654

NATURE

February 25, 1961 voL. 1a9

The Relativistic Theory of the Fresnel Drag Coefficient

IN this communication I give a somewhat more

general discussion of the Fresnel drag coefficient than

is usual, with the view of clarifying the relation

between two expressions for it, which occur in the

literature of special relativity. Theso expressions are

normally derived separately and their relationship has

recently been discussed by Synge1 • I take a further

step here, by showing how both results can be

obtained from a single argument. This procedure

directs appropriate attention to step (6) (see below),

which is the crucial point in both cases. The neces-

sary background may be outlined as follows :

Suppose a medium has velocity v parallel to the

x-axis of a reference system S. If light passes through

the medium parallel to this direction, let its phase
velocity be um relative to S. Assume that um =

c/n(vm), where n(vm) is the refractive index of light

of frequency Vm in the medium, provided the medium
is at rest in S (that is, v = 0). Then in the general

case (v ¥, 0) :

Um = c/n(vm) + kv

(1)

Here k is a coefficient which was once supposed to measure the extent to which the ::ether is dragged along by the medium (Fresnel, 1818). The special relativity theory also leads to (1). Fork, it yields the result:

k = 1 _ _l _ + ~ - ~ {dn(v)}

(2)

n(v 0) 2 v n(v 0)r dv v = v,

Here r = 1 in the case of water fl.owing at velocity v in a tube fixed in S, and r = 2 for a block of glass
moving with velocity v. These are the two cases
usually described. Consider now a uniform transparent medium in
uniform motion with velocity v parallel to the x-axis of an inertial system S. The medium is contained in
a transparent vessel the walls of which can unfold during the motion of the medium, so that the rear face of the container (which is perpendicular to the x-axis) moves with a velocity w ~ v. The optieal properties of the medium in the container are, however, kept homogeneous and constant in time by feeding additional material through a tube leading to the rear of the container. This means that we assume the following idealized situation : At any time t, contact between the rear surface of the container
and the medium is about to break due to the difference in velocity. But the gap (v - w)dt, which one would expect to develop in time dt, is filled in by additional material having already v as its x-component of velocity. Contact between the rear surface and the medium is assumed to be maintained in this way ; different layers of material are involved in the contact at different instants. This situation cannot be realized exactly, but my main interest is in the two limiting cases, which can be realized.
Let the reference system in which the medium is at rest be S' ; that in which the rear face is at rest, be S" ; the velocity of S" as observed in S' be w' ; the frequency of the light when moving in vacuum and observed in S" be vO" and the frequency when
moving in the medium and observed in S" be vm"· I shall adopt this type of notation also for the frames S and S' and for the phase velocity of the
light, which is denoted by u 0 for vacuum and by um
for the medium.

By the relativistic velocity addition :

Um = Um' + v = c/n(vm' ) + v 1 + um' v/c• 1 + v/cn(vm')

(3)

It has been assumed that um' = c/n(vm'), where
n(vm ' ) is the refractive index of the medium for waves
of frequency vm'· Suppose now that if g is a function of order v/c, we have, to first order in v/c :

(4)

It then follows t,hat :

1 n(v m')

=

1 n(v 0-v 0g)

=

1

{ d [ 1 ]}

n(v 0 )-gvo dv n(v) v "·" v..

= _l_ + ~ {dn(v)}
n(v 0 ) n(vo) 2 dv v = v,

Substituting in \3), one finds, to first order :

C
Um=n-(v-0)+ k v

= where k I - _l_ + ~ ~ g {dn(v)}

(5)

-

n(v 0) 2 v n(v 0) 2

dv ~ = ••

Here k has the form of a Fresnel drag coefficient. It
remains to calculate g. From the expressions for the radial Doppler effect :

Vm" y(l-(w'] 2/c 2) vo" _ y'(l-w2/c2 )
Vm' = I + w'/u,,." ' v"; - 1 + w/uo"

Now the rear wall of the container is at rest in S" so that the light entering it is unchanged in frequency for an observer at rest in S":
(6)

Hence to first order in v/c :

+ ~ - Vo - vm' _ vo"/vo = 1

~

Vm"/vm'

Um" Uo"

The last two terms are correction terms and need be evaluated only to zero order, so that:

w' '="' w-v, uin" = c/n(v 0" ) '="' c/n(vo)

Noting that uu'' = c, one finds that equation (4) is
valid, provided :

g = w__+_(v_-c_w)_n(_v 0)

(7)

Combination ef (5) and (7) yields an expression for
the drag coefficient. One notes that w = 0 corre-
sponds to the case of water fl.owing thr_ough a tube, while w = v corresponds to the movmg block of glass in accordance with (2). One can say qualitatively : The effect of dispersion on the drag coefficient goes up with the disparit)'.' between the velocity _of the medium and the velocity of the rear face of its container.
I am indebted to Prof. J . L. Synge for helpfnl correspondence.
P. T. LANDSBERG
Department of Applied Mathematics, University College, Cardiff.
• Synge, J. L., &latiintv: TM Speciar Theory, 142 (North-Holland: Amsterdam, 1956).

© 1961 Nature Publishing Group