On 1/5/2019 12:50 AM, agrayson2...@gmail.com wrote:
On Saturday, January 5, 2019 at 6:49:43 AM UTC, Brent wrote:
On 1/4/2019 9:20 PM, agrays...@gmail.com <javascript:> wrote:
*Still a little murky. Does coordinate time ever
differ from proper time? TIA, AG *
Of course. That's like asking does change in longitude
ever differ from distance sailed.
Brent
*Suppose I imagine a path in spacetime, say with loops,
returning to the same spatial location. *
Ok. Like the traveling twin.
*Some amount of proper time will have elapsed*
Along that path.
*, invariant for all observers, but the elapsed coordinate
time will in general be different, with proper time and
coordinate time initialized to identical but arbitrary values
as the path in spacetime is traversed. *
You can set proper time and coordinate time to the same value at one
event (the initial event). But I don't know what you mean by
"intialized...as the path is traversed".
*
*
*That's all I meant, as in your first sentence above. AG*
*The other imagined coordinate clocks can't be synchronized since
they relate to different events in spacetime, *
I don't know what this means. In generic spacetimes there are no
"coordinate clocks".
*OK, no coordinate clocks. The coordinate t is just the time label for
an event. AG*
Coordinates are just smooth functions that provide labels to each
point in 4-space. Since they don't have any physical significance, in
general there isn't any physical clock that keeps "coordinate time".
I don't know what you mean by "relate to different events in
spacetime". Clocks just mark intervals along their paths.
*so something is wrong with this model, specifically if the imagined
path in spacetime does not return to its initial spatial position.
TIA, AG*
But you hypothesized that it did. Now you're worrying that it
didn't?? Remember that clocks measure intervals between EVENTS
(things that have four coordinate values), not between PLACES (things
that have three coordinate values).
*This is my problem; maybe a non problem; for any path between two
events, the proper time interval is invariant, meaning the same for
all observers, but it will be different depending on the paths. But
the elapsed coordinate time intervals are the same, since the
endpoints represent the same pair of events. So there doesn't seem to
be any relationship between elapsed proper time and elapsed coordinate
time. AG*
The relation is provided by the metric. If you choose different
coordinate systems (e.g. cylindrical or spherical or whatever) then
there is different metric tensor. So the integral along the path of
g_ab dx^a dx^b is the same.
Brent
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