The Scaling Theory VIII: The Sagnac's Interference and

Michelson and Gale Experiment.

14.1. Introduction

Since its discovery in 1913 by Georges Sagnac many attempts to explain

the Sagnac effect had been forwarded. Some attempts followed a purely classical approach based on the relative velocity addition26,33, and some resorted to the ether model, or emission theory32,34-36. Relativists claim that this effect is explicable within the theory of relativity, but this claim is much disputed26,32. In an earlier work by the author6, the Sagnac’s effect corresponding to two pulses,

each making a complete equatorial round in opposite directions, was explained on

the basis of the scaling theory. The similarity in nature between this particular

type of Sagnac’s effect and that of two pulses covering equal distances in opposite direction on a line was also revealed10. In this work we employ the bound scaling

transformations, to present a neat and elaborate explanation of the general type of

this effect, as well as, of Michelson and Gale experiment. In a subsequent article37, the translational type of Sagnac effect will be discussed in detail.

14.2. The Sagnac Interferometer

The Sagnac's interferometer consists of n plane mirrors , , … ,

occupying the vertices of a regular polygon

…

with a center o,

radius a, side L, and a source of light capable to send simultaneously two beams

in opposite directions along the polygonal loop. The whole arrangement is set on a

turntable. For a more expound account of Sagnac interferometer we refer to Faraj32.

We assume that the enumeration of the mirrors increases counter clock-wise,

which is also the positive sense of rotation, and that the normals to the diagonals

point counter clock-wise. Some of the following easily obtained geometrical data

are useful in subsequent discussions:

`

For a regular polygon

- The angle of a regular polygon with n sides is

,

1 2⁄

- The angle between the diagonal and the corresponding side is

,
- The angles between the normal corresponding sides are

,

2

to the diagonal

and the

,

,

,

- The area of the polygon is

1 .2

where 2

2

1

1

2

2

2

2,

is the length of the polygon's side.

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- As n tends to infinity the area A tends to . - The area of a polygon (not necessarily regular) can be written in the form

1

14.1

2

,

where the vector

is identified by , and p is any point in the

plane of the polygon.

Now, two pulses of light that set out at the same time from the same point ,

to make closed trips in opposite directions along the perimeter of the polygon,

return to at the same time if the interferometer is at rest in the timed inertial

frame S. If the interferometer is rotating, the two pulses do not arrive back at

simultaneously. In fact the pulse moving against the rotation arrives earlier than

the pulse moving in its direction. Here we shall employ the bound scaling

transformations (BST) to calculate the time difference between the arrival of the

two pulses back at when the interferometer rotates in the positive sense at an

angular velocity about an axis through its center o and perpendicular to its

plane.

14.3. The Use of Bound Scaling Transformations

A frame attached to the terrestrial laboratory with respect to which the

interferometer is rotating can be considered during the short period taken by the experiment a timed inertial frame S of fixed stars38 . We assume that

2

/ , i.e. the angular velocity of the interferometer is much greater

than that of the earth about its axis. In the frame S the velocity of the mirror

when at

is perpendicular to the vector , and hence

,

,

,

,

where and

are the positions of the mirrors and

in S at the instant

of light emission.

If the system was stationary, light which has been reflected at a mirror

would take a duration

/ to reach either adjacent mirror. Now, the

characters of the trip

are the same whether the observer

was moving at his actual velocity or at a velocity which is the mirror image

of with respect to the line

. Since

, the conditions of the

applicability of the extended bound transformations are met thoroughly. Or we

may instead use directly the BST in the rigid frame attached to the interferometer

(see section (13.4)).

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Next, we rewrite the bound scaling transformations in a most suitable form to use. If and are the laboratory, or S-observers, conjugate to and at the instant light is emitted from , then

14.2

.,

,

,

1

,
,

where

,

|

|,

,

,

.

For

1, which is the case in Sagnac’s effect, the latter transformation is

approximated by

14.3

,

,

which is the length of the trip (

of the opposite trip (

.,

.

,

. Similarly, the optical length

is approximated by

14.4

,

.

.

14.4. Rotation About the Polygon’s Center

When the polygon rotates about its center o, the velocity of the mirror

will be

1

14.5

,

and the length of the counter rotation-wise trip ( by

,

.

.

is given then

14.6

.

.

Similarly, the optical length of the opposite trip (

is

14.7

,

.

.

.

The difference in paths lengths of the two trips for one side of the polygon is

14.8 ∆

,

,

.

2

.

.

The total difference in the durations of the two trips is therefore

14.9

∆

∆2 .

4 ,

where we identify

by , and A is the area of the polygon.

14.5. Rotation About an Arbitrary Point We consider now the case in which the turntable rotates about its center O
which is distinct from the center o of the polygon. During the short period taken by light to travel one side of the polygon the motion of the receiver does not deviate appreciably from uniformity. Also, since the velocities of all mirrors (i.e. receivers) are all much less than c, the conditions necessary for the applicability of the BST are met (see section (13.4)). [Note that these conditions are also fulfilled for rotation about the center].

52

Now, the velocity of the mirror is

14.10

.

By (14.3) and (14.4), the difference in path length corresponding to one side is

∆

,

,

.

.

.

.

14.11

.

.

The total difference between the durations of the two trips is therefore

14.12

∆2

∆

.

4 .

It is obvious, on geometrical bases, that the magnitude of the sum in the last equation is twice the area enclosed by the polygon, and it is easy also to verify this result. Indeed

, and hence

14.13

2.

14.6. General Remarks. (i)-It may be useful to mention that when light traces a loop in opposite directions, it is the area enclosed by this loop, but not its length, that determines the magnitude of Sagnac effect. Although it is true that a regular polygon’s area is expressible as a function of its circumference, but this function is not 1-1 correspondence. Indeed

, 4 and one can shape a given length (say a fiber optical loop) in a variety of polygons

that have the same circumference but differ in their areas. However, if n is fixed,

which means that the type of regular polygon is fixed, the latter formula

determines a 1-1 map between the area and the perimeter. The latter fact will be

illuminated through a broader treatment when discussing translational Sagnac

effect.

(ii) A fiber optics loop can assume a non-regular polygon, and the Sagnac effect,

given by (14.12) remains valid for any shape of polygon. Indeed, the derivation of

(14.9) or (14.12) do not make use at all of the regularity of the polygon, and the

area of a polygon is given by (14.1), whether it was regular or non-regular.

(iii) The magnitude of Sagnac effect can be expressed in the form

14.14

4

∆

.,

53

where is given by (14.13). We may bypass equation (14.13) which determines correctly the direction of the area vector, and consistently impose on to make an acute angle with . Under this convention the beam propagating rotation-wise rotates about in a positive sense, and ∆ will be positive as given by (14.14).
A non-regular polygonal interferometer rotating uniformly about an axis not perpendicular to the polygon’s plane. The Sagnac effect is given by (14.14 )
(iv) The optical circuit drawn below illustrates the idea of an optical coil, by which Sagnac effects add up.
2 The arrows in red represent the rotation-wise beam
14.6. The Michelson and Gale Experiment This experiment is a version of an interferometer placed on a turntable,
with the earth is the turntable. Here S is the frame of fixed stars and s is a frame attached to earth. The 640 320 rectangular interferometer is set up horizontally on earth’s surface. The rotation vector of the earth is in the direction of the (south pole north pole) axis, and its magnitude is 2 radian per sidereal day. The area vector points upwards (downwards) in the northern (southern) hemisphere, and makes with the rotation vector an angle , which is the angle between the equatorial plane and the interferometer plane.
Earth’s axis of rotation

earth’s center

640 320 interferometer placed at earth’s surface
The equatorial plane
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The Sagnac effect is given by

14.14

4

∆

.,

where:

.

projection of the area on the equatorial plane

.

The latter result constitutes an explanation of the Michelson and Gale experiment.

The relation (14.14) shows that Sagnac effect

(i)Assumes maximal values at the north and south poles, where 0.

(ii) Decreases towards the equator till vanishing at it.

(iii) Not influenced by the orientation of the interferometer in its horizontal site, as

well as, by the longitude of the later.

(iv) Is nil wherever the interferometer is placed parallel to the earth’s axis of

rotation.

(v) Has the same maximum magnitude measured at a pole if the interferometer is

placed vertically at the equatorial plane.

55