Time Dilation
In imaginary experiments aimed at
exploring Relativity we use a clock that comprises a laser and a mirror at
opposite ends of a glass tube. To ensure that we obtain valid results, results
beyond question, let's assume that only one company makes such clocks and that
they make only one model, so everyone uses identical clocks. If in each clock
the pulse of light from the laser must travel seven feet four inches between the
laser and the mirror, then one tick of the clock represents one tenth of a
second in our little fantasy world where light travels at a speed of 100 miles
per hour.
For the purpose of conducting the
appropriate imaginary experiment let's imagine that we have gone to Goshen
Junction, next to the Southern Pacific tracks that run straight from Bakersfield
to Fresno. Night has fallen and we can watch the pulse in our clock flittering
up and down like a hyper-kinetic firefly. A train goes by and we notice that
someone has mounted a clock on the outside of one of the cars. On impulse I pick
up my camera and snap a picture of the train as it passes, trusting the camera's
mechanism to set the exposure. When we develop the picture, we see that the
exposure was indeed set in accordance with the light available and was too long
for a good picture: the photo shows only a horizontal blur with a red sawtooth
pattern traced over it. That sawtooth pattern, resembling a series of red vees
strung together in the manner of paper dolls, was traced by light from the clock
mounted on the train and we see it as the only feature in the picture that makes
sense, the more so because each vee represents the path that one of the clock's
pulses followed in our inertial frame as it counted time in its clock.
Before we trash the photo (it's not the kind of thing that we want to stick into an album, after all), let's see if we can use it to learn something about how moving clocks tell time. I printed the picture large enough that the vees are several inches high, so we can easily draw two lines on one of them. We draw one straight line vertically through the bottom point of the vee, thereby bisecting the vee, and we draw one straight line horizontally in such a way that it just touches the upper points of the vee. Those two lines convert the vee into a pair of right triangles. Let's choose one of those triangles and list the things that we know immediately about the three lines that comprise it:
Line A; the length of the vertical line, from the bottom of the vee to the horizontal line simply corresponds to the length of the train clock's optical axis, the distance between its laser and mirror. By Lorentz Rule 1 it also equals the length of our clock's optical axis, because the train clock's optical axis is oriented perpendicular to the direction in which the train moves;
Line B; that part of the horizontal line that extends from the vertical line to one of the vee's upper points corresponds to the distance that the train moves in the time it takes the train clock's pulse to go from one end of the clock to the other; that is, the distance that the train moves in one half of its clock's cycle. To calculate that length we must multiply the speed of the train by the time elapsed on our clock during that half cycle of the train's clock;
Line C; the hypotenuse represents the line
traced by the train clock's pulse as it moves from one end of its clock to the
other. By Einstein's second postulate, it must move at the same speed of light
(100 miles per hour in our fantasy world) in both our frame and the train's
frame. Thus the length of the line equals the product of multiplying the speed
of light by the time elapsed on our clock, the same time that we used to
calculate the length of Line B.
We also know that for the passengers on
the train the pulse in their clock follows a path whose length corresponds to
the length of Line A. After all, in their frame the clock does not move, so the
pulse must follow a perfectly vertical straight line. They can calculate the
length of that line by multiplying the speed of light by the amount of time
that, in theory, elapses on their clock during that half cycle, one twentieth of
a second in this case, and we can accept that calculation because we make the
same calculation in regard to our own clock. But then we notice that Line C is
longer than Line A, which means that a pulse in our clock will complete its
traverse before a pulse in the train's clock completes its own traverse. That,
in turn, means that if we pick two events, such as the train's clock passing the
signal at a grade crossing and its crossing the points of a nearby switch, the
train's clock will count fewer seconds between those events than we would
calculate from the readings of our own well-synchronized clocks set up by the
signal and at the switch. The train clock appears to us to count seconds that
are dilated relative to the seconds that our clocks count.
Not only the laser clock slows down when
it moves. The Principle of Relativity tells us that the passengers on the train
can make no observation or experiment that would tell them that they're moving
and we're not. Thus everything on the train that they could conceivably use to
count time appears to us to slow down in the same proportion in which the laser
clock has slowed: the conductor's spring-wound pocket watch, the businessman's
wristwatch, which counts the vibrations of an electrically stimulated quartz
crystal, the rubber ball that a boy is bouncing off the floor of the coach, even
the rates at which the passengers' hearts beat. Indeed, if we could measure the
temperature of the train without touching it, we would discover that the coach
and everything in it is astonishingly cold: even the random atomic motions that
comprise heat have slowed down.
A temptation might come to us and urge us
to think that something in the fact of their motion has so altered all of the
objects mentioned above that they have slowed down. That's essentially what
Hendrick Lorentz assumed when he worked out time dilation himself. He assumed
that the aether blowing through a clock so altered the forces exerted among the
clock's atoms that the clock's parts moved more slowly. The Principle of
Relativity flatly contradicts that assumption, as we can see by the fact that if
it were true to Reality, then the passengers on our train would have to see our
clocks ticking faster than theirs do. But the passengers on the train will see
the pulses in our clocks tracing the same sawtooth pattern that we saw their
clock tracing in our frame and they will infer that our clocks are counting
dilated time relative to their clocks. From that proposition we must infer that
time dilation exists not in the objects but in the structure of the inertial
frames in which the objects are embedded.
Having thus deduced the existence of time dilation, we now want to determine the amount of dilation that we can expect to observe on a given moving clock. We can fill that want by using our descriptions of Lines A, B, and C and the fact that they comprise the sides of a right triangle. Quite a few centuries ago a fellow named Pythagoras figured out a rule about any right triangle: the square of the length of the hypotenuse equals the sum of the squares of the lengths of the sides. Let's use that rule in a slightly different form and say that the square of the length of Line C minus the square of the length of Line B equals the square of the length of Line A. We calculate the lengths of Line C and Line B from a time measured on our clocks and we calculate the length of Line A from the corresponding time measured on the train's clock, so we now have a mathematical relation between those times. When we apply some straightforward algebraic manipulation to that relationship (and it's not as difficult as you might think, as I demonstrate below), we obtain
LORENTZ RULE 2: The time that an observer measures elapsed on their own clock between two events that are both touched by a moving clock equals the time interval between those two events measured by the moving clock multiplied by the Lorentz factor between the inertial frames occupied by the moving clock and the observer.
We have the recipe for calculating the
Lorentz factor as follows: take the velocity between the two inertial frames,
square it, divide the square by the square of the speed of light, subtract the
resulting fraction from the number one, extract the square root of the result,
and divide that square root into the number one. Conversely, if you know the
Lorentz factor between two inertial frames and want to know the corresponding
speed, then you have the recipe for calculating the speed as follows: square
your Lorentz factor, divide that square into the number one, subtract the result
from the number one, and extract the square root of the resulting difference.
You will obtain the speed between your two inertial frames expressed as a
fraction of the speed of light.
Now let's do that analysis algebraically. Let's assume that we use lower-case letters to represent our variables (measurements) and that the passengers on the train use upper-case letters. Let's also assume that we have oriented our x-axis parallel to the railroad's track and our y-axis vertically. Thus we have an easy description of Line A in terms of Lorentz Rule 1:
(Eq'n 1)
The passengers on the train also see that as
(Eq'n 2)
Line B, which we measure in our frame as the distance that the train moves as measured by our clock, comes to us as
(Eq'n 3)
The length of Line C simply equals ct, so we can exploit Pythagoras' theorem directly as
(Eq'n 4)
in which equation I have progressively made the appropriate substitutions from the previous equations (for example, substituting Y for y in the second step). If I subtract the square of vt from both sides of that equation and then divide the result by the square of lightspeed, I obtain
(Eq'n 5)
I can extract the square root of that equation and then factor out the lower-case tee to obtain
(Eq'n 6)
Finally I divide both sides of that equation by the coefficient of lower-case tee and obtain the equation of time dilation,
(Eq'n 7)
That equation tells me that if the
passengers on the train measure so many seconds between two events, then I will
measure a larger number of seconds between the same events. That's what it means
to say that the train's clock counts dilated time.
So far I have shown you only deduction and
interpretation. But Science, properly understood, does not stand on logic alone.
Yes, the reasoning that I have shown you so far has its appeal and may seem
unassailable, but how can we tell if it went wrong somewhere in some subtle way
that eluded our notice? How can we know with any certainty that what we have
deduced is, in fact, true to Reality? We must refer back to the empirical
Scientific Method. Logic notwithstanding, this shows us the guild-mark of
Natural Philosophy, which sets Science apart from all other belief systems ever
conceived by human imagination, that the hypotheses of any theory must be put to
the proof against Reality itself; that is, they must be subjected to
experimental or observational tests that will either verify or falsify them.
To see what such a test involves consider
the demonstration of time dilation that David H. Frisch and James H. Smith
carried out in 1962. Because Frisch and Smith had no way in which they could
have accelerated anything like a clock to any speed close to the speed of light,
they chose to observe a naturally occurring phenomenon that can show an elapse
of time if read correctly: they chose to observe the decay rates of certain
cosmic-ray fragments.
Frisch and Smith chose to exploit the
knowledge that physicists had gained concerning the flux of particles carrying
extremely high energies into Earth's atmosphere from somewhere beyond the solar
system. When one such particle enters the atmosphere it inevitably strikes an
atom and shatters its nucleus, producing a jet of fragments, which goes on to
shatter more atoms, producing yet another shower of fragments until the energy
of the original particle has been used up. Because the original particle
typically travels at a speed exceptionally close to the speed of light
(physicists have inferred Lorentz factors from the millions to the hundreds of
billions from observations), the jet of fragments comes out of the collision in
a tight beam. The shattering of the protons and neutrons that comprise the
atomic nucleus, produce sprays of weird little whizbangers called pi mesons (or,
more compactly, pions). After flying some distance, the pions pop apart into mu
antineutrinos and mu mesons (or, more compactly, muons). Each of those muish
whizbangers then pops apart into a mu neutrino, an electron neutrino, and a
positron (or, rarely, a mu neutrino, an electron antineutrino, and an electron).
Those pop-aparts, technically called
decays, don't adhere to a strict schedule. Once a muon comes into existence, it
is not doomed to decay at a particular time; rather, in any given interval after
the muon's creation the muon has a certain probability of decaying. Such decays,
as well as the decays of radioactive nuclei, must obey the Law of the Half-Life,
the law which requires that the proportion of a given substance that decays is
the same in any given interval. From experiments performed with cyclotrons
physicists had, by 1962, determined that muons have a half-life of 1.523
millionths of a second. By the Law of the Half-Life, then, if you suddenly
create 1054 muons, then 1.523 microseconds later you will have 512 muons, give
or take a few; after another 1.523 microseconds elapse you will have 256 muons,
give or take a few; 1.523 microseconds later you will have 128 muons, give or
take a few; and so on. You can also carry out that calculation backward: if you
know how many muons were created and then count the number that you have at some
later instant, you can then calculate how much time elapsed in the muons'
inertial frame. That's what Frisch and Smith did.
Frisch and Smith used a transparent block
of doped polystyrene eleven inches thick as the window through which they hoped
to view relativistic muons. They so chose the dopant that the block would emit
flashes of light wherever and whenever a muon decayed. Thus, when a muon-laden
spray from a cosmic ray passed through the block, the light stimulated by the
spray triggered an oscilloscope, which then recorded the flashes from the muons
that had stopped in the block and decayed.
For their first data-collecting runs,
Frisch and Smith set up their apparatus on Mount Washington in New Hampshire.
There, 6265 feet above sea level, they built a barrier of iron two-and-a-half
feet thick over their plastic block and took their data over six intervals of
one hour each. Because of collisions with the atoms in the iron, the barrier
stopped all muons flying slower than 99.50 percent of lightspeed. All muons
flying faster than 99.54 percent of lightspeed would pass through both the iron
and the polystyrene, so all of the muons with speeds between 99.50 and 99.54
percent of lightspeed would come to a stop within the plastic block, where they
would decay, induce the emission of light, and be counted. With that setup
Frisch and Smith detected an average of 563 decays per hour.
What that means is that of muons with
speeds in that range, at 6265 feet above sea level, we can expect about 563 per
hour to pass through any horizontal surface whose area is equal to the area of
the top of Frisch and Smith's plastic block. For the purpose of applying the Law
of the Half-Life we can interpret that number as the number of muons with the
desired speeds being created at that altitude and sent downward. The flight to
sea level takes 6.3918 microseconds, which interval spans 4.197 consecutive muon
half-lives. If time in the muons' frame were not dilated relative to time
elapsed in our frame, then that flux of muons would diminish from 563 per hour
on Mount Washington (or at the same altitude above our sea level site) to about
31 per hour at sea level. On the other hand, if the muons' time is
dilated, then the Lorentz factor (for a relative speed of 99.52 percent of
lightspeed) equals 10.22, which dilates the muon half-life to 15.565
microseconds: the flux of the selected muons would then diminish at sea level to
423, give or take a few, per hour.
For their second set of six one-hour
data-collecting runs, Frisch and Smith rebuilt their detector on the campus of
the Massachusetts Institute of Technology, in Cambridge, which lies ten feet
above sea level. There they built their iron barrier with one foot less
thickness to compensate the shielding effect of the extra 6255 feet of
atmosphere between their plastic block and the source of the muon-laden sprays
that they observed. When they analyzed the data that they gathered, they found
that they had been detecting muons at an average rate of 408 per hour.
Given the relative crudity of the
experiment, that's excellent agreement between the fruit of observation and the
expectation of theory. And the wide difference between the two theoretical
possibilities, both in actual numbers of muons decaying at sea level and in
those decay rates as a proportion of the original decay rates on Mount
Washington, leaves little room for doubt: muons generated in cosmic-ray showers
choreograph their decays in dilated time.
References
1. Frisch, David H. and James H. Smith, "Measurement of the Relativistic Time Dilation Using Mu-Mesons", Am. Jour. Phys., Vol. 31, Pg. 342 (1963).
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