When I began to think about teaching special relativity
for my first modern physics course, I could usually guess which observer
in a given problem would see a clock running slow relative to his own. But
I wondered, couldn't we then call the phenomenon "time compression" from
the vantage point of the frame in which it runs fast?
In my desire to visualize this more concretely, I decided to illustrate the
process with an x-ct
diagram. Even though the sound byte "time dilation" was familiar to me,
and even though my intuition usually told me how to solve time dilation
problems, I didn't really understand the concept until this was done.
As Jonathon Swift's Gulliver might have said, "When I could hold the
fruit in my hand and taste it, the inadequacy of my prior wisdom became
apparent." This was one of my first object lessons in the sloppiness of
our own language for dealing with relativistic spacetime effects.
In special relativity terms, "time dilation" problems involve two events
- which we might refer to as start and stop events that take
place on one particular clock. As I looked at the diagrams I
realized that in time dilation problems, one and only one inertial frame,
namely the "undilated one", was accorded and deserved special status.
This special frame, inherent to the statement of the problem, is the frame
in which the clock is traveling. All other inertial frames see the clocks
in this privileged frame as traveling slower. Thus when you look at any
moving clock, or equivalently when any clock moves, it's time is "dilated"
and thus slowed down. Moving a clock never causes its time to
"contract" or move faster!
To pose a specific problem, imagine an inertial clock which times an
interval of one year, and an inertial frame that observes the
start and stop of this timing adventure from frame traveling at
constant coordinate-velocity of one third of a lightyear per map year
(i.e. v = c/3) to the right.
The rest frame of the clock is chosen as the one with orthogonal axes.
Imagine that the tick marks on the green grid denote times (vertical)
of one year, and distances (horizontal) of one light-year. (This would
work as well with seconds & light-seconds, respectively, if we wanted
to have the experiment over more quickly.)
The red line denotes the world line (path through time) of the clock.
The red boxes denote the start and stop events under consideration,
spaced one year apart in the frame of our clock.
The blue grid shows lines of constant inertial time b (about 18
degrees up from the horizontal right) and constant position (about 18
degrees rightward from the vertical) for an inertial observer moving
to the right at one third of a lightyear per map year
(i.e. v=c/3). The start event occurs at "time zero"
for each observer. The red dash illustrates the rightward moving
observer's measurement of the time of the stop event. Since a year
on the rightward frame's clock is marked by the dotted blue axis
crossing, the stop event happens a bit more than a year after the
start event in that frame. Since gamma for the rightward frame
is 1/Sqrt[1-(v/c)^2] = 1/Sqrt[1-1/9] = Sqrt[9/8] = 1.06, the time
interval measured in the blue frame between events is 1.06 years.
As mentioned on our x-ct
plotting page, the same information could be obtained for a plot that
uses orthogonal axes for the rightward moving frame. Then of course the
clock is moving to the left. As in other examples elsewhere on these
pages, the picture looks quite different even though the information it
provides is the same!
Check out also this
table-of-contents
of map-based anyspeed motion resources.
Mindquilts site
page requests ~2000/day approaching a million per year.
Requests for a "stat-counter linked subset of pages" since 4/7/2005:
.
Send ideas/ comments/ questions/ complaints, and register for updates
to these pages with an email subject-field containing "accel1",
to staff@newton.umsl.edu. This page might contain original
material. Hence if you echo, in print or on the web, a citation
would be cool. {Thanks. :) /pf}
