I also want to thank you Stephen for your detailed reply to leaking pen. I do
want to understand all of this as well, but it will take me a while to digest!
>You can view it that way, but it's a little hazardous, because time
dilation isn't really just a simple number.
So is time dilation a vector in space with direction and magnitude? That has
been my conclusion, but you clearly understand this at a level I do not.
>Thinking of it as a simple ratio leads to a lot of confusion. Time
dilation, expressed as a number, is dt/dtau for a particular observer,
"A", relative to a particular reference frame, "F". The "dt" value is
found by A, by looking at clocks which are stationary in frame F, as A
passes them by. The "dtau" value is found by "A" by looking at A's own
clock.
Could you please explain a little more what dtau represents? My understanding
of dt is that it represents the rate at which time moves forward in the frame
of reference of "A". Is that correct? Does dtau represent the time interval
elapsed in "A" between the observation of the first clock in "F" to the
observation of the second clock in "F"?
>Note well: "A" uses ONE clock in his/her own frame. "A" uses AT LEAST
TWO CLOCKS in frame "F", located at *different* points in frame "F".
You can't measure time dilation between two inertial frames without
using at least two clocks in one of the frames, because once the
observer has passed a clock, it's gone, and they can't see it any more
(except at a distance and using a telescope adds unnecessary hair
without changing the result).
Ok, I think I get this part.
>Thus, time dilation actually measures the rate at which time passes
along a *particular* *path*. Something that measures a rate of change
along a path is a directional derivative, or a "1-form". It's not a
simple number.
Sounds like I need to be educated about directional derivatives, or "1-form".
I'll do some googling, but any help you can give... How does it differ from a
simple vector?
Ok, I googled it - calc 3 - Ouch... Only made it partly through calc 2, and
that is very rusty, so this one is a little beyond my math abilities. But if I
understand what little I have read we are talking about the rate at which time
changes in a particular direction. That was my understanding already, so I
think I conceptually get this, or so I hope. So, knowing the rate at which time
moves forward in the direction of motion tells us nothing about dt in any other
direction, correct?
...
I'll need much more time to absorb the rest of this.
C. Michael Crosiar
