Relativistic Path-Dilation of the Light Emitted from a Source
in Inertial Motion and Dark Energy of the Universe
Giovanni Zanella
Dipartimento di Fisica e di Astronomia dell’Università di Padova and
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8,
35131 Padova, Italy
Abstract
In this paper it is demonstrated the dilation of the path of the light emitted
by a source in relativistic inertial motion in the empty space. Consequently
the measured distance of Type Ia Supernovae, which move at relativistic
velocities and are used as standard candles, is affected by this dilation.
Therefore, the correction of the Hubble diagram, fitting the experimental
data, reveals the validity of the Einstein-de Sitter model which involves an
universe of only matter.
1. Introduction
Using the methods of the Special Relativity [1], we can infer about the
dilation of the path of the light emitted by a source in inertial motion in the
empty space. As we will see, the dilation of the wavelength of the light
emitted from a receding source at relativistic velocities involves necessarily
the dilation of the path of the light in the opposite direction to the motion of
the source itself.
In our case, the light sources are Type Ia Supernovae, which are very
luminous objects used as standard candles, i.e. objects with identical
absolute luminosity to obtain a precise measurement of their distance, out
1000 Mpc.
Measurements of these great distances provided the first data which suggest
that the expansion rate of the universe is actually accelerating [2][3]. The
velocity of these sources is obtained instead by the relativistic relationship of
the measured z-parameter of red-shift.
In the following we demonstrate that the Hubble diagram pertaining
measurements on high-z Ia Supernovae [4], if corrected for the dilation of
the path of the light emitted from themselves, goes in accord with the
Hubble’s diagram pertaining the Einstein- de Sitter model.
1
2. Classical path of the light emitted from a moving source
Suppose in the empty space two Cartesian systems S and S’ in uniform
parallel translation along the direction of their x-axes. In particular, S’ have
velocity u in respect to S, along the increasing x co-ordinate.
Suppose also that an isotropic source, emitting light of wavelength λ, be put
in the origin of the S, thus spherical, concentric, λ-spaced wave-crests of
light will be emitted from the source, if viewed from an observer at rest in
the same reference, at the instant t. The observer is supposed ubiquitous in
the space, while the time t starts when the origin of S and S’ are
superimposed (Fig.1).
The second principle of the Special Relativity affirms the impossibility for
a single wave-front of light to be dragged by the source itself. So, if now the
light source is put in the origin of S’, the previous wave-crests will appear as
in Fig.2, if they are viewed from an observer at rest in S and at the instant t,
provided that the velocity of the source is approaching the velocity of the
light.
On the other hand, the co-ordinates x1,2’ of intersection with the x’-axis of
the wave-crest of the light emitted when the origin of S and S’ were
coincident, if measured from S at the time t, will be (Fig.2)
x1 '  x1  ut
x2 '  x2  ut
.
(2)
where x1,2 are the co-ordinates of intersection with the x-axis of S, of the
same wave-crest, at the time t.
It is interesting to note that in this non-relativistic view the wavelength of
the light emitted from the receding source, as viewed from an observer at
rest in S, can double at the most, supposing the source reaches the velocity
of the light.
In Fig.2 the boldfaced arrow represents the path d=ct of the light emitted
from the source at time t=0, when it reaches an observer put at the distance
x2 = ct on the x-axis.
3. Relativistic path of the light emitted from a moving source
Eq.s (1) do not represent the reality especially when the source of the light
reaches relativistic velocities, that is near to c. Indeed, Eq.s (1) obey to the
known Lorentz Transform, that is
2
X 1 '   ( x1  ut )
,
X 2 '   ( x 2  ut )
where   1 / 1 
(3)
u2
is the Lorentz factor, while the co-ordinates X1,2’ pertain
c2
the intersections with the x’-axis, measured from S at the time t, of the
relativistically distorted wave-crest of light emitted from the source put in
the origin of S’ when the origins of S’ and S were coincident (t=0).
Fig.3 represents, as an example, the previous wave-crests of the light in the
case of  = 1.38, where the wave-crests are viewed from an observer at rest
in the reference S and ubiquitous in the space, at the time t.
Looking to Fig.1 and Fig.3, putting t= x2/c, we derive the expression of the
z-parameter of red-shift. Indeed, being
u
c
u
1
c
1
X 2 '   ( x 2  ut )  x 2
then
 ' X 2 '  x 2
z



x2
,
u
c
1 ,
u
1
c
(4)
1
(5)
where λ’ is the wavelength of the light emitted by the source put in the origin
of S’ and λ is the wavelength of the same light viewed locally u.
It is evident in Eq.(5) that the wavelength λ’ can become infinite if the
source of light reaches the velocity c, differently from Fig.2 where such
wavelength can at the most double.
In conclusion the boldfaced arrow of Fig.3 represents the path D=c t of the
light emitted from the source, at time t=0, when it reaches at the time t an
observer put at the distance X2= -c t on the x-axis.
4. Relativistic deviation from the Einstein-de Sitter model
Hubble’s law [4] affirms that
z
H0D
c
,
(6)
3
where H0 is the Hubble constant at the present day and D the distance
travelled from the light from the instant of emission .
Now, the velocity of Ia Supernovae, as measured by the z-parameter, can
reaches relativistic velocities [2][3]. Indeed, the z-parameter derived from
the measurements of High-z Supernovae Team [3] and of Supernovae
Cosmology Project [4] ranges up the value z=0.3 where the relativistic effect
starts to acquire evidence (see Tab.1).
In Fig.4 it is reported the A diagram, concerning the Einstein–de Sitter
model, which involves an universe of only matter, and the B diagram,
which fits the experimental data concerning distances of Ia Supernovae vs
their z-parameters [5].
It is remarkable that the experimental B diagram starts to differ from A
diagram just when the Lorentz factor acquires significant values, that is for z
> 0.3. But it is amazing that correcting the D distances of Tab. 1, measured
on the B diagram at z=0.3, z=0.5, z=0.7, z=1 and z=1.5, that is dividing they
by the corresponding Lorentz’s factor, we find the distances d, measured on
the A diagram, at the same z-values, within a precision of  25 Mpc.
5. Conclusions
We have demonstrated the necessity to consider the relativistic dilation of
the path of the light emitted from high-z Ia Supernovae to interpret the
Hubble diagram which fits the experimental data [2][3][4]. Indeed, this
correction brought on the diagram itself reveals the validity of the model
Einstein-de Sitter also at high-z red-shifts with a precision of  25 Mpc. A
better validation will be possible operating on analytical expressions of the
Hubble diagrams, but in any case it is not possible to ignore the relativistic
effect of the dilation of the path of the light to interpret the experimental data
concerning the high-z Ia Supernovae.
Therefore, the view of an universe of only matter, tuned in such a manner
that it will expand at decreasing rate forever, remains still believable.
References
[1]
[2]
[3]
[4]
[5]
Einstein A., Ann. Phys. 17 (1905) 891.
Riess A. et al., Astr. J. 116 (1998) 1009.
Perlmutter J. et al., Astr. J. 517 (1999) 565.
Hubble E., Proc. Nat. Acad. of Sc. of U.S.A, 15, 3 (1929) 168.
Leibundgut B. and Sollerman J., Europh. N. 32, 4 (2001) 121.
4
Figure captions
Fig. 1 Spherical wave-front of the light emitted by an isotropic source put in
the origin of an S Cartesian reference, as viewed locally in the
empty space, elapsed the time t from the emission instant.
Fig. 2 Classical two-dimensional representation of wave-crests of light
emitted isotropically from a source in uniform translatory motion in
the empty space. The source (fixed in the origin of the Cartesian
system S’) moves with uniform velocity u along the direction of the
x-axis of the Cartesian system S, while the wave-crests are viewed
from S at the time t. The time started when the origins of S and S’
were coincident (see text).
Fig. 3 Relativistic view of the wave-crests which appear in Fig.2, supposing
a Lorentz factor  =1.38 (see text).
Fig.4 Hubble diagram concerning the fit of experimental data (B diagram)
deriving from measurements on distant Ia Supernovae [2][3] and
Hubble diagram (A diagram) concerning the Einstein-de Sitter model
(see text).
Tab.1 Measured distances ( 25 Mpc of precision) on the experimental B
diagram (D distances) and on A diagram (d distances), concerning
the Einstein-de Sitter model, at various z-parameters of red-shift (see
text).
5
y
-ct
ct
S
x1
x2
x
λ
Source of light
Wave-crests of light
Fig. 1
6
y’
y
-ct
x2 ’
x2
ut
S’
x1 ’
x1
d
u
x’
x
S
x
Light source elapsed the
time t
Visible source of
light
Wave-crests of light
Fig. 2
Wave-crests of
light
y
y’
 ut
- cyt
X2’
X2
S’
X1’
X1
D
u
x’
S
Light source elapsed the
time  t
Visible source of
light
Fig. 3
7
z
0.1
0.3
0.5
u/c
0.095
0.25
0.33
0.7
1
1.5
0.48
0.60
0.72

1.004
1.03
1.08
1.14
1.25
1.45
D (Mpc)
502
1750
2700
4200
6350
11600
500
1700
2500
3700
5100
8000
B diagram
d (Mpc)
A diagram
TAB.1
distance (Mpc)
10000
A
B
10000
B
A
1000
1000
100
100
.01
.01
.1
.1
1
.3
redshift z
1.5
.5
.7
1
1.5
FIG. 4
8