Revised

February 2024

Kerr-Newman and Electromagnetic Acceleration
Paul H<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica ”Ennio De Giorgi”,
Università del Salento and INFN-Lecce,
Via Arnesano, 73100 Lecce, Italy.

Previous discussions of charged dark matter neglected Primordial Black Hole
(PBH) spin and employed the Reissner-Nordstrom metric. In Nature we expect the
PBHs to possess spin which require use of the technically more challenging
Kerr-Newman metric. It is shown that the use of K-N metric retains the
principal properties already obtained using the R-N metric, in particular the
dominance of Coulomb repulsion requires super-extremality and the presence of
naked singularities. In this sense, the spin of the PBHs is not an essential
complication.

##  1 Introduction

In [1, 2], a cosmological model was introduced which departs from the
conventional wisdom that the energy pie for the visible universe is
approximately 5% normal matter, 25% dark matter and 70% dark energy. The term
dark energy was introduced in 1998 in association with the then newly-
discovered accelerated expansion. In the novel model, the energy pie is
revised to 5% normal matter, 95% dark matter and 0% dark energy shich is
replaced by what we may call charged dark matter.

We are discussing a toy model which is not intended to describe Nature
precisely but is intended to be sufficiently realistic that some general
suggestions about the origins of dark energy can be made. The dark matter
constituents are assumed to be PBHs with a spread of masses from one hundred
to at least one trillion solar masses.

We shall not endeavour to invent a constituent of dark energy directly but
shall conclude that dark energy, describable in an opaque manner by a
cosmological constant $\Lambda\sim(meV)^{4}$, and also possibly interpretable
as a vacuum energy, can be better understood as a result of electromagnetic
forces between extremely heavy PBHs which are a sector of dark matter but,
unlike lighter PBHs, carry electric charge.

The present article is concerned with one issue which is whether the spin of
the PBHs is an essential complication. To check this out, we must promote the
Riessner-Nordstrom metric used in [2] to the Kerr-Newman metric which is
technically more complicated and, again to anticipate our result, we shall
conclude that spin does not alter our main conclusions that the PBHs need to
be super-extremal and to possess naked singularities.

This maximally violates the cosmic censorship hypothesis, which is
nevertheless unproven.

The biggest assumption in this approach to replace dark energy by charged dark
matter is that Primordial Extremely Massive Black Holes (PEMBHs) with
unscreened electric charges exist. It is normally expected that electric
charges are screened from distant observers by accretion of material with
counterbalancing charge. While this is undoubtedly true for stars, planets and
galaxies,there is no observational evidence for or against this compensation
of charge happening for this heavier mass range.

Before PEMBHs are discovered, a promising precursor would be the discovery of
Primordlal Intermediate Mass Black Holes (PIMBHs) in the mass range 100 to
$10^{5}$ solar masses as constituents of the Milky Way dark matter. This may
well be possible in the foreseeable future using microlensing of the stars in
the Magellanic Clouds by, for example, the Vera Rubin Observatory (formally
LSST) in Chile.

##  2 Kerr-Newman metric

The most general metric for describing a black hole with non-zero charge and
spin is [3]

$ds^{2}=c^{2}d\tau^{2}=-\left(\frac{dr^{2}}{\Delta}+d\theta^{2}\right)\rho^{2}+\left(cdt-a\sin^{2}\theta
d\phi\right)^{2}\frac{\Delta}{\rho^{2}}-\left((r^{2}+a^{2})d\phi-
acdt\right)^{2}\frac{\sin^{2}\theta}{\rho^{2}}$ (1)

in which $(r,\theta,\phi)$ are spherical polar coordinates and

$a=\frac{J}{Mc};~{}~{}~{}\rho^{2}=r^{2}+a^{2}\cos^{2}\theta;~{}~{}~{}\Delta=r^{2}-r_{S}r+a^{2}+r_{Q}^{2}=r^{2}fr)$
(2)

with $f(r)=\left(1-r_{s}/r+r_{Q}^{2}/r^{2}\right)$ and

$r_{S}=\frac{2GM}{c^{2}};~{}~{}~{}r_{Q}^{2}=\frac{Q^{2}G}{4\pi\epsilon_{0}c^{4}}$
(3)

The coefficient of $dr^{2}$ is

$-\left(\frac{\rho^{2}}{\Delta}\right)=-\frac{r^{2}+a^{2}\cos\theta}{r^{2}f(r)}.$
(4)

To obtain the gravitational stress-energy tensor $T_{\mu\nu}^{(GRAV)}$
involves taking the second derivative of the metric by employing the
expression

$\displaystyle T_{\mu\nu}^{(GRAV)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi G}\left(G_{\mu\nu}+\Lambda g_{\mu\nu}\right).$
(5) $\displaystyle+\frac{1}{16\pi
G(-g)}\left((-g)(g_{\mu\nu}g_{\alpha\beta}-g_{\mu\alpha}g_{\nu\beta})\right)_{,\alpha\beta}$

where the final subscripts represent simple partial derivatives. We shall need
also to evaluate the electromagnetic counterpart

$T_{\mu\nu}^{(EM)}=F_{\mu\alpha}g^{\alpha\beta}F_{\mu\beta}-\frac{1}{4}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}.$
(6)

Let us begin with $T_{\mu\nu}^{(GRAV)}$. This calculation involves taking up
to the second derivatives of the KN metric in Eq.(1). For the function $f(r)$:

$\frac{\partial}{\partial r}f(r)\sim
O(1/r^{2})~{}~{}~{}~{}~{}\frac{\partial^{2}}{\partial^{2}r}f(r)\sim
O(1/r^{4})$ (7) $\frac{\partial}{\partial r}(f(r))^{2}\sim
O(1/r^{2})~{}~{}~{}~{}~{}\frac{\partial^{2}}{\partial^{2}r}(f(r))^{2}\sim
O(1/r^{4})$ (8) $\frac{\partial}{\partial r}(f(r))^{-1}\sim
O(1/r^{2})~{}~{}~{}~{}~{}\frac{\partial^{2}}{\partial^{2}r}(f(r))^{-1}\sim
O(1/r^{4})$ (9) $\frac{\partial}{\partial r}(f(r))^{-2}\sim
O(1/r^{2})~{}~{}~{}~{}~{}\frac{\partial^{2}}{\partial^{2}r}(f(r))^{-2}\sim
O(1/r^{4})$ (10)

It is not difficult to check, using these derivatives, that the 1st, 3rd and
4th terms in Eq.(5) all fall off as $O(1/r^{2})$ and that only the 2nd term
does not.

In $T_{\mu\nu}^{(EM)}$, we can straightforwardly see that both terms in Eq.(6)
fall off as $O(1/r^{2})$.

We deduce that

$T_{\mu\nu}^{(GRAV)}+T_{\mu\nu}^{(EM)}=-\left(\frac{\Lambda}{8\pi
G}\right)g_{\mu\nu}+O(1/r^{2}).$ (11)

Thus, the principal conclusions for spinless PBHs found in [2] carry over to
PBHs with spin. In particular, the 1/r expansion is unchanged and the EoS for
the dark energy remains $\omega=-1$. The requirement for super-extremality and
naked singularities, to allow Coulomb repulsion to exceed gravitational
attraction, remains the same as in the spinless case.

##  3 Varying cosmological ”constant” $\Lambda(t)$

The Freidmann expansion equation keeping only dark energy is

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{\Lambda_{0}}{3}$ (12)

which, normalizing to $a(t_{0})=1$, leads to a time dependence of $a(t)$ which
is exponentially growing

$a(t)=\exp\left(\sqrt{\frac{\Lambda_{0}}{3}}(t-t_{0})\right)$ (13)

In the EAU-model, the dark energy is replaced by charged dark matter and we
know from [2] that the equation of state for the dark energy is precisely
$\omega=-1$.

However, the time-dependence of $a(t)$ is no longer exponential. If we use the
dual dark energy description it is governed not by Eq.(12) but by a time-
dependent cosmological ”constant” $\Lambda(t)$

$\left(\frac{\dot{a}}{a}\right)^{2}\approx\frac{\Lambda(t)}{3}$ (14)

The time-dependence follows from the usual dilution of dark matter

$\Lambda(t)\approx a(t)^{-3}\approx t^{-2}$ (15)

so that adopting as the present value $\Lambda(t_{0})=(2~{}meV)^{4}$ we arrive
at the examples of future values for $\Lambda(t)$ provided in Table 1. In the
last row, for example, at a time which is $10^{5}$ times the present age of
the universe, the cosmological constant has decreased by a factor $\sim
10^{-9}$. This behaviour implies that the extroverse never expands to the very
large size discussed for the $\Lambda CDM$-model in [4].

Table 1: COSMOLOGICAL “CONSTANT”. time | $\Lambda(t)$
---|---
$t_{0}$ | $(2.0meV)^{4}$
$t_{0}+10Gy$ | $(1.0meV)^{4}$
$t_{0}+100Gy$ | $(700\mu eV)^{4}$
$t_{0}+1Ty$ | $(230\mu eV)^{4}$
$t_{0}+1Py$ | $(7.4\mu eV)^{4}$

##  4 Previous Work

There are in the literature, several papers on theoretical astrophysics and
cosmology that cast possible doubt on whether electrically-charged black holes
exist in Nature and on whether small charge asymmetries can exist within
galaxies and clusters of galaxies, and within the whole visible universe.
Since the existence of charged black holes is crucial to the EAU model [1, 2],
the present section addresses past work and attempts to convince the reader
that there is no known fatal flaw in EAU.

It is generally accepted that electrically neutral black holes exist in
Nature. Since the main result of the present article is that black hole spin
is an inessential complication in the EAU model, we may associate electrically
neutral BHs with the Schwartzschild (S) metric and electrically charged BHs
with the Reissner-Nordstrom (RN) metric. Both S and RN metrics are exact
solutions of general relativity. The RN metric is richer than the S metric in
that, depending on the electric charge Q, relative to the mass M, it has sub-
extremal (two horizons), extremal (one horizon) and super-extremal (no
horizon) varieties, Mathematically, however, both S and RN are solutions of
Einstein’s field equations and there appears to be no special reason why
Nature should simultaneously adopt the S solution and eschew the RN solution.

It is of considerable interest that in 1959 Lyttleton and Bondi [5] considered
electric charge as underlying the cosmological expansion, by introducing a
small difference between the proton and electron charges. Their theory did not
survive quantitative testing of the neutrality of the hydrogen atom. The LB
model is obviously different from the EAU model, although it does involve a
remarkably similar charge asymmetry $\epsilon_{Q}\sim 10^{-18}$ for the
universe as we have for clusters.

By studying the baryon budget[6, 7] and making reasonable assumptions, one can
show that at a scale of 2Mpc in the EAU model that, unlike in the LB model,
there is no charge asymmetry. This scale is larger than a galaxy. In the EAU
model, all processes conserve electric charge and so certainly the universe
remains electrically neutral unless there was, for an unknown reason, an
initial condition that imposed $|\epsilon_{Q}|\neq 0$ at the Big Bang.

One study of the cross-sections for capture of particles by an RN black holes
was made by Zakharov[8]. This study included, however, capture only of
electrically neutral particles and hence neglected the effects of Coulomb
forces,

In the EAU model, we have assumed that the fraction $f_{DM}$ of the dark
matter made up of PBHs is $f_{DM}=1$. The PBHs fall into different mass ranges
for PIMBHs, PSMBHs and PEMBHs and in the example we gave the PEMBHs contribute
$f_{DM}=0.5$. In [9] an upper limit $f_{DM}=0.01$ for mass
$M=10^{12}M_{\odot}$ which is contradictory. However, we have reason to
believe that the constraints in [9] are far too severe. This can be traced
back to the well known ROM paper[10] which introduced the popular $f_{DM}$
exclusion plot.

In the case of ROM, too severe limits were caused by the use, in numerical
analysis, of an oversimplified accretion model (Bondi model) which assumes
spherical symmetry and radial inflow. This naive model can overestimate
accretion, compared to Nature, not just by a factor 2 but even by some 4 or 5
orders of magnitude![11]. Subsequent exclusion plots appear to have been
overly influenced by ROM’s results. The fact that ROM imposed too severe
constraints has been confirmed to us by its senior author (J.P. Ostriker,
private communication).

There are other mechanisms proposed in the literature able to produce a
charged BH. For instance, in [12], the author argued that a rotating BH
embedded in a plasma may generate a magnetic field and may acquire an electric
charge. However, a recent investigation in [13] has shown that the charge in
[12] is in fact an upper limit and generally is quite small.

To summarise this section, previous work on these issues is undoubtedly
interesting but does not reveal a fatal flaw in the EAU model which replaces
dark energy by charged dark matter.

Dark energy can be simply and accurately parametrised by the cosmological
constant $\Lambda$ introduced by Einstein for a different, almost
antithetical, reason. It may equally be regarded as a type of vacuum energy.
In particle theory, however, vacuum energy is associated with the spontaneous
breaking of a symmetry and it is unclear which symmetry is involved.

The EAU model can provide a more physical picture than either. The model is
very speculative and cries out for more theoretical work to confirm its
internal consistency.

##  5 Discussion

Dark matter and dark energy are the two major unanswered questions in
cosmology. In this article we have discussed a model which suggests specific
answers. The model has the following ingredients.

In the Milky Way, the dark matter constituents are PIMBHs with masses between
100 and 100,000 solar masses. We cannot exaggerate the importance of detecting
these, in our opinion most easily by seeking microlensing light curves with
durations in excess of two years. Such a success would suggest that the idea
that DM constituents are PBHs is on the right track.

PBHs with higher masses between a million and a hundred billion solar masses,
PSMBHs, include the supermassive black holes know to exist at galactic
centres.The PIMBHs and PSMBHs are, like all stars and planets, electrically
neutral and so experience only the attractive gravitational force. The PSMBHs
may be formed from lighter seeds such as PIMBHs by merging and accretion.

At the next and final higher mass scale are the PEMBHs with at least a
trillion solar masses and carrying unscreened like-sign charges of order $\sim
10^{40}$ Coulombs, a charge which corresponds only to a $\sim 10^{-18}$ charge
asymmetry. These electric charges are not compensated by accreted halos of
opposite charge, and hence experience long-range electromagnetic repulsion.
This is the charged dark matter which replaces what was given the name dark
energy immediately after accelerated cosmic expansion was discovered.

The purpose of the present paper is to show that this model of dark matter and
energy [2] survives the generalisation to spinning black holes. We had
expected this to be far more difficult than it is, as we have shown rather
simply by extending the Reissner-Nordstrom metric to the Kerr-Newman metric.

If our EAU model is correct, the present decade promises to resolve the dark
matter and energy issues. The dark energy will have a dual description either
as charged dark matter or as a cosmological constant $\Lambda(t)$ with
constant equation of state $\omega(t)=P(t)/\rho(t)=-1$ with high accuracy. The
cosmological ”constant” itself falls in the future as $\Lambda(t)\propto
t^{-2}$.

This leads to quite a different prediction for the future fate of the universe
than in the conventional $\Lambda CDM$ model with constant $\Lambda$. Instead
of exponential superluminal expansion of the extroverse [4], the growth is as
a power and the extroverse does not grow within a trillion years to a
gargantuan size and, because of this far gentler rate of growth, billions of
other galaxies remain within the visible universe forever.

In the EAU model, we need not include dark energy as a slice of the energy
pie-chart which become simply 5% baryonic matter and 95% dark matter.With
irrational optimism, we may expect that by the end of this decade conventional
wisdom will change to where the accelerated expansion is ascribed to
electromagnetic repulsion rather than an unspecified form of repulsive
gravity.

Let us re-emphasize that the EAU-model is speculative and based on assumptions
which have already been entertained in the existing literature but which have
not been rigorously justified. This fact was already expressed ut supra. Our
aim in the EAU-model was to construct a cosmological theory in which dark
energy, as originally understood, is removed and replaced by charged dark
matter.

Before delving further into the rôle of the PEMBHs, let us first address the
simpler question of why are the PIMBHs electrically neutral. like stars and
galaxies, while the PEMBHs carry negative electric charge? This is due to the
fact that the PIMBHs are formed during the first second after the Big Bang
when the selective accretion of electrons over protons is unavailable. By
contrast the PEMBHs are formed much later, in the dark ages after the creation
of the Cosmic Microwave Background (CMB), when the difference in thermal
velocities of electrons and protons leads to the mechanism for acquisition of
a net negative electric charge.

In the EAU model, the same-sign-charged PEMBHs are so massive, at least a
trillion solar masses, that they are not associated with a specific galaxy or
cluster but move each on their own under the Coulomb repulsion by other
PEMBHs. An electric charge asymmetry
$\epsilon_{Q}\equiv(Q_{+}-Q_{-})(Q_{+}+Q_{-})^{-1}$ is induced in the
intergalactic and intracluster gas due to the remaining protons. It is
straightforward to estimate this charge asymmetry as $\epsilon_{Q}\sim
10^{-18}$ which survives in the intergalactic and intracluster gas due to the
residual protons. This charge-asymmetric gas is not in the PEMBH environment
so there is no opportunity to set up Debye screening of the PEMBH negative
charge and hence long-range inter-PEMBH electric repulsion exists.

There may have developed an incorrect prejudice, based on the observed
electric neutrality of stars and planets, of galaxies and of the dark matter
inside galaxies and clusters, that any macroscopic astrophysical object,
including a PEMBH, must be electrically neutral. If the EAU model is correct,
this prejudice must be discarded.

## Acknowledgement

We are grateful to an anonymous referee for providing a summary of past
research.

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