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Searchlight and Doppler Effects in the Visualization of Special Relativity: A Corrected Derivation of the Transformation of Radiance

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this article gives a comprehensive presentation of special relativistic rendering. We will demonstrate that a photo-realistic image would be dominated by the searchlight and Doppler effects, which are greatly underestimated when one is viewing the examples given by Chang et al. The Doppler effect causes a shift in wavelength of the incoming light, which results in a change of color. The searchlight effect increases the apparent brightness of the objects ahead when the observer is approaching these objects at high velocity. Amongst others, the Doppler effect, the relativistic aberration of light, and the time dilation contribute to the searchlight effect. 2. DERIVATION OF THE TRANSFORMATIONS 2.1 The Transformation of Radiance
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Searchlight and Doppler Effects in the
Visualization of Special Relativity:
A Corrected Derivation of the
Transformation of Radiance
DANIEL WEISKOPF, UTE KRAUS, and HANNS RUDER
University of Tübingen
We demonstrate that a photo-realistic image of a rapidly moving object is dominated by the
searchlight and Doppler effects. Using a photon-counting technique, we derive expressions for
the relativistic transformation of radiance. We show how to incorporate the Doppler and
searchlight effects in the two common techniques of special relativistic visualization, namely
ray tracing and polygon rendering. Most authors consider geometrical appearance only and
neglect relativistic effects on the lighting model. Chang et al. [1996] present an incorrect
derivation of the searchlight effect, which we compare to our results. Some examples are given
to show the results of image synthesis with relativistic effects taken into account.
Categories and Subject Descriptors: I.3.7 [Computer Graphics]: Three-Dimensional Graph-
ics and Realism—Color, shading, shadowing, and texture; J.2 [Computer Applications]:
Physical Sciences and Engineering—Physics
General Terms: Algorithms, Theory
Additional Key Words and Phrases: Aberration of light, Doppler effect, illumination, Lorentz
transformation, searchlight effect, special relativity
1. INTRODUCTION
Einstein’s Theory of Special Relativity is widely regarded as a difficult and
almost incomprehensible theory. One important reason for this is that the
properties of space, time, and light in relativistic physics are totally
different from those in classical, Newtonian physics. In many respects they
This work was supported by the Deutsche Forschungsgemeinschaft (DFG), and is part of the
project D4 within the Sonderforschungsbereich 382
Authors’ address: Theoretical Astrophysics, University of Tübingen, Auf der Morgenstelle 10,
Tübingen, D-72076, Germany
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ACM Transactions on Graphics, Vol. 18, No. 3, July 1999, Pages 278–292.
are contrary to human experience and everyday perception, which is based
on low velocities.
Mankind is limited to very low velocities compared to the speed of light.
For example, the speed of light is a million times faster than the speed of
an airplane and 40,000 times faster than the speed at which the space
shuttle orbits the earth. Even in the long term, there is no hope of
achieving velocities comparable to the speed of light. Computer simulations
are the only means of visually exploring the realm of special relativity, and
thus can help the intuition of physicists.
The visual appearance of rapidly moving objects shows intriguing effects
of special relativity. Apart from a previously disregarded article by Lampa
[1924] about the invisibility of the Lorentz contraction, the first solutions to
this problem were given by Penrose [1959] and Terrell [1959]. Various
aspects were discussed by Weisskopf [1960]; Boas [1961]; Scott and Viner
[1965]; Scott and van Driel [1970]; and Kraus [2000].
Hsiung and Dunn [1989] were the first to use advanced visualization
techniques for image shading of fast moving objects. They propose an
extension of normal three-dimensional ray tracing. Hsiung and Thibadeau
[1990] and Hsiung et al. [1990a] add the visualization of the Doppler effect.
Hsiung et al. [1990b] and Gekelman et al. [1991] describe a polygon
rendering approach based on the apparent shapes of objects as seen by a
relativistic observer. Polygon rendering was also used as a basis for a
virtual environment for special relativity [Rau et al. 1998; Weiskopf 1999].
Most authors concentrate their efforts on geometrical appearance and,
apart from the Doppler effect, neglect relativistic effects on the lighting
model. Chang et al. [1996], however, present a complete description of
image shading which takes relativistic effects into account. We agree with
most parts of their article, but would like to correct their derivation of the
relativistic transformation of radiance. We show how the correct transfor-
mation of radiance fits in their shading process. The combination of Chang
et al.’s work and this article gives a comprehensive presentation of special
relativistic rendering.
We demonstrate that a photo-realistic image is dominated by the search-
light and Doppler effects, which are greatly underestimated when we view
the examples given by Chang et al. The Doppler effect causes a shift in
wavelength of the incoming light, which results in a change of color. The
searchlight effect increases the apparent brightness of the objects ahead
when the observer approaches these objects at high velocity. The Doppler
effect, the relativistic aberration of light, and time dilation, among others,
contribute to the searchlight effect.
2. DERIVATION OF THE TRANSFORMATIONS
2.1 The Transformation of Radiance
The following derivation of the searchlight effect is based on a photon-
counting technique. A similar approach can be found in articles by Peebles
and Wilkinson [1968]; McKinley [1979; 1980]; and Kraus [2000].
Searchlight and Doppler Effects 279
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
Consider two inertial frames of reference S and S9, with S9 moving with
velocity v along the z axis of S. Suppose the observer O is at rest relative to
S and the observer O9 is moving with speed v along the z axis of S. The
usual Lorentz transformation along the z axis connects frames S and S9.
In reference frame S, consider a photon with circular frequency
v
,
wavelength
l
, energy E, and wave vector k
Y
5 ~
v
sin
u
cos
f
,
v
sin
u
sin
f
,
v
cos
u
!
/
c with spherical coordinates
u
and
f
, as shown in Figure 1.
In frame
S9, the circular frequency is
v
9, the wavelength is
l
9, the energy
is E9, and the wave vector is k9
Y
5 ~
v
9sin
u
9cos
f
9,
v
9sin
u
9sin
f
9,
v
9cos
u
9!
/
c.
The expressions for the Doppler effect and the aberration connect these two
representations, cf., McKinley [1979] and Møller [1972]:
l
95
l
D (1)
v
95
v
/
D (2)
E95E
/
D (3)
cos
u
95
cos
u
2
b
1 2
b
cos
u
(4)
f
95
f
(5)
D 5
1
g
~1 2
b
cos
u
!
(6)
where D is the Doppler factor,
g
5 1
/
Î
1 2
b
2
,
b
5 v
/
c, and c is the
speed of light.
Radiance is the radiant power per unit of foreshortened area emitted into
a unit solid angle. A detector at rest in S measures the energy-dependent
radiance
x
z
k
θ
y
φ
Fig. 1. A photon with wave vector k
Y
.
280 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
L
E
~
u
,
f
! 5
dF
dE dA
'
dV
where F is the radiant power or radiant flux, E is the energy, dV is the
solid angle, and
dA
'
is the area dA of the detector projected along the
radiation direction ~
u
,
f
!. The radiant flux F is the radiant energy per unit
time. Accordingly, the wavelength-dependent radiance is
L
l
~
u
,
f
! 5
dF
d
l
dA
'
dV
(7)
with the wavelength
l
.
In reference frame
S, consider a group of photons, dN in number, with
energies between E and E 1 dE and propagation directions in the element
of solid angle
dV around ~
u
,
f
!. Here, the energy-dependent radiance is
L
E
~
u
,
f
! 5
dN E
dE dA
'
dV dt
or
dN 5
L
E
~
u
,
f
!
E
dE dA
'
dV dt
We choose the area dA to be perpendicular to the z axis, so that
dA
'
5 dA cos
u
The z component of the velocity of the photons is ccos
u
. The photons
passing dA between time t
0
and time t
0
1 dt are contained in the shaded
volume dV in Figure 2:
dV 5 dA dt c cos
u
Consider another area dA
˜
with the same size and orientation as dA. Still
in reference frame
S, suppose dA
˜
is moving with velocity v along the z
axis. The photons passing dA
˜
between t
0
and t
0
1 dt are contained in the
shaded volume in Figure 3:
dV
˜
5 dA dt~c cos
u
2 v! 5
cos
u
2
b
cos
u
dV
The ratio of the number of photons passing dA
˜
in the time interval dt and
the number of photons passing dA is the same as the ratio of the volume
dV
˜
and the volume dV:
Searchlight and Doppler Effects 281
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
dN
˜
5
L
E
~
u
,
f
!
E
dE dV dt cos
u
dA
˜
cos
u
2
b
cos
u
(8)
Now consider the same situation in reference frame S9. The area dA
˜
is at
rest in S9. The time interval is
dt95dt
/
g
(9)
The number of photons counted does not depend on the frame of reference,
i.e.,
dN
˜
5 dN
˜
95
L9
E9
~
u
9,
f
9!
E9
dE9dV9dt9cos
u
9dA
˜
9 (10)
From Eqs. (8) and (10), we obtain
L
E
~
u
,
f
!
L9
E9
~
u
9,
f
9!
5
E
E9
dE9
dE
dV9
dV
dt9
dt
cos
u
9
cos
u
2
b
dA
˜
9
dA
˜
(11)
Since the area dA
˜
is perpendicular to the separating velocity, it is not
changed by Lorentz transformations:
dA
˜
95dA
˜
(12)
With Eqs. (4) and (5), the transformed solid angle is
dV9
dV
5
sin
u
9
sin
u
d
u
9
d
u
5
d
~cos
u
9!
d~cos
u
!
5
1
g
2
~1 2
b
cos
u
!
2
5 D
2
(13)
Using Eqs. (3), (4), (9), (12), (13), and (11), we obtain
dA
k
z
θ
dt c cosθ
dA
Fig. 2. Photons with propagation direction along the wave vector k
Y
. The area of the detector
is denoted dA and is perpendicular to the z axis; dA
'
is the projection of dA along the
radiation direction. The shaded volume dV contains the photons passing dA between time t
0
and time t
0
1 dt.
282 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
L
E
~
u
,
f
!
L9
E9
~
u
9,
f
9!
5 D
3
5
E
3
E9
3
With the relation between energy and wavelength,
l
5
hc
E
, d
l
52
hc
E
2
dE
and with
L
l
~
u
,
f
!?d
l
? 5 L
E
~
u
,
f
!?dE?
we get
L
l
~
u
,
f
! 5 L
E
~
u
,
f
!
E
2
hc
Ultimately, then, the transformation expression for the wavelength-depen-
dent radiance is
L
l
~
u
,
f
!
L9
l
9
~
u
9,
f
9!
5 D
5
(14)
The transformation law for the following integrated quantity is easily
obtained from this equation. With the use of Eq. (1), the transformed
radiance is
L~
u
,
f
! 5
E
0
`
L
l
~
u
,
f
!d
l
5 D
4
E
0
`
L9
l
9
~
u
9,
f
9! d
l
95D
4
L9~
u
9,
f
9! (15)
0
v dt
k
θ
0
z
~
~
dA(t )
dA(t +dt)
θdt c cos
Fig. 3. Photons with propagation direction along the wave vector k
Y
. The area dA
˜
moves with
velocity
v along the z axis. The shaded volume dV
˜
contains the photons passing dA
˜
between t
0
and t
0
1 dt.
Searchlight and Doppler Effects 283
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
2.2 Incident Irradiance from a Point Light Source
The measure for radiant power leaving a point light source in an element of
solid angle dV and in a wavelength interval is called the wavelength-
dependent intensity
I
l
:
I
l
5
dF
dV d
l
(16)
The wavelength-dependent irradiance E
l
i
is the radiant power per unit area
in a wavelength interval:
E
l
i
5
dF
dA d
l
(17)
For a surface patch on the object, the wavelength-dependent irradiance E
l
9
i9
coming from a moving point light source is
E
l
9
i9
5
1
D
5
cos
a
9
r9
2
I
l
(18)
with the angle
a
9 between the normal vector to the surface and the
direction of the incident photons and with the apparent distance r9 of the
light source from the surface patch. These quantities are measured in the
reference frame of the object, whereas the wavelength-dependent intensity
I
l
is measured in the reference frame of the light source. Accordingly, the
integrated, wavelength-independent irradiance is
E
i9
5
1
D
4
cos
a
9
r9
2
I (19)
The derivation of these equations is presented in the Appendix. Observe
that for an isotropic point source in one frame of reference, we get an
anisotropic source in the other frame of reference due to the implicit angle
dependency in the Doppler factor D.
3. COMPARISON WITH DERIVATION BY CHANG ET AL.
Chang et al. [1996] present a complete treatment of relativistic image
shading, which contains apparent geometry, the searchlight and Doppler
effects, and a detailed description of the shading process. However, their
derivation of the transformation properties of radiance is based on mis-
taken interpretations of the Theory of Special Relativity and leads to a
tremendous divergence from our correct results, presented above.
Chang et al. derive their expressions based on the assumption that the
same amount of radiant power is emitted from a surface patch on the object
and the corresponding surface patch on the apparent surface. Hence, they
284 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
compute the relation between the area of the surface patch on the object
and the area of the corresponding surface patch on the apparent surface, as
well as the relation between the respective normal vectors. They treat the
apparent surface as an object at rest with respect to the observer.
Their derivation is not correct for the following reasons:
Radiant power depends on time intervals and on the energy of photons,
both of which are subject to Lorentz transformations. These transforma-
tions are missing in Chang et al.’s work.
The observer is moving with respect to the surface patch of the object.
Approaching the object, the observer’s detector sweeps up photons so that
the rate at which radiant energy is received is increased by the observer’s
motion. This increase is absent for radiation from the apparent surface,
which is stationary in the observer’s rest frame. Chang et al. ignore this
effect as well.
In Chang et al.’s Eq. (36), the transformation of a solid angle is not
correct. The mistake is in their calculation of the partial derivatives
Q9
/
Q, Q9
/
F, F9
/
Q, and F9
/
F with the use of their Eq. (31) for
the transformation of the direction of the light ray. Equation (31) is valid
for the special case of polar angle
Q5
p
/
2 only, and cannot be used to
calculate partial derivatives.
Both wavelength and wavelength intervals are subject to Lorentz trans-
formations. When calculating the radiance per wavelength interval in their
Eq. (39) from Eq. (38), Chang et al. apply the Lorentz transformation to the
wavelength, but not to the wavelength interval.
In their Eq. (38), they ultimately end up with a factor of D in the
transformation of radiance, and also in in their Eq. (39) in the transforma-
tion of wavelength-dependent radiance, which differs from the correct
result by a factor of D
3
and D
4
, respectively. Similarly, the calculation of
irradiance in their Eq. (46) and of wavelength-dependent irradiance in
their Eq. (47) differs from the correct result by the same factor of
D
3
and
D
4
, respectively.
4. THE SHADING PROCESS
The searchlight and Doppler effects can be readily incorporated in the two
common techniques of special relativistic visualization, i.e., ray tracing and
polygon rendering.
Relativistic ray tracing as described by Hsiung and Dunn [1989] is an
extension of normal 3D-ray tracing. The ray starting at the eye point and
intersecting the viewing plane is transformed according to special relativ-
ity, i.e., the direction of light is turned due to relativistic aberration. At this
point the transformed properties of light can be included by calculating the
transformed radiance as well as the transformed wavelength.
In this framework it is not sufficient to consider only three tristimulus
values, such as RGB, but the wavelength-dependent energy distribution of
light has to be taken into account. The spectral energy distribution has to
be known over an extensive range so that the Doppler-shifted energy
Searchlight and Doppler Effects 285
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
distribution can be calculated for wavelengths in the visible range. For
final image synthesis, the tristimulus values can easily be obtained from
the wavelength-dependent radiance that gets to the eye point.
Relativistic polygon rendering is based on the apparent shapes of objects
with respect to the observer. The shading process is described by Chang et
al. in full detail. In this process, the expressions for irradiance in step
(2)(d)(iv) and for the transformation of radiance in step (2)(f) have merely
to be replaced by our Eqs. (18) and (14), respectively. The Doppler factor in
Eq. (18) depends on the direction of the photons that reach the object and
on the relative velocity of the frame of the point light source and the frame
of the object, whereas the Doppler factor in Eq. (14) depends on the
direction of the photons that reach the observer and on the relative velocity
of the frame of the object and of the frame of the observer.
5. EXAMPLES
The appearance of a scene similar to Chang et al.’s STREET in Figures 4 to
7 shows the tremendous effects of the transformation of radiance on image
synthesis. These pictures show the apparent geometry and the radiance
transformation, but neglect color changes due to the Doppler effect. Since
the spectral energy distribution of the light reflected by the objects in the
STREET is unknown, we only show gray-scale images that take the total
energy of the whole spectral energy distribution into account. In this case,
Eq. (15) is applied. If we used a speed as high as Chang et al.’s, 0.99c,we
would not be able to display the high intensities in Figure 7 properly. So we
choose a velocity of
0.8c. These images were generated by using the
ray-tracing method described above. The relativistic extensions are imple-
mented into RayViS [Gröne 1996], a normal 3D-ray-tracing program.
Figures 8 and 9 show the appearance of the sun moving at a speed of
0.5c to illustrate color changes due to relativistic lighting. We used the
polygon-rendering technique described above to produce these images. A
detailed presentation of the rendering software can be found in our previ-
ous work [Rau et al. 1998; Weiskopf 1999].
6. CONCLUSION
We have demonstrated that, aside from the apparent geometry, the search-
light and Doppler effects play dominant roles in special relativistic visual-
ization. Ray tracing and polygon rendering, two standard techniques in
computer graphics, can easily be modified and extended to take into
account the searchlight and Doppler effects.
The transformation of radiance could serve as an important element in
even more sophisticated shading algorithms in order to generate photo-
realistic and physically correct images of fast moving objects. For example,
radiosity could be extended to visualize relativistic flights through station-
ary scenes.
286 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
Fig. 4. Original appearance of the street.
Fig. 5. Appearance of the street with respect to a moving observer. The viewer is rushing into
the street with a speed of 0.8c. The light sources are at rest in the street’s coordinate system.
The searchlight and Doppler effects are ignored.
Searchlight and Doppler Effects 287
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
Fig. 6. Visualization of the searchlight effect based on the incorrect derivation by Chang et
al. The viewer is rushing into the street with a speed of 0.8c. The light sources are at rest in
the street’s coordinate system.
Fig. 7. Visualization of the searchlight effect based on the correct Eq. (15) for the transfor-
mation of radiance. The difference from Figure 6 is significant. The viewer is rushing into the
street with a speed of 0.8c. The light sources are at rest in the street’s coordinate system.
288 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
APPENDIX
A. INCIDENT IRRADIANCE
The derivation of Eqs. (18) and (19) is presented in this Appendix.
First, consider a finite light source which is at rest in frame
S. With Eq.
(7), the radiant flux emitted by the light source can be obtained in terms of
the wavelength-dependent radiance:
dF5L
l
dA
l
light
dV
obj
d
l
(20)
Fig. 8. Visualization of the Doppler effect only: the Doppler-shifted spectral energy distribu-
tion is shown with no further transformations. The sun passes by with a speed of 0.5c. The
sun is the only light source and emits blackbody radiation with a temperature of 5762 Kelvin.
Fig. 9. Visualization of the searchlight and Doppler effects based on Eq. (14). The sun passes
by with a speed of 0.5c. The sun is the only light source and emits blackbody radiation with a
temperature of 5762 Kelvin.
Searchlight and Doppler Effects 289
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
where dA
'
light
is the area of the projected surface patch of the light source
and
dV
obj
is the solid angle of the illuminated surface patch of the object as
seen from the position of the light source.
Now consider the same situation in frame S9 in which the object is at
rest. The radiant flux on the surface patch of the object is
dF9 5 L9
l
9
dA
'
obj9
dV
light9
d
l
9 (21)
with the projected area dA
'
obj9
on the object and the solid angle dV
light9
of
the surface patch of the light source as seen from the position of the object.
Using Eqs. (14) and (21), we obtain
dF9 5
1
D
5
L
l
dA
'
obj9
dV
light9
d
l
9
With the definition in Eq. (17), the incident irradiance emitted from the
small solid angle dV
light9
onto the surface patch of the object is
dE
l
9
i9
5
dF9
dA
obj9
d
l
9
5
L
l
D
5
dA
'
obj9
dA
obj9
dV
light9
(22)
The area dA
obj9
of the surface patch is related to the projected area dA
'
obj9
by
dA
'
obj9
5 dA
obj9
cos
a
9 (23)
with the angle
a
9 between the surface normal and the incident light.
With Eq. (13), the solid angle
dV
light9
is transformed into the frame S of
the light source. Furthermore, dV
light9
is expressed in terms of the projected
area on the light source and of the distance between the light source and
the surface patch, as shown in Figure 10:
dV
light9
5 D
2
dV
light
5 D
2
dA
'
light
r
2
5 dA
'
light
S
D
r
D
2
(24)
The light-like connection of the emission event at the light source and the
absorption event at the object has the same direction as the wave vector
that describes the photons. Therefore, the distance r is transformed in the
same way as the circular frequency
v
(see Eq. (2)). By following this
reasoning or by explicit Lorentz transformation of the separating vector
between the emission event and the absorption event, we get
r95r
/
D (25)
Using Eqs. (22), (23), (24), and (25), we obtain the incident wavelength-
dependent irradiance originating from a small area of the light source:
290 D. Weiskopf et al.
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
dE
l
9
i9
5
1
D
5
cos
a
9
r9
2
L
l
dA
'
light
By integrating over the area of the whole light source, we get the wave-
length-dependent irradiance produced by this finite light source:
E
l
9
i9
5
E
1
D
5
cos
a
9
r9
2
L
l
dA
'
light
(26)
Now consider a very small, yet finite, light source, described by its
wavelength-dependent intensity I
l
. With Eqs. (16) and (20), the wave-
length-dependent radiance and the wavelength-dependent intensity from
the area
dA
'
light
are related by
dI
l
5 L
l
dA
'
light
(27)
With Eq. (26) and after integrating over the area of the small light source,
we find the wavelength-dependent irradiance on the object
E
l
9
i9
5
E
1
D
5
cos
a
9
r9
2
L
l
dA
'
light
5
1
D
5
cos
a
9
r9
2
E
L
l
dA
'
light
5
1
D
5
cos
a
9
r9
2
I
l
This equation even holds for the limit of an infinitesimal light source.
Hence we obtain the wavelength-dependent irradiance due to a point light
source:
E
l
9
i9
5
1
D
5
cos
a
9
r9
2
I
l
object
r
light source
light
dA
dA
light
d
light
Fig. 10. Geometry of the surface patch of the light source in its rest frame S. The solid angle
is given by dV
light
5 dA
'
light
/
r
2
. The distance between the light source at emission time and
the surface patch of the object at absorption time is denoted r.
Searchlight and Doppler Effects 291
ACM Transactions on Graphics, Vol. 18, No. 3, July 1999.
Accordingly, the irradiance is
E
i9
5
1
D
4
cos
a
9
r9
2
I
where I is the intensity of the light source.
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