Time and Distance Measurement

The apparatus I will use is a light source, coupled with a detector, both orientated to use a mirror d units away. There is also a stop watch attached to the apparatus that measures the time t taken for light to go from the source to the receiver.

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 = t1 + 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 home’s 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