MMNRAS , 1–6 (2019) Preprint 6 August 2019 Compiled using MNRAS L A TEX style file v3.0
A sceptical analysis of Quantized Inertia
Michele Renda, (cid:63) Departament 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, formerlyknown with the acronym MiHsC, as presented by McCulloch (2007, 2013). Two majorflaws were found in the original derivation. We derive a discrete black-body radiationspectrum, deriving a different formulation for F ( a ) than the one presented in theoriginal theory. We present a numerical result of the new solution which is comparedagainst the original prediction. Key words: dark matter – galaxies: kinematics and dynamics – cosmology: theory
The discrepancy between observed galaxies rotation curvesand the prediction using the known laws of orbital kinemat-ics was initially observed by Rubin et al. (1980), and it isnow an accepted phenomenon.Several theories were developed to justify such discrep-ancies, such as the existence of a dark matter halo (Rubin1983), or the existence of a Modified Newtonian Dynamics,MoND (Milgrom 1983) at galactic scales.As today, no direct evidence of dark matter wasdetected, though many experiments such as XENON100(Aprile et al. 2012) and SuperCDMS (SuperCDMS Collab-oration et al. 2014) are looking for signal candidates. Somemodels support the idea that dark matter particles could becreated 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 a (cid:29) a m a a when a (cid:28) a [ N ] (1)This approach has been criticized due to the require-ment of an arbitrary parameter a and because it does notpredict 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’ rotationcurves. 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- (cid:63) E-mail: [email protected] nomena like the Pioneer anomaly (McCulloch 2007), theflyby anomalies McCulloch (2008), the Em-drive (McCul-loch 2015), opening the way for propellant-less spacecraftpropulsion (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 byeq. (7), are the absence of arbitrary tunable parameters (be-ing based on universal constants like the Hubble constantand the speed of light), its simple formulation and the widerange of phenomena it seems to explain. Its main weak pointis the fact it assumes the existence of the Unruh radiation(Unruh 1976), which is still not experimentally measured innature, although some recent simulations seems to confirmits existence (Hu et al. 2019).Quantized Inertia has collected some criticisms by main-stream press (Koberlein 2017), but, as today, no criticalanalysis was published in a peer-reviewed journal on thissubject.In the next sections we will present our analysis ofQuantized Inertia: in section 2 we will perform a brief re-capitulation of the theory as presented by McCulloch (2007,2013), in sections 3 and 4 we will present two major flawswe found in its derivation and we propose some correctionsand finally, in section 5, we will present our considerationsabout the validity of the whole theory.
Quantized Inertia states two important affirmation:(i) There exists a minimum acceleration any object canever have: a = c / Θ = × − m s − (see McCulloch 2017,sect. 2). Below such a value, the object’s inertia becomes zero © a r X i v : . [ phy s i c s . g e n - ph ] A ug M. Renda causing the object’s acceleration to increase to the minimumvalue.(ii) Inertia is caused by the Unruh radiation imbalancebetween the cosmic and the Rindler horizon (McCulloch2013, fig.1).The rationale behind the first point is this: according tothe Unruh radiation law (Unruh 1976), every acceleratingobject will feel a background temperature: T = (cid:126) a π ck [ K ] (2)where (cid:126) is the reduced Planck constant, c the speed of lightin vacuum, k the Boltzmann constant and a the object’s ac-celeration. It is important to notice that this temperatureis very tiny: for an object acceleration of − , the tem-perature will be T ≈ × − K , making very difficult anyexperimental detection.Planck’s law states such background will emit black-body radiation with spectrum: b λ ( λ, T ) = hc λ e hc / λ kT − [ W m − sr − m − ] (3)with a peak wavelength: λ peak = hca kT [ m ] (4)where a ≈ .
965 114 231 74 is the solution of the transcen-dental equation ( − e − x ) = x and h is Planck’s constant.We would like to remark that the Planck’s law describes an unconstrained system at equilibrium. This is the case in aclassic black-body radiation experiment where the radiationwavelength is much smaller than the cavity size, leading toa 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 Hubbledistance. We introduce so the Hubble diameter defined as: Θ = cH ≈ . × m (5)where H is the Hubble constant and c is the speed of lightin vacuum. In this case we can not consider the spectracontinuous any more, because, by principle, only the wave-lengths fitting twice the Hubble diameter can ever exist: λ n = Θ n where n = , , . . . ∞ [ m ] (6)The minimum acceleration arises from two phenomena:for lower accelerations, the object experiences lower Unruhbackground radiation due to a) the shift of the spectra be-hind the Hubble diameter and b) a more sparse sampling ofthe black body radiation spectra, as shown in fig. 1.The consideration expressed above were used to define aHubble Scale Casimir-like effect: using a not better specified direct calculation , McCulloch (2007, sect. 2.2) affirms thereis a linear relation between the continuous and the discretesampling of the Unruh radiation spectra. The ratio betweenthe two sampling, denoted with F ( a ) , is considered to be In this paper we assume H = . ± . × − s − , as used byMcCulloch (2007, sect. 2.1), for easier results comparison withthe original papers. Figure 1.
Plot of the Unruh radiation spectra for different objectaccelerations. The marks represent the λ peak for the given accel-eration. The last dashed line on the right represents the biggestwavelength fitting inside twice the Hubble diameter. a linear function. Assuming that for λ peak → we havethe classical case ( F = ), and for λ peak → Θ no Unruhradiation is sampled ( F = ), this relation was proposed byMcCulloch (2007): m I = F ( a ) m i = (cid:32) − βπ c a Θ (cid:33) m i [ kg ] (7)where β = / a ≈ . , a is the acceleration modulus, m i isthe classic inertial mass and m I the modified inertial mass. We focus our attention on the derivation of eq. (7), basedon the linear relation: m I = F ( a ) m i = B s ( a ) B ( a ) m i [ kg ] (8)where B s is the sampled (discrete) black body radiance and B is the classical one. The value of B can be found integratingeq. (3): B ( T ) = ∫ ∞ b λ ( λ, T ) d λ = π k h c T [ W m − sr − ] (9)while the determination of B s is more complex and will bediscussed in section 3.2. If we have a cubic cavity with side L , we can have an infinitenumber of independent radiation modes. Each mode can be MNRAS000
Plot of the Unruh radiation spectra for different objectaccelerations. The marks represent the λ peak for the given accel-eration. The last dashed line on the right represents the biggestwavelength fitting inside twice the Hubble diameter. a linear function. Assuming that for λ peak → we havethe classical case ( F = ), and for λ peak → Θ no Unruhradiation is sampled ( F = ), this relation was proposed byMcCulloch (2007): m I = F ( a ) m i = (cid:32) − βπ c a Θ (cid:33) m i [ kg ] (7)where β = / a ≈ . , a is the acceleration modulus, m i isthe classic inertial mass and m I the modified inertial mass. We focus our attention on the derivation of eq. (7), basedon the linear relation: m I = F ( a ) m i = B s ( a ) B ( a ) m i [ kg ] (8)where B s is the sampled (discrete) black body radiance and B is the classical one. The value of B can be found integratingeq. (3): B ( T ) = ∫ ∞ b λ ( λ, T ) d λ = π k h c T [ W m − sr − ] (9)while the determination of B s is more complex and will bediscussed in section 3.2. If we have a cubic cavity with side L , we can have an infinitenumber of independent radiation modes. Each mode can be MNRAS000 , 1–6 (2019) sceptical analysis of Quantized Inertia defined by three non-negative integers, l , m , n , such that thewave fits entirely in twice the cavity side: λ x = Ll λ y = Lm λ z = Ln [ m ] (10)Using this notation, it is possible to define a new quantitynamed wave-vector defined as: k x = πλ x = π lL k y = πλ y = π mL k x = πλ z = π nL [ m − ] (11)so any wave can be expressed as: A ( r , t ) = A sin ( k · r − ω t ) (12)Using this formalism, we can express each wave in a cav-ity using three non-negative integers, l , m , n : smaller integersrepresent longer wavelengths, while higher values representshorter ones. We can represent these points in a graph, asshown in fig. 2.Every point represents a wave-mode in the cavity: thepoints with the same modulo will have the same energy, or,more concisely, if we define p = l + m + n , for the samevalue of p , we have the same energy. The relation between λ and p now becomes: λ = Lp [ m ] (13)If we want to calculate the energy density, we have tosum the number of wave-modes with the same energy mul-tiplied by their average energy and divide by volume: U ( T ) = (cid:213) p u ( p , T ) = (cid:213) p N ( p ) E ( p , T ) L [ J m − ] (14)where N ( p ) is the number of independent modes with wave-mode p , E ( p ) is the average energy of that mode and thefactor reflects the fact that each wave can have two inde-pendent polarizations. E ( p ) can be found using the Boltz-mann distribution: E ( p , T ) = E p e − EpkT ∞ (cid:205) p = e − EpkT = hc λ p e hc / λ p kT − [ J ] (15)where λ p = L / p and E p = hc / λ p .For the determination of N ( p ) , we need to estimate thenumber of wave-modes for a given energy: for an uncon-strained system, when λ peak (cid:28) L , we can suppose the wave-modes are so dense we can estimate them as the volume ofa shell of a sphere with radius p and thickness d p (as shownin fig. 2a): N ( p ) d p =
18 4 π p d p (16)where the factor / reflects the fact we are only countingone octant ( l , m , n > ). We can now transform eq. (14) into a continuous sum, by frequency ( ν = cp / L ) or by wavelength( λ = L / p ): U ( T ) = ∫ u ( ν, T ) d ν = ∫ u ( λ, T ) d λ [ J m − ] (17)where u ( ν, T ) = π h ν c e h ν / kT − [ J Hz − m − ] (18) u ( λ, T ) = π hc λ e hc / λ kT − [ J m − m − ] (19) (a) Continuous integral(b) Discrete sum Figure 2.
Plane section of the l , m , n volume: each cross rep-resents a wave-mode. Points near to the origin will have longerwavelengths while points with the same modulus will share thesame wavelength and energy. Equations (18) and (19) are often presented in the formof power radiance, which can be found multiplying them by c / π : b ( ν, T ) = h ν c e h ν / kT − [ W Hz − sr − m − ] (20) b ( λ, T ) = hc λ e hc / λ kT − [ W m − sr − m − ] (21) MNRAS , 1–6 (2019)
M. Renda
In section 3.1 we discussed the derivation of Planck’s lawbecause now we will use the same principle to derive a similarequation for constrained cavities, where the wavelength sizesare comparable to the cavity dimensions. This time we cannot transform eq. (14) in a continuous sum, but we have tohandle it as an infinite discrete sum.The value of E ( p , T ) can be found using the Boltzmanndistribution: E ( p , T ) = E λ p = L / p ( λ p , T ) = hc λ p e hc / λ p kT − [ J ] (22)while the value N p are the number of modes where l + m + n = p , as shown in fig. 2b. Unfortunately, this value cannot be calculated analytically but, if we define n = p wecan find the value of N ( p ) in the sequence A002102 (Sloane& Plouffe 1995). Using this definition, eqs. (19) and (21)become, respectively: u s ( p , T ) = N ( p ) L hc λ p e hc / λ p kT − [ J m − ] (23) b s ( p , T ) = N ( p ) L hc πλ p e hc / λ p kT − [ W sr − m − ] (24)where λ p = L / p . Finally, we can find the sum for all themodes as: U s = ∞ (cid:213) p = u s ( p , T ) [ J m − ] (25) B s = ∞ (cid:213) p = b s ( p , T ) [ W sr − m − ] (26) B s and B Using the results from the previous section and eqs. (5), (8)and (13), now it is possible to find a new expression for thefunction F ( T ) : F ( T ) = H h π k T ∞ (cid:213) p = N ( p ) pe hp / H kT − (27)which can be expressed, using eq. (2), as a function of theobject’s acceleration: F ( a ) = π H c a ∞ (cid:213) p = N ( p ) pe p π cH / a − (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 fromeq. (7): while we can observe that for a > × − m s − , F = (classical case), and for a < a , F = , as predictedby McCulloch (2007), but we also have a critical point at a p ≈ . × − m s − where we have a maximum value for F ≈ . .This point would represent a stable point because, if weapply a small force to an object around this critical value,the shape of F ( a ) will stabilize the object’s acceleration. Atthe knowledge of the authors, no such behaviour was evermeasured or predicted theoretically by other models. Figure 3.
Plot of the F ( a ) = B s ( a )/ B ( a ) function for low accel-erations. We can observe that F ( a → ∞) = (classical case) and F ( a → ) = and a peak around a p ≈ . × − m s − . The marksrepresent 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 alongthe direction x , as shown in fig. 4, there will be an imbal-ance between the radiation pressure on the right, limited bythe 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 theclassical inertia. It is shown in McCulloch (2013) (and par-tially corrected by Gin´e & McCulloch (2016)), it is possibleto express the force F as: F = − π hA cV a [ N ] (29)where A is the object’s radiation cross-section, smaller thanthe physical cross-section, V is the object’s volume and a themodulus of the object’s acceleration.It is also shown that, if we assume the particle a cubewith size equal to the Planck’s length, l P = . × − m ,the model predicts an inertial mass of . × − kg , whichis
29 % greater than the Planck’s mass, m P = . × − kg (Gin´e & McCulloch 2016).Our main concern is how the energy density substitutionwas performed in both McCulloch (2013, eq. 9-10) and Gin´e& McCulloch (2016, eq. 7-8): u = EV = hc λ V [ J m − ] (30)In this substitution, the authors imply that only thepeak wavelength of the Unruh spectrum contributes to theenergy densities. In reality, this is deeply incorrect because MNRAS000
29 % greater than the Planck’s mass, m P = . × − kg (Gin´e & McCulloch 2016).Our main concern is how the energy density substitutionwas performed in both McCulloch (2013, eq. 9-10) and Gin´e& McCulloch (2016, eq. 7-8): u = EV = hc λ V [ J m − ] (30)In this substitution, the authors imply that only thepeak wavelength of the Unruh spectrum contributes to theenergy densities. In reality, this is deeply incorrect because MNRAS000 , 1–6 (2019) sceptical analysis of Quantized Inertia Figure 4.
Schematic representation of the Cosmic and Rindlerhorizon 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 willproduce a force against the direction of acceleration. for classical accelerations (i.e. a > × − m s − ), the peakwavelength contributes only a tiny part of the overall energydensity. We can define a new quantity G , expressing thecontribution of the peak wavelength to the radiation energydensity as: G ( a ) = u s ( p (cid:48) , a ) U s ( a ) = N ( p (cid:48) ) p (cid:48) e p (cid:48) π cH / a − ∞ (cid:205) p = N ( p ) pe p π cH / a − (31)where p (cid:48) is the wave-mode nearest to the peak wavelength,which can be calculated as: p (cid:48) = (cid:118)(cid:117)(cid:116) round (cid:32) a a π cH (cid:33) (32)In figure fig. 5, we can see the plot of G ( a ) over a rangeof different accelerations, and we can notice that for classicalaccelerations, eq. (30) it is wrong in principle. In this paper we analysed two main articles (McCulloch2007, 2013) describing Quantized Inertia. We found two ma-jor flaws on the derivation presented, and we propose somecorrections to address the found issues. Such flaws, if theydo not invalidate, at least will require a major rethinkingof the whole theory. In our article, we did not address theability of Quantized Inertia to match the observational data.We consider that speculative physics is fundamental forthe constant progress of science: Quantized Inertia was oftencriticized because it does go against well-established prin-ciples such as the equivalence principle. We consider thisshould not be the criterion used to establish the validityof 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 u s over the entire spectrum. For lower accelerations this value isnear to one because only a few modes are allowed but for higheraccelerations 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 shouldhave a stronger mathematical derivation and, eventually, astrategy 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 thanksthe Bokeh Development Team (2014) for the excellent toolused to create the plots of this paper and the reviewer ofthis article for his meaningful feedbacks and the intellectualintegrity shown during the review process.
REFERENCES
Abercrombie D., et al., 2015, Technical Report FERMILAB-PUB-15-282-CD, Dark Matter Benchmark Models for EarlyLHC Run-2 Searches: Report of the ATLAS/CMS Dark Mat-ter Forum. CERN ( arXiv:1507.00966 )Aguirre A., Schaye J., Quataert E., 2001, ApJ, 561, 550Aprile E., et al., 2012, Astropart. Phys., 35, 573Bokeh Development Team 2014, Bokeh: Python library for inter-active visualization.
Gin´e J., McCulloch M. E., 2016, Mod. Phys. Lett. A, 31, 1650107Hu J., Feng L., Zhang Z., Chin C., 2019, Nat. Phys., pp 1–9Koberlein B., 2017, Quantized Inertia, Dark Matter, The EM-Drive And How To Do Science Wrong, https://tinyurl.com/forbes-quantized-inertia
MNRAS , 1–6 (2019)
M. Renda
Liu J., Liu Z., Wang L.-T., 2019, Phys. Rev. Lett., 122, 131801McCulloch M. E., 2007, MNRAS, 376, 338McCulloch M. E., 2008, MNRAS Lett., 389, L57McCulloch M. E., 2012, Ap&SS, 342, 575McCulloch M. E., 2013, EPL, 101, 59001McCulloch M. E., 2015, EPL, 111, 60005McCulloch M. E., 2017, Astrophys. Space Sci., 362, 57McCulloch M. E., 2018, J. Space Expl., 7, 1Milgrom M., 1983, ApJ, 270, 365Mitsou V. A., 2015, J. Phys.: Conf. Ser., 651, 012023Rubin V. C., 1983, Science, 220, 1339Rubin V. C., Ford Jr. W. K., Thonnard N., 1980, ApJ, 238, 471Sanders R. H., 2003, MNRAS, 342, 901Sloane N., Plouffe S., 1995, The Encyclopedia of Integer Se-quences, 1st edition edn. Academic PressSuperCDMS Collaboration et al., 2014, Phys. Rev. Lett., 112,241302Unruh W. G., 1976, Phys. Rev. D, 14, 870This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000