Journal of Modern Physics, 2013, 4, 7-19
http://dx.doi.org/10.4236/jmp.2013.47A1002 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Cosmic Illusions
Bernard H. Lavenda
Università degli Studi, Camerino, Italy
Received April 17, 2013; revised May 21, 2013; accepted June 23, 2013
Copyright © 2013 Bernard H. Lavenda. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
A critique of black-hole-black-body radiation, black-hole thermodynamics, entropy bounds, inflation cosmology, and
the lack of gravitational aberration is presented. With the exception of the last topic, the common thread is the misuse of
entropy and, consequently, the second law. Hawking’s derivation of the entropy loss due to black hole emission rests on
Kirchhoff’s radiation law which equates the rates of absorption and emission of energy in any given frequency interval.
Black-body radiation cannot, therefore, be used as a mechanism for black-hole evaporation. A derivation of the Planck
factor from an exponential Doppler shift shows why the temperature cannot be proportional to the acceleration; accel-
erations do not cause Doppler shifts. Inflationary cosmology is based on a misconception that the adiabatic condition of
Einstein’s equations hold, and, yet, there can be an enormous increase in the entropy. The cause for the increase is a
negative pressure which contradicts the thermodynamic definition of positive pressure as the derivative of the entropy
with respect to the volume times the temperature: Increases in volume cause corresponding increases in the entropy. A
first-order phase transition cannot occur under adiabatic conditions, cannot generate entropy, and the latent heat cannot
be used to reheat the universe. Finally, a negative pressure is invoked to explain the absence of gravitational aberration,
assuming that gravity propagates at the speed of light.
It is the only physical theory of universal content which I am convinced will never be overthrown, within the fram-
work of applicability of its basic concepts.
Albert Einstein on Thermodynamics
Keywords: Black-Hole Radiation; Black-Hole Thermodynamics; Entropy Bounds; Inflationary Cosmology;
Gravitational Aberration
1. Introduction and Summary
The motto that history always repeats itself is not always
true. The comparison of Planck’s derivation of the black-
body spectrum and its associated entropy to the deriva-
tion of Hawking radiation and Bekenstein entropy of a
black-hole turns up vast differences. First, and foremost,
Planck always had an experimental verification of his
formulas. Planck’s assumption of the Wien distribution
was contested by the results of Ruben and Kurlbaum in
the long wavelength region, and led him to the search of
a new distribution that would make the spectral density
proportional to the absolute temperature in that region [1].
Moreover, the fitting parameters of that distribution led
to the discovery of not one, but two universal constants,
which along with c, the velocity of light, and G, the
gravitational constant, would be “independent of particu-
lar bodies or substances”, and would “necessarily retain
their significance for all times and for all cultures, in-
cluding extraterrestial and non-human ones”. Planck re-
ferred to them as “natural units”, and would “retain their
natural significance as long as the laws of gravitation and
the propagation of light in vacuum, and the two laws of
thermodynamics retain their validity” [2].
If Planck had thought through the natural units he was
considering he would have realized that they constituted
extremely high energies of 1.22 × 1019 Gev, extremely
short times of 5.4 × 10−44 sec, and extremely short
lengths of 1.6 × 10−35 m. It was only after the general
acceptance of the big bang that such a Planck scale could
have been realized immediately after the bang. On such a
Planck scale, G would be comparable to the other forces
of nature. Since there is no known theory that can probe
this scale, reliance must be made on the rest of what
Planck considered immutable, namely the two laws of
thermodynamics.
Hawking radiation is thermal radiation that is pre-
dicted to be spontaneously emitted by black-holes. It is a
consequence of the steady conversion of quantum vac-
C
opyright © 2013 SciRes. JMP

B. H. LAVENDA 8
uum fluctuations into pairs of particles, one of which is
ejected at the event horizon and escapes to infinity, while
the other is trapped within the event horizon. Since this
radiation reduces the mass of a black-hole, Hawking ra-
diation is said to be responsible for black-hole evapora-
tion.
A black-hole is thought to be the collapse of a star in
which its matter and energy is veiled to an external ob-
server behind an event horizon. Thus, a thermodynamic
description from an observer’s view cannot be based on
the mass and radiation before the black-hole was formed
because they are no longer observable. Associating an
entropy with a black-hole provides a handle on the ther-
modynamics. The question comes to mind: a thermody-
namics of what? Assuming there are many scenarios of
black-hole formation, the microscopic interpretation of
entropy as a measure of the number of complexions that
are all compatible with a single macro-state would quan-
tify the multiple ways in which a black-hole could be
formed. Once a black-hole is formed, information seems
to have disappeared. Since entropy is a measure of miss-
ing information, this would provide yet another reason of
associating entropy with a black-hole.
The second law guarantees that the entropy of an iso-
lated system will tend to increase until it reaches a maxi-
mum. But how large is this maximum? Not wanting to
appeal to statistical mechanical models, it should be pos-
sible to argue from thermodynamic principles how large
this maximum should be. Because maximum entropy
measures information, it should provide a bound on in-
formation capacity.
Entropy has also been implicated in resolving the
seemingly paradoxes associated with the standard model
of cosmology which refers to an adiabatically expanding,
radiation-dominated universe that is supposedly well-
described by a Robertson-Walker metric. The paradoxes
are due to 1) the horizon problem whereby causally dis-
connected regions could evolve into a homogeneous uni-
verse; and 2) the flatness problem whereby the only uni-
verse that could reach a critical energy in the present
epoch would be a flat universe since a closed universe
would have achieve it in Planck’s time while an open
universe would see its energy density rapidly dwindle
away. According to the inflationary scenario, if the adia-
batic assumption would be dropped in the standard model,
these paradoxes would disapper.
In all these cases there is the common thread of using
thermodynamic laws in the absence of experimental veri-
fication. More specifically, it is the second law and en-
tropy which hold center stage. But, the “entropy” used in
the first two cases has no relation to the second law, and,
hence, is not an entropy, and relinquishing the constancy
of the entropy while maintaining Einstein’s adiabatic
equations is a contradiction. It is the purpose of this pa-
per to redress these issues.
A simplified derivation of the black-body spectrum for
black-hole radiation and the Unruh temperature is marred
on, at least, two accounts: 1) acceleration does not cause
a Doppler shift; and 2) the integral of the spectral density
function over all frequencies diverges. Hence, absolute
temperature cannot be proportional to acceleration, in
general, and surface gravity, in particular. The expression
for the entropy of a black-hole violates both the second
and third laws of thermodynamics, and there is no bound
on the entropy-to-energy ratio, precisely because it is a
decreasing function of the energy: The entropy-to-energy
ratio can never be less than to the derivative of the en-
tropy with respect to energy. Black-body radiation can-
not be used as a mechanism of black-hole evaporation
precisely because it obeys Kirrchhoff’s radiation law
asserting that radiation cannot upset a state of thermal
equilibrium [1, p. 64].
Inflation cosmology violates the adiabatic nature of
Einstein’s equations. Since the Robertson-Walker metric
does not predict that the ratio of a circle to its radius will
be greater (less) than 2π for geometries of constant nega-
tive (positive) curvature, it cannot describe non-Euclid-
ean geometries of constant curvature. Thermodynami-
cally derived relations for the energy and pressure are
compared with those obtained from Einstein’s equations.
For space-time with constant Gaussian curvature, Ein-
stein’s equations require the pressure to be negative [3].
In the case of vanishing pressure, they reduce to the virial
theorem, and Newton’s law. Pressure has the effect of
causing deceleration which causes a compression of the
cosmological fluid, which, in turn, increases the pressure,
in violation of Le Châtelier’s principle. In general, nega-
ive pressures, invoked in expansion, must lead to a de-
crease in volume if entropy is to increase so that the
enormous increase in entropy predicted by inflation can-
not occur. The latent heat of a phase transition cannot be
used to re-heat a universe, and a first-order phase transi-
tion does not generate entropy.
Finally, the condition for the absence of gravitational
aberration leads to a negative pressure of exactly the
same magnitude, but of opposite sign, as the relativistic
equations of state of degenerate stars. These equations
show that ratio of the pressure to rest energy density ratio
is always proportional to the ratio of the Schwarzschild
radius to the radius of the star.
2. Do Black Holes Emit Black-Body
Radiation?
Much of modern cosmology is based upon an inexorable
chain of analogies [4]. A case in point is the conclusion
that black holes emit black-body radiation, which implies
that the temperature be proportional to surface gravity.

B. H. LAVENDA 9
A simplified derivation is based on one-dimensional
hyperbolic motion that was derived by Born in 1909, and
a year latter by Sommerfeld. It can be found in almost
any text on relativity [5]. For a particle moving under
uniform gravitational acceleration,
22
,
1
v
vc





00
d
d
at  (1)
will be constant so that integration, with initial conditions
at t, simply gives

0v

sinh ,cv c
22
1
v
at
vc

 (2)
where v is the hyperbolic measure of the velocity.
Rearranging results in

22
dtanh
d1
xat
vcvc
tat c
 

. (3)
Inverting (3) gives

1
tanhvc vc


1
ln .
21
cvc
vc





(4)
An expression for v can be obtained from what
appears as time-dilation,
2
2
1.
v
ddt
c
  (5)
Consider two coordinate systems,
K

v
, which is mov-
ing at a velocity with respect to system
K
. We
know that the two systems will be related by the Lorentz
transformation. If we consider the motion at the origin of
the
K
, its time

will be related to the time of the t
K
system by
2
2
1,
vt
c
  (6)
which expresses the well-known effect of time-dilation:
Clocks in motion appear to run slower. Sometimes, it is
expressed in its infinitesimal form, (5), but, because the
velocity is uniform it integrates simply to (6). Not so in
the case of uniform acceleration!
Since the velocity is not uniform, we must introduce (3)
into (6); integration then gives
 
1
sinh ,at c


022
d
1
tsc
a
as c
 


which, upon inverting, results in

sinh .a c  at  (7)
This is not time-dilation given by (6). In other words,
frequency is to be associated with inverse time, not the
inverse infinitesimal increment in time. Moreover, a
comparison of (2) and (7) leads to the identification
va, and once this identification is made, there results


1
e,
1
ac vc
vc



t
(8)
from (4). This is supposedly is an exponential Doppler
shift caused by uniform acceleration. Nothing could be
further from the truth!
Consider a light signal that is sent out at time o
t,
reflected at time tr, and returns in time tb. The laboratory
time, , is the arithmetic average,

1,
2ob
ttt (9)
while the distance covered is proportional to half of the
time difference,

1.
2bo
rct t (10)
The uniform velocity is the ratio of (10) to (9), vrt

.
Taking the sum and difference of (9) and (10), in turn,
give


1,and 1.
bo
tvct tvct (11)
Taking the product of the two expressions, and then the
positive square root of the two relations give
22
1.
ob
ttvc t (12)
Comparing (12) with (6), we conclude that the proper
time is the geometric mean time,
,
ob
tt   (13)
which can never be superior to the laboratory time, (9),
because of the arithmetric-geometric inequality. This is
mathematical explanation of time-dilation.
Now, divide the former time by the latter in (11) to get
1.
1
b
o
tvc
tvc


Take the square root of both sides and then their
logarithms to give
1
ln ln.
221
b
o
t
ccvc
tvc





(14)
On the strength of (4) and (8), (14) would necessarily
imply
ln ,
2
b
ob
o
t c
aatt t   (15)
which is evidently wrong. The Doppler effect requires a
uniform relative velocity between source and observer,
not one given by (3).

B. H. LAVENDA 10
In fact, (6) is a second-order Doppler shift discovered
by Ives and Stilwell back in 1938. Radiation traveling at
an angle
2
,
2sinh
22
ic
cc
a
aa


to the direction of a moving source has an
observed frequency

22
1.
1cos
vc
vc





The angle

, measured in the direction of the observer,
was set equal to 2, and the conclusion of Ives and
Stilwell was that a moving clock runs slower than a
stationary one.
This is not to say that uniform acceleration, as is
conventionally assumed, does not have an effect upon the
rate of a clock relative to a clock in an inertial system.
Instead of the reflection time being given by geometric
mean, (13), in the presence of uniform acceleration, it
will be given by harmonic mean

1
11
ob
tt
1
2





[5, p.
403]. Thus, clocks will run even slower in a uniformly
accelerating frame than in an inertial frame, as a conse-
quence of the geometric-harmonic mean inequality. How-
ever, this has nothing to do with the Doppler effect which
involves uniform velocities since time-dilation, (6), ap-
pears as a second-order Doppler effect. And if this no
longer holds, there will no longer be a Doppler shift in
general. Therefore, (8) is not an exponental Doppler shift
caused by acceleration. However, let this not deter us
from continuing the derivation of the spectral density of
black-body radiation involving (8).
On the strength of (8), the exponential Doppler shift
between emitted

and observed frequency


would
be
e.
ac 

 (16)
For small times, (16) would reduce to

1,ac   
 (17)
giving the wrong impression that the shift in frequency is
due to acceleration, and not velocity.
To draw an illusionary analogy with black-body radia-
tion, one [6] considers the frequency spectrum

2
e
2e .
ac
ica



de
i 




This is a double exponential integral which can be
obtained in closed form. To this end, set eac y

, and
get
 
e2
2
de ed
2
ac
icaic a
icyy
a
cici
aa







 








1
2
4
e
e.
icay
ca
ca
c
a



 



The relation for Gamma functions,










gives the spectral density as

2
e
21
de e.
e1
ac
ica
i
ca
c
a








 (18)
Apart from the minor point that an integral over all
frequencies of (18) diverges1, one considers


1e 1
ca 

to be a bona fide Planck factor, and this necessitates
defining the absolute temperature as [8]
.
a
Tkc 
a
(19)
Expression (19) is the well-known, and generally ac-
cepted, Unruh temperature, for which an accelerating
thermometer would measure black-body radiation, whereas
a stationary thermometer would measure no temperature
at all!
The temperature, (19), is also the Hawking temperature
[9] when we identify with gravitational acceleration,
g
,
2.
GM
TRkc 
R
(20)
One would therefore conclude that the absolute tem-
perature is proportional to the mass were it not for the
fact that is now the Schwarzschild radius,
2
2GM
Rc . (21)
Introducing (21) into (20) gives
3
4
c
TkGM 
2
. (22)
In contrast to (20), where the temperature appears to
be proportional to the mass M, (22) now has it inversely
proportional to it!
M
Confusing the rest energy, c, with the internal
energy and using the second law,
d14 ,
d
SkGE
ET c
  (23)
one obtains an absolute entropy [10],
2
Pl
2,
E
Sk
E
 (24)
upon integrating and setting the arbitrary constant of
5
Pl
EcG
integration equal to zero, where is the
1The integral,
 
1
0
d1 ,
e1
x
xx








0 exists for

and

[7]. 1




B. H. LAVENDA 11
Planck energy.
Since the entropy is the square of the energy, or mass,
and the mass is proportional to the Schwarzschild radius,
the entropy is proportional to the area,
A
, of the event
horizon. Prior to making the identifications, and in units
where all the universal constants are equal to one, the
infinitesimal Euler relation would read [11]
d,
8
M
dA d




J


(25)
for a neutral, rotating black hole at an angular velocity
, angular momentum
J
. The surface gravity, which
is subsequently set equal to the absolute temperature is
14
M
 .
We will now follow Page’s [12] presentation of
Hawking’s calculation of black-hole emission for free
fields, since it has a hindsight of thirty years. Hawking
found the expected number of particles, n, emitted in a
wave mode of frequency,

, angular momentum, , as j

1,
e1




j
1
2
n
 (26)
where is the absorption probability for an in-
coming wave.
For the entropy of radiation, Hawking took the von
Neumann entropy for the thermal density matrix of each
mode2,

rad ln1 ln nn
n
Sppn   
1ln ,nnn  (27)
since the probabilities are

1.
1
n
n
n
n

n
p
The expected loss in the black-hole entropy from the
emission mode, due losses in energy, n

, and angular
momentum, nj, is

1
bh
Sn 


  

2ln.
n
nn





j (28)
The second law is satisfied since the total change in
entropy from radiation plus angular momentum loss due
to emission is
 
ln1 0,n 
rad bh
1
ln1ln 1
SS S
nn n
  


 



(29)
“with the extreme right inequality being saturated only if
there is no emission, 0n
0
0T
” [12].
“Thus, the Hawking emission from a black-hole into
empty space obeys the second law of thermodynamics,
and it actually produces entropy from all modes with
nonzero emission. This is as one would expect, since the
emission from a black-hole with bh
T into empty
space with

is an out-of-equilibrium process” [12].
Rather than using the second law of thermodynamics,
the effective temperature is obtained from the Boltzmann
factor,
1
0
e,
1
T
pn
pn
 


which upon taking logarithms of both sides yields
1
ln .
n
Tn 




(30)
For a Schwarzschild black-hole, or “s-waves of neutral
particles in any Kerr-Newman geometry”, the Planck fac-
tor becomes


2
1e 1
 

, and “when , so that the
classical incoming wave is totally absorbed by the black-
hole, then
1
bh 2TT




, the Hawking temperature of
the hole. But, otherwise, the effective temperature T
for the mode generically depends on the mode” [12].
Parenthetically, the form of the inverse temperature
given by (30) is incompatible with the black-hole expres-
sion (23). This becomes apparent when we write the av-
erage energy as En

in (30), 
ln 1,
TEE





 (31)
for 1n, in the Rayleigh-Jeans limit. Expression (31)
asserts that the absolute temperature is proportional to
the average energy, and not inversely proportional to it as
(23) claims. Moreover, if we attempted to apply the sec-
ond law to (28) we would not come out with (30) be-
cause (28), or its negative, is not an entropy, but only
part of one. Even worse, (28) has nothing whatever to do
with the Bekenstein expression (24) for the entropy of a
black-hole. For if it did, (31) could be integrated to give
that expression in the same way that (23) is integrated to
give (24).
By the conservation of energy and angular momentum,
the rate that the black-hole loses them is equal to the rate
that radiation gains them. No so for the total entropy,
(29), which gains more than it loses. The rate of change
of black-hole entropy is given as

1
bh
0
d12d
d2
Sn
t

. 





j
dd dETS
(32)
Let us begin our critique with the first law, (25), which
black-hole thermodynamicists would write as
.


J

E
(33)
2Hawking makes allowance for fermions in all his expressions. However,
fermions, or for that matter any particles other than photons, do not
have a pure black-
b
ody spectrum, so the generalization and particle
emission are illusory.
Distinction must be made between the energy, , in a
fixed coordinate system, and the energy,

B. H. LAVENDA 12
,EE
 d
J
 (34)
in a coordinate system rotating with the body [13], in this
case a Kerr black-hole. Since the adiabatic definition of
angular momentum is
,
S
E 






J

.
TdS dJ
,
(35)
it follows from (35) that the differential for the energy in
a rotating coordinate system is
dE (36)
Differentiating (34) and introducing it into (36) gives
(33).
Likewise, the differential for the Helmholtz free en-
ergy,
F
ETS


,SdT d J
.SdT d 
(37)
in a rotating coordinate system is
dF  (38)
while that in a fixed coordinate system is
dF 
J

e
(39)
Kirchhoff made use of the general thermodynamic law
that radiation cannot upset a state of thermal equilibrium.
For black-body radiation, Kirchhoff showed that, at ther-
mal equilibrium and in each frequency interval, there is
an equality between emitted,

, and absorbed, aK


,
radiant energy, where a

is the fraction of absorbed
energy in the frequency interval,

and d

 , and
K
d


is the intensity of radiation in the same inter-
val. This is expressed by Kirrchhoff’s radiation law [1, p.
64]:
.eaK



1a
(40)
Hawking calculates the left-hand side of (40) and
claims that this energy per unit frequency interval is
dumped into empty space with zero temperature, consti-
tuting an out-of-equilibrium process. This is tantamount
to calculating a reaction in the forward direction and
forgetting to equate it with the rate of the reverse reaction
in the law of mass action which secures chemical equi-
librium. Likewise, at thermal equilibrium, the total emis-
sive power must be equal to the total absorptive power.
Kirchhoff was able to extend this equality to each and
every frequency interval by considering cavities made of
different materials. Moreover, Hawking considers a black-
hole as a perfect black-body by setting

, but
fails to take into account the form of the emissive power
K

,

3,
T  
u
4
uK
c


 (41)
dictated by Wien’s displacement law, where

is the
energy density,
K

is the intensity in the frequency
range from

to



, and

is an unkown func-
tion of the single argument, T

. It is precisely the
pre-factor that is missing in (26) that is required in order
to obtain Stefan’s law upon integrating over all frequen-
cies. The cavity may even have perfectly reflecting walls,

0a


u
provided it contains a speck of charcoal dust
that, through absorption and emission, will allow an arbi-
trary distribution of energy to approach the equilibrium
distribution,

. Therefore, Hawking is not calculating
the emission of energy and angular momentum, but,
rather, the equilibrium distribution of the average number
of photons, provided the number of oscillators m

in
the frequency interval from d

to

  is inserted
into the numerator of (26).
The expression used by Hawking to calculate the
entropy of black-body radiation, (27), in the frequency
interval from

to d



is the classical Shannon-
Gibbs entropy that is compatible with Boltzmann statis-
tics, and not the negative binomial distribution of black-
body radiation. At most, (27) accounts for the Wien limit.
The entropy of black-body radiation in the interval from
d

to



is not (27), but [1, p. 77 eqn (2.24)]

rad ln ln
nm nm
Snn m
nm










. (42)
23
8mc   Just as

is the number of Planck oscil-
lators in the frequency interval from d

to



, so
(42) is the entropy density in the same range. It is also
not the change in entropy. This makes (32) blatantly
wrong since an integration over all frequencies does not
give the time rate of change of the black-hole entropy!
In view of the definition of the energy in a rotating
coordinate system, (34), the definition of the change in
the black-hole entropy, (28), is also wrong for it would
require
1
bh 2,SE


  (43)

where

En 

j
d

, is just the average energy in
the coordinate system rotating with the Kerr black-hole.
The expression of the black-hole entropy (28) was
thought to be justified by the fact that it gave back the
first law. But, (43) shows that it gives only part of that
law. Missing is the second expression in (42) which is
the negative of the ratio of the average free energy to
temperature, in the frequency interval

, [cf. eqn (37)],
in the rotating coordinate system, viz.,

ln 1e,
T
FmT





(44)
where


 j


. Expression can easily be verified
by calculating its derivative with respect to the tempera-
ture,
rad ,
FnFEFS
TT T


 

 (45)

B. H. LAVENDA 13
which is (37).
Furthermore, the temperature, (30), can be obtained
directly from the second law which is [1, p. 137]
radrad rad
dd
dd
SSS
E
nnEnT


 
 ln ,
nm
n






R
(46)
since black-body radiation has zero chemical potential.
Whereas the putative expression of a black-hole (28) is
not related to the emissive power of a black-body, the
entropy density (42) is. The appeal to Boltzmann statis-
tics is an artifice since that statistics does not, in general,
apply to black-body radiation as Planck realized. Fur-
thermore, the black-hole entropy, (28), will not give the
black-hole (body) temperature, (30), only (42) will when
the degeneracy factor is properly accounted for.
3. Are There Bounds on Entropy?
Faced with the enormous numerical value of the entropy
(24), Bekenstein [14] attempted to show that (24) was an
upper limit for all entropies. For according to him, “There
is no gap in magnitude between black-hole entropy and
ordinary entropy. This comes about because of the exis-
tence of a hitherto unnoticed upper bound to the entropy-
to-energy ratio of non-black-hole systems of given effec-
tive radius ”,
. (47)
SR
k
Ec

SE
2
E
R
2E S2 E
“For systems of negligible self-gravity, inequality (47)
keeps from growing faster than , a well-known
property of ordinary bodies which is responsible for the
seeming gap between this entropy and black-hole entropy
(which grows as ). However, as one compresses a
body to its gravitational radius, becomes of order
, and can begin growing as thus ‘catching
up’ with black-hole entropy”. No matter what the effect
of gravitational compression may be, it cannot change a
concave function into a convex one.
“Evidently, systems composed of nonrelativistic parti-
cles are not very interesting from the point of view of the
bound” [14]. Therefore, the characteristic radius can be
taken as the relativistic thermal wavelength, RckT

,
so that (47) is actually
d,
d
SS
EE

S
E
k
SE
1k
(48)
on the strength of the second law. Then, the whole
question boils down to is whether is an increasing or
decreasing function of . Because of the universality of
the entropy it cannot be both. According to a theorem on
twice differentiable functions [15], will be
convex in

if or , and concave if
. Hence,

0,

0k
0k 1
11
d,
d
kk
SS
EkE
EE



1k
(49)
precisely because

.
The defining property of the entropy is its concavity [1,
16]. We consider the entropy as a sole function of the
energy, and use primes to denote derivatives. If

d
0and0,
d
ES SES SES
E


 (50)
then SE E E decreases. If 1 and 2 are any two val-
ues of the energy, the concavity of the entropy requires:


   
1212
11,SEE SESE   
01
(51)
for



E
1
EE
. In particular, the slope of a straight line
from the origin to any point on the entropy curve, say ,
cannot be inferior to the tangent of the curve at that point.
We thus set


20SE  in (51), and obtain

and


,
SE SE
EE



E
(52)
where we have divided both sides of the inequality by
EE

. Since

, (52) shows that that the ratio SE
decreases.
As a prime example of where to apply his inequality
(47), Bekenstein considers black-body radiation for which
1
4
44
.
33
SV
EE T




T
(53)
He then realizes that the ratio of entropy-to-energy “can
be made as large as we please simply by lowering T
sufficiently. But in fact the thermodynamic description of
radiation on which [(53)] is based breaks down when
is no longer large compared to the reciprocal of the
characteristic size of the system (typical wavelength not
small compared to cavity size). Boundary effects make
themselves felt. These can be expected to arrest the
growth of SE T
as is lowered further”. On the con-
trary, black-body radiation makes no such demands: the
integral is over all wavelengths from 0 to ∞!
The first expression in (50) is the Massieu transform of
the entropy with respect to energy, which is the product
of the Helmholtz free energy, F, and the inverse tem-
peratue

,


ln 0,ES SF 


 
 
(54)
where is the partition function. For an ideal gas
is a monotonic decreasing function of

, whereas for
black-body radiation is the same decreasing func-
tion of
ln 

,

3
1
ln ,
3V


 (55)

is the Stefan-Boltzmann constant, and V is where

B. H. LAVENDA 14
the volume of the cavity containing the radiation. Ac-
cording to Bekenstein, “the wanted distribution is the ca-
nonical one whose inverse temperature is just the peak
value of SE
ln ,SE
for the system”. Since
(56)



0
ln 0, 
(which is a rearrangement of (54)) an equality in (48),
would be given by the condition that
(57)




since 0max
Unfortunately, there is no finite value of 0
SE 
.

that
would satisfy (57) for black-body radiation where the
logarithm of the partition function is given by (55). If
such a value could be found, (54) would accomodate
both concave entropies, for which , and convex
entropies, for which . Since is a monotonic
decreasing function of
0

F
ln0F

, the conclusion does not
follow that “the problem is thus superfically simple; the
maximal SE is just that

for which the partition
function is unity”.
The second inequality in (50) ensures that the heat
capacity at constant volume is positive, viz.,

1
20.
V
STC

  (58)
Inequality (58) is violated by the Bekenstein entropy
expression, (24), since
2
Pl
40. k
SE
 
(59)
The heat capacity at constant pressure cannot be de-
fined for black-body radiation because black-body radia-
tion is a phase equilibrium insofar as the number of pho-
tons is not conserved [1,17]. The pressure, p, is a sole
function of the temperature and obeys the Carnot-
Clapeyron equation,
d,
d
ph
TT

h
(60)
where is the enthalpy density, or the latent heat of
sublimation. In fact, the Carnot-Clapeyron relation (60)
leads at once to Stefan’s law of black-body radiation
once the thermal equation of state, 1
3
p

, is intro-
duced where EV




, the energy density.
Hawking [9] claims that tiny black-holes will radiate
away at a temperature (20) leading to complete evapora-
tion in an explosion that would result in an intense burst
of photons in the x-ray region. It is well-known that
saturated vapor pressure depends only on the temperature,
and not upon volume. Changing the volume of vapor
pressure at constant temperature will result in evapora-
tion or condensation so as to leave the vapor pressure
constant. This is the physical content of (60). Evapora-
tion would mean the disappearance of the photon phase
leading to a rupture in the phase equilibrium. For the
processes of evaporation or condensation to proceed ir-
reversibly, the pressure of the two phases cannot be equal
but differ by the capillary pressure. An increase in vol-
ume of cavity radiation can be thought of as new radia-
tion being evaporated from the walls of the cavity [1,17].
It is precisely the job of the latent heat h to maintain the
walls at constant temperature. If a black-body would ab-
sorb more radiation than it emits, it would heat up and
the radiation would no longer be (iso) thermal radiation.
4. Does Inflation Violate Einstein’s
Equations?
It has been said that inflation is an elegant way of ex-
plaining why the universe is so homogeneous on the
Hubble scale [18]. In this section, we show the contrary.
The Robertson-Walker (RW) metric,


22222 2
2sin ,
sin
dsR t ddd    (61)

where

Rt
sinrR
, the expansion, or scale, factor, is said to
describe a space that is both homogeneous and isotropic.
The question is: Why is space homogeneous and iso-
tropic? It is this question that inflationary cosmology
hoped to answer. But before we get to that scenario, there
are criticisms to be lodged against the RW metric, (61).
The RW metric (61) is a metric of constant curvature.
Introducing


[19] in (61) gives

2
22222
22 sin ,
1
dr
dsr dd
rR








(62)
2
1R1 where the scalar curvature can be chosen as

,
1

, or 0 for positive, negative, or zero spatial curvature,
respectively. It is well-known that the transition from
spherical to hyperbolic geometry is allow the radius to
become imaginary, . This would transform the
RW metric (61) into
iRR 




2222222
sinhsinh ,dsR t ddd   
sinhrR
(63)
but


, as it is assumed in almost all texts on
cosmology [19, eqn (111.12)].
The hyperbolic measure of distance is well-known to
be given by 1 tanh rR 

, so that when it is intro-
duced into (63) it will give the well-known Beltrami
metric [5, p. 494]
 
22
2222
222
22
d
ddsin d
1
1
rr
srR
rR
.



 


iRR
(64)
With the substitution , the transformation r =
Rtanχ would not give the RW metric, (61). The deviation
from Euclidean geometry can be seen very clearly from

B. H. LAVENDA 15
(64) by considering , and constr2


2
. For then,
the ratio of the periphery of a circle to the radius will not
be , but3
22
22,
rR

1
since 22
1rR
2
sinhRr  . In constrast, for elliptic
space the ratio would be less than ,
22
22,
rR

1
because 22
1rR
iRR
sin  Rr . It is rather curious that
since the metric (61) bears the name of Robertson that
the only place where it is mentioned in his book, Relativ-
ity and Cosmology, is in the forward by Fowler. What
Robinson refers to as the cosmological metric is the
Beltrami metric (64) [3, p. 342, eqn (14.12)], and its
elliptical counterpart, obtained as usual by .
The starting point for the inflationary scenario is the
pair of Einstein equations that have been derived from
the RW metric, (61), which are, quite remarkably, the
same for the Beltrami metric, (64). The expressions for
the energy density [3, p. 372],
2
2
3,
Rk
R



 (65)
and pressure
2
2,
R k
R



2RR
p    (66)
where 2

,p
8Gc   , are given in terms of the cosmic
time derivatives of the expansion factor R, k, the constant
space curvature, and , the cosmological constant.
Between the energy density and pressure there is the
thermal equation of state [20],



(67)
where normally 12 ,
33




,TdS pdV
, the limits being the ultra-
relativistic and non-relativistic non-interacting particle
limits, respectively.
From thermodynamics we have the infinitesimal Euler
relation,
dE (68)
where S is the entropy. Choosing V, T as the independent
variables, differentiating with respect to the volume, V,
and using a Maxwell relation result in [21,22]
.
TV
Ep
Tp
VT









Employing the thermal equation of state, (67), to elimi-
nate the pressure gives linear partial differential equation
of first-order,
.
VT
EVE
TE
TV 

 


 

  (69)
Lagrange [23] reduced the method of solving the
linear partial differential Equation (69) to one of solving
the auxiliary set of ordinary differential equations,
dd d
,
TEV
TEV
 
whose solutions, the characteristic curves, fill the (T, V,
E)-space. The vector,


,,,TV E  v (70)
will be tangent to the family of characteristic curves.
The general solution to (69) is


12
,0,cc  
TV c

where 1

and 2, are the two independ-
ent solutions, and
EV c



is an arbitrary function. The gen-
eral solution can also be written as:

,EV TV
 
 (71)
where

is arbitrary function, which must be further
conditioned by the thermodynamic stability conditions,
e.g.,

cannot be chosen as an inverse function for that
would have energy decreasing with temperature. The
solution (71) to (69) can be represented as a surface in


,,TV E-space. Starting at any point on surface, (71), a
curve can be traced out in the direction of the tangent
vector (70). The surface, (71), can be thought of as being
formed by the family of such characteristic curves.
Returning to (68), we can write it as:
,
E
TdSdEpdVdEdV
V

  (72)
with the aid of (67). It is readily seen that V

is an
integrating factor for (72) since

.VTdSd EV

 (73)
This allows us to express the entropy as


,STV

 (74)
where

is an arbitrary function, again with the caveat
that it must not violate thermodynamic stability criteria.
The Einstein equations, (65) and (66), can be combined
into [3, p. 373]
33
dd
,
dd
RpR
tt
  (75)
3For a history of the uniformly rotating disc in relativity consult in [5,
§9.1]. which, on the strength of (68), is the condition of adia-

B. H. LAVENDA 16
baticity. The fundamental error [24] in both the old [25],
and new [26,27], inflationary cosmologies is to consider
(75) as a separate condition distinct from
3
d0,
dsR t

(76)
where is the entropy density. As can be seen from
(68), (76) is not independent of (75).
s
So given any expression for the energy density

, the
expression for the pressure can be found from (75),
and Einstein’s expressions for the energy density and
pressure are consistent with (75). But, there is a new
element now, the thermodynamic relation (71), which is
independent of time derivatives. Although Einstein’s equa-
tions express the energy density and pressure in terms of
the time derivatives of the scale factor, they must neces-
sarily be compatible with (71) if they are to have any
thermodynamic relevance whatever.
p
3constTR 

Since Einstein’s theory demands an adiabatic universe,
it follows from (74) that
. (77) TV
Imposing this on (71) results in

31 ,
EC
VR
 

C
 (78)
where is a constant. Such a system is said to be
repulsive [21,22], because will not condense:
0.
T
E
V





 (79)
Expression (78) is exactly the solution to the energy
conservation equation,

3,
Rp
R
 


,
kk
ii
uup

 (80)
which is obtained when the energy-momentum tensor,
k
i
Tp

is substituted into the Bianchi identity,
  (81)
0.
k
i
kT
x

 (82)
Thus, the Bianchi identities are consistent with an
adiabatic universe4.
Introducing (78) into (65) gives



31
312
3const.RRkR





0k
(85)
Since the expansion factor depends on time, we must
necessarily set


312 const.RR  

constR
, because (85) must be homo-
geneous of order zero in time. Then (85) reduces to
(86)
The perfect cosmological principle asserts that the uni-
verse looks the same to any observer in motion [3, p.
347]. It requires

., or constRR

constp
, with k = 0.
The latter is the de Sitter model in which an empty
universe expands exponentially. According to (65) and
(66), the energy density is a constant, and the pressure is


 . This equation of state is said to charac-
terize the vacuum [20]. The same conclusion can be
reached by considering solely the cosmological constant
in those expressions, and that is precisely what de Sitter
did [29]. The problem lies with Einstein’s equations for
they cannot accomodate an energy density or pressure in
a stationary universe.
Alternatively, in the standard model, the Hubble pa-
rameter, RR1
H
falls off as t, which means that
RAt
A
is a constant. This requires that we , where
take 1
3
 in (86) which then reduces to , constRR 


which not the virial theorem. Moreover, although the
expansion factor increases in time, it will be decelerating
because 3 4RAt

0
. In short, the condition (86) is
entirely ad hoc since it does not correspond to any
known theorem, apart from


0
, which is the virial
theorem.
The expression for the energy density (65) can be
made to look like a generalization of the relativistic virial
theorem, and would seem to satisfy (86) for


. For
upon introducing
4However, when (81) is introduced into (82) there results,

23
43Mc R   into (65), there
would result
0,
ki
kk
u p
u
xx
 


kk
i
i
kk
T hu
uh
xx
 

 (83)
where h = ε + p is heat content, or the enthalpy, density. It is unclear
how by multiplying a zero-vector like (83) by the 4-velocity uithat it
can it be projected on the direction of the 4-velocity [28]. In any event,
since the 4-velocities satisfy uiui = −1 so that
2
1,
2
GM
RR 

0k
0p
(87)
which is the relativistic virial5. The second of Einstein’s
equations is given by (66), again with , but
this time with
0
ik
i
uu x, the en-
ergy balance condition,

, viz.,
2,
GM
RR 

0p
0,
k
k
kk k
hu p
u
xx x

 

k
i
i
T
u

 (84)
is correct. Yet, the two conditions, (83) and (84) are not one and the
same, for when (84) is introduced into (83) there results
(88)
which is Newton’s law. Therefore, will cause
0.
k
i
ki k
u pp
hu x



k
i
uu xx

 (85)5The nonrelativistic virial would see the constant κ halved.

B. H. LAVENDA 17
deviations from Newton’s law.
Differentiation of (65) with respect to cosmic time ,
and use of the adiabatic condition in the form (80) give
t
 
13.R R

3
66
Rp


  
 (89)
The equation of motion (89) can be written as

213 ,
GM
R
R


 
0
(90)
so that

 appears as a correction term to Newton’s
law. In fact,
2
32
3,
pGM
Rc

  (91)
is equation of state for degenerate matter. The ratio of the
pressure to rest energy density is the same order as the
ratio of the Schwarzschild radius to the radius of a star,
which is of the same order of magnitude as the red shift,
the mass defect, and the deflection of light around white
dwarfs [30]. For normal relativistic stars,

is of the
order , while for white dwarfs it is , reaching
unity in the case of a black-hole. Therefore, (67) can
hardly be considered a thermal equation of state where
6
104 10
12
,
33




2n
.
This is because (91) has nothing to do with the thermal
equation of state (67), although many would contest this
fact [20]. There is a big distinction between the rest
energy density, , where is the number density,
and the internal energy
nmc

. The distinction is blurred by
writing the energy density and pressure as [19, p. 87]
2
22 and 3
1
c
nm p
vc
 

2
22
,
1
nmv
vc
(92)
where the bar means an average over all particles. Then,
in the non-relativistic limit, where terms are retained to
order 22 vc, the thermal equation of state, (67), is re-
produced with 2
3
 . An average over all particles is
confused with an average over a distribution of velocities,
and the rest energy density has been omitted. As the
above orders of magnitude show, the rest energy density
is by far greater than the internal energy. In fact, the
equation of state of (91) is [16, p. 41]
22
2pG
nmc c,
C
T
M
R







(93)
where Cmc

 is the Compton wavelength, and T

is the thermal wavelength. In the non-relativistic limit
2

 and T, while, in the ultra-relativistic
limit,
mkT 
1

 and TckT. In both cases, (93) be-
comes an expression for thermal-gravitational equilibrium.
Finally, using (65) to eliminate the energy density
gives



2
113
2
RR
R
  


13 0
.
The condition for acceleration, therefore, is [18, p. 395]
.

 (94)

Under this condition, (86) shows that the kinetic energy
will increase with the radius instead of decreasing with it
as the virial theorem dictates. Moreover, Le Châtelier’s
principle is violated because (89) implies that due to a
decrease in


Rt , the cosmic fluid will undergo com-
pression, leading to an increase in the pressure which, in
turn, aids the compression [3].
Because of the adiabatic condition, (77), there will be
no type of phase transition envisioned by the inflationary
cosmology. If we impose on (71) that the energy be
extensive, we will get
11 .EVT


 (95)
For the standard model, 1
3
 , so that this model
would identify (95) with Stefan’s law of black-body
radiation. Such a system is attractive and show signs of
wanting to condense [21,22]:
0,
T
E
V






p


(96)
as opposed to our adiabatic system which will not
condense because of (79). Black-body radiation needs a
heat source to keep the walls of the cavity at a constant
temperature. So it cannot occur in an adiabatic universe,
unless part of that universe can be singled out as being in
thermal equilibrium with the rest of the universe. The
same distinction is made between the canonical (iso-
thermal) ensemble which is embedded in a larger micro-
canonical (adiabatic) ensemble. This could explain CMBR,
which would then not be a relic of a big bang.
Likewise, we require to be an intensive variable.
Using (67) to evaluate (71) we obtain
1
11
1.
TV
pT
V








T
(97)
Differentiating (97) with respect to we get


11
d,
d
p
ph
TTT
 

h
T
(98)
where is the enthalpy density, or the latent heat. The
latent heat can be used to create a structural change in the
system at the temperature ; if the volume is varied at
constant temperature the liquid will either condense or

B. H. LAVENDA 18
evaporate so that the pressure will remain constant at that
temperature. Latent heat is required, but it cannot be
used to re-heat the system to just below the critical
temperature after the phase transition ended [25] because
the temperature remains constant. In short, latent heat
cannot heat, so that inflationary cosmology must look for
some other source to reheat the universe. Moreover, en-
tropy “can[not] be generated in any phase transition”
[25].
The Carnot-Clapeyron equation, (98), describes a phase
equilibrium, and as such it violates the adiabatic condi-
tion that (74) be constant. It was first derived by Lord
Rayleigh in an attempt to generalize Stefan’s law to 1

dimensions [1, §6.2]. Thus, the phase transitions which
the early universe was thought to have undergone could
not have happened.
Be that as it may, the inflationary scenario confuses a
first-order phase transition, in which there is latent heat,
with a symmetry-breaking phase transition in which there
is none. In the latter type of transition, discontinuities oc-
cur in the derivatives of thermodynamic potentials like
the specific heat. New inflation [26,27] employs a scalar
field,

, which was at one time displaced from the mini-
mum of its potential, 0


0
(the “false vacuum”), to a
new symmetry-breaking minimum


4
4.T
[20, §8.2]. The
two states have a different symmetry. This is not a phase
transition of the first kind where two different states are
in equilibrium; a symmetry change occurs at the critical
point so it is possible to say in which of the two phases
the system belongs at any point of the transition.
The total energy consists of black-body radiation and
an energy (zero-point) associated with the scalar field,


  (99)
The free energy,
F
ETS, will have the same form
as (71), namely,


,TV
 


const
FV (100)
where is an arbitrary function. Like (71), (100) ex-
presses a dichotomy: Either


0T
., and the system
has a zero-point energy and will not condense, or it is not
constant and the system will condense as . If the
system has a zero-point energy,
0,
V
p
T






showing that it will not condense. The entropy [21],
d,
VV
Fp
SVC
TT
 


C


 



will then be a constant, , or zero, in accordance with
the third law. Yet, the zero-point energy in (99) leads to a
condensation even at absolute zero, for
0
0,
T
E
V









EV
since


. The conclusion to be drawn is that a
zero-point ene rgy in (99) (repulsion) is incompatible with
the simultaneous presence of black-body radiation (at-
traction).
5. Is There Gravitational Aberration?
It is known that the absence of gravitational aberration
requires a diminution in the magnitude of gravitational
acceleration, just as if the electric field is to point in the
direction of the charge, and not in its past position [31].
This is contested by the fact that light propagation is not
collinear with its gravitational force [32], and the delayed
time of propagation of the gravitational force would
cause a couple to form between celestial bodies that
would destroy their otherwise stable orbits. The reason
why the field points to the instantaneous position of the
source is thought to be a consequence of an exact can-
cellation of, at most, linear terms in the velocities. What
is changed is the magnitude of the field, not its direction.
In the case of the Schwarzschild metric, the accelera-
tion will be directed to the instantaneous position of a
moving source of mass
M
and velocity . Thus, no
aberration occurs, but there will be a decrease in magni-
tude of the acceleration by an anount [31]
v
5
2
22
22
11
22
2
1,
GM
R
GMGM R
cR cR
GM GM
RcR





 




to linear order terms in the ratio, vc. In comparison
with (90), the last expression shows that the decrease in
the magnitude of acceleration is caused by a negative
pressure,
2
2
3.
GM
Rc
  (101)
which is the equation of state, (91), with the sign re-
versed. Curiously, the same effect has been attributed to
gravitational aberration which would increase the angular
momentum of an orbiting body thereby causing an ex-
pansion of the universe as a whole [33].
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