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[help] Special relativity, reference frames and timekeeping
So I’m noodling a bit more of special relativity into my poor, ageing, liberal arts educated brain, via (among other things), posts such as this one. (The first few dozen comments are excellent, btw. After that the thread starts breaking down.)
What got me rattling down this track was the problem of timekeeping across astronomical distances. Since reference frame is a critical concept, is it reasonable to have your clock so far away that it stands outside your reference frame?
For example, let us posit a pulsar 30,000 light years distant. Let us further posit that its periodicity is slowly lengthening, by a degree both measurable and predictable. Say that I know from a certain zero point, say, Earth on a given day and time, what the periodicity is. Then, some arbitrary amount of time later, having moved at relativistic or supraluminal speeds to an astronomical distance, I measure my beacon star’s periodicity again. I then calculate backwards (or, potentially forwards) from my known zero point reference.
Have I in my new, arbitrary location now established the time at my zero point? Ie, have I synchronized my clocks?
I am absolutely certain there is at least one serious error here, but I can’t see it yet. I’d appreciate comments that might correct this idea, or references to more successful examples of this kind of thinking.
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Posted: 7:12 pm Wed December 01 2010 |
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Relativity doesn’t prevent you from synchronizing clocks over arbitrarily long distances. It just says that, if your clock is synchronized with mine in a given reference frame (that of our galactic arm, say), then they are out of sync when viewed by an observer who is moving relative to both of us.
Seems like a complicated way to keep time. The usual way would be to receive a message from a person a known distance away who knows what time it is, and then account for time lag due to distance.
In the scheme you suggest, you need to also (1) know how far away the pulsar is, and (2) know how fast you are moving with respect to it. In principle you also need to know your gravitational potential relative to the pulsar, but I’ll assume you’re not simultaneously falling into a black hole, since if you were, probably “what time is it, exactly” is not the most important thing you are currently worrying about.
Ah. So, for example, at interstellar differences, one could aim a sufficiently powerful signal at a target star, and the people at the other end just read off the “time on the tone” and do the required math for the time/distance lag. I suppose the value of the pulsar method would be if one wandered off somewhere that wasn’t on the receiving end of a timekeeping broadcast.
Also, your comment about the black hole made me laugh out loud. Thank you.
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In addition to the delta in distance to the chronometric pulsar that Geoffrey mentioned (since the pulsar’s period is changing, albeit predictably, it’s conceivable that the pulse that hit Earth at time T reaches the destination at time T+X where X is directly proportional to the Cartesian distance between Earth and the destination), one also would need to account for Doppler shift changing the frequency of the pulsar dependent upon the velocity of your vessel relative to the pulsar (toward/away).
And then also account for time dialation and space compression for the observers on the ship moving at relativistic speeds… which is calculable relative to a Cartesian/stationary reference frame based on the velocity of the ship… So the ship’s computer, given a starting Earth time reference at departure, and input from the nav system on velocity and ship’s time at speed, it could calculate accurate earth time without all that pulsar stuff.
Which means there would be some pretty severe reverse jet lag as the occupants of the ship would age slower than the folks on earth, but find themselves wondering where the time went…
The other problem is the precision with which one needs to “synchronize the clocks” — it is severely limited by both instrumentation in general and quantum effects in particular… and it’s no better than one’s calculations of how close one is to at rest relative to the reference pulsar in the first place.
For example, imagine that the pulsar’s periodicity is lengthening by one Planck-unit (approximately 5.4E(-44) seconds) every 3602 seconds (one hour and two seconds). If you want to synchronize your clocks to the precision of an hour, that requires your measuring apparatus to have precision greater than one Planck-unit… which is not legal under current understandings of fundamental and quantum physics, as the Planck-unit is fundamental and indivisible. Thus, the maximum precision for such a system would be somewhere around ten to twelve hours, with perfect instrumentation. I’ve spent waaaaaay too much time in laboratories to believe in perfect instrumentation… and half a day doesn’t seem like one’s clocks are “synchronized” anyway.
Conversely, if the pulsar’s period is changing too fast — so that relatively less precise instruments can be used and/or the change is discernable at shorter intervals — its behavior will become less predictable over time as other effects build up. My grasp of statistical mechanics is too rusty to predict “how much” this would be… but there’s a distinct possibility that the amount of data that it would take to recognize the pulsar’s regularity, and establish it as a standard, would be so vast that the pulsar would be moving out of the realm of regularity. Warning! This is a potential plot complication! Warning!