Variation of Planck’s quantum of action in Entangled Relativity

Abstract Entangled Relativity is a recent non-linear reformulation of General Relativity that does not include Planck’s constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℏ</mml:mi> </mml:mrow> </mml:math> or Newton’s gravitational constant G in its fundamental structure. One of its key predictions is that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℏ</mml:mi> </mml:mrow> </mml:math> emerges as a dynamical field, potentially varying across space and time. In this study, we estimate the magnitude of such variations in three different astrophysical environments: the weak gravitational fields of the Sun and Earth, the intermediate regime of white dwarfs, and the strong fields found in neutron stars. In the Solar System, the relative change in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℏ</mml:mi> </mml:mrow> </mml:math> is minimal, reaching at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> <mml:mn>2.5</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . In white dwarfs, depending on central density, variations range from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> <mml:mn>7</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . For neutron stars, the variation can be as high as 1.5% at the surface relative to a remote observer, and up to 5.7% at the center. These results suggest that, if Entangled Relativity accurately describes gravity, spatial variations of Planck’s constant could become an observable signature, particularly in the context of dense stellar objects.