Redshift and Energy Conservation
- 1 Redshift
and Energy Conservation
Alasdair Macleod§
ABSTRACT
It has always been considered a serious error to treat the cosmological redshift as a Doppler
velocity effect rather than the result of space expansion. It is demonstrated here that in
practical terms this is not the case, and that the apparent distance - redshift relation derived
from a Doppler interpretation is reasonably consistent with supernova data (though not as
good as standard cosmology with dark energy). The normal Doppler effect is examined in
detail and shown to conserve energy as expected. Because of the equivalence between the
general relativistic space expansion paradigm and the Doppler effect (as demonstrated) the
long-standing problem of energy loss associated with the expansion of the Universe is
treated in a similar manner to the normal well-behaved Doppler effect. The mechanism by
which energy is conserved with the normal Doppler shift is applied to the cosmological
redshift and the energy violation disappears. However, an additional luminosity-dependent
recession factor is introduced. The effect on astronomical objects is examined and it is found
to add only a small additional redshift to a body generating power by nuclear means but can
be very large for objects powered by gravity. A possible connection to the claimed
anomalous redshift of quasars is considered.
V2.0 Changes: Figure 4 altered and conclusions drawn from it modified somewhat;
discrepancies with the modelling of Davis and Lineweaver investigated – Figure 5 added.
1. Introduction
It is well established that the cosmological redshift originates from a general-relativistic
process with the spectral ratio being a measure of the scale factor of the Universe at the epochs
of observation and emission1
. The apparent motion resulting from the expansion of space
cannot be rightly considered as a proper velocity, but there is still a tendency to erroneously
apply the relativistic Doppler formula to the measured redshift and derive an equivalent
recession velocity, effectively transferring the process into the special relativistic domain¶
.
When the geometrical effect of the space expansion is correctly transformed into an apparent
velocity, the value is actually equal to the Doppler velocity when distances are short relative to
the dimensions of the Universe because both formulae approximate to the linear Hubble law
under that constraint 2
. However it would seem the equivalence can not possibly hold at large
distances. In the expanding space paradigm the apparent velocity is permitted to exceed the
speed of light (with the point of equality defining the observer horizon). This is not possible in
the context of special relativity. It is perhaps because of this obvious and significant divergence
that no careful analysis of observational data has yet taken place to quantify the discrepancy at
high z when the Doppler formula is incorrectly applied in place of the Robertson-Walker
metric¶
.
In this paper, the redshift-apparent distance relationship of distant supernovae is examined.
Perhaps surprisingly, it is found that the luminosity distance obtained by deriving the Doppler
equivalent velocity from z and applying special relativity to the propagation process is
reasonably consistent with observation (similar to the ΩM=1, ΩΛ=0 prediction of standard
cosmology), with the simplest assumption of a flat and empty Universe and constant expansion;
in other words if we are consistently special relativistic. In practical terms therefore, the use of
the Doppler velocity expression is an acceptable approximation (and simplification) of
§
University of the Highlands and Islands, Lews Castle College, Stornoway, Isle of Lewis, Scotland
email: Alasdair.Macleod@lews.uhi.ac.uk
¶
Perhaps the reason why this interpretation is so deeply ingrained is the visual appeal of the model of the
expanding Universe as a hypersurface with an expansion velocity (and equivalent kinetic energy) working
against gravity and large-scale cosmological forces. The model makes it difficult to conceptualise the
difference between proper and expansion velocities.
¶
The paper of Davis and Lineweaver37
will be considered in the text, but their analysis is believed to be
erroneous.
Redshift and Energy Conservation
- 2 Friedman-Robertson-Walker
(FRW) cosmology so long as one is careful with the form of the
Hubble relation.
One problematic aspect of the cosmological expansion is the apparent loss of energy
associated with the redshift. The effect is particularly bad with cosmological background
photons received in the current epoch – they are received with only about 0.1% of their
emission energy. Attempts to account for the missing energy within the framework of general
relativity have met with severe problems because of the difficulty in defining a local
gravitational energy density (gravitational energy cannot be expressed in tensor form). As a
result, it is widely accepted that energy is not locally conserved in general relativity3
, although
claims are made that energy is globally conserved during expansion. This is in stark contrast to
the normal Doppler shift where, as demonstrated in the text, energy is conserved on a photonby-photon
basis. We are surely entitled to demand that photons redshifted by the cosmological
expansion similarly conform. Having established a working equivalence between a Doppler
velocity and the cosmological expansion, an attempt is made here to apply the Doppler energy
conservation ‘recipe’ to the cosmological shift to explore concepts that may later be
transferable to general relativity to recover energy conservation. We avoid the normal
unproductive approach of associating the missing energy to unknown field properties; instead,
the ‘missing’ energy is accounted for by an apparent kinematic change at the source to
guarantee energy conservation in the observer frame.
Though speculative, the analysis does suggest that energy conservation is possible through
the introduction of an additional luminosity-dependent redshift term that evolves according to
the mass-to-light ratio of the bound system. It is important to emphasise that this is not strictly
speaking an ‘intrinsic’ redshift because the value varies with observer distance. When applied
to cosmological entities, the luminosity redshift is seen to contribute only a small amount to the
total cosmological redshift of systems generating energy through nuclear processes but can
overwhelm the cosmological effect in systems powered by gravitation such as active galactic
nuclei. The actual proportion in this latter case is strongly dependent on the mass of the central
black hole, the accretion rate and the radiative efficiency.
2. The Relativistic Doppler Effect
The Doppler shift will be analysed in some detail to demonstrate that energy is always
conserved, both globally and locally (i.e. for each photon). Comprehensive calculations are
included as the results are important to the analysis of the cosmological redshift that is the main
focus of the paper.
Consider two masses A and B separated by a distance r with A moving at velocity v with
respect to B (Figure 1).
Figure 1 Relative motion between emitter A and absorber B
Redshift and Energy Conservation
- 3 If
A emits a photon in its rest frame of wavelength λo, the wavelength measured at position B
is
.)cos1( oλϕβγλ −= (1)
β is defined as |v|/c and γ = (1-β2
)-1/2
. This is the Doppler shift. Visible light may have the
wavelength increased towards the red end of the spectrum or decreased towards the blue. The
relation is readily derived by Lorentz transforming the system four-vectors and using the
invariance of the scalar product of the source and photon four-momenta5
.
The ratio of received to emitted wavelengths in the special case ϕ =π defines the redshift
factor z (=∆λ/λo):
),1(1 βγ +=+ z (2)
or equivalently,
.
1)1(
1)1(
2
2
++
−+
=
z
z
cv (2a)
The relativistic Doppler shift has two components: the Doppler term (1-βcosϕ) is analogous
to the acoustic Doppler effect and is dependent upon angle; the transverse term γ, the time
dilatation factor, has no acoustic analogue and arises because, unlike sound in air, the velocity
of light is a constant independent of reference frame. The transverse term is sometimes referred
to as the anomalous, perpendicular or second order Doppler effect. The correctness of equation
(1) has been established to an extraordinary level of accuracy by a recent experiment6
.
Consider the case of an isotropic light source of luminosity L radiating photons of
wavelength λ. Assume a set of stationary observers on the spherical surface at distance r. The
flux, f, measured by each observer is
.
4 2
r
L
f
π
= (3)
If the source is instantaneously accelerated to a velocity v as shown in Figure 2, then each
observer notes a change in the flux after a due propagation delay.
Figure 2 Light source accelerated to a velocity v
Redshift and Energy Conservation
- 4 Consider
first the limited case where ϕ =π ¶
. The energy flux measured by the observer is
reduced by a factor (1+z)4
in comparison with equation (3) (see, for example, Misner, Thorne
and Wheeler1
, sections 22.6 and 29.4). The four powers of (1+z) include a combination of local
and global effects. The energy of each photon is reduced by (1+z). The received photon count is
reduced by the same factor. There is an additional (1+z)2
diminution in flux because the
absorber in the moving frame lies on the surface of a sphere of radius equal to the rest-frame
separation times (1+z) - the effective photon journey time (distance) is longer by this factor. Of
course, this analysis assumes a point source. A real source will appear extended and energy
over the full extent of the source must be integrated for the (1+z)4
flux reduction to apply.
The analysis is slightly different for extended sources where the surface brightness is the
appropriate parameter rather than the luminosity, for example, the cosmic microwave
background radiation (CMBR). In Appendix A, it is demonstrated that a blackbody spectrum of
a source at temperature T in the rest frame is transformed to a blackbody spectrum of lower
temperature T/(1+z) if a source recession velocity as defined by equation (2a) is applied. The
fact that the CMBR data is verified to have the form of a blackbody spectrum to high accuracy7
shows that, for this specific example, the general-relativistic space expansion can be equated to
an equivalent proper velocity and treated with special relativity.
Referring to Figure 2, the diminution of flux can be generalised for observers at all angles ϕ
as [γ(1-βcosϕ)]4
. We will determine the luminosity of the moving source by integrating the
emitted radiation over the closed surface on which the observers reside. The distortion of the
sphere by the Lorentz boost makes the integration a bit messy as the surface is no longer
spherical. However, if two of the powers are dropped, the absorption surface is transformed
back to a sphere (converting to angular distance). The total luminosity in the moving frame L is
therefore
.
)cos1(
sin2
'
0
22
2
ϕ
ϕβγ
ϕππ
d
r
fL
−
= (4)
Using the substitution, u = cos ϕ, the integral simplifies;
.
)1(
12
'
1
1
22
2
du
u
fr
L
−
−
−
=
βγ
π (5)
Integrating and evaluating
L
fr
fr
u
fr
L
=
=
−
=
−
−
=
−
2
22
2
1
1
2
2
4
1
22
)1(
12
'
π
βγ
π
ββγ
π
(6)
The last line of simplification follows from equation (3).
Although individual photon energy alters as a result of the velocity boost, the luminosity of
the source does not vary. Energy is globally conserved. This does not mean that energy
conservation is maintained by some form of collusion between blue-shifted photons with a
surplus of energy and red-shifted photons with a deficit: we can show that energy is locally
conserved for each photon by a simple kinematic derivation of the Doppler shift expression.
For each photon the difference in energy associated with the wavelength shift is actually
¶
This is the angle at which the Doppler effect mimics the cosmological redshift.
Redshift and Energy Conservation
- 5 balanced
by a kinetic energy change at the source. Of course, this automatically follows from
the transformation of the four-vectors used to derive equation (1) but the alternative derivation
below is more illuminating.
Let the source have an equivalent rest mass M and let the photon emission in the rest frame
correspond to a loss of mass of ∆M. The equivalent photon energy, Eλo is ∆Mc2
and the
momentum is ∆Mc. From the observer frame the total energy of the source is
.42222
cMcpE += (7)
Differentiating with respect to pµ,
.42
µ
µ
µ p
M
Mccp
p
E
E
∂
∂
+=
∂
∂ (8)
Since the momentum in this case is in the x direction only, only the partial equation in x need
be considered. If Eλ is the photon energy measured from the moving frame, ∂E = - Eλ. and
because ∂M = -Eλo/c2
, equation (8) becomes
.
γ
λ
λ
o
x
E
pvE +∂−= (9)
In the observer frame, the change in momentum in the x direction is -Eλ cos ϕ /c. Note that
the momentum projection onto the y axis can be ignored because vy is instantaneously 0 hence
has no effect on the source energy (this is why the y and z partials of equation (8) were ignored;
see Appendix B for further discussion of this point). The equation becomes
)cos1( ϕβγ
λ
λ
−
= oE
E (10)
consistent with equation (1) after taking into account that E ∝ λ-1
. In effect the photon recoil
changes the velocity and kinetic energy of the source, taking up the energy difference. Of
course the source must be isotropic if the velocity in the y direction is to remain zero. For an
isotropic source, the velocity in the x direction will also remain globally constant from
symmetry considerations and the principle of relativity (a Lorentz boost cannot introduce an
acceleration). Non-isotropic sources can be created using mirrors but it is shown in Appendix B
that anisotropic sources also respect energy conservation. The analysis concluded here verifies
the Doppler effect conserves energy on a photon-by-photon basis.
3. The Cosmological Redshift
The systematic redshift of distant galaxies is interpreted as the expansion of the space
between galaxies. If a(t) is the scale factor or size of the Universe at epoch t, it can be
demonstrated1
that the cosmological redshift is related to the scale factor by
.
)(
)(
1
emitted
received
o ta
ta
z =+=
λ
λ (11)
The bolometric flux is given by the following expression
22
)1(4 zR
L
f
+
=
π
(12)
Redshift and Energy Conservation
- 6 where
R is radius of curvature of the 2-sphere surrounding the emitter and passing through the
receiver at the time of reception. An analysis of the WMAP cosmic background radiation
(CMBR) data suggests that the Universe is completely flat8
, hence
.
)'(
'
)(=
received
emitted
t
t
received
ta
dt
taR (13)
In principle, the form of the function a(t) can be deduced from a comparison of the measured
flux and luminosity of standard candles over a range of z values. This is objective of the High-Z
Supernova Search Team9
and the Supernova Cosmology Project10
.
Robertson11
confirmed the (1+z)4
dimming with redshift with the general-relativistic
expansion model previously derived by Tolman which is consistent with the present nature of
the CMBR¶
.
4. Doppler Velocity-Distance Relation for Type Ia Supernovae
It is commonplace to interpret the cosmological redshift as arising from proper motion and
to determine the velocity using the Doppler expression of equation (2a). This is clearly
incompatible with the general-relativistic description of the cosmological expansion. Harrison 2
in his section 6.1 states that it is not possible to extract the velocity from z and relate this to the
linear Hubble law V(z) = H(t)de (where de is the proper distance). However, his reason for
rejecting the procedure is based on the incompatibility of the FRW and Doppler velocity
models not observational data. In practice, is it really such a bad assumption to make? Are the
errors serious? Narlikar12
and Longair13
claim the error is fundamental and significant but again
fail to quantify the comments. It is difficult to make a simple comparison because the Doppler
effect is dependent only on the conditions at the photon emission and absorption points; in the
FRW model, changes in the rate of space expansion during propagation affect the redshift
hence an integration over history is required to get a parametric solution.
In this section we will take a pragmatic approach to the dispute and try to quantify the error
that is actually introduced when the Doppler velocity is assumed by making a comparison with
observational data. In this way, we may determine the z-domain over which the assumption is
acceptable.
We can initially make a rough theoretical calculation and show that for small z, the two
formalisms are equivalent. Equating the Doppler and general relativistic derivations of z:
.
)(
)(
1
1
1
r
e
ta
ta
z
c
v
c
v
=+=
−
+ (14)
The size of the universe at emission time is therefore related to the apparent velocity by
.
)()(
)()(
22
22
er
er
tata
tata
cv
+
−
= (15)
If we let a(tr) –a(te) = de << a(tr) then
,
)(
eo
r
e
r
e
dH
t
d
ta
cd
v ≅≅≅ (16)
¶
Robertson in the same reference derives the (1+z)4
effect in special relativity as discussed in our Section
2.
Redshift and Energy Conservation
- 7 which
is the Hubble law. We have not been particularly careful with the proper distance or even
the current value of the Hubble constant (simply equating it to the inverse of the current age of
the Universe) but the approximate equivalence at low z is clear.
We are more interested in what is happening at high z where a clear divergence is expected
and this will be investigated by considering the relationship between the apparent brightness
and redshift of type Ia supernovae. The procedure followed is to take the same set of
supernovae currently being used to determine the form factor of the Universe in FriedmannRobertson-Walker
cosmology9, 10
and derive a Doppler velocity from each measured z using
equation (2a). The photon-travel distance¶
, or retarded distance dr, is found by dividing the
apparent or luminosity distance dL derived from the apparent magnitude by a single power of
(1+z) to correct for the photon energy reduction and dilution effects (the source is treated
special relativistically). We are then interested in the relationship between v and dr. At the low
velocity limit, it should be reduce to the linear Hubble law.
If the data is corrected for host galaxy extinction and the K-correction is applied, the
luminosity distance (in units of Mpc) is obtained using the magnitude-distance formula:
[ ] [ ] 25)1(5255 ++=+=− zdLogdLogMm rL
(17)
M is the absolute magnitude and m is the apparent magnitude. Riess and his team9
have
provided both new and recalculated m-M (distance modulus) values for a large number of
supernovae. Knop and the Supernova Cosmology project team10
give only the apparent
magnitude. Data from both groups is used in our analysis, specifically:
High-Z Supernova Search Team9
(a) A set of 23 low z supernovae with K, galactic, stretching and host-galaxy extinction
corrections applied. Column (e) in Table 5 of the reference.
(b) A set of 24 recalculated supernovae data from 1995 to 1997 with K, galactic, stretching and
host-galaxy extinction corrections applied. Column (d) in Table 4 of the reference.
(c) A set of 10 HST supernovae from 1997-2000 with K, galactic, stretching and host-galaxy
extinction corrections applied. Column (d) in Table 3 of the reference.
Supernova Cosmology Project10
(d) A set of 46 supernovae where z > 0.600 with m-M calculated. K and host-galaxy extinction
corrections applied. Table 5 of the reference. Note that a ‘zero calibration’ on the distance
modulus was not performed.
There is a certain amount of overlap (SN 1995ax, SN 1996cl, SN 1887R, and SN1997ap)
that illustrates how the data analysis of each group results in a different final value. The data is
plotted in Figure 3. A small MB-MV correction has been applied.
¶
The reason for using the photon travel time is that the normal Doppler effect relies only on the relative
velocity at emission and reception. For consistency, an equivalent expansion velocity should rely on the
parameters that relate to the photon emission and absorption. In the case of the cosmological expansion, the
significant parameter is time. We do not perform a second division by (1+z) because the rest separation is
not considered relevant.
Redshift and Energy Conservation
- 8 -
Redshift and Energy Conservation
- 9 To
model the data, the most straightforward try is to extrapolate the Hubble law relationship
and assume the function is linear over the entire domain (although there is no compelling
reason why this should be the case). In Figure 3, this proposition that the velocity is
proportional to the retarded distance, dr, is tested against the dataset. The current value of Ho is
taken to be 1/T (T is equivalent to the tr of equation (14)). The Hubble relationship converted
to interaction time is therefore
T
tTc
v e )( −
= (18)
where the age of the Universe is taken to be 13.7 Gyr old (the value that is used for T in the
equation). This is shown as the straight line in Figure 3. The two points that fall on the
extremities of this straight line are easily identified: (0,0) - the velocity is 0 at distance 0; (1, -
13.7Gyr) - this is data from the CMBR – the equivalent Doppler velocity is almost c and the
retarded time is approximately the age of the Universe.
Equation (18) actually represents a good fit for the data up to v/c = 0.1, but supernovae at v/c
= 0.6 (z = 1) appear about 30% further than expected. The prediction is almost identical to that
of the Friedmann equation with ΩM=1, ΩΛ=0. Of course, it was this same failure of the
Einstein–de Sitter Universe at high z that required the cosmological term to be reintroduced
into standard cosmology.
Figure 4 A comparison of the special relativity (SR) Doppler model and standard cosmology.
The results should be compared to Figure 5 of Davis and Lineweaver37
who place the SR line 23
standard deviations below the ΛCDM concordance fit (0.3, 0.7) and thus exclude SR as a viable
cosmological model.
The peak absolute magnitude is taken to be –19.33 as estimated by Wang14
(who gives a Vband
magnitude of –19.33 ± 0.25). We can conclude that it is legitimate to represent the
cosmological expansion with an equivalent Doppler velocity up z = 1 with the caveat that there
is a significant error beyond z = 0.1.
The only adjustable factor in this special relativistic treatment of cosmology is the form of
equation (18). It must reduce to the standard Hubble law for te ~ T, and v should not exceed c as
te → 0 (only general relativistic cosmology permits the recession velocity to exceed the speed
of light). Although we will not investigate further, a modification to the special relativistic
Hubble law is justified as equation (18) is not relativistic, and there are many suitable functions
that could be explored to enhance the fit (following the precedent of the modification to the
basic Friedmann equation to match high redshift data).
Redshift and Energy Conservation
- 10 Davis
and Lineweaver37
attempt to demonstrate that equations (2a) and (18) do not match
observational data. They correctly highlight the difficulty relating the recession velocity to
observational features, but plump for the same relationship as our equation (18) here. However,
the plot of the special relativity prediction (their Figure 5) is inconsistent with Figure 3 and
appears to be incorrect§
. Figure 4 reproduces their calculation with the normalised luminosity
distance HodL/c converted to familiar units of magnitude. The special relativity expression (as
described above) is (1+z)[(1+z)2
-1]/[(1+z)2
+1]. The standard model curves use the expression
z+z2
/2[1-ΩM/2+ΩΛ] from Perlmutter38
. It is clear that the (1,0) case and special relativity are
very similar.
Of course, equation (18) has to be largely consistent with all types of observational data, not
just distant supernovae. As a further example we can consider the variation of angular width of
standard rods with redshift. This is important because proper distance is more difficult to
reconcile with the retarded frame used here. For this reason we would expect differences
between the Doppler and general-relativistic model predictions.
Consider a ‘standard rod’ of length l = 10/h pc¶
(measured with rod and observer at relative
rest). There is some doubt as to whether standard rods genuinely exist but some data on radio
sources is available15
and whilst the scatter is large, an averaging process can produce data
against which the reasonableness of a cosmology can be tested (although it is unlikely a
specific cosmology can be selected).
The angular width in the stationary emission frame is
ed
l
=φ , (19)
where de is proper separation distance. Following a velocity boost (in the Doppler formalism),
the observer maps the stationary circumference 2πde to the circumference 2πde(1+z) onto which
objects are projected hence the apparent length is scaled by a factor (1+z). The distance is also
scaled by (1+z) so the apparent angular width remains unchanged. The distance de can be
derived from observables using a modified equation (18): de = vT/(1+z). Converting the
angular measurement units from radians to mas, and taking l to be 14.0 (= 10/0.714) pc,
.
/
)1(671.0
cv
z+
=φ (20)
This function is plotted in Figure 5 below along with the binned data from 330 compact
radio sources gathered by Gurvits, Kellermann and Frey15
.
§
It is probable the (1+z) term in the special relativity model to convert from effective to luminosity
distance has been neglected.
¶
Note that h is derived by expressing the Hubble constant as 100 h km s-1
Mpc-1
, the value being 0.714 for
a Universe of age 13.7 billion years
Redshift and Energy Conservation
- 11 Figure
5 The red line shows the apparent angular width of a standard rod of length 14 pc as a
function of redshift. The match with the data is similar to that of the FRW model.
The fit is reasonable. Equation (20) can be rewritten
.
)1)1((
)1)1)((1(
2
2
−+
+++
=
z
zz
oφφ (21)
There exists a minimum (dφ/dz=0) at z = 1.06. In contrast, the FRW cosmology functions
are typically of the form
zz
z
+−+
+
≈
1)1(
)1( 2
φ (22)
with a minimum at z = 1.25. This is shown for comparison in Figure 4. Of course, for the poor
quality of data available, the expressions cannot be distinguished and the equivalence between
the Doppler and general relativistic formalisms still holds.
5. The Cosmological Redshift and Energy Conservation
The cosmological expansion appears not to conserve energy. Photons are received with a
lower energy than when emitted and because the source appears to be moving away from all
the distant absorbers, there is no obvious mechanism by which energy can be conserved – all
photons lose energy. Ideas involving a variation in gravitational energy or zero-point energy as
space expands tend to fail because the expansion is determined by matter density and to balance
the energy loss we would have to postulate local variations in expansion based on photon
density. Whilst these ideas are very appealing and evolve fascinating space-time topologies,
they ultimately fail to satisfy energy conservation requirements.
Exploiting the working equivalence between a Doppler velocity and the expansion and using
equation (18), a previous paper16
argued that energy could be conserved from the viewpoint of
the emitter if a distant absorber subject to the cosmological expansion were attributed an
‘Hubble Energy’. It was shown that this could give rise to anomalous accelerations similar to
that experienced by the Pioneer probes. In this paper, we will instead consider the situation
from the viewpoint of the absorber and explore the possibility that the supposed energy loss
associated with the cosmological redshift can be dealt with in a manner analogous to the
Doppler shift described earlier.
Redshift and Energy Conservation
- 12 Consider
an object subject to the cosmological expansion. The radiation power loss is
proportional to the luminosity L and the observer distance. If we consider the source
‘surrounded’ by a set of observers on the surface of a sphere of radius r, the total integrated
power loss is zL/(1+z). The apparent momentum gain at the source is L/c(1+z) away from the
observer. The assumption is made that the momentum associated with all the photons can be
added in a scalar fashion, even though they are emitted in different directions – this means that
even if the output is beamed, the cosmological momentum change is the same for all observers
regardless of the angle of orientation with respect to the beam. By analogy with the Doppler
equation, we can postulate an increase in the recession velocity equivalent to the momentum
change. Consideration of equation (8) shows this procedure conserves both energy and
momentum. The increase in cosmological velocity, per second, is
).1(
)1( c
v
Mc
L
zMc
L
v −=
+
=∆
γ
(23)
All velocities are assumed to be cosmological. We can get an idea of the accumulated effect
by assuming the source has had the same mass-to-light ratio from T = 0 up to the time the
photon was emitted, Te. Multiplying ∆v by Te and using equation (18), the accumulated increase
in velocity is
.)1(
280 12 −
−
Π
= skm
c
v
vl
(24)
The subscript l indicates an ‘luminosity’ velocity which gives rise to a ‘luminosity redshift’.
The value of To is taken as 13.7 Gyr and Π is the mass-to-light ratio in units of M /L .
This type of crude calculation would be appropriate to relatively near galaxies. The mass-tolight
ratio by galaxy type is remarkable constant. Faber and Gallagher17
give estimates for a
range of galaxy morphologies. For example, the value of Π for a type Scd galaxy is 3.9. Slotting
this into equation (24) and assuming v/c is small, the increase in velocity is about 72 km/s. S0galaxies
by comparison give a value of about 28 km/s. This might offer an explanation for the
tendency of smaller galaxies (Sc I) to have a systematically larger redshift18
. Note that the
effect cannot be related to the observation that young, blue stars to have an anomalous
redshift19
– stars in our galaxy do not have a cosmological velocity component therefore there
can be no luminosity velocity component. It should be noted that luminosity velocity
component can decay with time as the galaxy attracts non-luminous matter such as hydrogen
gas or dark matter – the velocity drops through conservation of (cosmological) momentum:
m1v1 = [m1 + δm][v1 - δv] where δm and δv are both positive.
To summarise, energy and momentum conservation require that distant objects have an
additional cosmological velocity component proportional to their integrated luminosity. The
effect on galaxies is small but measurable and is generally distance independent. The effect
would show up as a discrepancy between the redshift and Tully-Fisher or supernova calculated
distances.
The additional velocity cannot be associated with to a physical change in position (radial
distance) otherwise the ordering of events can be changed in an acausal way. It therefore
follows that the source is at the position indicated by the cosmological element of the recession
velocity and that this additional luminosity component be intrinsic to the observer (not the
source). In what sense can it therefore be considered a velocity? This question is left open for
the moment. Remember the luminosity component represents energy conservation and the
actual mechanism is unimportant – any mechanism with these conservation properties will have
the same observable effect.
6. Cosmological Redshift and Quasars (QSOs)
Redshift and Energy Conservation
- 13 For
some systems it more appropriate to consider the total energy loss over the lifetime of
the source rather than on the basis of small time intervals. Let M be the original total mass of
the source, and ξ the fraction of mass eventually converted into energy. The total increase in
intrinsic velocity is of the order
).1(
1 c
v
c
vl
−
−
=
ξ
ξ (25)
For a compact or extended source (i.e. a star or galaxy) powered by nuclear reactions, ξ has
a value of about 0.001 and the redshift change is of the same order (consistent with the normal
galaxy mass-to-light ratio applied to equation (24)). The situation is very different in the case
of an accreting black hole (BH). The power generation mechanism is then gravitational and ξ
can then be identified as the radiative efficiency, a factor whose value is typically believed to
be about 0.1 but can approach 1, the latter case being where all incident mass is converted to
energy. As an example, if we consider a BH accreting with a radiative efficiency of 0.9, and
with a cosmological value of z of 0.1 (based on distance), the measured redshift would be z ≅ 3.
It is therefore possible that active galactic nuclei may have a large redshift associated with
luminosity mixed in with the cosmological redshift¶
.
The most visible evidence of galactic core activity is QSOs. There has been a long running
dispute over the nature of QSOs and whether they are really located at the redshift distance.
The main points of each side of the argument will be considered and we will judge whether the
notion of a luminosity redshift is consistent with observational data and can help resolve the
dispute with the standard cosmological model§
.
The first point that strongly favours the standard interpretation of AGNs in general and
QSOs in particular is that there exists a reasonable theory that not only explains many of the
observations but actually unifies apparent disparate objects into a single schema20
. However,
the observational data available is limited, and the model has not been rigorously tested. It is
rather difficult to make redshift comparison over the full extent of an active galaxy: the jet
frequently associated with AGNs is often the most visible component but generally produces
radiant energy by synchrotron radiation - there is no spectral information from which to obtain
a redshift. Certainly the redshift can be extracted from the core emission, the source being the
central accretion zone, but only occasionally can light from the host galaxy be resolved. This
fact that the host galaxy can sometimes be resolved is again strongly suggestive of QSOs being
truly distant objects. On the occasions when resolution is possible, there is no example of the
host galaxy having a different redshift to the central active region. This indicates there is little
gravitational redshift associated with emission from around the central black hole and severely
restricts alternative explanations. A QSO lifetime of 106
– 108
years is estimated21
, activity
presumably beginning as gas coalesces to form the galaxy and finishing when the nearby matter
has been consumed. The supply of additional material through galactic collisions and mergers
is thought to reactivate the process later in the lifetime of a galaxy.
We can immediately discount alternative models that consider all QSO redshift to be
intrinsic with the sources actually stars in the galaxy - the idea presumably arises from the
similarity to jets from young stars and micro-quasars in the galaxy. The idea fails because the
source distribution is generally isotropic with no correlation with the galactic plane. Arp has
suggested an alternative mechanism for QSO activity based on observed QSO-galaxy
associations. In Arp’s model22
, QSOs are ejected from parent galaxies, initially with a high
intrinsic redshift, but with the redshift declining with time as the quasar evolves into a normal
¶
Note that it is not just the active nucleus that has the additional redshift. It affects all the bound mass,
including the host galaxy. The conservation redshift cannot be determined by comparing the host and core
redshifts.
§
Care is taken to avoid the definition of an intrinsic redshift as the value is relative to the observer and is
thus not strictly intrinsic.
Redshift and Energy Conservation
- 14 galaxy.
The idea is given some theoretical credence by the notion of particles being created at
galactic cores with initially zero mass but acquiring their normal mass over time. Part, but not
all the redshift is intrinsic by this model.
Comparing the standard and Arp’s idea, an HST survey of 20 near-QSOs23
has resolved the
host galaxies. Most galaxies are normal and do not show evidence of the mergers that would be
expected to trigger late QSO activity. However many had galactic companions close by (at
about the same redshift), which is largely consistent with Arp’s model, but the authors
interpreted the associations as a merging rather than diverging process. Luminous QSOs are
associated with a young host stellar population24
and there is a tendency for the host to be
brighter than normal galaxies, again in accordance with Arp’s observations. Recent attempts25
failed to resolve host galaxies of quasars at z ~ 2. This would be expected from Arp’s model
where these are considered very young objects that have not yet formed galaxies.
One problem with Arp’s model is that if this is the way galaxies are produced then all QSOs
should be associated with galaxies. In fact the anomalous associations affect only a very small
proportion of objects. Is it reasonable to expect there to be two very different mechanisms for
quasar production to be at work? Another problem is that the supporting theory of mass
creation is far removed from established physics with no rationale for the process in the
standard particle model or indications of it from high-energy experiments. As such there is no
corroborating evidence.
However Arp’s observations do appear to be reasonable and should not be dogmatically
rejected, and in fact the limited anomalous data is readily incorporated into a standard model
augmented with luminosity redshift. We may consider one process whereby galaxies may form
to show how the anomalies might arise. Active nuclei eject huge volumes of material through
the jet and counter jet often present. As the ejecta is slowed by the interstellar medium, it is
natural to assume that gravity will draw the matter together. We would expect a BH to rapidly
form and efficiently accrete, appearing as a QSO. Peripheral gaseous material is gravitationally
attracted but not initially gravitationally bound; the mass-to-light ratio of the luminous source
will be core dominated and therefore very low, resulting in an anomalously large redshift (by
equation (25)). As peripheral material loses kinetic energy through collisions, the bound mass
will increase (without a corresponding increase in energy output) hence the mass-to-light ratio
will fall. We might therefore expect galaxies to begin as QSOs with an initially high anomalous
redshift that decays to that of a normal older galaxy over time as the body of the galaxy forms
and stars begin to shine - perfectly consistent with Arp’s observations. The short duration of
QSO activity would indicate that it contributes little to the overall luminosity redshift over the
lifetime of the galaxy and we would expect to see little residual redshift in older nearby
galaxies. A second phase of QSO activity would not result in a significant anomalous redshift.
By this process, we might expect to observe a small number of quasars caught in the early act
of galaxy creation with a significant anomalous redshift but the vast majority in the later stage
of quasar activity or a later triggering with little or no anomalous redshift. This is consistent
with observation and for the most part QSOs will be at the redshift distance.
This is very speculative of course, but the notion of galaxies ejecting material, the ejection
material forming galaxies which in turn eject, and so on, may help explain the large-scale
structure of clusters and super-clusters. In this scenario the collective streaming motion of
galaxies is the residual velocity of the jet from which the cluster formed. The QSO peak at
around z=2 and the varying intrinsic luminosity can presumable be modelled in a manner
analogous to standard growth-supply equations.
Other evidence has been presented for QSOs being at the redshift distance. One of the best
examples is a study by Qin, Xie, Zheng and Liang to compare the emission and absorption lines
of 400 quasars26
. They found that all quasars had one or more absorption lines with a lesser
redshift. This is interpreted as absorption by intervening galaxies or gas clouds and is strong
evidence for QSOs being at the redshift distance. The analysis does not and cannot disprove the
possibility of a fractional non-cosmological redshift in a small number of cases. As with all
Redshift and Energy Conservation
- 15 QSO
surveys, the analysis turned up a surprise – 16% of the QSOs had absorption lines at a
greater redshift then the emission line. The maximum z difference was 0.1450 and the mean
was 0.0137. One of the few ways to account for this is to postulate that QSOs have a high
proper velocity relative to the local neighbourhood, presumably as a result of the material
ejection. The QSOs displaying this effect will be those moving towards the observer, the
percentage being amplified by the selection effect.
Following this line of reasoning, it should not be surprising to find QSOs associated with
superluminal motion and proper motion. In the standard model, QSOs are not supposed to
exhibit proper motion (and indeed this assumption forms the basis of an effective search
technique), but if they are indeed associated with galactic ejection, limited proper motion might
be expected in many QSOs, even accepting the huge distance. An analysis of 580 radio sources
from 1980 to 200227
concluded that most exhibited proper motion, one as large as 0.4 mas/yr
but typically about 0.05 mas/yr¶
. At z=2, where the redshift is cosmological, 0.05 mas/yr
corresponds to 2.6 light years per year. This is more than expected, but the paper does not make
it clear how the QSO core is distinguished from ejected material (which is known to emit at
radio frequencies and move superluminally) from which the radio energy may really originate.
QSO output is known to vary significantly over periods as short as hours. If there is no
evolution of variability, one would expect the time scale to show a (1+z) dependence. This is
true even if part of the redshift is a result of the conservation mechanism described - the special
relativistic dilation and (1+z)4
dimming are derived from the nett velocity. The finding of
Hawkins28
that there appears to be no (1+z) dependence is at odds with the standard model and
the conservation modification. A larger survey of 25,000 quasars from the Sloan Digital Sky
Survey29
also failed to detect a (1+z) dependence.
Gravitational lensing is a strong indicator that QSOs are at the redshift distance (e.g. QSO
0959+561). In many cases, the intervening lensing galaxy or matter concentration has been
detected.
One other intriguing aspect of QSOs is the claim that the redshift is quantized. There are two
schools of thought; the first believe that the redshift difference between galaxies and nearby,
presumably ejected, QSOs is a multiple of 0.06. This seems to be discredited by an independent
survey that searched for a Log(1+zdifference) periodicity in 1647 QSO-galaxy pairs identified in
the 2dF quasar survey30
and found none. Bell proposes a much more complex scheme involving
sums of series (but still based on the z value 0.062) and is supported by reasonable evidence31,
32
.
If is incredible to think that quantization could be possible. The observed redshift may have
a continuous cosmological component, a proper motion component (continuous because of the
angle of orientation with respect to the observer) and possibly even a gravitational
component33
. Added to the mixture is the possibility of a luminosity redshift component. How
can quantization possibly occur? Rather than reject the possibility out of hand, we can postulate
that quantization is possible only if one of the elements is quantized and dominates all other
factors. The cosmological and proper motion components cannot be quantized. It is difficult to
accept that the gravitational redshift is significant because the core and host galaxy would have
different redshifts so the last possibility is the luminosity redshift. Its value depends on BH
dynamics, growth and accretion rates and possible preferred rates may occur, but the problem is
that the luminosity redshift is not intrinsic but relative to the observer. For this reason, we must
be dubious that it can offer an explanation for the claimed quantized redshift observations.
A redshift – apparent magnitude plot of 64,866 QSOs, BLac objects and Active Galaxies
taken from an online catalogue34
is shown in Figure 6. There is no obvious indication of
¶
The observations in the reference are in the range z = 0 – 4.4. These observations will be subject to the
angular width relation of Figure 4 and with the assumption of a consistent proper motion could form
another test of the cosmological models.
Redshift and Energy Conservation
- 16 quantization,
but it does highlight the problem many of the dissenters have with the standard
model of QSOs. From z = 0.4 to z = 3, the apparent magnitude and the dispersion is uniform.
There is no hint of the diminishing brightness associated with Hubble’s law and obeyed
reasonably well by galaxies and supernovae. Note that the diagram includes a (1+z)4
flux
reduction hence the absolute magnitude increases dramatically with redshift. Presumably, the
critics question the coincidence of the z-independent apparent magnitude but standard theory
explains this by evolution and it is consistent with the notion of galaxies formed from ejecta –
the volume of material is divided with each step of the cycle and hence the luminosity will
decline. As such, we might find any luminosity redshift mixed in with many QSOs at the actual
redshift distance with huge luminosity and simply add to the distribution but the factor is
probably too small to significantly skew the distribution.
Although there are many questions, the standard model is the best description of active
nuclei currently available and is generally adequate. There are significant anomalies, and while
they represent only a fraction of the complete evidence, an explanation is still required for each
and in the most part this are not forthcoming. The introduction of a luminosity redshift helps to
explain some of the anomalous data but by no means all. An analysis of the detailed properties
of BHs and accretion during galaxy formation is required to make detailed quantitative
predictions with which the idea can be properly tested. In conclusion, we will cite the case of
galaxy NGC 7603 and its surroundings to show that the dissenters do have some fair
ammunition against the standard model and the objections will not simply go away¶
.
Galaxy NGC 7603 (z = 0.029) was investigated by Sharp in 198635
. An extended spiral arm
ends on a smaller, higher surface brightness galaxy NGC 7603B with a redshift of 0.057. It was
commented that the simplest explanation is of interaction, but because of the disparate redshifts
this is clearly impossible. Sharp concluded that it is a chance projection effect. A more detailed
investigation by López-Corredoira and Gutiérrez in 200436
showed two faint galaxies with z =
0.245 and z = 0.394, one at each end of the narrow connecting filament. The filament had a
redshift consistent with that of the main galaxy over its entire length. The arrangement of the
group is striking because of the agreement with the notion of galaxies forming at knots in jets,
initially with a high redshift because of energy conservation.
Figure 6 A plot of the apparent magnitude against redshift for all the objects in the Véron-Cetty & Véron
catalogue of QSOs and active galaxies (11th
Edition, 2003)34
. All the 2dF and the first part of the SLOAN
survey results are included. The plateau arises from the 21 magnitude sensitivity limitation of some of the
surveys, and the weak striation is a rounding effect.
¶
There are many examples give by Arp, E M Burbridge, G R Burbridge, Hickson and many others of
apparently anomalous redshift associations between galasies or QSOs and galaxies. There is a considerable
literature on the subject.
Redshift and Energy Conservation
- 17 -
7. Conclusion
In this paper the issue of energy conservation associated with the normal Doppler shift of
light was examined in some detail and was found to hold. The cosmological redshift was
examined and it was shown that it could be treated as a Doppler effect arising from an
equivalent velocity. Having established a working equivalence, an attempt was made to reestablish
energy conservation with the cosmological expansion by treating it like a Doppler
effect with an additional recoil ‘cosmological velocity’. The new redshift may explain some
reported anomalies in the redshift of QSOs.
References
1
Misner W, Thorne K S, Wheeler J A, “Gravitation”, W H FREEMAN, SAN FRANSCISCO
(1973)
2
Harrison E, Ap. J. 403, 28 (1992)
3
Carlip S, Scranton R, Mod. Phys. Lett. 14, 71 (1999)
4
Bak D, Cangemi D, Jackiw R, Phys. Rev. D 49, 5173 (1994)
5
Muirhead H, “The Special Theory of Relativity”, MACMILLAN PRESS, LONDON (1973)
6
Saathoff G et al, Phys. Rev. Lett. 91, 190403 (2003)
7
Lachièze-Rey M, Gunzig E, “The Cosmological Background Radiation: Echo of the Early
Universe”, CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE (1999)
8
Bennett C L et al, Ap. J. S. 148, 1 (2003)
9
Riess A G et al, Preprint astro-ph/0402512v2 (2004)
10
Knop R A et al, Preprint astro-ph/0309368 (2003)
11
Robertson H P, Z. Astrophys. 15 69 (1938)
12
Narlikar J V, “Introduction to Cosmology”, CAMBRIDGE UNIVERSITY PRESS,
CAMBRIDGE (1993)
13
Longair M S, “The physics of background radiation” in The Deep Universe, ed. Binggeli B,
Buser R, SPRINGER-VERLAG, BERLIN (1995)
14
Wang Y, Ap. J. 536, 531 (2000)
15
Gurvits L I, Kellermann K I, Frey S, A. & A. 342, 378 (1999)
16
Macleod A, Preprint physics/0403124 (2004)
17
Faber S M, Gallagher J S, Annu. Rev. Astron. Astrophys. 17, 135 (1979)
18
Clark S, “Towards the Edge of the Universe: A Review of Modern Cosmology”, JOHN
WILEY & SONS, CHICHESTER, p109 (1997)
19
Trumpler R J, Weaver H F, “Statistical Astronomy”, UNIVERSITY OF CALIFORNIA
PRESS, LOS ANGELES (1953)
20
Robson I, “Active Galactic Nuclei”, ”, JOHN WILEY & SONS, CHICHESTER (1996)
Redshift and Energy Conservation
- 18 -
21
Martini P, “QSO lifetimes” in Coevolution of Black Holes and Galaxies, ed. Ho L C,
CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE (2003)
22
Arp H, Preprint astro-ph/9812144 (1998)
23
Bahcall J N, Kirhakos S, Saxe D H, Schneider DP, Ap. J. 479, 642 (1997)
24
Kauffmann G et al, M. N. R. A. S. 346, 1055 (2003)
25
Croom S M et al, Ap. J. 606, 126 (2004)
26
Qin Y-P, Xie G-Z, Zheng X-T, Liang E-W, Preprint astro-ph/0005006 (2000)
27
MacMillan D S, Preprint astro-ph/0309826 (2003)
28
Hawkins M R S, Ap. J. 553, 97 (2001)
29
Vanden Berk D E et al, Ap. J 601, 692 (2004)
30
Hawkins E, Maddox S J, Merrifield M R, M. N. R. A. S. 336, 13 (2002)
31
Bell M B, Preprint astro-ph/0403089v1 (2003)
32
Bell M B, Comeau S P, Preprint astro-ph/0305060v1 (2003)
33
Corbin M R, Ap. J. 447, 496 (1995)
34
Véron-Cetty M-P, Véron P, “A Catalogue of Quasars and Active Nuclei”, 10th
Edition (2003)
WEB LINK: http://www.obs-hp.fr/www/catalogues/veron2_11/veron2_11.html
35
Sharp N A, Ap. J. 302, 245 (1986)
36
López-Corredoira M, Gutiérrez C M, A. & A. 421, 407 (2004)
37
Davis T M, Lineweaver C H, Publications of the Astronomical Society of Australia, 21 97
(2004)
38
Perlmutter S, Schmidt B P, Preprint astro-ph/0303428 (2003)
Redshift and Energy Conservation
- 19 Appendix
A
The Cosmic Microwave Background Radiation (CMBR)
The CMBR can be given a Doppler interpretation in a way that is consistent with the
observation that the transformed blackbody source remains a blackbody when viewed today.
Referring to Figure 2, the cosmic background clearly cannot be treated as a point source as it
surrounds the observer. The picture has to be reversed. Previously the source was at the centre
and the observers were on the surface of the sphere. In this case, the observer is in the centre
and the emitters are on the surface. When there is no relative velocity, the emitters appear on a
continuous surface of a sphere of radius r. The moving observer is centred in a circle of radius
(1+z)r, assuming a special relativistic treatment of the recession velocity. The radiance or
surface flux is the total energy emitted in unit time from unit surface area at the emission
'surface'in the normal direction (units: Js-1
m-2
). For a black body, the radiance is found to be
related to the surface temperature by Stefan’s Law:
.4
TR σ= (A1)
This is the actually the integral of the blackbody energy density spectrum:
.
1
.
8
)( 5
−
=
kT
hcT
e
dhc
d
λ
λ
λ
π
λλρ
(A2)
σ is Stefan’s constant and the integration of equation (A2) gives
,
15
2
32
45
hc
kπ
σ = (A3)
as has been experimentally established.
If the observer is assumed stationary with respect to the CMBR surface of last scattering, the
measured radiance at a detector is the same as the surface radiance because the observer is
bathed in the equilibrium photon gas. If the observer is moving at velocity v with respect to the
scattering surface we can show that the blackbody spectrum of temperature T is transformed to
another blackbody spectrum with a lower temperature T/ (1+z).
The measured radiance will be reduced by (1+z)4
because of the reduction in photon energy
and photon number and the reduced energy density at the emitting surface (the last scattering
surface is projected onto a greater surface area hence the surface density of emitters is reduced,
not exactly the same argument as for the point source). The measured energy density between
wavelengths λ and λ+dλ is
.
)1(
8
)1(
)1(8
)1(
1
)1()1( 5
)1(
)1(5
5
54
)1(
)1/(
−
=
−
+
+
=
++
=
+
+
+
+
kT
hc
zkT
zhc
Tz
zT
e
dhc
e
dzhc
zz
d
z
d
λλ
λ
λ
λ
λ
πλ
λ
πλρ
λρ
(A4)
This calculation shows that the observed energy is that of a black body of temperature T.
The energy is consistent with having originated from a blackbody of higher temperature (1+z)T
Redshift and Energy Conservation
- 20 receding
from the observer at the velocity extracted from z as in equation (2a). The result is
consistent at all wavelengths ¶
.
Appendix B
Anisotropic Doppler Source
It is worth examining if it is possible to violate energy conservation by creating a source that
is not isotropic. If, for example, a mirror is placed behind the source and accelerated with it
(Figure B1): Is energy still conserved?
Figure B1 Light reflected from moving mirror
Note first that if the mirror is located at position B, stationary, then to observer A, looking at
the source through the mirror will see the source moving away at velocity v, and thus with the
same energy that an observer at B would measure. Note that the photons will exert a force on
the mirror through their momentum, but we will assume the mass is so great that no energy is
transferred. If this mirror at position B (or any place between B and the source) is
instantaneously accelerated to velocity v, an observer at, for example, position C will see the
source moving towards then through the mirror. The photons that were emitted with a red shift
should be measured to have a blue shift. The concept of the mirror ‘doing work’ on the photons
is fraught with conceptual difficulties hence we will simply conduct a kinematic analysis
(energy and momentum conservation). The analysis differs from the analysis of the source
because the rest mass of the mirror is not altered – it can gain or lose momentum and kinetic
energy – that’s it. Differentiating the energy equation for the mirror (equation (7)), we obtain
vv
dp
dE
mirror
mirror
mirror
== (B1)
Consider that the incident photon of energy Erest /γ(1-βcosϕ) which hits the mirror and
emerges with energy E’, imparting the change of momentum dp on the mirror in the process.
In the x direction, we can work out the final energy and the change in momentum
,
)cos1(
' mirror
rest
dE
E
E −
−
=
ϕβγ
(B2)
¶
The transformation is consistent with Stefans law and also with Wien’s Law which states that λmax * T is a
constant.
Redshift and Energy Conservation
- 21 -
.
cos'
)cos1(
cos
c
E
c
E
dp rest
photon
ϕ
ϕβγ
ϕ
+
−
= (B3)
Rearranging equation (B2) and dividing by the negative of equation (B3),
.
)cos1(
'
cos
)cos1(
'
v
E
E
c
E
E
dp
dE
rest
rest
mirror
mirror
=
−
+
−
−
=
ϕβγ
ϕ
ϕβγ (B4)
This simplifies to
.
)cos1(
'
ϕβγ +
= restE
E (B5)
This is consistent. The mirror is slowed down supplying KE to increase the photon
frequency. The emitter appears to be moving towards the observer at C when viewed through
the mirror. With ϕ =π, the change of a red shift to a blue shift is consistent with this viewpoint.
Specifically, the emitter appears to be moving away from the observer when viewed through
the mirror. Note that the mass of the mirror is not relevant to the calculation. There is a tiny
momentum transfer in the y direction because the incident and reflected photon energies are not
the same, but the effect cancels over the whole apparatus. Looking again at Figure B1, if the
mirror is part of the moving assembly, it is clear the now directed output beam will slow down
the assembly ensuring energy conservation. There is approximately –2p of momentum
associated with the mirror, +p associated with the escaping photon and +p from the recoil of the
original emission which has been neglected from the calculation by presuming Erest
incorporates any energy loss associated with the emission recoil. Note also the important
kinematic difference between absorption and reflection. You may be concerned that the photon
wavelength changes without being absorbed by the mirror but we can consider it to be
associated with the path length and how it changes.
This is all relevant to the idea of the ‘solar sail’, a propulsion concept to extract energy from
solar photons. There has been some dispute as to whether the concept is feasible but it clearly
is: the mirror- emitter apparatus must recoil with a balancing momentum to that of the direct
photons. Probably the source of confusion is when we consider the situation from rest frame of
the source. The mirror is initially stationary with respect to the emitter therefore the photon
reflected back must have the same frequency as that emitted (emitter looks stationary in the
mirror hence no redshift). There is no energy transfer so how can the mirror ever begin to
move? The energy transfer from a single ‘standard’ photon of energy Eγ as a function of mirror
velocity is
.
2
γE
cv
v
E
+
=∂ (B6)
Initially the mirror velocity is 0 hence no energy is transferred. The problem is related to
Zeno’s paradox of motion and the resolution is to recognise that motion is not a continuous but
a discrete process. Substituting equation (B6) into equation (8) and noting that the change in
internal energy is 0 for a reflection process we obtain
.
2
γE
cv
p
+
=∂ (B7)
The efficiency of the mirrored solar sail doubles as the velocity approaches c. For the
alternative of a solar sail made up of perfectly absorbing material the equivalent expression is
Redshift and Energy Conservation
- 22 -
.
1
γE
c
p =∂ (B8)
The change in internal energy is not 0 in this case hence an absorbing sail will heat up and
radiate isotropically but with no effect on the nett velocity. If a reflector is placed in the
forward direction of the absorbent surface, some of the radiation can be used to increase trust
but the efficiency will only approach, not exceed, the performance of a perfect mirror as
defined by equation (B7). Note that as the velocity approaches c, an absorber and a mirror
become equally efficient.
As a final thought on mirrors, consider an observer sending out photons to an orthogonal
mirror moving radially at velocity v (the positive direction being towards the observer). The
kinematic calculation following the lines of the ones above shows that the emitted and received
wavelength are related by
.
)1(
)1(
emittedreflected λ
β
β
λ
−
+
= (B9)
Effectively the observer appears to himself to be moving at a velocity v plus v added
relativistically : 2v /(1+v2
/c2
). Substituting this velocity into equation (1), with ϕ = 0 gives
equation (B9). Note that if the mirror is moving transversely, v is 0 and from equation (B1) the
energy change at the mirror must be 0 – there is no redshift; because the photon path is not
altered.