Chrisc said:
One last appeal to your ...
10/20 = 1/2
If I gave you 10 dollars over twenty days or 1 dollar over 2 days, I would in each case give you the same thing, a ratio of money/days that is a constant $0.50/day.
I think you will agree the ratio of money/days remains constant and if all you could ever look at was the money/day you would argue (as your are now) that there is no difference in the money/day in either case.
I am simply arguing that since you can look at more than the money/day in that you can see that the time of A is different than the time of B, you must consider the magnitude of the ratio they both agree on is different between them. One gets a total of 10 dollars, the other a total of 1 dollar.
Oi, is someone using my economic theory without attribution?
Chrisc, maybe the problem here is that you are only thinking of one factor. I discussed a sort of economic theory of spacetime with DaleSpam a while back.
Think of it this way, you have a constant income of spacetime dollars - ct spacetime dollars per t. You can spend it just sitting around doing nothing, which means you spend it in terms of time (c.t_{you}). You might notice that I split mine up and spend part of my income on moving from one place to another (distance of v_{me}.t_{me}) and the remainder goes up in time (c.t_{me}) . The exchange rate depends on how long I take to move from one place to another, but the end result is that the spacetime dollars I spent my distance traveled and my time elapsed will equal the spacetime dollars you spent on your time elapsed (and your distance traveled which was zero (v_{you} = 0) ):
\sqrt{v.t_{me}^{2} + c.t_{me}^{2}} = c.t_{you}
The same sort of equation can be used for a rod at rest. A rod at rest has simultaneous ends, the value of the simultaneous ends of a rod which has a length of x is x spacetime dollars.
Relative to you, that rod can convert some of that length into motion (this is not rational economics here) - this is giving the rod a time component, and will make the ends of the rod non-simultaneous to you. The magnitude of the non-simultaneity (MNS) and the length of the rod will vector sum to the resting length of the rod:
x_{at rest} = \sqrt{x_{in motion}^{2} + MNS_{in motion}^{2}} ... (1)
Finally, the same sort of equation can be used for a clock. A clock at rest, with colocal ticks and tocks, has a time value of ct_{at rest} - where t_{at rest} is the number of ticks and tocks, not the period between each tick and tock. The clock can convert some of that, relative to you, into motion but the ticks and the tocks will not be colocal. The vector sum of the values, time value in motion plus the extent of non-colocality (ENC) of the clock in motion will add up to the rest time value:
ct_{(at rest)} = \sqrt{ct_{(in motion)}^{2} + ENC_{in motion}^{2}} ... (2)
...
Equations (1) and (2) can be more balanced, if one considers a zero MNS_{at rest} and a zero ENC_{at rest}:
\sqrt{x_{at rest}^{2} + MNS_{at rest}^{2}} = \sqrt{x_{in motion}^{2} + MNS_{in motion}^{2}}
and
\sqrt{ct_{(at rest)}^{2} + ENC_{at rest}^{2}} = \sqrt{ct_{(in motion)}^{2} + ENC_{in motion}^{2}}
Leaving these out is possibly where a lot of confusion comes in.
Anyways, in conclusion, if you "make" the ticks and tocks of a clock non-local or you "make" the ends of a rod non-simultaneous, by giving them motion relative to you, then you will reduce the number of tick and tocks or contract the rod to an extent equivalent to the extent to which the ticks and tocks are non-local or the ends of the rod are non-simultaneous.
...
No-one but me may understand this "economic theory", but the by the state of the world economy, no-one understands the real thing either
cheers,
neopolitan
PS In terms of the quoted section, you could say that one guy got $10 for just sitting there, the other got only $1 but did get to do some very fast, mind broadening travel (since the traveling one spends less spacetime money on time, he could arrive back home after twenty days of the $10 guys time having only 2 days on his clock, qualifying for only 2 days' pay).