July 2024

Avoidance of Schwinger Mechanism in Electromagnetically Accelerating Universe
Paul H<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica ”Ennio De Giorgi”,
Università del Salento and INFN-Lecce,
Via Arnesano, 73100 Lecce, Italy.

In quantum electrodynamics, Schwinger showed in 1951 that in a uniform
electric field the vacuum may spontaneously produce electron-positron pairs.
For this to occur, the required electric field is exceptionally large, $\sim
10^{18}$ N/C. In the Electromagnetically Accelerating Universe (EAU) model,
such an electric field would be undesirable because it discharges the
Primoridal Extemely Massive Naked Singularities (PEMNSs) whose mutual
repulsion causes the acceleration. This leads to a constraints on the charges
and masses of the PEMNSs which must be satisfied in an EAU model.

##  1 Introduction

In the present short note, we address the issues of the mass and charge of the
Primordial Extremely Massive Naked Singularities (PEMNSs) hypothesised in
using the novel Electromagnetically Accelerating Universe (EAU) model.

In the EAU model [1, 2] the main idea is that electromagnetism dominates over
gravitation in the explanation of the accelerating cosmological expansion.
This idea takes us beyond the first paper[3] which applied general relativity
to theoretical cosmology. This is not surprising, since in 1917 that author
was obviously unaware of the fact [4, 5] discovered only in 1998 that the rate
of cosmological expansion is accelerating.

We study in turn the avoidance of the Schwinger mechanism, the condition for
super extremality and dominance of Coulomb repulsion.

##  2 Avoidance of Schwinger Mechanism

In 1951, Schwinger [6] showed in the context of QED that in a strong enough
constant electric field, electron positron pairs are created from the vacuum.
This could lead to the discharge of the PEMNSs by attracting the positron.

According to [6], to avoid the Schwinger mechanism requires the the electric
field $|E|$ satisfies

$|E|<10^{18}N/C$ (1)

Let is consider a PEMNS with mass $10^{p}M_{\odot}$ and electric charge
$|Q|=10^{q}$ Coulombs. A characteristic scale is the Schwarzschild radius
$R_{S}\sim 3\times 10^{p+3}$ metres at which the electric field is

$|E|=\frac{k_{e}|Q|}{R_{S}^{2}}\sim 10^{q-2p+3}N/C$ (2)

using the electric constant $k_{e}=9\times 10^{9}nm^{2}C^{-2}$.

Consistency of Eq.(2) with Eq.(1) requires that

$(q-2p)<15$ (3)

Using the value $p=12$ and $q=32$ used in the EAU model [8] we see that the
electric field associated with the Schwinger mechanism is avoided by several
orders of magnitude.

##  3 Condition for Super Extremality

In the Reissner-Nordstrom metric [7] describing an electrically charged black
hole, there are two independent lengths depending respectively on the mass and
the charge.

$r_{S}=\frac{2GM}{c^{2}}$ (4)

and

$r_{Q}^{2}=\frac{k_{e}Q^{2}G}{c^{4}}$ (5)

Extremality depends on the relative sizes of $r_{S}$ and $(2r_{Q})$. If
$r_{S}>2r_{Q}$, the solution is sub extremal with two horizons. When
$r_{S}=2r_{Q}$, it is extremal with one unique horizon at the Schwarzschild
radius, $r_{Q}$.

Super extremality occurs when $2r_{q}>r_{S}$. From Eqs. (4) and (5) this
requires that

$\sqrt{k_{e}}Q\geq\sqrt{G}M$ (6)

which, for mass $M=10^{p}M_{\odot}$ and $Q=10^{q}$, gives

$(q-p)\geq 20$ (7)

With Eq. (7) there is no horizon and hence a naked singularity.

##  4 Dominance of Coulomb Repulsion

Let us consider two identical PEMNSs each with mass $M=10^{p}M_{\odot}$ and
electric charge $Q=-10^{q}$ Coulombs, separated by distance $R$. Then the
Coulomb repulsion will dominate provided that

$\frac{k_{e}Q^{2}}{R^{2}}\geq\frac{GM^{2}}{R^{2}}$ (8)

Using the values $k_{e}=9\times 10^{9}Nm^{2}C^{-2}$, $G=6.67\times
10^{-11}m^{3}kg^{-1}s^{-2}$, $M_{\odot}=2\times 10^{30}kg$ Eq.(8) translates
into

$(q-p)\geq 20$ (9)

For the EAU model [8], Eq.(9) is satisfied. As the PEMNQ mass is increased
beyond $10^{12}M_{\odot}$, we learn from Eq.(9) that the electric charge must
increase at least linearly in the mass.

##  5 Summary

The EAU model is sufficiently revolutionary that it is easy to suspect that
there must be a fatal flaw. Therefore it is well worth discussing at length
the physics resulting from our three previous short sections.

In the EAU model, since a central rôle is played by the PEMNSs, it is
important to attribute to them masses $M=10^{p}M_{\odot}$ and electric charges
$10^{q}$ Coulombs. The values of $p$ and $q$ exponents are our principal topic
here. Values were chosen in 2022 [1] and 2023 [2].

We assume that the PEMNSs have a mass at least equal to the Milky Way,
including it dark matter halo, i.e. a mass of one trillion $10^{12}$ Suns.
Because they are black hofes, or better naked singularities, they are far more
concentrated and have a physical size of about one light year, compare to the
hundred thousand light years of the Milky Way.

If the Schwinger mechanism had been operative in the vicinity of the PEMNSs,
it would provide a fatal flaw of the EAU model because pair production in the
vacuum would enable the PEMNSs to lose their electric charge and thus fail to
drive the accelerating expansion. But the mechanism requires a large electric
field, too large to be generated by the PEMNSs contained in the EAU model[8]
which used the parameters $p=12$ and $q=32$.

If we increase the mass, keeping to a linear increase in charge, for exmple to
$p=23$ and $q=43$, the pair creation is suppressed according to Eq.(3) by
several further orders of magnitude. Thus the biggest risk of the triggering
the Schwinger mechanism comes with the minimal value $p=12$ and even there is
avoided by an electric field at least five orders of magnitude too small.

In summary, the Schwinger mechanism does not provide a fatal flaw for the EAU
model.

The subsequent two sections about respectively the condition for super
extremality and the dominance of Coulomb repulsion lead to similar, actually
identical, calculations. This is in itself very interesting.

This is one of the motivations for naked singularities, since we learn that
two RN extremal black holes exert no force on one another yet when they
increase their charge to become super extremal, assuming all the PEMNS s have
the same sign electric charge, the Coulomb repulsion dominates over the
gravitational attraction.

If we increase the PEMNS mass, it leads to two changes., . Firstly, the total
entropy increases. As we increase $p$ from $p=12$ to, say, $p=23$ the number
of PEMNSs, assuming the total mass is $10^{23}M_{\odot}$, the number decreases
from $n=10^{11}$ to $n=1$. The total entropy increases from $10^{113}$ to
$10^{124}$ eventually saturating the maximal holographic bound.

Meanwhile, by making these changes, the charge asymmetry $\epsilon_{Q}$
remains unaltered assuming that, as suggested earlier, $q$ is increased
linearly with $p$. The incremental increase to the total charge is $\Delta
Q=10^{43}$ Coulombs compared to the total charge of electrons, or protons, in
the visible universe $Q\sim 10^{61}$ Coulombs and hence $\epsilon_{Q}\sim
10^{-18}$.

There are at least two possibilities regarding the charge asymmetry. It could
be an initial condition at the Big Bang or, if not, we must live with
$\epsilon_{Q}\neq 0$ at present.

##  6 How to test the EAU model

Ideally, one would like to find evidence for the existence of PEMNSs. One
valiant attempt to do this was presented in [9] but it is unclear whether it
is practicable.

To support the general approach, it would be encouraging if the PIMBHs in the
Milky Way could be found using the Rubin Observatory by microlensing of stars
in the Magellanic Clouds.

## References

  * [1] P.H. Frampton,
Electromagnetic Accelerating Universe.
Phys. Lett. B835, 137480 (2022).
arXiv:2210.10632[physics.gen-ph].

  * [2] P.H. Frampton,
A Model of Dark Matter and Energy.
Mod. Phys. Lett. A38, 2350032 (2023).
arXiv:2301.10719[physics.gen-ph].

  * [3] A. Einstein,
Cosmological Considerations in the
General Theory of Relativity.
Sitz.Preuss.Akad. 1917, 142 (1917).

  * [4] S. Perlmutter, et al. (Supernova Cosmology Project Collaboration)
Astrophys.J. 517, 565 (1999).
arXiv:astro-ph/9812133 [astro-ph].

  * [5] A.G. Riess, et al. (Supernova Search Team),
Astron.J. 116, 1009 (1998).
arXiv:astro-ph/9805201 [astro-ph].

  * [6] J.S. Schwinger,
On Gauge Invariance and Vacuum Polarization.
Phys. Rev. 82, 664 (1951).

  * [7] H. Reissner,
Über die Eigengravitation des Elektrischen Feldes
nach der Einsteinschen Theorie.
Annalen Physik 355, 106 (1916).
G. Nordstrom,
On the Energy of the Gravitational Field
in Einstein’s Theory.

  * [8] P.H. Frampton,
Status of Electromagnetic Accelerating Universe.
2404.08999[gr-qc]

  * [9] J. Wagner,
Observables for Moving, Stupendously Charged and Massive Primordial Black
Holes.
arXiv:2301.12210[astro-ph.CO]
J.Wagner,
Observables for Super-Extremal Black Holes Challenging Cosmic Censorship to
Comprehend the Cosmological Constant.
arXiv:2405.08057[gr-qc].