, it makes it only a few seconds faster that plenty of others.

Only the particle itself doesn't experience it like this. You can either say, that time flows much slower in the frame of the particle, or that the milky way is much shorter (again, as seen by the particle only!).

These are two equivalent formulations, and you will find them explained in more breadth on the first pages of every relativity book.

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    $\begingroup$ It took 220,000 years for the photon to gain a 1cm lead, but the "Oh-my-God" particle has to experience the speed of light as the speed of light, not as 1cm/220000years. So its relative timeframe is slowed relative to ours. $\endgroup$
    – wizzwizz4
    Mar 28 '16 at 8:26
  • $\begingroup$ I dont see your point, what are you commenting? Or do you mean my second possibility of explanation? - that's also okay, you can say that it thinks that it is not moving and experiences the widths of the milky way as 1cm $\endgroup$
    – Ilja
    Mar 28 '16 at 8:39
  • $\begingroup$ I was adding the explanation for why it is the length of time it is. $\endgroup$
    – wizzwizz4
    Mar 28 '16 at 9:10
  • $\begingroup$ @wizzwizz4 that's a pretty intuitive and nice way of looking at things. No wonder special relativity was referred to by Einstein himself to be a theory of invariants. $\endgroup$ Apr 1 '16 at 10:31

There is a common misunderstanding about special relativity related to this. What we in physics call a velocity is distance/time where distance and time are measured by the same observer. This one is always smaller than the speed of light. So... if we measure the galaxy and we measure the time, we see the particle needing quite a lot of time to get from there to here.

What the particle itself sees is a VERY relativistically contracted space. The galaxy for the particle looks like a bright (doppler-shifted) and extremely thin pancake. So naturally, it takes it almost no time to travel across. That's the 10 seconds the article is talking about. That is, if the particle itself had a watch on its wrist, it would measure that time. Again, dividing these two quantities gives you the hairline below speed of light.

But the speed "limit" doesn't mean you can't get very far in your own time. It actually makes travel easier. If you say things naively: measure the distance in earth coordinates, then accelerate and measure your own time while traveling, and divide the two, you can get any number you want, this isn't really a physical quantity at all, the quantities are measured from different reference frames and what you get isn't really a velocity. You can get anywhere you want in almost no time (sure, people on earth will be long dead when you get there, but for you it'll be a short journey).

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    $\begingroup$ Note: such a particle would also 'see' relativistic aberration; the universe would look mostly black but with very bright spots directly in front and behind. The pancake wouldn't unfold until the particle was almost right on it. $\endgroup$
    – M.M
    Mar 28 '16 at 23:53

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