Covariant theory of Bose-Einstein condensates in curved spacetimes with electromagnetic interactions: The hydrodynamic approach

Abstract

We develop a hydrodynamic representation of the Klein-Gordon-Maxwell-Einstein equations. These equations combine quantum mechanics, electromagnetism, and general relativity. We consider the case of an arbitrary curved spacetime, the case of weak gravitational fields in a static or expanding background, and the nonrelativistic (Newtonian) limit. The Klein-Gordon-Maxwell-Einstein equations govern the evolution of a complex scalar field, possibly describing self-gravitating Bose-Einstein condensates, coupled to an electromagnetic field. They may find applications in the context of dark matter, boson stars, and neutron stars with a superfluid core.1 As reported by Bhaumik [4], "Langevin, who was de Broglie's thesis advisor, was very skeptical about de Broglie's theory and contacted Einstein to have his opinion. Einstein strongly supported de Broglie's work and suggested to physicists to look for an evidence of the matter wave. A proof was furnished soon by the accidental discovery of electron waves by Davisson and Germer [5] in observing a diffraction pattern in a nickel crystal. Later, Einstein mentioned to Rabi that he had thought about the equation for matter waves before de Broglie but did not publish it because there was no evidence for it at that time." 2 Here and in the following, the dates correspond to the dates of submission. 3 In this brief review, we stick to what was originally called the "wave (or undulatory) mechanics" initiated by de Broglie [2] and developed by Schrödinger [6][7][8][9]. We shall not discuss in detail what was originally called the "quantum mechanics" (a term introduced by Born) based on the matrix theory of Heisenberg [12] further developed by Born and Jordan [13,14]. In Ref. [15] (18 March 1926), Schrödinger showed the equivalence between his wave mechanics and the Heisenberg-Born-Jordan quantum mechanics.[8], Schrödinger applied his theory to the perturbation of the hydrogen atom caused by an external homogeneous electric field (Stark effect). In his fourth paper (21 June 1926) [9], he obtained for the first time the time-dependent equation (F23) that now bears his name. From this equation, he introduced a (charge) density and a current (of charge) [see Eqs. (174) and (176)] and derived a local conservation equation for ρ = |ψ| 2 [see Eq. (175)]. This gives an interpretation to the wave function in the sense that |ψ| 2 (r, t) characterizes the "presence" of the particle at some point. Schrödinger thought that the wave function represents a particle that is spread out, most of the particle being where the modulus of the wave function |ψ| 2 is large. For example, according to Schrödinger's view, the charge of the electron is not concentrated in a point, but is spread out through the whole space, proportional to the quantity |ψ| 2 . On the other hand, Born (21 July 1926, 16 October 1926 developed a probabilistic interpretation of the wave function. 4 Born proposed that the magnitude of the wavefunction ψ(r, t) does not tell us how much of the particle is at position r at time t, but...

Dynamical boson stars

“…Recent work by Chavanis with Newtonian gravity solves the Gross–Pitaevskii equation, a variant of Eq. ( 58 ) which involves a pseudo-potential for a Bose–Einstein condensate, to model either dark matter or compact alternatives to neutron stars (Chavanis 2011 , 2012 , 2015 ; Chavanis and Harko 2012 ; Chavanis and Matos 2017 ). However, see a rebuttal to some of this work (Mukherjee et al.…”

“…However, note that as mentioned in Sect. 1.1 , ground state BSs can be considered as condensed states of bosons without invoking any analogy (Chavanis 2015 ; Chavanis and Matos 2017 ).…”

Bose–Einstein condensation and symmetry breaking of a complex charged scalar field

“…In this sense, the present work is taken to be complementary to those results reported in [13][14][15]. Particularly, this work studies the Klein-Gordon equation of a complex and charged scalar field (with a U(1) symmetry) inside a thermal bath.…”

“…(16b) through the third and last terms, which now not only contain factors accounting for the quantum potential, but they also contain factors related to the electromagnetic field and to the relativistic contributions. Also, from (15) it shown that when electromagnetic fields are present, the fluid in question is not longer irrotational since the relation ∇ × v = 0 for the fluid is not longer true.…”

A new logotropic model based on a complex scalar field with a logarithmic potential

“…6 See, e.g., Refs. [20][21][22][28][29][30][31] and Sec. XV for the general expression of the KG equation valid for possibly inhomogeneous systems.…”