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I was thinking lately about relativistic time dilation and time passing differently on other planets (which both, eventually turned out to be caused by the same principle).

Now, quite naturally (as it seems to me, at least =) I came to the question stated in the question title. Here is what I mean: time is measured in different units, which all stem from a second (which is a SI base unit). According to wikipedia, as one (probably) would expect (italic is mine):

... the historical definition of the unit (second) was based on this division of the Earth's rotation cycle ...

It is quite logical, since initially most of the things we measured time for were directly related to the Earth and it's rotation (i.e. ships leaving at the exact time etc.).
Following the quote above:

... the formal definition in the International System of Units (SI) is a much steadier timekeeper: 1 second is defined to be exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" (at a temperature of 0 K). Because the Earth's rotation varies and is also slowing ever so slightly, a leap second is periodically added to clock time to keep clocks in sync with Earth's rotation.

With that said, moving to my questions now:

  1. What does it actually mean to say that on planet A time is twice as fast as on planet B? Does it mean that while 9,192,631,770 "periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom (at a temperature of 0 K)" will occur on planet B — on planet A 18,385,263,540 such radiations will occur?
  2. Are the radiations in Caesium-133 are consistently relative to the properties of all other elements? Basically what I am asking here is: if we were to change the said radiation of caesium-133 in the whole universe, then the physical world, as we know it, would completely change (if not cease to exist), is this correct?
  3. Any stable (under given conditions) property of any element could have been selected as a basis for second, and Caesium-133 was picked up, pretty much, arbitrary, am I right in stating this?
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What does it actually mean to say that on planet A time is twice as fast as on planet B?

It is a statement about observers, for the observer in B time is twice as fast on planet A. Planet A does not know it, i.e. the definition of a second is the standard one on both planets.

Are the radiations in Caesium-133 are consistently relative to the properties of all other elements?

In the frame of all involved nothing changes. It is observations on the others frames that show a different time .

Any stable (under given conditions) property of any element could have been selected as a basis for second,

Yes , any stable radiative transition uniquely identifiable .

You might get a better intuition if you read up on the GPS system which needs special and general relativity too to keep correct distances on earth.

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Does it mean that while 9,192,631,770 "periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom (at a temperature of 0 K)" will occur on planet B - on planet A 18,385,263,540 such radiations will occur?

The crucial word is "while". It's absolutely essential to give it an observational meaning. To do this many more words are needed.

We must assume that on both planets advanced physics labs have been built, and that previous communication established that the same physics holds on them. E.g. they will have identified chemical elements, among them caesium. They will have built atomic clocks, following quite similar protocols. (Just as we have here on Earth, in several labs all world around - and here we have quite a similar problem: how can we know all those clocks run at the same rate? Actually they don't.)

Then a cross checking campaign is started. From lab on A e.m. signals are sent to B at intervals of 1 seconds of A's clocks. On B arrival times (according B's clocks) are recorded. For a best comparison, the reverse procedure is also actuated, from B towards A.

Data analysis, jointly executed on all data, can show that in direction A$\,\to\,$B a factor $k_{AB}$ is observed, i.e. signals sent at 1 s intervals from A are received in B at $k_{AB}\,$s. In reverse direction a factor $k_{BA}$ is measured.

This unmistakebly shows a difference between both planets. Should we say that "time runs faster (or slower, according as $k_{AB}>k_{BA}$ or vice versa) on A than on B"? I wouldn't, as a matter of personal taste. But what really matters is the observational result.

A question for you. Why both signals are required? Wouldn't A$\,\to\,$B have been sufficient?

Are the radiations in Caesium-133 are consistently relative to the properties of all other elements?

I already answered in the affirmative. I said physics is the same on both planets.

Basically what I am asking here is: [...]

Here I have no answer, mainly because I can't understand the question.

Caesium-133 was picked up, pretty much, arbitrary, am I right in stating this?

Not quite. I'm no metrology expert, but I believe the choice was made in function of some favourable properties of various kinds. Above all, I would say, on better quality of atomic clocks built with that element.

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  • $\begingroup$ What I meant by Basically what I am asking here is: [...] is this: if we hypothetically, could tweak the speed of Caesium-133 radiation, say "set" it to be twice as fast as it is now, would it affect "well-being" of the universe (i.e. it's physical laws) or every characteristic will change (relative to new speed of Caesium-133 radiation) and no physical laws will be affected? Hope this makes sense. $\endgroup$ – Filipp W. Dec 20 '18 at 16:52
  • $\begingroup$ The best idea I have as for "Why both signals are required? Wouldn't A→B have been sufficient?": I think it is the matter of double checking (e.g. if signal sent every second from A to B, reaches B twice a second, we would expect signal sent from B to A to reach A every two seconds; otherwise — something wrong with our measurements and we should recheck everything). $\endgroup$ – Filipp W. Dec 20 '18 at 17:05
  • $\begingroup$ @FilippW. As to your hypothesis of setting a caesium clock twice as fast, still I can't understand. If you could do this, why in the world would such manoeuvre influence other things? Up to affect physical laws? But I'm afraid there's a hidden idea you aren't able to express clearly. $\endgroup$ – Elio Fabri Dec 20 '18 at 19:57
  • $\begingroup$ @FilippW.Redshift. I believe you don't see what I'm thinking because there's something you are taking for granted (with no reason). You talked about two planets. Now it's very unlikely they are stationary relative to each other. So you had to be prepared to a Doppler effect. But if it were present it would be symmetrical. Instead a gravitational redshift can be asymmetrical. A cross check would reveal this. $\endgroup$ – Elio Fabri Dec 20 '18 at 19:59
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    $\begingroup$ @FilippW. I'm becoming to see some light ;-) Caesium-133 hyperfine splitting (and transition frequency depending thereon) is due to nuclear magnetic moment. So it's certainly sensitive to nuclear force and then to strong interaction. Your quotation of Dawkins quoting Rees brings in a well known direction. You may find some further information looking for "anthropic principle". I don't want to say more, as we would go OT. Just a telegraphic note: I've always been rather skeptical about that principle, even in its most moderate form. To me it's chatter physics. $\endgroup$ – Elio Fabri Dec 22 '18 at 16:46

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