Einstein's Theory of Relativity## versus

## Classical Mechanics

by Paul Marmet

Where to get a Hard Copy of this BookChapter NineSimultaneity and Absolute Velocity of Light.

The problem of simultaneity has been much studied in relativity. According to Einstein, simultaneous events in one frame cannot be simultaneous in another. This is known as Einstein's principle of relativity of simultaneity.

When two events take place at the same time, we say that they are simultaneous. We know that Einstein always considered that time is what clocks show. Therefore when he writes that two events are simultaneous in two different frames, he means that they occur at the moment when the clocks of observers in both frames show the same display. Since we understand that time does not flow more slowly because clocks run more slowly, Einstein's statement brings much confusion. Instead of saying that two events simultaneous in one frame are not simultaneous in another, he should have said that there is no identity of clock displays between clocks in different frames. Two clocks moving independently at different velocities do not maintain identical clock displays after a time interval. This means that even if both observers see the events at the same absolute time they will record different clock displays. Einstein's relativity of simultaneity becomes understandable only if he means that the clocks can show different displays at one given time.

**9.2 - Thought
Experiment on Clocks Synchronization.**

In
order to study this problem in more detail, let us consider
figure 9.1 illustrating Einstein's thought experiment.

*
*Figure 9.1

Identical clocks labeled A and B are located at rest at each end
of a station A-B having a length *l*_{o}[rest].
There is no gradient of gravitational potential in this
experiment. In front of the station A-B, a moving train a-b has a
length such that when in motion, the clock labeled a located at one end of the train passes
in front of clock A at the same time as clock b, located at the other end of the train,
passes in front of clock B. Clocks a,
b, A and B were built identically on
the station. Clocks a and b were later put in motion. The
synchronization of the clocks is described below.

**9.3 - Synchronization
of Clocks A and B.**

**9.3.1 - Method #1.**

Clocks
A and B on the station are synchronized in the following way. A
pulse of light is emitted from A and reflected on a mirror at B
toward A. The observer in A records on his clock a difference of
clock displays DCD_{A} for
the return trip of the light.

When
the traveling clock a passes near A,
we arbitrarily synchronize clocks a
and A together at zero. At that moment, the absolute time t[rest] on the frames is defined as zero:

t[rest] = 0 and CD_{A}
= CD_{a} = 0 |
9.1 |

9.2 |

t[rest] = 0 when CD_{B}
= 0 |
9.3 |

**9.3.2 - Method #2.**

Nobody
ever
proved
experimentally
that
the
velocity of light is the same when moving from A to B than when
moving from B to A. Michelson's experiment has shown that the
time taken for light to make a return trip between two points
oriented in a different direction in space is the same. However,
there is an error in the
Michelson-Morley demonstration. His experiment has
nothing to do with the measurement of any difference of transit
time during each half of the trip. Some researchers wishing to
investigate more deeply this problem have realized that the
method of synchronization described in section 9.3.1 is not
appropriate if the velocity of light is not identical in both
directions. Consequently, other methods of synchronization have
been suggested in hopes of taking into account the possibility
of a non constant velocity of light in different directions. A
very original method [1]
consists in using a new reference clock labeled m, which carries the display shown by A
at a very small velocity e (of the
order of 10^{-}^{9} of
the velocity of light) on the station from A to B and later from
B to A. In this way, the stationary clocks A and B can be
synchronized independently in each direction with the traveling
clock m. This method of
synchronization is quite interesting since, as we will now show,
any shift of display on clock m due
to its passage from A to B (or B to A) is negligible at very low
velocity.

The
time taken by clock m to move from A
to B is:

9.4 |

9.5 |

9.6 |

9.7 |

DCD_{A-}m = DCD_{A}-DCD_{m}
= 0 |
9.8 |

**9.4 - Loss of
Synchronization of Clock a on the
Moving Frame.**

Let us
calculate the difference of clock displays on clock a moving across distance *l*_{o}[rest]
from A to B as shown on figure 9.2.

Since
the train moves at velocity v[rest] and the distance traveled by
a is *l*_{o}[rest], the
time interval Dt_{1}[rest]
for clock a to reach B will be:

9.9 |

9.10 |

However, the moving clock a runs at a slower rate than clock A. From equation 3.10 we find that after the time interval Dt

9.11 |

9.12 |

9.13 |

**9.5 - Synchronization
between Moving Clocks a and b (Method #1).**

In
section 9.3.1, we described the synchronization of clock B with
clock A. It consists in setting clock B when light is received
at B, to one half of the interval DCD_{A}
taken by light to go from A to B then back to A. We now
calculate the consequences of applying the same synchronization
method inside a moving frame. Let us consider a pulse of light
emitted from x on figure 9.3 at time t[rest]
= 0. At that moment, we have:

t[rest] = 0, CD_{a} = CD_{A} = CD_{B}
= 0 |
9.14 |

**Figure 9.3**

We see
that light approaches clock b at a
relative velocity of c-v. For the observer in the moving frame,
the distance to be traveled is *l*_{o}[rest]. The
absolute time interval Dt_{2}[rest] to reach clock b
is:

9.15 |

9.16 |

9.17 |

Dt[rest](A ®b®a) = Dt |
9.18 |

9.19 |

9.20 |

9.21 |

9.22 |

9.23 |

_{9.24} |

**9.6 - Asymmetric
Relative Velocity of Light.**

We have
seen that the time interval Dt2[rest]
(equation 9.15) for light to go from a
to b is larger that the time
interval Dt_{3}[rest] (equation 9.17) for the return from b to a.
However, the locations a and b between which light moves, are always
separated by the constant distance *l*o[rest].

Because
we used the synchronization method #1 on clocks a and b, the
differences of clock displays recorded on those local clocks
when light travels from a to b and from b
to a are identical. Consequently,
Einstein's synchronization method leads to a difference of
synchronization between clocks a and
b such that it prevents the moving
observer from being able to detect that the absolute time for
light to move from a to b is different from the time to move from
b to a.
It is this difference of synchronization between clocks a and b that
prevents the observers in a and b to realize that the light that
approaches them has a relative velocity different from c. The
expression "velocity of light" is too vague. It is much more
significant to describe the velocity at which light approaches
an observer or recedes from him. Using that description, the
velocity of light with respect to an observer can be different
from c.

We see
that this constant number representing the absolute velocity of
light in any frame (in [frame] units) is just a mathematical
illusion. We have shown that it is due to the different clock
rate on the moving frame and to the clock synchronization of the
moving observer. In fact, the velocity of light is an absolute
constant in an absolute frame at rest but due to the different
clock rate on the moving frame and to the synchronization, it ** appears
constant** in any frame.

One must conclude that inside a moving frame, a difference of clock displays always exists at one given instant between two clocks (a and b) located on that frame. Consequently, synchronization method #1 inside a moving frame satisfies the condition of an

**9.7 - Synchronization
of Clocks a and b (Method #2).**

We have
seen in sections 9.5 and 9.6 that inside the moving frame,
synchronization method #1 does not lead (at a given time t[rest]) to the same clock display on
clocks a and b,
even if they are attached to the same frame. A moving observer
might believe that he could detect this difference of clock
displays using synchronization method #2 which consists in
moving a third clock m at low
velocity from a to b. We have seen in section 9.3.2 that
there is no drift of clock display on clock m
when it moves slowly across a frame at rest from A to B. Let us
study now what happens when we move clock m
within the moving frame a-b.

_{ }
Figure
9.4
illustrates
a
train
moving at velocity v with respect to the station. Its length is
*l*o[rest]. Clock m inside the
train moves at a very small velocity with respect to the train
(using rest units). The observer on the station measures the
velocity of clock m to be e[rest] larger than the velocity v[rest]
of the train. The total velocity u[rest] of clock m with respect to the station is then:

u[rest] = v[rest]+e[rest] | 9.25 |

_{9.26} |

9.27 |

l_{3}[rest] = l_{2}[rest]
+ l_{o}[rest] |
9.28 |

9.29 |

9.30 |

9.31 |

9.32 |

9.33 |

9.34 |

9.35 |

9.36 |

To study the case when clock m moves in the opposite direction, we just have to substitute v+e in equation 9.33 by v-e and replace DCD

9.37 |

**9.8 - References.**

**
[1] **This method is often used by F. Selleri, Universita
di Bari, Dipartimento di Fisica, Sezione, INFN, Via Amendola,
173, I70126 Bari, Italy.

CD_{A} |
clock display on clock A |

CD_{a} |
clock display on clock a |

CD_{B} |
clock display on clock B |

CD_{b} |
clock display on clock b |

l_{o}[rest] |
length of the station and the moving train in rest units |

t[rest] | absolute time (in rest units) |

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