Quantized inertia

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]

Quantized inertia (QI) 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, 2014 and 2019, 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][23][24]

Quantized inertia is directly related to other theories of modified gravity.[25] 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.[26][27]

The theory of quantized inertia has been criticized in articles online as being pseudoscience.[2][28][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.[29]

In 2019, a Romanian particle physicist performed a derivation of quantized inertia in which he claims to have found two errors in McCulloch's original work prior to 2013. He subsequently provides a new derivation showing different predictions.[30]

• Physics from the Edge: Mike McCulloch's blog about quantized inertia.
• quantizedinertia.com: List of peer-reviewed published papers and preprints about quantized inertia.
• McCulloch, Michael Edward (15 September 2014). Physics from the Edge: A New Cosmological Model for Inertia. World Scientific Publishing. Bibcode:2014pecm.book.....M. doi:10.1142/9122. ISBN 978-9814596251.
• Mike McCulloch (March 2018). How quantised inertia gets rid of dark matter | Mike McCulloch | TEDxPlymouthUniversity on YouTube. TEDx talk, University of Plymouth.