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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

ABSTRACT

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

1 INTRODUCTION

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-

erations:

F=

m a when aa0

ma2

a0

when aa0

[N](1)

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-

?E-mail: michele.renda@cern.ch

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

subject.

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.

2 RECAPITULATION ON QUANTIZED

INERTIA

Quantized Inertia states two important aﬃrmation:

(i) There exists a minimum acceleration any object can

ever have: a0=2c2/Θ=2×10−10 m s−2(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

value.

(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:

T=

~a

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 s−2, the tem-

perature will be T≈4×10−21 K, making very diﬃcult any

experimental detection.

Planck’s law states such background will emit black-

body radiation with spectrum:

bλ(λ, T)=2hc2

λ5

1

ehc/λk T −1[W m−1sr−1m−2](3)

with a peak wavelength:

λpeak =hc

a5kT [m](4)

where a5≈4.965 114 231 74 is the solution of the transcen-

dental equation 5(1−e−x)=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:

Θ=2c

H0

≈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:

λn=2Θ

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±0.9×10−18 s−1, 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= 1−βπ2c2

aΘ!mi[kg](7)

where β=1/a5≈0.2,ais the acceleration modulus, miis

the classic inertial mass and mIthe modiﬁed inertial mass.

3 CORRECTION 1

We focus our attention on the derivation of eq. (7), based

on the linear relation:

mI=F(a)mi=

Bs(a)

B(a)mi[kg](8)

where Bsis the sampled (discrete) black body radiance and

Bis the classical one. The value of Bcan be found integrating

eq. (3):

B(T)=∫∞

0

bλ(λ, T)dλ=2π4k4

15h3c2T4[W m−2sr−1](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

cavities

If we have a cubic cavity with side L, we can have an inﬁnite

number of independent radiation modes. Each mode can be

MNRAS 000,1–6(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:

λx=

2L

lλy=

2L

mλz=

2L

n[m](10)

Using this notation, it is possible to deﬁne a new quantity

named wave-vector deﬁned as:

kx=

2π

λx

=

πl

Lky=

2π

λy

=

πm

Lkx=

2π

λz

=

πn

L[m−1](11)

so any wave can be expressed as:

A(r,t)=A0sin(k·r−ωt)(12)

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:

λ=

2L

p[m](13)

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:

U(T)=Õ

p

u(p,T)=Õ

p

2N(p)E(p,T)

L3[J m−3](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:

E(p,T)=Ep

e−Ep

kT

∞

Í

p2=1

e−Ep

kT

=hc

λp

1

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):

N(p)dp=1

84πp2dp(16)

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

(λ=2L/p):

U(T)=∫u(ν,T)dν=∫u(λ, T)dλ[J m−3](17)

where

u(ν, T)=

8πhν2

c3

1

ehν/kT −1[J Hz−1m−3](18)

u(λ, T)=

8πhc

λ5

1

ehc/λk T −1[J m−1m−3](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

c/4π:

b(ν, T)=

2hν3

c2

1

ehν/kT −1[W Hz−1sr−1m−2](20)

b(λ, T)=

2hc2

λ5

1

ehc/λk T −1[W m−1sr−1m−2](21)

MNRAS 000,1–6(2019)

4M. Renda

3.2 Derivation of Planck’s law for constrained

cavities

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

distribution:

E(p,T)=Eλp=2L/p(λp,T)=

hc

λp

1

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:

us(p,T)=2N(p)

L3

hc

λp

1

ehc/λpk T −1[J m−3](23)

bs(p,T)=2N(p)

L3

hc2

4πλp

1

ehc/λpk T −1[W sr−1m−2](24)

where λp=2L/p. Finally, we can ﬁnd the sum for all the

modes as:

Us=

∞

Õ

p2=1

us(p,T)[J m−3](25)

Bs=

∞

Õ

p2=1

bs(p,T)[W sr−1m−2](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):

F(T)=

15H4

0h4

128π5k4T4

∞

Õ

p2=1

N(p)p

ehp /4H0kT −1(27)

which can be expressed, using eq. (2), as a function of the

object’s acceleration:

F(a)=

30π3H4

0c4

a4

∞

Õ

p2=1

N(p)p

epπ2cH0/a−1(28)

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>1×10−8m s−2,

F=1(classical case), and for a<a0,F=0, as predicted

by McCulloch (2007), but we also have a critical point at

ap≈1.20 ×10−9m s−2where we have a maximum value for

F≈2.17.

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(a→0)=0and a peak around ap≈1.20 ×10−9m s−2. The marks

represent the values in which the calculation was performed.

4 CORRECTION 2

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:

F=−π2hA

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 ×10−35 m,

the model predicts an inertial mass of 2.799 ×10−8kg, which

is 29 % greater than the Planck’s mass, mP=2.176 ×10−8kg

(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):

u=

E

V

=

hc

λV[J m−3](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 000,1–6(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>1×10−8m s−2), 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:

G(a)=

us(p0,a)

Us(a)=

N(p0)p0

ep0π2c H0/a−1

∞

Í

p2=1

N(p)p

epπ2c H0/a−1

(31)

where p0is the wave-mode nearest to the peak wavelength,

which can be calculated as:

p0=v

u

tround a5a

π2cH0!2

(32)

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.

5 CONCLUSIONS

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.

ACKNOWLEDGEMENTS

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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