Generalized uncertainty principle and white dwarfs redux: How the cosmological constant protects the Chandrasekhar limit

Abstract

It was previously argued that generalized uncertainty principle (GUP) with a positive parameter removes the Chandrasekhar limit. One way to restore the limit is by taking the GUP parameter to be negative. In this work we discuss an alternative method that achieves the same effect: by including a cosmological constant term in the GUP (known as "extended GUP" in the literature). We show that an arbitrarily small but nonzero cosmological constant can restore the Chandrasekhar limit. We also remark that if the extended GUP is correct, then the existence of white dwarfs gives an upper bound for the cosmological constant, which -while still large compared to observation -is approximately 86 orders of magnitude smaller than the natural scale. I. GENERALIZED UNCERTAINTY PRINCIPLE AND WHITE DWARFSThe generalized uncertainty principle (GUP) is a quantum gravity inspired correction to the Heisenberg's uncertainty principle, which reads (in the simplest form)where L p = 1.616229×10 −35 m denotes the Planck length, and α is the GUP parameter typically taken as an O(1) positive number in theoretical calculations, i.e. one expects that the GUP correction becomes important at the Planck scale. GUP is largely heuristically "derived" from Gedanken-experiments under a specific quantum gravity theory (such as string theory [1-4]), or general considerations of gravitational correction to quantum mechanics [5][6][7][8]. GUP is useful as a phenomenological approach to study quantum gravitational effects. From phenomenological point of view, the GUP parameter can be treated as a free parameter a priori, which can be constrained from experiments [9][10][11][12]; α as large as 10 34 is consistent with the Standard Model of particle physics up to 100 GeV [9], while a tunneling current measurement gives α 10 21 [13]. See also [14][15][16].It turns out that GUP has a rather drastic effect on white dwarfs. This is somewhat of a surprise, since we do not usually expect GUP correction to be important for scale much above the Planck scale. A standardthough hand-wavy -method to obtain the behavior of degenerative matter is to consider the uncertainty principle ∆x∆p ∼ , and then take ∆x ∼ n −1/3 , where n is the number density n = N/V = M/(m e V ) of the white dwarf (here modeled as a pure electron star), where N is the total number of electrons, whereas V, M are, respectively, the volume and the total mass of the star, and m e Electronic address: ycong@yzu.edu.cn the electron mass. Then, the total kinetic energy in the non-relativistic case iswhere R is the radius of the star. Equating this with the magnitude of the gravitational binding energy |E g | ∼ GM 2 /R, one obtains the radius as a function of the mass:Thus the more massive a white dwarf is, the smaller it becomes. A similar derivation using the relativistic kinetic energy, and assuming that the momentum dominates over the rest mass of the electrons, allows one to obtain the Chandrasekhar limit up to a constant overall factor (see [17] for details). That is, the ultra-relativistic curve i...

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