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We perform an analysis of the derivation of Quantized Inertia (QI) theory, formerly known with the acronym MiHsC, as presented by McCulloch (2007, 2013). Two major flaws were found in the original derivation. We derive a discrete black-body radiation spectrum, deriving a different formulation for $F(a)$ than the one presented in the original theory. We present a numerical result of the new solution which is compared against the original prediction.
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A sceptical analysis of Quantized Inertia
Michele Renda,1?
1Departament of Elementary Particle Physics, IFIN-HH, Reactorului 30, P.O.B. MG-6, 077125, M˘agurele, Romania
Accepted XXX. Received YYY; in original form ZZZ
We perform an analysis of the derivation of Quantized Inertia (QI) theory, formerly
known with the acronym MiHsC, as presented by McCulloch (2007,2013). Two major
flaws were found in the original derivation. We derive a discrete black-body radiation
spectrum, deriving a different formulation for F(a)than the one presented in the
original theory. We present a numerical result of the new solution which is compared
against the original prediction.
Key words: dark matter – galaxies: kinematics and dynamics – cosmology: theory
The discrepancy between observed galaxies rotation curves
and the prediction using the known laws of orbital kinemat-
ics was initially observed by Rubin et al. (1980), and it is
now an accepted phenomenon.
Several theories were developed to justify such discrep-
ancies, such as the existence of a dark matter halo (Rubin
1983), or the existence of a Modified Newtonian Dynamics,
MoND (Milgrom 1983) at galactic scales.
As today, no direct evidence of dark matter was
detected, though many experiments such as XENON100
(Aprile et al. 2012) and SuperCDMS (SuperCDMS Collab-
oration et al. 2014) are looking for signal candidates. Some
models support the idea that dark matter particles could be
created at LHC and efforts in this direction are in progress
(Abercrombie et al. 2015;Mitsou 2015;Liu et al. 2019).
MoND models remove the necessity for dark matter can-
didates introducing a modified law of motion for low accel-
m a when aa0
when aa0
This approach has been criticized due to the require-
ment of an arbitrary parameter a0and because it does not
predict the dynamics of galaxy clusters (Aguirre et al. 2001;
Sanders 2003). A new theory, by McCulloch, proposes a so-
lution to the discrepancies observed in the galaxies’ rotation
curves. This theory, named Modification of inertia result-
ing from a Hubble-scale Casimir effect (MiHsC) or Quan-
tized Inertia (QI), may give a model for the galaxies’ rota-
tion curves (McCulloch 2012) and explain some other phe-
nomena like the Pioneer anomaly (McCulloch 2007), the
flyby anomalies McCulloch (2008), the Em-drive (McCul-
loch 2015), opening the way for propellant-less spacecraft
propulsion (McCulloch 2018). In addition, this theory pro-
vides also an intuitive explanation for objects’ inertia (Mc-
Culloch 2013).
The main strong points of this theory, as shown by
eq. (7), are the absence of arbitrary tunable parameters (be-
ing based on universal constants like the Hubble constant
and the speed of light), its simple formulation and the wide
range of phenomena it seems to explain. Its main weak point
is the fact it assumes the existence of the Unruh radiation
(Unruh 1976), which is still not experimentally measured in
nature, although some recent simulations seems to confirm
its existence (Hu et al. 2019).
Quantized Inertia has collected some criticisms by main-
stream press (Koberlein 2017), but, as today, no critical
analysis was published in a peer-reviewed journal on this
In the next sections we will present our analysis of
Quantized Inertia: in section 2we will perform a brief re-
capitulation of the theory as presented by McCulloch (2007,
2013), in sections 3and 4we will present two major flaws
we found in its derivation and we propose some corrections
and finally, in section 5, we will present our considerations
about the validity of the whole theory.
Quantized Inertia states two important affirmation:
(i) There exists a minimum acceleration any object can
ever have: a0=2c2/Θ=2×1010 m s2(see McCulloch 2017,
sect. 2). Below such a value, the object’s inertia becomes zero
©2019 The Authors
arXiv:1908.01589v1 [physics.gen-ph] 1 Aug 2019
2M. Renda
causing the object’s acceleration to increase to the minimum
(ii) Inertia is caused by the Unruh radiation imbalance
between the cosmic and the Rindler horizon (McCulloch
2013, fig.1).
The rationale behind the first point is this: according to
the Unruh radiation law (Unruh 1976), every accelerating
object will feel a background temperature:
2πck [K](2)
where ~is the reduced Planck constant, cthe speed of light
in vacuum, kthe Boltzmann constant and athe object’s ac-
celeration. It is important to notice that this temperature
is very tiny: for an object acceleration of 1 m s2, the tem-
perature will be T4×1021 K, making very difficult any
experimental detection.
Planck’s law states such background will emit black-
body radiation with spectrum:
bλ(λ, T)=2hc2
ehc/λk T 1[W m1sr1m2](3)
with a peak wavelength:
λpeak =hc
a5kT [m](4)
where a54.965 114 231 74 is the solution of the transcen-
dental equation 5(1ex)=xand his Planck’s constant.
We would like to remark that the Planck’s law describes an
unconstrained system at equilibrium. This is the case in a
classic black-body radiation experiment where the radiation
wavelength is much smaller than the cavity size, leading to
a continuous spectrum.
However, for very low temperatures, the associated ra-
diation wavelength may become bigger than the cosmic hori-
zon, defined as the sphere with radius equal to the Hubble
distance. We introduce so the Hubble diameter defined as:
2.607 ×1026 m(5)
where H0is the Hubble constant1and cis the speed of light
in vacuum. In this case we can not consider the spectra
continuous any more, because, by principle, only the wave-
lengths fitting twice the Hubble diameter can ever exist:
nwhere n=1,2,3. . . [m](6)
The minimum acceleration arises from two phenomena:
for lower accelerations, the object experiences lower Unruh
background radiation due to a) the shift of the spectra be-
hind the Hubble diameter and b) a more sparse sampling of
the black body radiation spectra, as shown in fig. 1.
The consideration expressed above were used to define a
Hubble Scale Casimir-like effect: using a not better specified
direct calculation,McCulloch (2007, sect. 2.2) affirms there
is a linear relation between the continuous and the discrete
sampling of the Unruh radiation spectra. The ratio between
the two sampling, denoted with F(a), is considered to be
1In this paper we assume H0=2.3±0.9×1018 s1, as used by
McCulloch (2007, sect. 2.1), for easier results comparison with
the original papers.
Figure 1. Plot of the Unruh radiation spectra for different object
accelerations. The marks represent the λpeak for the given accel-
eration. The last dashed line on the right represents the biggest
wavelength fitting inside twice the Hubble diameter.
a linear function. Assuming that for λpeak 0we have
the classical case (F=1), and for λpeak 4Θno Unruh
radiation is sampled (F=0), this relation was proposed by
McCulloch (2007):
mI=F(a)mi= 1βπ2c2
where β=1/a50.2,ais the acceleration modulus, miis
the classic inertial mass and mIthe modified inertial mass.
We focus our attention on the derivation of eq. (7), based
on the linear relation:
where Bsis the sampled (discrete) black body radiance and
Bis the classical one. The value of Bcan be found integrating
eq. (3):
bλ(λ, T)dλ=2π4k4
15h3c2T4[W m2sr1](9)
while the determination of Bsis more complex and will be
discussed in section 3.2.
3.1 Derivation of Planck’s law for unconstrained
If we have a cubic cavity with side L, we can have an infinite
number of independent radiation modes. Each mode can be
MNRAS 000,16(2019)
A sceptical analysis of Quantized Inertia 3
defined by three non-negative integers, l,m,n, such that the
wave fits entirely in twice the cavity side:
Using this notation, it is possible to define a new quantity
named wave-vector defined as:
so any wave can be expressed as:
Using this formalism, we can express each wave in a cav-
ity using three non-negative integers, l,m,n: smaller integers
represent longer wavelengths, while higher values represent
shorter ones. We can represent these points in a graph, as
shown in fig. 2.
Every point represents a wave-mode in the cavity: the
points with the same modulo will have the same energy, or,
more concisely, if we define p2=l2+m2+n2, for the same
value of p, we have the same energy. The relation between
λand pnow becomes:
If we want to calculate the energy density, we have to
sum the number of wave-modes with the same energy mul-
tiplied by their average energy and divide by volume:
L3[J m3](14)
where N(p)is the number of independent modes with wave-
mode p,E(p)is the average energy of that mode and the
factor 2reflects the fact that each wave can have two inde-
pendent polarizations. E(p)can be found using the Boltz-
mann distribution:
ehc/λpk T 1[J](15)
where λp=2L/pand Ep=hc/λp.
For the determination of N(p), we need to estimate the
number of wave-modes for a given energy: for an uncon-
strained system, when λpeak L, we can suppose the wave-
modes are so dense we can estimate them as the volume of
a shell of a sphere with radius pand thickness dp(as shown
in fig. 2a):
where the factor 1/8reflects the fact we are only counting
one octant (l,m,n>0). We can now transform eq. (14) into a
continuous sum, by frequency (ν=cp/2L) or by wavelength
U(T)=u(ν,T)dν=u(λ, T)dλ[J m3](17)
u(ν, T)=
ehν/kT 1[J Hz1m3](18)
u(λ, T)=
ehc/λk T 1[J m1m3](19)
(a) Continuous integral
(b) Discrete sum
Figure 2. Plane section of the l,m,nvolume: each cross rep-
resents a wave-mode. Points near to the origin will have longer
wavelengths while points with the same modulus will share the
same wavelength and energy.
Equations (18) and (19) are often presented in the form
of power radiance, which can be found multiplying them by
b(ν, T)=
ehν/kT 1[W Hz1sr1m2](20)
b(λ, T)=
ehc/λk T 1[W m1sr1m2](21)
MNRAS 000,16(2019)
4M. Renda
3.2 Derivation of Planck’s law for constrained
In section 3.1 we discussed the derivation of Planck’s law
because now we will use the same principle to derive a similar
equation for constrained cavities, where the wavelength sizes
are comparable to the cavity dimensions. This time we can
not transform eq. (14) in a continuous sum, but we have to
handle it as an infinite discrete sum.
The value of E(p,T)can be found using the Boltzmann
ehc/λpk T 1[J](22)
while the value Npare the number of modes where l2+m2+
n2=p2, as shown in fig. 2b. Unfortunately, this value can
not be calculated analytically but, if we define n=p2we
can find the value of N(p)in the sequence A002102 (Sloane
& Plouffe 1995). Using this definition, eqs. (19) and (21)
become, respectively:
ehc/λpk T 1[J m3](23)
ehc/λpk T 1[W sr1m2](24)
where λp=2L/p. Finally, we can find the sum for all the
modes as:
us(p,T)[J m3](25)
bs(p,T)[W sr1m2](26)
3.3 Ratio between Bsand B
Using the results from the previous section and eqs. (5), (8)
and (13), now it is possible to find a new expression for the
function F(T):
ehp /4H0kT 1(27)
which can be expressed, using eq. (2), as a function of the
object’s acceleration:
The term N(p)make very difficult any analytical so-
lution of eq. (28), but it is possible to solve numerically,
as shown in fig. 3. We can observe it is different from
eq. (7): while we can observe that for a>1×108m s2,
F=1(classical case), and for a<a0,F=0, as predicted
by McCulloch (2007), but we also have a critical point at
ap1.20 ×109m s2where we have a maximum value for
This point would represent a stable point because, if we
apply a small force to an object around this critical value,
the shape of F(a)will stabilize the object’s acceleration. At
the knowledge of the authors, no such behaviour was ever
measured or predicted theoretically by other models.
Figure 3. Plot of the F(a)=Bs(a)/ B(a)function for low accel-
erations. We can observe that F(a→ ∞) =1(classical case) and
F(a0)=0and a peak around ap1.20 ×109m s2. The marks
represent the values in which the calculation was performed.
In this section, we will discuss the radiation imbalance be-
tween the Cosmic and Rindler horizon. In McCulloch (2013),
it is shown how applying eq. (7) to an object moving along
the direction x, as shown in fig. 4, there will be an imbal-
ance between the radiation pressure on the right, limited by
the cosmic horizon, and the radiation pressure on the left,
limited by the nearer Rindler horizon.
This radiation pressure imbalance will cause a force re-
acting against any acceleration similar in behaviour to the
classical inertia. It is shown in McCulloch (2013) (and par-
tially corrected by Gin´e & McCulloch (2016)), it is possible
to express the force Fas:
48cV a[N](29)
where Ais the object’s radiation cross-section, smaller than
the physical cross-section, Vis the object’s volume and athe
modulus of the object’s acceleration.
It is also shown that, if we assume the particle a cube
with size equal to the Planck’s length, lP=1.616 ×1035 m,
the model predicts an inertial mass of 2.799 ×108kg, which
is 29 % greater than the Planck’s mass, mP=2.176 ×108kg
(Gin´e & McCulloch 2016).
Our main concern is how the energy density substitution
was performed in both McCulloch (2013, eq. 9-10) and Gin´e
& McCulloch (2016, eq. 7-8):
λV[J m3](30)
In this substitution, the authors imply that only the
peak wavelength of the Unruh spectrum contributes to the
energy densities. In reality, this is deeply incorrect because
MNRAS 000,16(2019)
A sceptical analysis of Quantized Inertia 5
Figure 4. Schematic representation of the Cosmic and Rindler
horizon as presented by McCulloch (2013). If the object is ac-
celerating to the right a Rindler horizon is formed on the left,
disallowing some Unruh waves on that side and, consequentially,
a lower radiation pressure. The radiation pressure imbalance will
produce a force against the direction of acceleration.
for classical accelerations (i.e. a>1×108m s2), the peak
wavelength contributes only a tiny part of the overall energy
density. We can define a new quantity G, expressing the
contribution of the peak wavelength to the radiation energy
density as:
ep0π2c H0/a1
epπ2c H0/a1
where p0is the wave-mode nearest to the peak wavelength,
which can be calculated as:
tround a5a
In figure fig. 5, we can see the plot of G(a)over a range
of different accelerations, and we can notice that for classical
accelerations, eq. (30) it is wrong in principle.
In this paper we analysed two main articles (McCulloch
2007,2013) describing Quantized Inertia. We found two ma-
jor flaws on the derivation presented, and we propose some
corrections to address the found issues. Such flaws, if they
do not invalidate, at least will require a major rethinking
of the whole theory. In our article, we did not address the
ability of Quantized Inertia to match the observational data.
We consider that speculative physics is fundamental for
the constant progress of science: Quantized Inertia was often
criticized because it does go against well-established prin-
ciples such as the equivalence principle. We consider this
should not be the criterion used to establish the validity
of a theory: history teaches us that many scientific break-
throughs, encountered, in the beginning, strong resistance
Figure 5. Plot of G(a): it shows the contribution of the highest
usover the entire spectrum. For lower accelerations this value is
near to one because only a few modes are allowed but for higher
accelerations its contribution tends to zero.
from the scientific community because they were against ex-
isting principles. For this reason, it is of fundamental im-
portance that any new iteration of quantized inertia should
have a stronger mathematical derivation and, eventually, a
strategy for a practical experimental verification.
This work was supported by the research project
PN19060104. We would like to thank our ATLAS group col-
leagues for the supportive working environment and in par-
ticular Prof. C˘alin Alexa for his guidance and support dur-
ing the writing of this article. We would like also to thanks
the Bokeh Development Team (2014) for the excellent tool
used to create the plots of this paper and the reviewer of
this article for his meaningful feedbacks and the intellectual
integrity shown during the review process.
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