*Michael Fowler*

*UVa Physics*

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The object of this exercise is to show explicitly how it is possible
for two observers in inertial frames moving relative to each other
at a relativistic speed to each see the other's clocks as running
slow and as being unsynchronized, and yet if they both look at
the same clock at the same time from the same place (which may
be far from the clock), they will *agree* on what time it
shows!

Suppose that in frame *S* we have two synchronized clocks
*C _{1}* and

Suppose *C'* is synchronized with *C _{1}* as
they pass, so both read zero.

As measured by an observer *O* in *S* - the "ground
frame" --* *the spaceship will take just 10 seconds
to reach *C _{2}*, since the distance is 6 light seconds,
and the ship is traveling at 0.6

What does clock *C*' (the clock on the ship) read as it passes
*C _{2}*?

The time dilation factor ,
so *C*', the ship's clock, will read 8 seconds.

Thus if both *O*, *O'* are at *C _{2}* as

**How, then, can O' claim that the clocks C _{1}, C_{2}
are the ones that are running slow?**

To *O'*, *C _{1}*,

Therefore, *O'* will conclude that since *C _{2}*
reads 10 seconds as she passes it, at that instant

Of course, *O*'s assertion that as she passes the second
"ground" clock *C*_{2} the first "ground"
clock *C*_{1} must be registering 6.4 seconds is
not completely trivial to check! After all, that clock is now
a million miles away!

Let us imagine, though, that both observers are equipped with Hubble-style telescopes attached to fast acting cameras, so reading a clock a million miles away is no trick.

To settle the argument, the two of them agree that as she passes
the second clock, the ground observer will be stationed at the
second clock, and at the instant of her passing they will both
take telephoto digital snapshots of the faraway clock *C _{1}*,
to see what time it reads.

*O*, of course, knows that *C _{1}* is 6 light
seconds away, and is synchronized with

What does *O' *'s digital snapshot
show? It must be identical -- two snapshots taken from the same
place at the same time must show the same thing! So, *O' must
also* gets a picture of *C _{1}* reading 4 seconds.

**How can she reconcile a picture of the clock reading 4 seconds
with her assertion that at the instant she took the photograph
the clock was registering 6.4 seconds?
**

The answer is that she can if she knows her relativity!

*First point: length contraction*. To *O'*, the clock
*C _{1}* is actually only 4/5 x 18 x 10

*Second point: The light didn't even have to go that far!*
In her frame, the clock *C _{1}* is

Having established that the clock image she is seeing as she takes
the photograph left the clock when it was only 9 x 10^{8}
meters away, that is, 3 light seconds, she concludes that she
is observing the first ground clock as it was three seconds ago.

*Third point: time dilation*. The story so far: she has
a photograph of the first ground clock that shows it to be reading
4 seconds. She knows that the light took three seconds to reach
her. So, what can she conclude the clock must actually be registering
at the instant the photo was taken? If you are tempted to say
7 seconds, you have forgotten that in her frame, the clock is
moving at 0.6*c* and hence *runs slow* by a factor 4/5.

Including the time dilation factor correctly, she concludes that in the 3 seconds that the light from the clock took to reach her, the clock itself will have ticked away 3 x 4/5 seconds, or 2.4 seconds.

Therefore, since the photograph shows the clock to read 4 seconds, and she finds the clock must have run a further 2.4 seconds, she deduces that at the instant she took the photograph the clock must actually have been registering 6.4 seconds, which is what she had claimed all along!

The key point of this lecture is that at first it seems impossible for two observers moving relative to each other to both maintain that the other one's clocks run slow. However, by bringing in the other necessary consequences of the theory of relativity, the Lorentz contraction of lengths, and that clocks synchronized in one frame are out of synchronization in another by a precise amount that follows necessarily from the constancy of the speed of light, the whole picture becomes completely consistent!

Index of Lectures and Overview of the Course

Copyright © Michael Fowler, 1996