Reissner–Nordström Expansion

Emil M. Prodanov

Dublin Institute of Technology

On a large scale, the Universe is isotropic and homogeneous (Robertson–Walker) with energy-momentum sources modeled as a perfect fluid, specified by an energy density and isotropic pressure in its rest frame.

This applies for matter known observationally to be very smoothly distributed. On smaller scales, such as stars or even galaxies, this is a poor description.

We propose a microscopic description, modeling the Universe in the radiation-dominated epoch as a two-component gas with preserved global Robertson–Walker geometry, but with local Reissner–Nordström geometry.

The first component is a gas of ultra-relativistic particles described by an equation of state of an ideal quantum gas of massless particles (the “normal” fraction).

The second component consist of very massive charged particles.

The expansion mechanism is based on the interaction between the “normal” fraction of the Universe and the very massive charged particles: the interaction is described purely classically in terms of the Reissner–Nordström geometry of the charged massive particles.

The candidates:

…

The Model

Hawking’s arguments:

The ultramassive charged particles are viewed general-relativistically as naked singularities and the interaction mechanism is the gravitational repulsion of the naked singularities.

Naked singularities are particles of charge Q greater than their mass M (in geometrized units G = 1 = c) described by Reissner–Nordström geometry.

For the electron, the charge-to-mass ratio Q/M is of the order of 1021. In view of this, in the 1950s, the Reissner–Weyl repulsive solution served as an effective model for the electron.

We confine our attention to the local spherical neighbourhood of a single naked singularity and consider the Universe as multiple copies – fluid – of such neighbourhoods (local Reissner–Nordström geometry).

We show that the radii of the repulsive spheres of the naked singularities grow in inverse proportion with the temperature: the ultraheavy charged particles “grow” as the temperature drops and drive away the “normal” fraction of the Universe.

For a naked singularity of unit charge, the mass cannot exceed 10-8 kg.

This repulsion results in power law expansion with scale factor a() ∼ 1/2, corresponding to the expansion during the radiation-dominated era.

If charge non-conservation of the naked singularities is involved, then the gravitational repulsion could be powerful enough to achieve accelerated expansion that solves the horizon problem and thus accounts for the inflation the Universe: a() ∼ eH or a() ∼ n, with n > 1.

For temperatures below 1029K, quantum effects do not play a role in the interaction between the two fractions of the Universe.

We determine the “radius” of an ultraheavy charged particle by calculating the turning radius of a radially moving incoming (charged) test particle of the “normal” fraction of the Universe, having ultra-high energy kT >> mc2 (where m is the test particle’s rest mass).

The proposed model is simplified significantly by considering the incoming particles as collisionless probes rather than involving their own gravitational fields and by not considering the more general and physically more relevant Kerr–Newman geometry.

We thus ignore the magnetic effects caused by rotation of the centre, which drags the inertial reference frames, and we also ignore the particles’ spins.

The Kerr–Newman metric in Boyer–Lindquist coordinates and geometrized units is given by:

where:

The motion of a particle of mass m and charge q in gravitational and electromagnetic fields is governed by the Lagrangian:

Where λ – proper time per unit mass m: λ = /m, and A – the vector electromagnetic potential, determined by the charge Q and specific angular momentum a of the centre:

The equations of motion for the particle are:

Where:– conserved energy of the particle

– conserved projection of the particle’s angular momentum on the axis of the centre’s rotation

K is another conserved quantity given by:

With the -component of the particle’s four-momentum.

We consider the radial motion of a particle in Reissner–Nordström geometry:

The radial motion is described by:

Where = E/m is the specific energy of the particle. Motion is possible only if – non-negative.

Thus the radial coordinate of the test particle must necessarily be outside the region (r- , r+) where the turning radii are given by:

The loci of the event horizon and the Cauchy horizon for the Reissner–Nordström geometry are:

The centre r = 0 however, can be reached by a suitably charged incoming particle satisfying:

For example, a positively charged center (Q > 0), can be reached by an incoming probe with specific charge q/m 1.

The naked singularity can be destroyed if sufficient amount of mass and opposite charge are fed into it.

We assume that the naked singularities have survived such annihilation.

A positively charged naked singularity will never be reached by incoming particles of small negative charge (i.e. −1 < q/m < 0), neutral particles (q = 0) and all positively charged particles (q > 0).

For these particles the centre is surrounded by an impenetrable sphere of radius r 0(T ) = r + .

The radius r0(T) of that impenetrable sphere depends on the energy of the “normal” particles or the temperature T of the Universe. For very high energies, the centre’s “radius” can be written as:

With the drop of the temperature, the superheavy charged particles increase their “size” in inverse proportion and drive apart the “normal” fraction of the Universe.

The Universe increases its size in inverse proportion with the temperature:

a() ∼ r0() ∼ 1/T () ,

where a( ) is the scale factor of the Universe.

We therefore get the usual relation: aT = const or:

Consider the expansion rate equation without cosmological constant:

The main contribution in the energy density comes from the electrostatic field of the ultraheavy charged particles.

The energy density of the electrostatic field is proportional to the square of the intensity of the electrstatic field, that is, the main contribution to ρ comes from a term proportional to Q4 /r0

4(T).

Therefore: and

The solutions are: and

At recombination ( ∼ 300 000 years), the free ions and electrons combine to form neutral atoms (q = 0) and this naturally ends the Reissner–Nordström expansion: a neutral “normal” particle will now be too far from a superheavy charged particle to feel the gravitational repulsion (the density of the Universe will be sufficiently low).

At recombination the volume of the Universe is:

Assuming that the repulsive spheres of the ultraheavy charged particles are densely packed during radiation domination, at recombination the “radius” of such sphere would be:

The pseudo-Newtonian potential that describes the interaction between particles of the two fractions is:

We now cut off the potential of the interaction between a naked singularity and a “normal” particle with the same sign of their charges when r0(T) reaches Rc.

For “normal” particles with specific charges satisfying 0 ≥ sign(Q) q/m ≥ −1, we cut off the potential at the point where gravitational attraction and repulsion interchange:

Namely:

Atoms are surrounded by imaginary hard spheres and the molecular interaction is strongly repulsive in close proximity, mildly attractive at intermediate range, and negligible at longer distances.

This is possible in the light of the deep analogies between the physical picture behind the Reissner–Nordström expansion model and the classical van der Waals molecular model:

We are now in a position to model the Universe as a van der Waals gas.

The pre-van der Waals equation is:

Note that only in the limit N/V 0, this equation reduces to the van der Waals equation:

The correction parameter is due to the extra pressure, while the correction parameter accounts for the non-zero volumes of the “atoms”.

For the parameters and we get:

The equation of state is:

Here = const and tends to zero at Recombination.