RELATIVISTIC REFLECTION OF LIGHT
Antonio Leon
Interciencia, Salamanca, Spain
interciencia.es
Abstract. In spite of the fact that the reflection of light by a mirror is involved in many
relativistic arguments, the relativity of the reflection itself has not been adequately addressed. In a previous work on the relativistic refraction of light we found an inconvenient
involving Snell law: by assuming the refractive index does not change with relative motion, the refraction of a visible laser beam does not follow Snell law when observed in
relative motion [1]. Now it is the second law of the reflection of light which seems not to be
satisfied when the reflection is observed in certain relativistic conditions. The argument
is developed in basic geometry terms, without making additional physical assumption as
in the case of Snell law.
1. A conflicting reflection
1. As Figure 1 shows, the mirror M is attached to the inclined surface of a right angled structure whose orthogonal
sides have the same length Lo , being one of them parallel
to the Xo axis and the other to the Yo axis of its proper
reference frame RFo . In consequence, M and its normal No
are both inclined at an angle of π/4 radians with respect
to Yo .
2. In these conditions, a photon that reaches the mirror
through a vertical trajectory will be reflected in a horizontal direction. In fact, according to the Second Law of the
Reflection of Light1, and being π/4 the angle of incidence
io , the angle of reflection ro will also be π/4. Therefore,
Figure 1. The mirror M in its
the reflected trajectory will be horizontal.
proper reference frame RFo . A
photon moving through a vertical
3. The argument that follows could also be applied to trajectory will be horizontally reflected.
a rigid ball of mass m > 0 fired vertically towards a rigid
surface inclined by an angle of π/4 on which the ball collides elastically.
4. Let us begin by analyzing the reflection of a photon ea emitted by its source in the
vertical direction towards the mirror M in the proper reference frame RFo of M . Let B
be the point where ea hits the mirror and assume it is at a vertical distance yo from the Xo
1The
angle of incidence is equal to the angle of reflection
1
2 —— Relativistic reflection of light
axis. At point C , placed at the same vertical distance yo from Xo and in the same plain
of the photon incident-reflected trajectory, it is placed a sensor of photons that fires a
visible red flash when it is activated by a photon coming through a horizontal trajectory,
and only by a photon coming through a horizontal trajectory (Figure 2, left).
Figure 2. Reflection of the photon e
a by the mirror M as seen in the proper frame RFo of the
mirror (left) and in RFv (right), from which RFo moves from left to right at a velocity v (see text).
5. From the perspective of RFo the photon ea is reflected according to the laws of reflection
so that the angle of incidence is equal to the angle of reflection and being the mirror
inclined at an angle π/4 the incidence trajectory AB will be perpendicular to the reflected
trajectory BC . Thus the reflected trajectory BC will be horizontal. Consequently, the
sensor is activated and the red flash is fired. Obviously, this red flash will be seen in all
references frames.
6. Assume that RFo moves from left to right at a velocity v = kc (0 < k < 1) with respect
to the frame RFv and in such a way that its spacetime diagram coincides with that of
RFv at the precise instant at which the photon e
a is emitted by its source. In accordance
with Lorentz transformation the observers in RFv will come to the following conclusions
regarding the reflection of the photon ea by the mirror M (Figure 2, right):
(1) Since there is no contraction in the direction perpendicular to the relative motion,
the points B (at which ea hits M ) and C (where the sensor of the red flash is placed)
are at the same vertical distance (yv = yo ) from the Xv axis.
(2) The red flash is fired. So, the endpoint C of the reflected trajectory is at the same
vertical distance from the Xv axis as its start point B .
(3) Therefore, the reflected trajectory BC of ea is a horizontal line.
(4) The photon ea takes a time tv in going from its source to the mirror.
(5) During tv the mirror moves a horizontal distance kctv towards the right.
(6) Therefore, the incident trajectory AB of the photon ea will not be vertical but
inclined by an angle:
sin−1
kc
= sin−1 k
c
(1)
Relativistic reflection of light —— 3
(7) The horizontal side of the right angled structure to which the mirror is fixed is
√
√
contracted by a factor γ −1 = 1 − k2 . Therefore, this side has a length 1 − k2 Lo .
(8) The vertical side of the right angled structure maintains its proper length Lo .
(9) Therefore, the mirror is not inclined from the vertical direction by an angle π/4
but by an angle:
p
γ −1 Lo
= tan−1 γ −1 = tan−1 1 − k 2
(2)
Lo
(10) Consequently, the normal Nv to the mirror is not inclined from the vertical by an
angle π/4, as is the case of the normal No observed in RFo .
tan−1
7. As seen from the frame RFv , the reflected trajectory of the photon ea is horizontal
as in RFo , but its incident trajectory is not vertical; thanks to the relative velocity kc of
RFo , the incident trajectory is inclined at an angle sin−1 k . Now then, since the normal
Nv is not inclined at an angle π/4 as it is in RFo , it could be the case that both differences
compensate each other so that the second law of reflection is still fulfilled. As we will see
next, it is not. And we will prove it is not by elementary geometry.
Figure 3. Geometry of the reflection of the photon e
a by the mirror M in RFo (left) and in RFv
(right).
8. It is immediate to prove the angle of incidence iv is always greater than the angle of
reflection rv and that that difference increases as the relative velocity kc increases. Basic
geometry and trigonometry allow us to write (see Figure 3):
(1) In the place of π/4, the mirror is now inclined at an angle tan−1 γ −1 .
(2) The angle α between the incident trajectories in RFo and in RFv is:
kc
= sin−1 k
c
(3) The angle β between the normals No and Nv is:
α = sin−1
β = π/4 − tan−1 γ −1
(3)
(4)
4 —— Relativistic reflection of light
(4) The angle of incidence iv is:
π
+β
4
p
π π
= sin−1 k + + − tan−1 1 − k 2
4
4
p
π
= + sin−1 k − tan−1 1 − k 2
2
(5) The angle of reflection rv is:
p
rv = tan−1 1 − k 2
iv = α +
(5)
(6)
(7)
(8)
Figure 4. In RFv the difference iv − rv between the angle of incidence and the angle of reflection
increases as k increases within the interval (0, 1).
9. In accordance with 8, we can conclude that as the relative velocity v = kc increases:
(1)
(2)
(3)
(4)
k increases within the real interval (0, 1).
√
√
1 − k 2 and tan−1 1 − k 2 decrease.
α = sin−1 k increases.
√
β = π/4 − tan−1 1 − k 2 increases.
Therefore, as v increases
(1) The angle of incidence iv = π/4 + α + β increases.
√
(2) The angle of reflection rv = tan−1 1 − k2 decreases.
In consequence, from the perspective of RFv , the second law of the reflection of light is
not observed. Far from being equal, as that second law states, the difference between the
angle of incidence and the angle of reflection:
p
π
+ sin−1 k − 2 tan−1 1 − k 2
2
increases as the relative velocity v = kc increases (Figure 4).
iv − rv =
(9)
2. Conclusions
10. According to the above discussion, FitzGerald-Lorentz contraction can only be
apparent. Otherwise it would be incompatible with the laws of optics, or the laws of
Relativistic reflection of light —— 5
elastic collisions if in the place of the reflection of a photon we would have considered the
case of the massive ball suggested in 3.
11. The alternative would be that some physical laws that hold in the proper frame of
the events governed by those laws are not satisfied when the same events are observed in
relative motion.
12. By symmetry, time dilation, phase difference in clocks synchronization and increment
of mass with relative motion would also be apparent. Simply because of all of them are
direct consequences of the same Lorentz transformation.
13. Thus, Lorentz transformation does not transform properly a physical legality into
another physical legality. Or to put it another way, the physical laws are not always
preserved when the events governed by those laws are observed in relative motion.
14. Conclusion 13 means the analog interpretation of Lorentz transformation distorts
reality until the point of making it unrecognizable from the perspective of the physical
laws. This could be the proof that the relativistic distortions of space and time are rather
apparent, and that not always these distortions can be used to draw conclusions about
what happens in the reference frame in which those distortions seem to take place.
References
[1] Antonio Leon, Digital relativity, Printed by Bubok Publishing, Madrid, 2013 (pdf avalaible at interciencia.es).