Reparametrization Invariance and Its Relation to the Dark Matter Phenomenon

Abstract The existence and nature of Dark Matter (DM) remains an enigma. The existence of a non-luminous matter is based on the continuation of the matter paradigm. As such it is hypothesized to be the reason for the flat rotational curves in galaxies. Modified Newtonian Dynamics (MOND) is an alternative view point on the flat rotational curves phenomenon. This work builds upon the recently introduced Reparametrization Invariant Scaling Symmetry (RISS), that introduces scale factor λ ( t ) relevant for cosmic reparametrization, which can be used to remove the Einstein cosmological constant Λ E from the relevant extended equations of General Relativity (GR) and therefore defines the Scale Invariant Vacuum (SIV) gauge for λ ( t ). Furthermore, by insisting on a reparametrization symmetry for the equation of motion, the paper demonstrates how to address the missing mass problem at galactic and extragalactic scales, where unproper (non co-moving) time parametrization gives rise to forces needed for appropriate reparametrization symmetry considerations. This approach leads to the derivation of the MOND fundamental relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>~</mml:mo> <mml:msqrt> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>g</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> within the RISS paradigm, where g is the gravitational acceleration, a 0 is the MOND fundamental acceleration, and g N is the Newtonian gravitational constant. Remarkably, the derived value for a 0 ≈ 10 −10 m/s 2 is found to be consistent with its observed order of magnitude for MOND applications.