# Quantized inertia

Quantized inertia (QI), previously known as the acronym MiHsC (Modified Inertia from a Hubble-scale Casimir effect), is a controversial theory of inertia.[1][2][3][4] The concept was first proposed in 2007 by physicist Mike McCulloch, a lecturer in geomatics at the University of Plymouth,[5] as an alternative to general relativity and the mainstream Lambda-CDM model.[6][7][8][9]

According to McCulloch, quantized inertia would also be able to explain various anomalous effects such as the Pioneer and flyby anomalies,[5][10] as well as controversial propellantless propulsion experiments sometimes called "horizon drives".[11][2][12][13][14][15] In August 2018, planned experiments to test quantized inertia were funded by DARPA with a grant of 1.3 million dollars over a four-year study.[3][4]

## Unruh radiation and horizon mechanics

There is an event horizon in the universe where light (and therefore any information) cannot and will never be able to reach an object, because the cosmic acceleration outpaces the speed of light: the cosmological comoving horizon. If the object accelerates in one direction, a similar event horizon is produced: the Rindler horizon. Anything beyond these horizons is outside the observable universe, and therefore can't affect the object at the center of the Rindler space.

The Rindler event horizon is effectively the same as the event horizon of a black hole, where quantum virtual particle pairs are occasionally separated by gravity, resulting in particle emissions known as the Hawking radiation. For a Rindler horizon produced by an accelerating object, a similar radiation is predicted by quantum field theory: the Unruh radiation. Due to the difficulty of measuring such tiny quantum background radiation seen only from the reference frame of an accelerated object, Unruh radiation has not been definitely observed so far, although some evidence may exist.[16]

Quantized inertia posits that Unruh radiation is the origin of inertia: as a particle accelerates, the Rindler information horizon expands in the direction of acceleration, and contracts behind it. Although being different in essence, this is a macroscopic analogy of the Casimir effect: a non-fitting partial wave would allow an observer to infer what lies beyond the event horizon, so it would not be a horizon anymore. This logical assumption disallows Unruh waves that don't fit behind an accelerating object. As a result, more Unruh radiation pressure (which acts through the volume of the mass, not only on its surface like the electromagnetic radiation pressure) hits the object coming from the front than from the rear and this imbalance pushes it back against its acceleration, resulting in the effect observed as inertia.[17][18]

There is another event horizon much farther away: the Hubble horizon. So even in front of an accelerating object, some of the Unruh waves are disallowed, especially the very long Unruh waves that exist if the object has a very low acceleration. Therefore, quantized inertia predicts that such an object with very low acceleration would lose inertial mass in a new way.[6]

This loss of inertia occurs more gradually than the empirical relation proposed by MOND. In quantized inertia, the inertial mass is modified according to the relation:[5][10][6][7][8][9]

${\displaystyle m_{i}\;=\;m_{g}\left(1-{\frac {\beta \,\pi ^{2}c^{2}}{|a|\,\Theta }}\right)\;\approx \;m_{g}\left(1-{\frac {2c^{2}}{|a|\,\Theta }}\right)}$

where ${\displaystyle m_{i}}$ is the inertial mass, ${\displaystyle m_{g}}$ the gravitational mass, ${\displaystyle \beta =0.2}$ part of Wien's displacement law, ${\displaystyle c}$ the speed of light, ${\displaystyle |a|}$ the modulus of the acceleration, and ${\displaystyle \Theta }$ the diameter of the cosmological comoving horizon.

The minimum acceleration threshold allowed for any object in the universe is then:[9]

${\displaystyle a'\;=\;{\frac {2\,c^{2}}{\Theta }}\;=\;2\;\pm 0.2\times 10^{-10}\,{\text{m.s}}^{-2}}$

An uncertainty of about 0.18 arises from uncertainties in the Hubble constant of 9%.[19]

Using the constant ${\displaystyle a'}$, quantized inertia predicts the empirical Tully–Fisher relation, the rotational velocity of galaxies being:[7][8][9]

${\displaystyle v^{4}\;=\;{\frac {2\,G\,M\ c^{2}}{\Theta }}}$

where ${\displaystyle G}$ is the gravitational constant and ${\displaystyle M}$ the baryonic mass of the galaxy (the sum of its mass in stars and gas).

This relation is in good agreement with available observational data at various scales, without the need to introduce dark matter. Quantized inertia indeed reduces the inertial mass of outlying stars (whose acceleration ${\displaystyle a=v^{2}/r}$ becomes low enough) and allows them to be bound by the gravitational attraction from visible matter only.[7][8][9]

## Comparison with related theories

Quantized inertia is an alternative to the Lambda-CDM model. Among the main differences between them, QI has no free parameter and explains the cosmic acceleration without dark energy,[6] and galaxy rotation curves as well as residual velocities in galaxy clusters without resorting to dark matter.[7][8] As of 2018, two kinds of observations that seem to be incompatible with dark matter are proposed by McCulloch to be explained by quantized inertia:

• Globular clusters: in 2006, ESO researchers confirmed Mordehai Milgrom's main point, i.e. that the dynamics of stars becomes non-Newtonian when their gravitational acceleration drops below a critical threshold of about ${\textstyle 2\times 10^{-10}{\text{m}}/{\text{s}}^{2}}$, however they also showed that such peculiar behavior does not only occur at the periphery of large galaxies but also in much smaller structures such as globular clusters, a phenomenon impossible to explain by dark matter (which has a large and smooth distribution over the whole galaxy).[20]
• Wide binaries: in 2012 and 2014, UNAM researchers published results of the study of a particular type of wide binary star system. When such a pair of stars is separated by more than 7000 AU, so that their gravitational acceleration drops below the threshold of ${\textstyle 2\times 10^{-10}{\text{m}}/{\text{s}}^{2}}$, their behavior also becomes non-Newtonian, i.e. their observed orbital speed becomes so large that the centripetal acceleration should produce centrifugal forces overcoming their gravitational attraction, so that they should separate, but they do not do so. The behavior of such a small system remains unexplainable by dark matter.[21][22]

Quantized inertia is directly related to other theories of modified gravity. Modified Newtonian dynamics (MOND) for example modifies Newton's law with an adjustable parameter ${\textstyle a_{0}}$ whose value is tuned arbitrarily to fit observed intermediate sized systems such as average galaxies (MOND typically implies different values of the parameter ${\displaystyle a_{0}}$ in the range of ${\displaystyle 1.2\times 10^{-10}}$ to ${\displaystyle 2\times 10^{-10}{\text{m}}/{\text{s}}^{2}}$), an empirical relation that however fails with smaller or bigger systems like dwarf galaxies or galaxy clusters. Unlike MOND, the inertial law of quantized inertia does not have a tunable parameter, and better explains the anomalous behavior of globular and galaxy clusters, wide binaries and dwarf galaxies.[9]

The holographic principle in quantum gravity has been suggested by Jaume Giné to provide a potential link between quantized inertia and entropic gravity.[23][24]

## Criticism

The theory of quantized inertia has been criticized in articles online as being pseudoscience.[2][25][3] Some of the problems it was initially proposed to solve have since been solved by conventional physics, in particular the Pioneer Anomaly is explained by thermal recoil from the spacecraft's power source. Furthermore experiments to measure the thrust of resonant cavity thrusters have recorded values much lower than originally predicted that are likely explained by interactions with the Earth's magnetic field.[26]

## References

1. ^ Clarke, Stuart (19 January 2013). "Sacrificing Einstein: the keystone of relativity that just has to go" (PDF). New Scientist. 217 (2900): 32–36. doi:10.1016/S0262-4079(13)60180-3.
2. ^ a b c Koberlein, Brian (15 February 2017). "Quantized Inertia, Dark Matter, The EMDrive And How To Do Science Wrong". Forbes. Retrieved 5 November 2018.
3. ^ a b c Oberhaus, Daniel (2 October 2018). "DARPA is Researching Quantized Inertia, a Theory Many Think is Pseudoscience". Vice Motherboard. Retrieved 2 October 2018.
4. ^ a b c McCulloch, M. E. (21 March 2007). "Modelling the Pioneer anomaly as modified inertia" (PDF). Monthly Notices of the Royal Astronomical Society. 376 (1): 338–342. arXiv:astro-ph/0612599. doi:10.1111/j.1365-2966.2007.11433.x.
5. ^ a b c d McCulloch, M. E. (20 May 2010). "Minimum accelerations from quantised inertia". EPL. 90 (2): 29001. arXiv:1004.3303. doi:10.1209/0295-5075/90/29001.
6. McCulloch, M. E. (December 2012). "Testing quantised inertia on galactic scales". Astrophysics and Space Science. 342 (2): 575–578. arXiv:1207.7007. doi:10.1007/s10509-012-1197-0.
7. McCulloch, M. E. (March 2017). "Low-acceleration dwarf galaxies as tests of quantised inertia". Astrophysics and Space Science. 362 (3): 57. arXiv:1703.01179. doi:10.1007/s10509-017-3039-6.
8. McCulloch, M. E. (September 2017). "Galaxy rotations from quantised inertia and visible matter only". Astrophysics and Space Science. 362 (9): 149. arXiv:1709.04918. doi:10.1007/s10509-017-3128-6.
9. ^ a b McCulloch, M. E. (September 2008). "Modelling the flyby anomalies using a modification of inertia" (PDF). Monthly Notices of the Royal Astronomical Society. 389 (1): 57–60. arXiv:0806.4159. doi:10.1111/j.1745-3933.2008.00523.x.
10. ^ Rodal, José (May 2019). "A Machian wave effect in conformal, scalar--tensor gravitational theory". General Relativity and Gravitation. 51 (5): 64. doi:10.1007/s10714-019-2547-9. ISSN 1572-9532.
11. ^ McCulloch, M. E. (January 2015). "Can the Emdrive Be Explained by Quantised Inertia?" (PDF). Progress in Physics. 11 (1): 78–80.
12. ^ McCulloch, M. E. (1 October 2015). "Testing quantised inertia on the emdrive". EPL. 111 (6): 60005. arXiv:1604.03449. doi:10.1209/0295-5075/111/60005.
13. ^ McCulloch, M. E. (7 July 2017). "Testing quantised inertia on emdrives with dielectrics". EPL. 118 (3): 34003. doi:10.1209/0295-5075/118/34003.
14. ^ McCulloch, M. E. (20 December 2018). "Propellant-less Propulsion from Quantized Inertia" (PDF). Journal of Space Exploration. 7 (3).
15. ^ Smolyaninov, Igor I. (24 November 2008). "Photoluminescence from a gold nanotip in an accelerated reference frame". Physics Letters A. 372 (47): 7043–7045. arXiv:cond-mat/0510743. doi:10.1016/j.physleta.2008.10.061.
16. ^ McCulloch, M. E. (March 2013). "Inertia from an asymmetric Casimir effect". EPL. 101 (5): 59001. arXiv:1302.2775. doi:10.1209/0295-5075/101/59001.
17. ^ Giné, J.; McCulloch, M. E. (2016). "Inertial mass from Unruh temperatures" (PDF). Modern Physics Letters A. 31 (17): 1650107. doi:10.1142/S0217732316501078. hdl:10459.1/58386.
18. ^ Freedman, Wendy L.; et al. (20 May 2001). "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant". The Astrophysical Journal. 553 (1): 47–72. arXiv:astro-ph/0012376. doi:10.1086/320638.
19. ^ Scarpa, Riccardo; Marconi, Gianni; Gilmozzi, Roberto (27 March 2006). "Globular Clusters as a Test for Gravity in the Weak Acceleration Regime". AIP Conference Proceedings. 822 (102): 102–104. arXiv:astro-ph/0601581. doi:10.1063/1.2189126.
20. ^ Hernandez, X.; Jiménez, M. A.; Allen, C. (February 2012). "Wide binaries as a critical test of classical gravity". The European Physical Journal C. 72 (2): 1884. arXiv:1105.1873. doi:10.1140/epjc/s10052-012-1884-6.
21. ^ Hernandez, X.; Jiménez, M.; Allen, C. (2014). "Gravitational anomalies signaling the breakdown of classical gravity" (PDF). In Moreno González, C.; Madriz Aguilar, J.; Reyes Barrera, L. (eds.). Accelerated Cosmic Expansion: Proceedings of the Fourth International Meeting On Gravitation and Cosmology. Astrophysics and Space Science Proceedings, Vol. 38. Springer Science+Business Media. pp. 43–58. arXiv:1401.7063. doi:10.1007/978-3-319-02063-1_4. ISBN 978-3-319-02062-4.
22. ^ Giné, Jaume (10 November 2012). "The Holographic Scenario, the Modified Inertia and the Dynamics of the Universe" (PDF). Modern Physics Letters A. 27 (34): 1250208. doi:10.1142/S0217732312502082.
23. ^ Giné, Jaume (January 2013). "Cosmological Consequences of the Holographic Scenario" (PDF). International Journal of Theoretical Physics. 52 (1): 53–61. doi:10.1007/s10773-012-1298-0.
24. ^ McCulloch, Mike (2017-02-18). "Physics from the edge: My response to the Forbes article". Physics from the edge. Retrieved 2018-12-26.
25. ^ "How The Pioneer Anomaly Was Solved". 17 August 2018. Cite magazine requires |magazine= (help)