# Variation of Planck's constant for different astrophysical bodies

Thomas Chehab, Olivier Minazzoli, Aurélien Hees

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Thomas Chehab, Olivier Minazzoli, Aurélien Hees. Variation of Planck's constant for different astrophysical bodies. Rencontres de Moriond 2025, Mar 2025, La Thuile, France. 2025. hal-05068377

HAL Id: hal-05068377

https://hal.science/hal-05068377v1

Submitted on 19 May 2025

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# Variation of the Planck constant near compact objects

Thomas Chehab¹, Olivier Minazzoli ¹,² Aurélien Hees³

OBSERVATOIRE
DE LA CÔTE D'AZUR
UNIVERSITÉ CÔTE D'AZUR

<figure><img src="image_3.png" /></figure>

1: thomas.chehab@oca.eu; Artemis, Université Côte d'Azur, CNRS, Observatoire Côte d'Azur, BP4229, 06304, Nice Cedex 4, France

2: olivier.minazzoli@oca.eu; Bureau des Affaires Spatiales, 2 rue du Gabian, 98000 Monaco

3: aurelien.hees@obspm.fr LTE, Observatoire de Paris, Université PSL, Sorbonne Université, Université de Lille, LNE, CNRS 61 Avenue de l'Observatoire, 75014 Paris, France

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Recent arguments have put forth the idea that a novel general theory of relativity known as **Entangled Relativity** [1,2] which encompasses Einstein's General Relativity as a limiting case while offering a more parsimonious formulation, predicts notable variations in Planck's quantum of action ħ. These variations may notably be significant within the universe's densest objects that are not concealed by an event horizon: white dwarfs and neutron stars.

## I Entangled Relativity

$$ \Theta = -\frac{1}{2\epsilon^2} \int d_g^4 x \frac{\mathcal{L}_m^2(f,g)}{R(g)} \quad \begin{array}{c} \text{Matter density Lagrangian} \\ \text{Ricci scalar} \end{array} $$

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

Equivalent formulation :

$$ \Theta_{eq} = \frac{1}{\epsilon^2} \int d_g^4 x \frac{1}{\kappa} \left( \frac{R(g)}{2\kappa} + \mathcal{L}_m(f,g) \right) $$

<figure><img src="image_7.png" /></figure>

Quantum
of energy
squared

<figure><img src="image_8.png" /></figure>

Spacetime
volume
element

<figure><img src="image_9.png" /></figure>

Dimensionful
scalar field

Comparison with Standard
Model of particle physics

$$ \Theta_{SM} = \frac{1}{c\hbar} \int d_g^4 x \left( \frac{R(g)}{2\kappa_{GR}} + \mathcal{L}_m^{sm}(f,g) \right) $$

By neglecting gravity :

<figure><img src="image_10.png" /></figure>

$$ \kappa \epsilon^2 = c\hbar $$

Here, $ħ$ is not a constant

* Modify the coupling between spacetime and matter

* General relativity = limit of entangled relativity

* More **parsimonious** than general relativity

* Potential bridge between **quantum and gravity**

* Predicts **variation of ħ** at high energy density

<figure><img src="image_11.png" /></figure>

The relative variation of ħ is expected to range from $10^{-6}$ to $10^{-10}$ between the star's surface and a distant observer, depending on the central density [3].

White dwarfs and neutron stars are perfect places to test theories of **modified gravity** because they exhibit a gravitational potential at their surface that is **much stronger** than at the Earth's surface. If a theory predicts a variation of a fundamental constant depending on gravity, it might be observable possibly thanks to **spectroscopy**. Indeed, on white dwarf, numerous photospheric absorption features are detected, making precision studies of white dwarf atmospheres a **new and unique laboratory** for fundamental physics that cannot be replicated on Earth.

In the context of Entangled Relativity, a variation of the fine structure constant α could be explained by a variation of ħ, the Planck constant, as α is defined as follows :

$$ \alpha = \frac{e^2}{4\pi\epsilon_0\hbar c} $$

With :

* e the elementary charge

* ε₀ the vacuum permittivity

<figure><img src="image_12.png" /></figure>

$$ \frac{\Delta\alpha}{\alpha} = -\frac{\Delta\hbar}{\hbar} $$

Assuming a simple degenerate Fermi gas in a **Tolman-Oppenheimer-Volkoff** simulation, this modelisation has been made for **250 different central densities**, from $10^9$ to $10^{13}$ kg/m³.

Current constraints $|\frac{\Delta\alpha}{\alpha}|$ < a few $10^{-5}$ [4]

## III Variation of ħ - Neutron star

We modeled neutron stars using a **generic polytropic equation of state** in a **Tolman-Oppenheimer-Volkoff** simulation. Here is shown results for densities between **100 and 2000 MeV/fm³**. We retained only those where the speed of sound remains **below** the conservative limit of $c/\sqrt{3}$.

In entangled relativity, neutron stars are predicted to be **systematically heavier** than in general relativity [5]. The **variation of ħ** is expected to range from **0.1% to 1.5%** between the star's surface and a distant observer, depending on the central density. The masses and radii for the simulations in general relativity are given in the dashed-line for comparison.

<figure><img src="image_13.png" /></figure>

Plot of the relative variation of ħ between the surface of a neutron star and a remote observer for various central densities.

## IV Backreaction of ħ

<figure><img src="image_14.png" /></figure>

Plot of the variation of ħ versus the distance for a neutron star with a central density of 1000 MeV/fm³.

Here we assume an equation of state that **explicitly depends on ħ**, analogous to a **degenerate Fermi gas**, to investigate the **backreaction of ħ variation** on the equation of state.

Accounting for the backreaction of ħ leads to an **11% error** in its evaluation, both at the **center and at the surface** when comparing cases with and without backreaction.

The backreaction is thus a **perturbative effect**

The blue (top) curve shows the variation of ħ when **taking into account** its variation into the equation of state, whereas the orange (bottom) curve is obtained by **neglecting it**.

[1] : L.Hendrik et al. Physics letter B 1506.03278

[2] : O.Minazzoli 2206.03824

[3] : T.Chehab et al. hal-04996227

[4] : C.Lee et al. 2410.01849

[5] : D.Arruga et al. Physical review D 2011.14629