Suppose this device moves past you at velocity `v` and orientated at
`A` degrees to the direction of motion.

The following event is timed:

A ray leaves the source, is reflected by the mirror and is detected at the receiver.

The diagram shows the object at three stages in the event:

1. When the light is emitted from the source.

2. When it is reflected at the mirror.

3. When it is collected at the receiver.

* (see diagram above)*

The heavy line indicates the path taken by the successful light rays collected at the receiver.

Let:

`t1` = the time to reach the mirror from the source

`t2` = the time to reach the receiver from the mirror

`t` = `t _{1} + t2` the time for the whole transaction

`c` = the speed of the photons

The distance travelled to the mirror is `ct1` and from the mirror to the
receiver is `ct2`. The respective horizontal motions of the device are
`vt1` and `vt2`. Using the cosine rule:

rearranging the equation involving `t1` we have a quadratic equation
which is solved to find `t1`

Similarly

`
`

Since the times taken are inherently positive we will use the positive instance
for both `t1` and `t2`, noting that

`
`

the total time for the event is

`
`

This result indicates that the time taken for the event depends on both its speed and orientation.

Michelson and Morley proposed to detect the motion of the Earth through space
by using a rotating configuration of two identical, perpendicular light paths
with mirrors at the ends to reflect light from a common source to a receiver at
which the interference of the rays could be observed. The time difference for
the paths could be derived from the observations of changing interference
patterns. No change in the interference patterns was detected. Hence if
`t` is measured for my apparatus at various orientations there should be
no variation either. Let us assume that this is the case.

Poincare suggested that not only the Michelson Morley experiment, but any experiment would be unable to determine the absolute velocity or direction of motion of an object, this is principle 2.

The second principle indicates that the time `t`, recorded on the watch
attached to the apparatus, for the event should be the same independent of the
velocity of the apparatus, relative to a stationary observer, when the event
was recorded. Note this does not mean that the value of `t`, measured
by a stationary observer with a stationary watch, is the same for any value of
`v`. In fact, the theory of relativity holds that this is not the
case.

If we suppose that the value of `t`, recorded on the watch attached to
the apparatus, does not alter with the orientation or velocity of the device
then there is an apparent contradiction with the algebra. Suppose that we let
`d` change with the orientation and velocity of the apparatus so as to
allow `t` to be constant. This may seem like a fairly desperate attempt
to fix the algebra but there are common sense precedents for the idea. For
example think of a string of marbles separated by springs, turn it into the
flow of a current and the string will shorten. Matter is made up of atoms held
by forces and perhaps it may be effected by some cosmic flow coming from a
particular direction. Since all measuring sticks would be similarly effected
you could not directly measure this alteration of length.

If `d` is to vary then so should `t` so that the speed of light
remains constant as supposed in principle 1. What then of our measurements of
`t` being constant? Recall that `t` is measured by a watch
moving with the apparatus, if the passage of time is altered for the event then
so will it be for the watch in such a way that no discrepancy will be detected.
The value of `t` may however be different for an observer moving with
respect to the apparatus. I will only suppose that the value of `t`
depends on the relative velocity of the apparatus but not on its orientation.

To indicate that `t `depends on `v` we use `t(v)`

and that `d` depends on `v` and `A` we use
`d(v,A)`

So an hypothesis to reconcile the algebra with the experimental results is to
rewrite the equation for `t `as:

`
*EQ1`

or changing the subject

`
*EQ2`

NOTE: Henceforth assume that the general concepts of time `t` and
distance `d` depending on `v` or `A` are true for any
events and not just the elements of my apparatus (as Poincare did).

Consider events with the same velocity `v` and orientations of `A =
90` degrees and `A = 0` degrees in ` *EQ1`.

`
`

equating the two expressions and rearranging:

`
`

This factor will appear often, it is called the Lorentz contraction factor.
Note that `L(v) < 1` when `v < c`.

Hence `d(v,90).L(v) = d(v,0) *EQ3`

Which means that an observer should see the apparatus shortened in the direction of motion compared to its vertical dimension.

Since `L(0) = 1, d(0,90) = d(0,0) *EQ4`

Which means that orientation has no effect on the distance measurement when the apparatus is stationary with respect to the observer.

Suppose that `d(v,90)` is constant for all values of `v` since
the object being measured has a velocity of `0` at `90` degrees
to the direction of motion, and is therefore stationary in this aspect.

in particular `d(v,90) = d(0,90)` **EQ5*

and also `d(v,90) = d(0,0)` (using ` *EQ4`)

Considered with **EQ*

` d(0,0).L(v) = d(v,0) *EQ6`

For `v < c` the distance measure of an object in the direction of
motion is less as measured by a stationary observer than the measure made by an
observer travelling with the object, or as measured afterwards when the object
is stationary relative to an observer.

Using ` *EQ2`:

since the left hand sides are equal (using ` *EQ5`)

` t(0) = L(v).t(v) *EQ7`

For `v < c` the time taken for an event occurring in a moving
apparatus is greater as measured by a stationary observer than the measure made
by an observer travelling with the apparatus, or as measured afterwards when
the apparatus is stationary relative to an observer. This time dilation effect
is supported by observations that some short-lived particles which result from
cosmic collisions in the upper atmosphere live much longer than can be
accounted for without supposing that their time rate is slower than ours. It
is rather intriguing to realise that our time rate also looks slower to them.

Einstein proposed The twins "paradox" to popularise this effect, but lets use triplets. Two of the triple go on identical journeys in fast spacecraft in opposite directions, eventually returning to the homebound member, each observes the other pairs time to be going slower than their own by virtue of their high relative velocities. Upon returning home the two travellers are equally aged but the homebound one is older. The pair who journeyed, experienced identical accelerations and would have noticed that the account of ages was finalised while decelerating to home. Part of the "paradox" is resolved if you take into account distance dilation as well, the distance travelled by the journeying pairs is reduced by a factor of L(v) compared to the stay at homeÕs measurement of the distance and yet all parties would agree on their respective speeds. A more detailed explanation is provided elsewhere

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This document was created on 23 August 1995

last modified on 23 August 1995

and is written by and copyright to btaylor@taylormade.com.au