Study of reflection of light by a moving mirror

Indian J. Phys. 83 (2) 233-240 (2009)

© 2009 IACS*Corresponding Author

Study of reflection of light by a moving mirror

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Abstract : The law of reflection of light by a moving mirror in the case of special Lorentz transformation iswell known. We have derived the law of reflection of light by a moving mirror in the case of the most generalLorentz transformation. It has been observed that in the case of special Lorentz transformation, the law ofreflection of light by a moving mirror is very simple but in the case of the most general Lorentz transformation, thelaw of reflection of light by a moving mirror is very complex.

Keywords : Reflection of light, special Lorentz transformation and Most general Lorentz transformation.

PACS Nos. : 30.30.+p, 42.25.-p

When the velocity v of S ′ frame with respect to S frame is not along X-axis i.e. thevelocity v has three components vx, vy and vz, the relation between the coordinates ofS and S ′, which is called the most general Lorentz transformation, can be written as[1]

t2

( 1)γ γ⎡ ⎤⎛ ⎞⋅ ⎟⎜⎢ ⎥′ = + − −⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦

r vr r v

v , (1a)

t t( )γ′ = − ⋅r v (1b)

wherevc

2

2

1

1

γ =⎛ ⎞⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜⎝ ⎠

and c = 1.

Note

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The inverse most general Lorentz transformation can be written as [1]

tv 2

( 1) ,γ γ⎡ ⎤⎛ ⎞′ ⋅ ⎟⎜⎢ ⎥′ ′= + − +⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

r vr r v (2a)

t t( )γ′ ′ ′= + ⋅r v (2b)

If vx, vy and vz denote the components of the velocity of the system s ′ relative to s,then eqs. (1a) and (1b) can be written as

( ) x yx x zx

v vv v vx x y z v t

v v v

2

2 2 21 1 ( 1) ( 1)γ γ γ γ⎧ ⎫⎪ ⎪⎪ ⎪′ = + − + − + − −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

, (3a)

( )x y y y zy

v v v v vy x y z v t

v v v

2

2 2 2( 1) 1 1 ( 1)γ γ γ γ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪′ = − + + − + − −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭, (3b)

( )y zx z zz

v vv v vz x y z v t

v v v

2

2 2 2( 1) ( 1) 1 1γ γ γ γ⎧ ⎫⎪ ⎪⎪ ⎪′ = − + − + + − −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

, (3c)

x y zt v x v y v z tγ γ γ γ′ = − − − + (3d)

and eqs. (2a) and (2b) can be written as

( ) x yx x zx

v vv v vx x y z v t

v v v

2

2 2 21 1 ( 1) ( 1)γ γ γ γ⎧ ⎫⎪ ⎪⎪ ⎪ ′ ′ ′ ′= + − + − + − +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

, (4a)

( )x y y y zy

v v v v vy x y z v t

v v v

2

2 2 2( 1) 1 1 ( 1)γ γ γ γ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪′ ′ ′ ′= − + + − + − +⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭, (4b)

( )y zx z zz

v vv v vz x y z v t

v v v

2

2 2 2( 1) ( 1) 1 1γ γ γ γ⎧ ⎫⎪ ⎪⎪ ⎪′ ′ ′ ′= − + − + + − +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

, (4c)

x y zt v x v y v z tγ γ γ γ′ ′ ′ ′= + + + . (4d)

Let us consider two systems s and s′, the later moving with velocity v relative toformer along X – X ′ axis. Now consider a mirror M to be fixed in the Y ′ – Z ′ planeof system S ′. For simplicity consider the mirror to be perfectly reflecting, moving in thedirection of its normal relative to system S ′.

Let a ray of light in X ′ – Y ′ plane be incident at angle φ′ at O′ in system S ′.As mirror M is stationary in system S ′ ordinary laws of reflection hold good andtherefore the angle of reflection will be φ ′ and the reflected ray lies in X ′ – Y ′ plane.

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Let the corresponding angles of incidence and reflection as measured byobserver in system S be φ1 and φ2, we get [2]

221

1 2

sin (1 )sincos cos

φ βφφ β φ β

−=

+ −. (5)

Usingv

ccv

c

22

2

1 1, 1 and ,

(1 )1

γ ββ

= = = =⎛ ⎞ −⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜⎝ ⎠

we get from eq. (5)

( )v v1 2

1 2

sin sincos cos

φ φφ γ φ

=+ − (6)

This eq. (6) is the law of reflection of light by a moving mirror [2] in the case ofspecial Lorentz transformation.

Let us consider two systems S and S′ where the system S ′ is moving with thevelocity v with respect to the system S along any arbitrary direction i.e. the velocityof v has two components vx and vy. Now consider a mirror M to be fixed in theY ′ – Z ′ plane of system S′. For simplicity consider the mirror to be perfectly reflecting,moving in the direction of its normal relative to system S′.

Let the corresponding angles of incidence and reflection as measured byobserver in system S be φ1 and φ2. For measuring the angles, we shall use theconvention that all angles between positive X-axis and the positive direction of travel oflight are positive. According to this convention angle of incidence is (π + ϕ′) in systemS′ and (π + ϕ1) in system S; while the angle of reflection is (2π – ϕ′) in system S′and (2π – ϕ2) in system S.

Figure 1. Reflection of light by a moving mirror where a mirror M is fixed in the Y ′ – Z ′ plane of system S ′.

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Let the incident ray in system S and S′ be represented by ψ and ψ ′ where

x yAiv t

c1 1cos ( ) sin ( )

exp 2π φ π φ

ψ πγ

⎡ ⎤⎧ ⎫+ + +⎪ ⎪⎪ ⎪⎢ ⎥= −⎨ ⎬⎢ ⎥⎪ ⎪⎪ ⎪⎩ ⎭⎣ ⎦(7)

and

A x yiv t

ccos ( ) sin ( )

exp 2π φ π φ

ψ πγ

⎡ ⎤⎧ ⎫′ ′ ′ ′ ′+ + +⎪ ⎪⎪ ⎪⎢ ⎥′ ′ ′= −⎨ ⎬⎢ ⎥⎪ ⎪′ ⎪ ⎪⎩ ⎭⎣ ⎦(8)

where v and v ′ are the frequencies of the wave as observed from systems S and S ′respectively.

As phase is a Lorentz invariant quantity, using eqs. (7) and (8) we must have

x y x yv t v t

c c1 1cos( ) sin ( ) cos ( ) sin ( )π φ π φ π φ π φ⎧ ⎫⎧ ⎫ ′ ′ ′ ′+ + + + + +⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪′ ′− = −⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

or

x y x yv t v t

c c1 1cos sin cos sin

.φ φ φ φ⎧ ⎫⎧ ⎫ ′ ′ ′ ′+ +⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪′ ′+ = +⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

(9)

Using c = 1 we get from eq. (9) can be written as

{ } { }v t x y v t x y1 1cos sin cos sinφ φ φ φ′ ′ ′ ′ ′ ′+ + = + + . (10)

Now using eqs. (3a), (3b) and (3d) in eq. (10), considering the velocity of v have twocomponents vx and vy, we get.

{ } xx y

vv t x y v v x v y t x x

v

2

1 1 2cos sin cos ( 1) cosφ φ γ γ γ φ γ φ

⎡⎢′ ′ ′+ + = − − + + + −⎢⎢⎣

Figure 2. The system S ′ is moving in any arbitrary direction with velocity v relative to the system S where amirror M is fixed in the Y ′ – Z ′ palne of system S′.

X

S

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x y x z xy z

v v v v v tv v2 2

( 1) cos ( 1) cos cosγ φ γ φ γ φ′ ′ ′+ − + − −

yx y

vxv v y y

v v

2

2 2( 1) sin sin ( 1) sinγ φ φ γ φ′ ′ ′+ − + + −

y z yz

v v v tv 2( 1) sin sinγ φ γ φ

⎤′ ′ ⎥+ − −

⎥⎦

[ ]zv z0 and 0= =∵ . (11)

Equating coeffients of t in above eq. (11), we get

{ }x yv v v v1 cos sinγ φ φ′ ′ ′= − − . (12)

Now equating coefficient of x in eq (11), we get

xx x y

vv v v v v

v v

2

1 2 2

1cos cos ( 1) cos ( 1) sinφ γ φ γ φ γ φ

⎧ ⎫⎪ ⎪⎪ ⎪′ ′ ′ ′= − + + − + −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭. (13)

Using eq. (12), we get from eq. (13)

{ }x yv v v 11 cos sin cosγ φ φ φ′ ′ ′− −

xx x y

vv v v v

v v

2

2 2

1cos ( 1) cos ( 1) sinγ φ γ φ γ φ

⎧ ⎫⎪ ⎪⎪ ⎪′ ′ ′ ′= − + + − + −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

or

{ }

xx x y

x y

vv v v

v vv v

2

2 2

1

1cos ( 1) cos ( 1) sin

cos1 cos sin

γ φ γ φ γ φφ

γ φ φ

′ ′ ′− + + − + −=

′ ′− −. (14)

Now equating coeffiecients of y in eq. (11), we get

yy x y

vv v v v v

v v

2

1 2 2

1sin ( 1) cos 1 ( 1) sinφ γ γ φ γ φ

⎡ ⎤⎧ ⎫⎪ ⎪⎪ ⎪⎢ ⎥⎪ ⎪′ ′ ′= − + − + + −⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎪ ⎪⎩ ⎭⎣ ⎦. (15)


{ }x y y x yv v v v v v vv

1 2

11 cos sin sin ( 1) cosγ φ φ φ γ γ φ

⎡′ ′ ′ ′ ′⎢− − = − + −

⎢⎣

yv

v

2

21 ( 1) sinγ φ

⎤⎧ ⎫⎪ ⎪⎪ ⎪ ⎥⎪ ⎪ ′+ + −⎨ ⎬ ⎥⎪ ⎪ ⎥⎪ ⎪⎪ ⎪⎩ ⎭ ⎦

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or,

{ }

yy x y

x y

vv v v

v vv v

2

2 2

1

1sin ( 1) sin ( 1) cos

sin1 cos sin

γ φ γ φ γ φφ

γ φ φ

′ ′ ′− + + − + −=

′ ′− −. (16)

Combining eqs. (14) and (16), we get

yy x y

xx x y

vv v v

v vv

v v vv v

2

2 2

1 2

2 2

1sin ( 1) sin ( 1) cos

tan1

cos ( 1) cos ( 1) sin

γ φ γ φ γ φφ

γ φ γ φ γ φ

′ ′ ′− + + − + −=

′ ′ ′− + + − + −

or

y y x y

x x x y

v v v v v v

v v v v v v

2 2 2

1 2 2 2

sin ( 1) sin ( 1) costan


γ φ γ φ γ φφ

γ φ γ φ γ φ

′ ′ ′− + + − + −=

′ ′ ′− + + − + − . (17)

The eq. (17) gives the exact relativistic aberration formula.

Now using eqs. (4a), (4b) and (4d) in eq. (10), considering the velocity of v havetwo components vx and vy, we get

x y x yv x y

v v x v y t x v vv v

2

1 12 21 ( 1) cos ( 1) cosγ γ γ γ φ γ φ⎡ ⎧ ⎫⎪ ⎪ ′⎪ ⎪⎢ ′ ′ ′ ′+ + + + − + −⎨ ⎬⎢ ⎪ ⎪⎪ ⎪⎢ ⎩ ⎭⎣

yx x y

vxv t v v y

v v

2

1 1 12 2cos ( 1) sin 1 ( 1) sinγ φ γ φ γ φ

⎧ ⎫⎪ ⎪′ ⎪ ⎪⎪ ⎪′ ′+ + − + + −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

{ }yv t v t x y1sin cos sinγ φ φ φ⎤′ ′ ′ ′ ′ ′ ′+ = + +⎥⎦

[ ]zv z0 and 0= =∵ . (18)

Equating coefficients of t ′ in above eq. (18), we get

x yv v v v1 1cos sinγ γ φ γ φ⎡ ⎤′ = + +⎢ ⎥⎣ ⎦ . (19)

Now equating coefficients of x ′ in eq. (18), we get

xx x y

vv v v v v

v v

2

1 1 12 2

1cos cos ( 1) cos ( 1) sinφ γ φ γ φ γ φ

⎡⎢′ ′ = + + − + −⎢⎢⎣

. (20)


xx y x x y

vv v v v v v v

v v

2

1 1 1 1 12 2

11 cos sin cos cos ( 1) cos ( 1) sinγ φ φ φ γ φ γ φ γ φ

⎡ ⎤⎢ ⎥⎡ ⎤ ′+ + = + + − + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

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or

xx x y

x y

vv v v

v vv v

2

1 1 12 2

1 1

1cos ( 1) cos ( 1) sin

cos1 cos sin

γ φ γ φ γ φφ

γ φ φ

+ + − + −′ =

⎡ ⎤+ +⎢ ⎥⎣ ⎦. (21)

Now equating coefficients of y ′ in eq. (18), we get

yy x y

vv v v v v

v v

2

1 1 12 2

1sin sin ( 1) cos ( 1) sinφ γ φ γ φ γ φ

⎡ ⎤⎢ ⎥′ ′ = + + − + −⎢ ⎥⎢ ⎥⎣ ⎦

. (22)


yy x y

x y

vv v v

v vv v

2

1 1 12 2

1 1

1sin ( 1) cos ( 1) sin

sin1 cos sin

γ φ γ φ γ φφ

γ φ φ

+ + − + −′ =

⎡ ⎤+ +⎢ ⎥⎣ ⎦. (23)

Combining eqs. (21) and (23), we get

y x y y

x x x y

v v v v v v

v v v v v v

2 2 21 1 1

2 2 21 1 1

sin ( 1) cos ( 1) sintan


γ φ γ φ γ φφ

γ φ γ φ γ φ

+ + − + −′ =

+ + − + −. (24)

Similarly for reflected ray the relation between angle of reflection in systems S and S ′is obtained as

y x y y

x x y x

v v v v v v

v v v v v v

2 2 22 2 2

2 2 22 2 2

sin ( 1) cos ( 1) sintan

cos ( 1) sin ( 1) sin

γ φ γ φ γ φφ

γ φ γ φ γ φ

− − + − − −′ =

− + − − − . (25)

From eqs. (24) and (25), we get

y x y y

x x x y

v v v v v v

v v v v v v

2 2 21 1 1

2 2 21 1 1

sin ( 1) cos ( 1) sin


γ φ γ φ γ φ

γ φ γ φ γ φ

− + − + −

+ + − + −

=y x y y

x x y x

v v v v v v

v v v v v v

2 2 22 2 2

2 2 22 2 2



γ φ γ φ γ φ

γ φ γ φ γ φ

− − + − − −

− + − − −. (26)

This eq. (26) is the law of reflection of light by a moving mirror in the case of mostgeneral Lorentz transformation.

Table 1. Comparison of reflection of light by a moving mirror in the special and most general Lorentz transformation.

Law of reflection Special Lorentz Most general Lorentz transformation

transformation

of light by a v v1 2

1 2

sin sincos (cos )

φ φφ γ φ

=+ −

y x y y

x x x y

v v v v v v

v v v v v v

2 2 21 1 1

2 2 21 1 1



γ φ γ φ γ φ

γ φ γ φ γ φ

+ + − + −

+ + − + −

moving mirror = y x y y

x x y x

v v v v v v

v v v v v v

2 2 22 2 2

2 2 22 2 2



γ φ γ φ γ φ

γ φ γ φ γ φ

− − + − − −

− + − − −

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We have derived the formula for the law of reflection of light by a moving mirrorusing Special Lorentz transformation and the most general Lorentz transformation. Wehave observed that in the case of Special Lorentz transformation, the formula of the lawof reflection of light by a moving mirror is simpler than that in the case of the mostgeneral Lorentz transformation (Table 1). This formula will be helpful in further studiesof reflection of light by a moving mirror when the mirror moves in any arbitrary directioninstead of X-axis.

Acknowledgment

We are grateful to Mushfiq Ahmad, Department of Physics, University of Rajshahi,Rajshahi, Bangladesh and Prof. Habibul Ahsan, Department of Physics, ShahjalalUniversity of Science and Technology, Sylhet, Bangladesh, for their help and advice.

References

[1] C Moller The Theory of Relativity (London : Oxford University Press) (1972)

[2] S Prakash Relativistic Mechanics (Pragati Prakash) (1993-1994)

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Study of reflection of light by a moving mirror