Two Clocks PuzzleBurt Jordaan posted on December 25, 2011 |
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We know that two ideal clocks permanently at rest relative to each other can be synchronized by means of the Einstein method.
^{(1) }Simply stated it means: measure the distance (
d) between the clocks (Alpha and Beta); send a light pulse from Alpha to Beta at time
t; when that pulse reaches clock Beta, set its time to
t+d/c.
An equivalent method is to start with the two clocks in close proximity, set both to the same time and then slow-transport either Alpha or Beta (or both) to end up a distance d apart. Their synchronization can then be verified in both directions, using the above method, ensuring that they are indeed "Einstein-synchronized".
An equivalent method is to start with the two clocks in close proximity, set both to the same time and then slow-transport either Alpha or Beta (or both) to end up a distance d apart. Their synchronization can then be verified in both directions, using the above method, ensuring that they are indeed "Einstein-synchronized".
Suppose these two clocks were of Vulcan origin and Starship Enterprise do a flyby at a constant (mild-warp) 0.6c relative to the clocks. When adjacent to clock Alpha, Scotty records its time and also the bridge clock's time. He does the same when adjacent to clock Beta in order to obtain the elapsed time between the two flybys, as given by clocks Alpha and Bravo. To his surprise, Scotty finds this elapsed time to be larger than what the bridge clock has shown it to be.^{(2)}
At the debriefing Captain Kirk requests: "Right Mr. Scott, did we learn anything about the Vulcan clocks in this experiment?
Scotty: "Ay, Captain, we found that Vulcan clocks tick faster than our bridge clock".
Spock: "How can you say so, Mr. Scott? All Vulcan clocks conform to the United Federation of Planets standards of time. What is more, Starfleet-approved relativity theory says exactly the opposite. The Vulcan clocks were moving relative to our ship, so to us they will appear to tick slower than the bridge clock".
Scotty: "Sorry Mr. Spock, but I measured it and to measure is to know...".
Kirk: "I spot a discrepancy here, Mr. Spock. Will you be so kind as to give us your Vulcan solution to this?"
If you were Spock, and assuming that Scotty measured properly, how would you have resolved the apparent paradox for the crew?
Burt Jordaan (http://www.relativity-4-engineers.com)
Notes
- "Permanently at rest relative to each other" implies free-fall in zero (or at least extremely weak) gravity and also not at 'cosmological distances' apart.
- Take c as one foot per ns, the distance (d) between the clocks as 1000 ft in the Vulcan frame. Clocks Alpha and Bravo will then show an elapsed time of 1667 ns between the flybys, while the bridge clock will record the elapsed time as 1333 ns.