6 Milgrom’s Law as a Modification of Classical Dynamics: MOND

Thus, it appears that many puzzling observations, that are difficult to understand in the ΛCDM context (and/or require an extreme fine-tuning of the DM distribution), are well summarized by a single heuristic law. Therefore, it would appear natural that this law derives from a universal force law, and would reflect a modification of dynamics rather than the addition of massive particles interacting (almost) only gravitationally with baryonic matter20. However, applying blindly Eq. 7View Equation to a set of massive bodies directly leads to serious problems [150, 293Jump To The Next Citation Point] such as the non-conservation of momentum. In a two-body configuration, as the implied force is not symmetric in the two masses, Newton’s third law (action and reaction principle) does not hold, so the momentum is not conserved. Consider a translationally invariant isolated system of two such masses m1 and m2 small enough to be in the very weak acceleration limit, and placed at rest on the x-axis. The amplitude of the Newtonian force is then FN = Gm1m2 ∕(x2 − x1)2, and applying blindly Eq. 7View Equation, would lead to individual accelerations |a | = ∘F---a-∕m-- i N 0 i. This then immediately leads to

∘ ------√ --- √ --- ˙p = a0FN ( m1 − m2 ) ⁄= 0 if m1 ⁄= m2, (12 )
meaning that for different masses, the momentum of this isolated system is not conserved. This means that Eq. 7View Equation cannot truly represent a universal force law. If Eq. 7View Equation is to be more than just a heuristic law summarizing how dark matter is arranged in galaxies with respect to baryonic matter, it must then be an approximation (valid only in highly symmetric configurations) of a more general force law deriving from an action and a variational principle. Such theories at the classical level can be classified under the acronym MOND, for Modified Newtonian Dynamics21. In this section, we sketch how to devise such theories at the classical level, and list detailed tests of these theories at all astrophysical scales.

 6.1 Modified inertia or modified gravity: Non-relativistic actions
  6.1.1 Modified inertia
  6.1.2 Bekenstein–Milgrom MOND
  6.1.3 QUMOND
 6.2 The interpolating function
 6.3 The external field effect
 6.4 MOND in the solar system
 6.5 MOND in rotationally-supported stellar systems
  6.5.1 Rotation curves of disk galaxies
  6.5.2 The Milky Way
  6.5.3 Disk stability and interacting galaxies
  6.5.4 Tidal dwarf galaxies
 6.6 MOND in pressure-supported stellar systems
  6.6.1 Elliptical galaxies
  6.6.2 Dwarf spheroidal galaxies
  6.6.3 Star clusters
  6.6.4 Galaxy groups and clusters

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