derivative, the second clock effect does not actually arise in gravity
theories with non-metricity.

Up to this point our aim may have already become clear. We will examine
whether $\hat{R}_{ab}$ satisfies Maxwell equations, and whether there is a
correspondence between homothetic curvature and the Maxwell field. In
particular, it is the goal of this Chapter to present the observation that
there is indeed a (metric-affine) curvature component field which actually
turns out to present an equivalence with the Maxwell field. In face of this
finding, we put into question our conventional perspective regarding the way
we envisage macroscopic electromagnetic fields and their relation to gravity.

####  2.2 Homogeneous (metric affine) Maxwell equations and the implication
for gravito-electromagnetic equivalence

Let us start from the expression
$\nabla_{[a}F_{bc]}=(1/3)(\nabla_{a}F_{bc}+\nabla_{c}F_{ab}+\nabla_{b}F_{ca})$,
within a Riemannian framework. According to the homogeneous Maxwell equations,
it has to be equal to zero. Taking thus into account that the Faraday tensor
comes from a potential 4-vector, we follow the operations:

$\displaystyle\nabla_{[a}F_{bc]}$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\left[2\nabla_{[a}\nabla_{b]}A_{c}+2\nabla_{[c}\nabla_{a]}A_{b}+2\nabla_{[b}\nabla_{c]}A_{a}\right]=\frac{1}{3}\left(R_{abcd}+R_{cabd}+R_{bcad}\right)A^{d}$
(2.1) $\displaystyle=R_{[abc]d}A^{d}=0\,.$

In other words, we have recalled that if a second-rank antisymmetric tensor
field can be written as the gradient of a 4-vector field, then the homogeneous
Maxwell equations are a consequence of two geometric properties of the
Riemannian spacetime333Besides, the homogeneous Maxwell equations can be
derived theoretically in Minkowski spacetime [13] through variation of the
action
$\mathcal{S}=\int{\left(-\sum{m_{i}}\sqrt{\eta_{ab}\dot{x}_{(i)}^{a}\dot{x}_{(i)}^{b}}-\frac{1}{4}F_{cd}F^{cd}-\sum{e_{i}}A_{a}\dot{x}_{(i)}^{a}\right)\,d\tau}$
with respect to the particles’ coordinates $x^{a}(\tau)$ ($\tau$ is the
particle’s proper–time, its world–line parameter). Subsequently, the
homogeneous Maxwell equations are generalised to curved (Riemannian) spacetime
via the so–called minimal substitution rule.; these are the Ricci identities
in the form of (2) and the first Bianchi identities (i.e.
$R_{[abc]d}=0$).444For an arbitrary vector field $A_{a}$ the aforementioned
properties imply that $\nabla_{[a}\nabla_{b}A_{c]}=0$. Inversely, if the
homogeneous Maxwell equations are satisfied, the second-rank antisymmetric
tensor field can be written as the gradient of a 4-vector field in Riemannian
spacetime. Therefore, it is clear that $\nabla_{[a}\hat{R}_{bc]}=0$, within
the geometric framework in question. It is worth noting that the above well-
known conclusion can be generalised to (non-Riemannian) geometries which
possess non-metricity555It can be shown that both the Ricci and the first
Bianchi identities maintain their Riemannian form when the relativistic
background is modified by the additional non-metricity requirement. In fact,
non-metricity is incorporated into the Riemann tensor. (e.g. see eqs. (1.152)
and (1.158) in [8], corresponding to the metric-affine version of the Ricci
identities and of the first Bianchi identities respectively). Nevertheless, in
a general metric affine geometry, possessing torsion as well, the homogeneous
Maxwell equations cease to be valid (once again see eqs. (1.152) and (1.158)
of [8], in combination with (2.1)). In this case, homothetic curvature
satisfies the following generalised version of Bianchi identities (known as
Weitzenbock identities-see eq (1.169) in [8]):

$\nabla_{[a}\hat{R}_{bc]}=2\hat{R}_{d[a}S_{bc]}{}^{d}\,.$ (2.2)

Observe that in the absence of torsion, $\hat{R}_{ab}$ satisfies the
homogeneous set of Maxwell equations (i.e. $\nabla_{[a}\hat{R}_{bc]}=0$).
Besides, it is known that Einstein-Hilbert action666In fact, there is a
generalised action (known under the name quadratic theory [14]), containing
the Einstein-Hilbert, which has as a consequence the property
$S_{ab}{}^{c}=-(2/3)S_{[b}\delta_{a]}{}^{c}$. implies that
$S_{ab}{}^{c}=-(2/3)S_{[b}\delta_{a]}{}^{c}$ (with $S_{a}\equiv S_{ab}{}^{b}$
being one of the torsion vectors)-see [8]. Given the aforementioned property,
let us stress out the observation that homothetic curvature satisfies (recall
eq. (2.2)) the following homogeneous set of Maxwell-like equations, namely

${\color[rgb]{1,0,0}\hat{\nabla}_{[a}\hat{R}_{bc]}=0}\,,\hskip
14.22636pt\text{where}\hskip
14.22636pt\hat{\nabla}_{a}=\nabla_{a}-\frac{4}{3}S_{a}\,,\hskip
14.22636pt\text{for}\hskip
14.22636ptS_{ab}{}^{c}=-\frac{2}{3}S_{[b}\delta_{a]}{}^{c}\,.$ (2.3)

We note once again that the above turns out to hold for a generalised action
(quadratic theory [14]), a part of which is the Einstein-Hilbert. A possible
correspondence between the Faraday tensor and the Maxwell potential with the
homothetic curvature and the non-metricity vector is apparent. In particular,
let us focus on the correspondence $A_{a}\rightarrow Q_{a}$ and
$F_{ab}\rightarrow\hat{R}_{ab}$. Taking into account that in geometrised
units, $A_{a}$ and $g_{ab}$ are dimensionless, a coupling constant $k$ of
length dimension is needed so that dimensional equivalence is established,
i.e.

$A_{a}=kQ_{a}\hskip 14.22636pt\text{and}\hskip
14.22636ptF_{ab}=k\hat{R}_{ab}\,,$ (2.4)

where $Q_{a}$ obviously has inverse length dimension. Thus, a potential
equivalence between the homogeneous Maxwell equations and (2.3) makes its
appearance via the correspondence: $F_{ab}\rightarrow k\hat{R}_{ab}$ and
$\nabla\rightarrow\hat{\nabla}$. The question is: Is there an action
reproducing both Einstein and Maxwell field equations, and satisfying the
condition $S_{ab}{}^{c}=-\frac{2}{3}S_{[b}\delta_{a]}{}^{c}$ (appearing in
(2.3)) as well? On finding such an action, the above assumed equivalence will
be established.

####  2.3 Inhomogeneous Maxwell equations: From electrodynamics in curved
spacetime to metric affine (gravitational) equivalent of Maxwellian
electrodynamics

In contrast to the homogeneous set of Maxwell equations, which springs from a
purely geometric principle, the inhomogeneous one is known to be a consequence
of an action principle (involving the electromagnetic field’s strength and its
coupling with matter.

#####  2.3.1 Maxwellian action in curved (relativistic) spacetime

Before answering the question stated in the end of the previous subsection,
let us recall that the action for electrodynamics in curved (Riemannian-
relativistic) spacetime, reads (e.g. see [15] and [16]):

$\mathcal{S}_{CEM}=\int{\left(R_{ab}g^{ab}+\mathcal{L}_{\text{m}}-\frac{1}{4}F_{ac}F_{bd}g^{ab}g^{cd}-A_{a}J_{b}g^{ab}\right)\sqrt{-g}\,d^{4}x}\,,$
(2.5)

where $J^{a}$ is the current 4-vector, $\mathcal{L}_{\text{m}}$ is the
Lagrangian density of matter and $g$ the determinant of the metric tensor. In
the aforementioned combined action, the electromagnetic field couples with the
metric tensor of the gravitational field to form the scalar (Lorentz
invariant) inner products $F_{ab}F^{ab}=F_{ac}F_{bd}g^{ab}g^{cd}$ and
$A^{a}J_{a}=A_{a}J_{b}g^{ab}$. Note that in the above, there are two
fundamental fields, the spacetime geometry or gravitation, and the Maxwell
gauge potential. In this context, the metric tensor acts as a mediator between
fields-sources of gravity-with geometric nature (vectors, tensors) and their
energy content (i.e. Lagrangian densities). On the one hand, variations with
respect to the potential $A_{a}$ lead to Maxwell equations of the form:

${\rm
D}_{b}F^{ba}\equiv\frac{1}{\sqrt{-g}}\nabla_{b}\left(\sqrt{-g}F^{ba}\right)=J^{a}\,,\hskip
14.22636pt\text{where}\hskip
14.22636pt\frac{1}{\sqrt{-g}}\nabla_{a}\sqrt{-g}=-\frac{1}{2}Q_{a}\,.$ (2.6)

The above formula reduces to $\nabla_{b}F^{ba}=J^{a}$ in Riemannian spacetime,
where $Q^{a}$ vanishes. Variations with respect to the metric field, on the
other hand, lead to Einstein’s equations (1) and the energy-momentum tensors
for the matter and Maxwell fields.

#####  2.3.2 Metric affine (gravitational) equivalent of the Maxwellian action
and field equations

We have seen that General Relativity accommodates separate field equations for
gravity and electromagnetism, which are derived by a common combined (or
‘coupled’) action. Let us now return to our question regarding the search for
an action reproducing inhomogeneous Maxwell-like equations for $\hat{R}_{ab}$,
under the condition: $S_{ab}{}^{c}=-(2/3)S_{[b}\delta_{a]}{}^{c}$ (so that the
homogeneous set (2.3) is also satisfied). Motivated by (2.5), the simplest
action we can imagine, consists of the Einstein-Hilbert and a gravitational
analogue of the Maxwellian-electromagnetic action-based on the correspondence
(2.4). Besides, our action (aside from the term $Q_{a}J^{a}$), is a particular
case of a general model, known as quadratic theory (e.g. see [14], [17] and
[2]). We consider the following:

$\mathcal{S}_{GEM}=\int{\left(R+\mathcal{L}_{\text{(m)}}-\frac{k^{2}}{4}\hat{R}_{ab}\hat{R}^{ab}-\frac{k}{2}Q_{a}J^{a}\right)\sqrt{-g}\,d^{4}x}\,,$
(2.7)

where $Q_{a}J^{a}$ represents a coupling between charged currents and the non-
metricity vector (in analogy with the coupling $A_{a}J^{a}$ between matter and
electromagnetic fields777The electric charge can be envisaged as a kind of
coupling constant between matter and electromagnetic fields.). Within the
spirit of our work, unlike electromagnetic fields, we do not envisage matter
(and therefore the current $J^{a}$) as a geometric quantity888Our
consideration, regarding the non-geometric origin of matter, differs from the
historical effort for gravito-electromagnetic unification in a metric-affine
framework, started by Eddington and developed by Einstein [9]. Therefore, the
term $Q_{a}J^{a}$ expresses a coupling between charged matter and an element
of metric affine curvature. It is worth noting that no new-unknown fields are
introduced, just a gravitational analogue of the classical electromagnetic
action. Moreover, all action terms are invariant under general coordinate
transformations (in contrast to e.g. [3]. Note that in the aforementioned
paper the homogeneous set of Maxwell equations is not satisfied). In eq. (2.7)
both $Q_{a}$ and therefore $\hat{R}_{ab}$ depend on the metric tensor as well
as on the connection (for details see [8]). Also, we shall keep in mind that
the metric appears in the Lagrangian inner products and scalars (i.e.
$\hat{R}_{ab}\hat{R}^{ab}=\hat{R}_{ac}\hat{R}_{bd}~{}g^{ab}g^{cd}$,
$Q_{a}J^{a}=Q_{a}J_{b}~{}g^{ab}$ and $R=R_{ab}~{}g^{ab}$).
First of all, let us consider metric variations of (2.7). Taking into account
the auxiliary relations in the appendix 2.A, we arrive at Einstein field
equations (of the form (1)) with stress-energy tensor
$T_{ab}=T^{\text{(m)}}_{ab}-(k^{2}/4)\hat{R}_{cd}\hat{R}^{cd}g_{ab}-k^{2}\hat{R}_{ac}\hat{R}^{c}{}_{b}$.
Note that $R_{ab}$ and $R$ contain now contributions from torsion and non-
metricity, while $T^{\text{(m)}}_{ab}$ refers to the energy-momentum tensor
for matter.
Regarding variations with respect to the connection (see the appendix), we
receive the following field equations:

$\frac{1}{2}Q_{c}g^{ab}-Q_{c}{}^{ab}-\frac{1}{2}Q^{a}\delta^{b}{}_{c}+q^{a}\delta^{b}{}_{c}+2S_{c}g^{ab}-S^{a}\delta^{b}{}_{c}+g^{ad}S_{dc}{}^{b}+k^{2}\delta^{b}{}_{c}{\rm
D}_{d}\hat{R}^{da}-kJ^{a}\delta^{b}{}_{c}=0\,,$ (2.8)

where ${\rm D}_{a}\equiv(1/\sqrt{-g})\nabla_{a}(\sqrt{-g}...)$. Note that the
first four terms represent the so-called Palatini tensor. Moreover, all the
first seven terms originate from the Einstein-Hilbert action, allowing for
non-vanishing torsion and non-metricity (see chapter 2 of [8]). Subsequently,
taking the three traces of (2.8), leads to the relations:

$\displaystyle-\frac{3}{2}Q^{a}+3q^{a}+$ $\displaystyle 4k^{2}{\rm
D}_{b}\hat{R}^{ba}$ $\displaystyle-4kJ^{a}-4S^{a}=0\,,\hskip
28.45274pt\frac{1}{2}Q_{a}+q_{a}+k^{2}{\rm
D}_{b}\hat{R}^{b}{}_{a}-kJ_{a}+4S_{a}=0$ (2.9) $\displaystyle\text{and}\hskip
14.22636pt{\color[rgb]{1,0,0}k{\rm
D}_{b}\hat{R}^{ba}=J^{a}}~{}~{}(\text{with}~{}{\color[rgb]{1,0,0}{\rm
D}_{a}J^{a}=0})\,.$

Note that eq. (2.9c) represents the inhomogeneous set of Maxwell equations.
Within the same (metric-affine) framework, action (2.5) would lead to the same
equations for the Faraday field, namely $\boldsymbol{{\rm
D}}_{\boldsymbol{b}}\boldsymbol{F}^{\boldsymbol{ba}}=\boldsymbol{J}^{\boldsymbol{a}}$.
Let us point out that eq. (2.9c) is essentially a consequence of two basic
mathematical properties and one physical property. In detail, the two
mathematical properties are: firstly, the similar mathematical construction
between the homothetic curvature $\hat{R}_{ab}$ and the Faraday $F_{ab}$
tensor field (i.e. written as the gradient of a vector field); secondly, the
linear dependence of the non-metricity vector $Q_{a}$ on the connection, so
that
$\delta_{\Gamma}Q_{a}=2\delta_{a}{}^{d}\delta_{c}{}^{b}\delta\Gamma^{c}{}_{bd}$.
The aforementioned physical property is associated with the action (2.7)
itself.

#####  2.3.3 Constraints

We observe that charge conservation is expressed in the form ${\rm
D}_{a}J^{a}=0~{}(\Leftrightarrow\nabla_{a}J^{a}=(1/2)Q_{a}J^{a})$. Moreover,
taking the nabla divergence of (2.9c), we come up with the constraint:

$\nabla_{a}J^{a}=\frac{k}{2}\left(\hat{R}_{ab}\hat{R}^{ab}+Q_{a}\nabla_{b}\hat{R}^{ba}\right)\hskip
14.22636pt\text{or}\hskip
14.22636ptk\hat{R}_{ab}\hat{R}^{ab}=Q_{a}\left(J^{a}-k\nabla_{b}\hat{R}^{ba}\right)\,.$
(2.10)

In other words, we have figured out that the last two terms in the action
(2.7) are actually related with each other through the above expression.
Subsequently, considering various combinations of the three traces in (2.9)
with the initial field equations (2.8) (this involves some lengthy but
straightforward algebra)999Note that due to the non-metricity requirement,
raising indices is no-longer a trivial operation. For instance, raising
indices in (2.9b) leads to:
$\frac{1}{2}Q^{a}+q^{a}-kQ_{b}{}^{ca}\hat{R}^{b}{}_{c}+\frac{k^{2}}{2}Q_{b}\hat{R}^{ba}+4S^{a}=0\,.$
, we eventually arrive at the constraints:

$Q^{a}=4q^{a}=-\frac{16}{3}S^{a}\,.$ (2.11)

Namely, within the framework of the action (2.7), the non-metricity and
torsion vectors are linearly dependent, so that they all together correspond
to only one degree of freedom. The same thing generally happens when
considering only the Einstein-Hilbert action (e.g. see [8]). In particular, it
is well-known that Einstein-Hilbert action does not reproduce general
relativity. Instead, it leads to Einstein’s field equations along with an
additional degree of freedom expressed by (2.11). As a consequence of the
latter, relation (2.4) recasts into:

$A_{a}=kQ_{a}=4kq_{a}-\frac{16}{3}kS_{a}\hskip 14.22636pt\text{and}\hskip
14.22636ptF_{ab}=k\hat{R}_{ab}=4kq_{ab}=-\frac{16}{3}kS_{ab}\,,$ (2.12)

where $q_{ab}\equiv\partial_{[a}q_{b]}$ and $S_{ab}\equiv\partial_{[a}S_{b]}$.
In other words, the vectorial degree of freedom expressed by (2.11) and
allowed by the Einstein-Hilbert action, provides a gravitational equivalent
for the Maxwell field. Furthermore, following some lengthy operations,
involving eqs. (2.8) and (2.9) (see Chapter $2$ of [8]), it can be shown that
the torsion and non-metricity tensors are related with the associated vectors
via

$S_{ab}{}^{c}=-\frac{2}{3}S_{[b}\delta_{a]}{}^{c}\hskip
14.22636pt\text{and}\hskip 14.22636ptQ_{abc}=\frac{1}{4}Q_{a}g_{bc}\,.$ (2.13)

The above constraints hold exactly the same for action (2.7), given that eq
(2.8) reduces to Einstein-Hilbert $\Gamma$-field equations under (2.9c).
Therefore, the homogeneous set of Maxwell equations in the form of (2.3), is
also satisfied by the $\hat{R}_{ab}$ field in the case of the action we
examine.

####  2.4 Closing remarks-Questions for further research

Although the present work was initially motivated by the problem of classical
gravito-electromagnetic unification, our study points out more a potential
equivalence between the Maxwell field and a metric affine component of the
gravitational field (i.e. homothetic or length curvature), analogous to mass-
energy equivalence. If someone would like to place the present effort within
the unified theories context, then it would belong somewhere between the lines
of Weyl and Eddington-Einstein. It shares some similarities with both the
aforementioned approaches but it essentially differs from both. In particular,
envisaging electromagnetism as a component of metric-affine gravity, dates
back to the efforts of Weyl, Eddington and Einstein [9], [18] (refer to the
aforementioned reviews for any information concerning past efforts and
failures of unification). Despite that unifying theories are widely regarded
by the modern scientific community as a vain dream (presumably because of a
long history of failures), history of physics tends to favour an
antidiametrically opposite point of view. Let us recall for instance, the many
new paths opened through unification of electricity and magnetism, as well as
of electromagnetic and weak interactions, in the distant and recent past.

Overall, we have shown that the antisymmetric part of the Ricci tensor, namely
the homothetic curvature, satisfies all of Maxwell equations. This finding
points out the fundamental question: Is it possible to exist two different
kinds of fields both satisfying Maxwell equations and describing different
things? If not, should electromagnetism be envisaged as a form, instead of a
source, of gravity on large scales? Alternatively, are electromagnetic fields
equivalent to gravitational fields, and which is the equivalence relation?.
Our work shows that there must be such an equivalence, taking the form of
(2.4), so that the Maxwell field can be calculated from a given metric. The
aforementioned relation implies that a given electromagnetic field has a
gravitational equivalent determined via the conversion constant $k$. It is
worth noting that there is a remarkable analogy between gravito-
electromagnetic (eq. (2.4)) and mass-energy equivalence, i.e. $E=mc^{2}$ ($k$
is the counterpart of $c^{2}$). Presenting and supporting the idea of a
potential gravito-electromagnetic equivalence is essentially the contribution
of the present piece of work. Therefore, two crucial questions arise.

Firstly, which is the nature of the conversion constant $k$ and how can it be
determined? Let us make a conjecture. On the one hand, we observe that the
action term $J_{a}Q^{a}$, introduced in (2.7), establishes a coupling between
matter and non-metricity, mediated by the electric charge. On the other hand,
within classical electrodynamics, the electric charge is known to act as a
coupling constant between matter and electromagnetic field (see $J_{a}A^{a}$
in (2.5)). The aforementioned double coupling potentially implies an
equivalence relation between non-metricity and the Maxwell field, where the
electric charge plays the role of the coupling constant. Besides, we take into
account that the electric charge has length dimension in geometrised units.
Therefore and in other words, we state the following question: Could the
coupling constant $k$ (with length geometrised dimension) be identified as the
total electric charge of a given charged distribution? If this is the case, it
would appear that the electric charge behaves on large-scales as a quantity
which translates a given electromagnetic field into its gravitational
equivalent. Furthermore, according to (2.4), with $k\rightarrow\mathcal{Q}$,
opposite charges correspond to homothetic curvature of opposite sign. Could
the macroscopic interaction between a positive and a negative charge
distribution be envisaged as a consequence of an ‘interaction’ between
opposite kinds of homothetic curvature?

Secondly, how are the properties of the Maxwell field
($A_{a}=kQ_{a}=(16/3)kS_{a}$, via (2.11)) reconciled with the geometric
significance/properties of non-metricity and torsion? The aforementioned
properties are respectively the change to a vector’s magnitude under its
parallel transport along a given curve, and the impossibility to form a closed
(small) parallelogram under parallel transport of one vector to the direction
of another [8].
Addressing the above exposed questions/problems is left to the future.

###  Appendix 2.A Metric and connection variations

In deriving the field equations within the main text, we make use of the
following relations for metric and connection variations [8], [14]. Concerning
the former, we have:

$\delta_{g}Q_{a}=\partial_{a}\left(g_{bc}\delta g^{bc}\right)\,,\hskip
11.38109pt\delta_{g}\hat{R}_{ab}=\partial_{[a}\delta_{g}Q_{b]}=0\,,\hskip
11.38109pt\delta_{g}\sqrt{-g}=-(1/2)\sqrt{-g}g_{ab}\delta g^{ab}\,,$
$\delta_{g}\left(\hat{R}_{ab}\hat{R}^{ab}\right)=\delta_{g}\left(\hat{R}_{ab}\hat{R}_{cd}g^{ac}g^{bd}\right)=-2\hat{R}_{ac}\hat{R}^{c}{}_{b}\delta
g^{ab}$

and

$\delta_{g}\left(\hat{R}_{ab}\hat{R}^{ab}\sqrt{-g}\right)=\left(-2\hat{R}_{ac}\hat{R}^{c}{}_{b}-(1/2)\hat{R}_{cd}\hat{R}^{cd}g_{ab}\right)\sqrt{-g}\\\
\delta g^{ab}\,.$

As for the latter, we deploy:

$\delta_{\Gamma}Q_{a}=2\delta_{a}{}^{d}\delta_{c}{}^{b}\delta\Gamma^{c}{}_{bd}\hskip
11.38109pt\text{and}\hskip
11.38109pt\delta_{\Gamma}(\hat{R}_{ab}\hat{R}^{ab})=-4\nabla_{b}\hat{R}^{ba}\delta\Gamma^{c}{}_{ca}=-4\nabla_{d}\hat{R}^{da}\delta^{b}{}_{c}\delta\Gamma^{c}{}_{ab}\,,$

with $\delta_{a}{}^{b}$ being the Kronecker symbol.

### Bibliography

  * [1] P. Mavrogiannis, Could electromagnetism be envisaged as a form of gravity in a metric affine framework?, arXiv e-print: 2308.00466 (submitted for publication) (2023).
  * [2] Y. N. Obukhov, E. J. Vlachynsky, W. Esser and F. W. Hehl, Effective Einstein theory from metric-affine gravity models via irreducible decompositions, Phys. Rev. D 56, 7769 (1997).
  * [3] N.V. Kharuk, S.A. Paston, A.A. Sheykin, Classical electromagnetic potential as a part of gravitational connection: ideas and history, Grav. & Cosm. 24, 219 (2018).
  * [4] R.W. Tucker and Ch. Wang, Black holes with Weyl charge and non-Riemannian waves, Class. Quantum Grav. 12, 2605 (1995).
  * [5] C.G. Tsagas, Magnetic tension and the geometry of the universe, Phys. Rev. Lett. 86, 5421 (2001).
  * [6] R. Maartens and B.A. Bassett, Gravito-electromagnetism, Class. Quantum Grav. 15, 717 (1998).
  * [7] R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  * [8] D. Iosifidis, Metric–Affine Gravity and Cosmology/Aspects of Torsion and non–Metricity in Gravity Theories, arXiv:gr-qc/1902.09643, Doctoral Thesis (2019).
  * [9] H.F.M. Goenner, On the History of Unified Field Theories, Living Rev. Rel. 7, 2 (2004).
  * [10] T. A. T. Sanomiya, I. P. Lobo, J. B. Formiga, F. Dahia and C. Romero, Invariant approach to Weyl’s unified field theory, Phys. Rev. D 102, 124031 (2020).
  * [11] M.P. Hobson and A.N. Lasenby, Weyl gauge theories of gravity do not predict a second clock effect, Phys. Rev. D 102, 084040 (2020).
  * [12] M.P. Hobson and A.N. Lasenby, Note on the absence of the second clock effect in Weyl gauge theories of gravity, Phys. Rev. D 105, L021501 (2022).
  * [13] R. De Luca, M. di Mauro, S. Esposito and A. Naddeo, Feynman’s different approach to electromagnetism, Eur. J. Phys. 40, 065205 (2019).
  * [14] D. Iosifidis, Metric-Affine Vector-Tensor Correspondence and Implications in F(R, T, Q, T, D) gravity, Physics of the Dark Universe 37, 101094 (2022).
  * [15] L.D. Landau and E.M. Lifshitz, A course in theoretical physics: Vol. II (4th ed.) (Butterworth-Heinemann, Amsterdam, 1975).
  * [16] F. De Felice and C. Clark, Relativity in Curved Manifolds (Cambridge University Press, Cambridge, 1990).
  * [17] F. W. Hehl and A. Macias, Metric affine gauge theory of gravity. 2. Exact solutions, Int. J. Mod. Phys. D 8, 399 (1999).
  * [18] L. O’Raifeartaigh, The Dawning of Gauge Theory (Princeton University Press, Princeton, New Jersey, 1997).

## Part II

####  2.2 Magnetised fluids in astrophysical and cosmic environments

Large-scale magnetic fields form an unambiguously existing component of the
universe’s energy content, which potentially contributes to cosmic dynamics
and structure formation (via its effects on density inhomogeneities). Although
magnetic fields are widely present in the universe, in both astrophysical and
cosmological scales (i.e. compact stellar objects, interstellar medium,
galaxies, galaxy clusters, intergalactic space), their origin, evolution and
role have not been adequately explained.

#####  2.2.1 The Magnetohydrodynamics (MHD) approximation

The unambiguous detection of large-scale magnetic fields along with the
parallel absence of large-scale electric fields, motivates the study of
magnetised fluid models. Within the theoretical framework of
magnetohydrodynamics (MHD) it is possible to isolate the magnetic component of
the Maxwell field by adopting a highly conducting fluid model (e.g. see [1]
for the covariant description). In particular, according to Ohm’s law applied
in the fluid’s rest frame,

$\mathcal{J}_{a}=\varsigma E_{a}\,,$ (2.14)

non-zero spatial currents arise for $E_{a}\rightarrow 0$ at the MHD limit
(i.e. $\varsigma\rightarrow\infty$, where $\varsigma$ is the conductivity of
the medium). Then, according to Alfvén’s theorem, the magnetic field lines of
a highly conducting fluid behave as being frozen into the fluid; they move
with it and always connect the same matter particles.
Under the MHD approximation, the magnetic field itself can be envisaged as a
viscous fluid with energy-momentum tensor:

$T^{\text{(magn)}}_{ab}=\frac{1}{2}B^{2}u_{a}u_{b}+\frac{1}{6}B^{2}h_{ab}-B_{\langle
a}B_{b\rangle}\,,$ (2.15)

where $\rho_{B}=B^{2}/2$, $P_{B}=B^{2}/6$ and $\pi^{B}_{ab}=-B_{\langle
a}B_{b\rangle}$ encode the magnetic energy density, isotropic pressure and
anisotropic stress respectively. Similarly, Maxwell equations reduce to the
following propagation equation:

$\dot{B}_{\langle a\rangle}=\left(-\frac{2}{3}\Theta
h_{ab}+\sigma_{ab}+\epsilon_{abc}\omega^{c}\right)B^{b}\,,$ (2.16)

also known as the magnetic induction equation, which is the MHD version of
Faraday’s law; and the three constraints

$\mathcal{J}_{a}=\epsilon_{abc}\dot{u}^{b}B^{c}+\text{curl}B_{a}\,,$ (2.17)
$2\omega^{a}B_{a}=\mu\quad\text{and}\quad{\rm D}^{a}B_{a}=0\,,$ (2.18)

where according to (2.17) the magnetic field lines remain frozen-in with the
matter, in the form of currents. In accordance with eq. (2.16), the magnetic
field is not sourced by currents but instead by purely kinematic effects. It
is worth noting that the multiplication of (2.16) by $B^{a}$ leads to the
following evolution formula for the magnetic energy density:

$\dot{\left(B^{2}\right)}=-\frac{4}{3}\Theta
B^{2}-2\sigma_{ab}\pi^{ab}_{B}\,.$ (2.19)

The above shows that in the absence of anisotropy, magnetic density changes as
$B^{2}\propto a^{-4}$ (with $a$ being the scale factor).
For an ideal fluid, the zero divergence of the total energy-momentum tensor
(matter plus magnetic field), $\nabla^{b}T_{ab}=0$ splits into the continuity
equation: $\dot{\rho}=-\Theta(\rho+P)$, where there is no magnetic
contribution; and the equation of motion (i.e. Euler’s equation):

$(\rho+P)\dot{u}_{a}=-{\rm D}_{a}P+\epsilon_{abc}\mathcal{J}^{b}B^{c}\,,$
(2.20)

where the pressure gradients and the magnetic Lorentz force are the remaining
causes of non-geodesic motion. Substituting the current from (2.17) into the
last term in the above relation and following the operations we arrive at:

$\epsilon_{abc}\mathcal{J}^{b}B^{c}=-B^{2}\dot{u}_{a}+\dot{u}^{b}B_{b}B_{a}-\frac{1}{2}{\rm
D}_{a}B^{2}+B^{b}{\rm D}_{b}B_{a}\,.$ (2.21)

With the aid of the last expression, Euler’s equation (2.20) recasts into:

$(\rho+P+B^{2})\dot{u}_{a}=-{\rm D}_{a}P+\dot{u}^{b}B_{b}B_{a}-\frac{1}{2}{\rm
D}_{a}B^{2}+B^{b}{\rm D}_{b}B_{a}\,.$ (2.22)

The last two terms in the right-hand side of the above relation split the
magnetic Lorentz force into its pressure and tension component respectively.

#####  2.2.2 Magnetic field evolution formulae under expansion/contraction

Let us focus our interest on Faraday’s formula (2.16) and the search of
solutions for the magnetic field. In particular, let us introduce the unit
spacelike vector field $n^{a}$, along the magnetic direction (i.e.
$u_{a}n^{a}=0$, $n_{a}n^{a}=1$ and $B^{a}=\mathcal{B}n^{a}$)101010For details
regarding the kinematics on the surface normal to $n^{a}$, see the following
chapter and its appendix section 3.C.. Therefore, projecting Faraday’s
equation along the magnetic direction $n^{a}$, it transforms into:

$\dot{\mathcal{B}}=\left(\Sigma-\frac{2\Theta}{3}\right)\mathcal{B}\,,$ (2.23)

with $\Sigma\equiv\sigma_{ab}n^{a}n^{b}$ the shear scalar component, a
quantity associated with volume expansion/contraction
($\Sigma=\Theta_{ab}n^{a}n^{b}-\Theta/3$ -see the comments below)111111In
fact, note that $\Sigma\equiv\sigma_{ab}n^{a}n^{b}\equiv({\rm D}_{\langle
b}u_{a\rangle})n^{a}n^{b}=({\rm
D}_{b}u_{a})n^{a}n^{b}-(\Theta/3)h_{ab}n^{a}n^{b}=u^{\prime}_{a}n^{a}-\Theta/3$,
where $u^{\prime}_{a}\equiv n^{b}{\rm D}_{b}u_{a}$.. Note that in deriving the
above we have taken into account that $\dot{B}_{\langle
a\rangle}=h_{a}{}^{b}\left(\dot{\mathcal{B}}n_{b}+\mathcal{B}\dot{n}_{b}\right)=\dot{\mathcal{B}}n_{a}+\mathcal{B}\alpha_{a}$
where $\dot{n}_{a}=\mathcal{A}u_{a}+\alpha_{a}$
($\mathcal{A}\equiv\dot{u}_{a}n^{a}$ and $\alpha_{a}$ its component lying on
the $2$-dimensional surface normal to $n^{a}$). Considering that
$\Sigma=\lambda\Theta$ ($\lambda$ being a real number), equation (2.23)
recasts into the solvable form:

$\dot{\mathcal{B}}=\left(\lambda-\frac{2}{3}\right)\Theta\mathcal{B}\,.$
(2.24)

From the above it is clear that only solutions with $\lambda<2/3$ predict a
decrease for the magnetic field during expansion (note that $\lambda=2/3$
implies constant magnetic density under volume changes). Therefore, we
envisage as physically senseful those solutions which satisfy the
aforementioned constraint. Moreover, in reference to the problem of magnetised
gravitational collapse, the requirement for pure contraction translates into a
negative volume scalar $\Theta<0$ and a negative generalised Hubble parameter
(see [16]), i.e. $\Theta_{ab}n^{a}n^{b}<0$. In detail, the generalised Hubble
relation reads:

$\Theta_{ab}n^{a}n^{b}=\Sigma+\frac{\Theta}{3}=\left(\lambda+\frac{1}{3}\right)\Theta<0\,,\hskip
22.76219pt\text{namely}\hskip 22.76219pt\lambda>-\frac{1}{3}\,,$ (2.25)

where $\Theta_{ab}\equiv\nabla_{(b}u_{a)}\equiv\sigma_{ab}+(\Theta/3)h_{ab}$
is the expansion tensor and $n^{a}$ is an arbitrary spatial ($3$-D) and
unitary vector. Overall, substituting $\Theta=3\dot{a}/a$ ($a$ being an
average scale factor) into eq. (2.24), leads to the following law of variation
for the magnetic field:

$\mathcal{B}\propto a^{3\lambda-2}\hskip 17.07164pt\text{with}\hskip
17.07164pt-\frac{1}{3}<\lambda<\frac{2}{3}\,,$ (2.26)

where its lower boundary is imposed by the problem of magnetised gravitational
contraction. The above fundamental expression (a new result within the
literature-as far as we know) is generally valid within the context of
magnetohydrodynamics. Throughout this chapter, we make use of it to study
magnetic fields in astrophysical and cosmological frameworks.
It is worth mentioning the particular condition under which eq. (2.26) holds
in orthogonal Bianchi models. Following [2], the kinematics of the
aforementioned models is constrained by:

$-\frac{\Theta}{3}\leq\Theta_{ab}n^{a}n^{b}<\Theta\hskip
28.45274pt\text{or}\hskip
28.45274pt-\frac{2\Theta}{3}\leq\Sigma<\frac{2\Theta}{3}\,,$ (2.27)

where $\Theta_{ab}\equiv\nabla_{(b}u_{a)}\equiv\sigma_{ab}+(\Theta/3)h_{ab}$
and $n^{a}$ is a vector tangent to the hypersurfaces of homogeneity. In
deriving eq. (2.27b), we have considered that
$\Theta_{ab}n^{a}n^{b}=\Sigma+\Theta/3$ (with
$\Sigma\equiv\sigma_{ab}n^{a}n^{b}$). Consequently, eq. (2.26) reduces, for
orthogonal Bianchi models, to:

$\mathcal{B}\propto a^{3\lambda-2}\hskip 17.07164pt\text{with}\hskip
17.07164pt-\frac{2}{3}\leq\lambda<\frac{2}{3}\,.$ (2.28)

The last expression introduces a lower boundary to the rate of
decrease/increase of the magnetic field during expansion or contraction
respectively.

###  Chapter 3 Gravito-magnetic elasticity and the problem of magnetised
gravitational collapse

Magnetic fields are a very special form of elastic medium. Within cosmic and
astrophysical environments (magnetised stars and protogalaxies) they
counteract shear and rotational distortions as well as gravitational collapse.
Their vector nature allows for their extraordinary coupling with spacetime
curvature in the framework of general relativity. This particular coupling
points out the way to study magnetic elasticity under gravitational
deformation. In the context of magnetohydrodynamics, we provide a covariant
description of a self-gravitating magnetised fluid; reveal the law of gravito-
magnetic elasticity and derive a parametric (kinematics dependent) evolution
formula for the magnetic field. Subsequently, we arrive at a non-collapse
criterion for a magnetised fluid, and address the question whether the
magnetic forcelines are able to prevent contraction towards a singularity. Our
answer depends on the explicit value of the parameter determining our magnetic
evolution formula. Finally, our argumentation peaks by suggesting a
calculation for the fracture limit of magnetic fieldlines under gravitational
contraction. Two illustrative applications, in a neutron star and a white
dwarf, accompany the results.

####  3.1 Introduction

It is well known (mainly from astrophysical studies of magnetised fluids, e.g.
see [3] and [4], but from relativistic as well [5]) that magnetic forcelines
behave like an elastic medium under their kinematic (shear or rotational)
deformation. Namely, in analogy with a spring under pressure, they develop
tension stresses resisting their deflection. However, it may be less known
that magnetic fieldlines present a similar behaviour under spacetime
distortions [6]-[10]. In particular, the aforementioned studies have shown
that the magnetic elasticity in question is expressed through a magneto-
curvature tension stress, coming from the Ricci identities. Therefore, from a
relativistic point of view, magnetic fields acquire particular interest due to
their direct coupling, as vectors, with spacetime curvature (see the
associated sections in chapters 1 and 2).

In detail, previous independent relativistic studies have supported the
following basic ideas regarding the behaviour of magnetic fields in curved
spacetimes. First, magnetic fields have the impressive ability not to self-
gravitate; in other words, not to contract or collapse under their own gravity
independently of the latter’s strength [11, 12]. Second, in the presence of an
external gravitational field, magnetic forcelines tend to stabilise themselves
by developing naturally curvature related tension stresses (monitored by Ricci
identities for the magnetic field) which resist their gravitational
deformation [11, 12]. Third, the key factor giving rise to such an
unconventional behaviour in both cases is the aforementioned magnetic
elasticity [7, 8, 9]. In a sense, it appears as though the elastic properties
of the magnetic forcelines are transferred into the fabric of the host space,
which seems to act like an elastic medium under tension.

In accordance with the above data, magnetic fieldlines counteract
gravitational implosion of a highly conducting fluid, and potentially hold it
up [7]-[14]. The question on such a possibility primarily motivated the work
of this chapter (see [15]). In reference to this problem and given the elastic
behaviour of magnetic fields, one can raise another associated question
concerning the existence of a possible elastic and more crucially a fracture
gravito-magnetic limit (i.e. an amount of gravity under which magnetic
forcelines lose their coherence-cease to exist). Moreover, if such a fracture
limit exists, could we have an estimation of it for a given magnetised
(collapsing) star? Crucially, could magnetic forcelines manage to disrupt
gravitational collapse before reaching their fracture limit [10]? Besides, it
is worth noting that a systematic theoretical description for gravito-magnetic
elasticity is elusive. In addressing all of the above questions, one needs an
evolution formula for the magnetic field of a highly conductive, self-
gravitating fluid. In alignment with the magnetic elasticity, we introduce an
approximation leading to such a law of variation, which we subsequently apply
to the aforementioned problems.

####  3.2 Kinematically induced magnetic tension stresses

To begin with, let us consider the decomposition of the magnetic 3-D gradient
$\rm D_{b}B_{a}$ into its symmetric (trace-free), antisymmetric and trace
part. In other words,

${\rm D}_{b}B_{a}={\rm D}_{\langle b}B_{a\rangle}+{\rm
D}_{[b}B_{a]}+\frac{1}{3}({\rm D}^{c}B_{c})h_{ab}\,,$ (3.1)

which reveals the individual tension111Actually the magnetic tension force
vector refers to the directional derivative along the field itself. See the
following analysis. components triggered by, and resisting to shape (i.e.
$\sigma^{(B)}_{ab}={\rm D}_{\langle b}B_{a\rangle}$), rotational (i.e.
$\omega^{(B)}_{ab}={\rm D}_{[b}B_{a]}$) and volume distortions (i.e.
$\Theta^{(B)}={\rm D}^{a}B_{a}$) of the magnetic forcelines respectively.
Besides, at the magnetohydrodynamic limit (MHD), the tension component
opposing to volume expansion/contraction (last term) vanishes (i.e. ${\rm
D}^{c}B_{c}=0$ from Gauss’s law). In the above, ${\rm
D}_{a}=h_{a}{}^{b}\nabla_{b}$ is the projected (3-D) covariant derivative
operator and $h_{ab}=g_{ab}+u_{a}u_{b}$ (with $g_{ab}$ being the spacetime
metric and $u^{a}$ being a timelike $4$-velocity vector) an operator
projecting upon the observer’s (3-D) rest-space. The covariant kinematics of
the magnetic tension stresses is monitored by the Ricci identities for the
magnetic field:

$2\nabla_{[a}\nabla_{b]}B_{c}=R_{abcd}B^{d}\,,$ (3.2)

where $R_{abcd}$ is the Riemann spacetime tensor. In particular, the timelike
part of the above leads to propagation equations for the magnetic shear
$\sigma^{(B)}_{ab}$ and vorticity
$\omega^{(B)}_{a}=\epsilon_{abc}\omega^{bc}$. On the other hand, its spacelike
part leads to divergence conditions (constraints) for the aforementioned
quantities. The equations in question, appearing here for the first time-as
far as we know-, could prove useful when studying the kinematics of magnetised
fluids in various contexts. However, as we do not make any use of those in the
present thesis, we have chosen to place them in the brief appendix 3.A.

####  3.3 Gravitationally induced magnetic tension stresses

In analogy with their deflection due to kinematic effects, associated with the
fluid’s motion, magnetic forcelines counteract their gravitational distortion.
Where does the corresponding magneto-curvature tension stress come from? The
answer lies in the direct coupling of magnetic fields (as vectors) with
spatial curvature via the (3-D projected) Ricci identities (e.g. see [14] or
[15]),

$2{\rm D}_{[a}{\rm D}_{b]}B_{c}=-2\omega_{ab}\dot{B}_{\langle
c\rangle}+\mathcal{R}_{dcba}B^{d}\,,$ (3.3)

where $\mathcal{R}_{abcd}$ represents the 3-D counterpart of the Riemann
tensor. Note that the aforementioned coupling manifests itself at the second
differentiation order.

#####  3.3.1 Describing the magneto-curvature tension stress

Let us consider the 3-gradient of the magnetic tension force vector
$\tau_{a}=B^{b}{\rm D}_{b}B_{a}$ (the non-zero tension force implies that the
magnetic fieldlines are not spacelike geodesics). Employing the 3-D Ricci
identities (3.3) we arrive at:

${\rm D}_{c}\tau_{a}={\rm D}_{c}B^{b}{\rm D}_{b}B_{a}+B^{b}{\rm D}_{b}{\rm
D_{c}}B_{a}+2\omega_{bc}B^{b}\dot{B}_{\langle
a\rangle}+\mathcal{R}_{dabc}B^{b}B^{d}\,.$ (3.4)

The first three terms involve kinematic effects through eq (3.1) whilst the
last one can be envisaged as the magneto-curvature tension stress (or the
gradient of the magneto-curvature tension component). If $n^{a}$ is the
magnetic field direction (i.e. $B^{a}=\mathcal{B}n^{a}$), the term in question
can alternatively be written as:

$s_{ac}=\mathcal{R}_{dabc}B^{b}B^{d}=\mathcal{B}^{2}\mathcal{R}_{dabc}n^{b}n^{d}=-\mathcal{B}^{2}u_{ac}\,,$
(3.5)

where $u_{ac}\equiv-\mathcal{R}_{dabc}n^{b}n^{d}$ can be envisaged as a kind
of strain tensor222Typically, within conventional elastic mechanics the strain
tensor is defined to be a dimensionless symmetric quantity [17]. However, here
we allow for a non-vanishing anti-symmetric part taking into account any
torsional deformation. Furthermore, our strain tensor has inverse square
length dimensions in geometrised units., describing spatial distortions of the
magnetic forcelines (for a commentary on the law of magnetic elasticity under
volume gravitational distortions refer to section 3.5). Our definition for the
strain tensor is metric independent and thus essentially differs from its
counterpart (4.9) in [18]. As for the stress tensor $s_{ab}$, it includes
those forces which act against (see the following discussion on the problem of
gravitational collapse) spatial curvature and tend to restore the forcelines
to their initial state. Overall, the meaning of (3.5) is the following. Due to
spatial curvature, the magnetic fieldlines are bent and twisted. In analogy
with an elastic rod under pressure, they react via the restoring stress
$s_{ab}$ which increase in proportion to the amount of deformation $u_{ab}$
(Hooke’s law of elasticity) and the magnetic density. In fact, when appearing
in the kinematic equations for a magnetised fluid, it turns out that the
magneto-curvature tension stress depends on the ratio of the magnetic density
over the total system’s density (i.e. matter and magnetic fields-see eq.
(3.18) in the following).

Let us recall that any kind of deformation can be reduced into a sum of a pure
shear ($u_{\langle ab\rangle}=\mathcal{R}_{d\langle ab\rangle c}n^{c}n^{d}$),
a torsional one (or twisting)333For the sake of accuracy, vorticity or
rotational deformations are included in the shear/shape type of distortions as
well. ($u_{[ab]}=\mathcal{R}_{d[ab]c}n^{c}n^{d}$) and a hydrostatic
compression ($(u^{c}{}_{c}/3)h_{ab}=(1/3)\mathcal{R}_{cd}n^{c}n^{d}h_{ab}$).
Hence, on splitting the magneto-curvature tension stress into its symmetric
trace-free ($s_{\langle ac\rangle}$), antisymmetric ($s_{[ac]}$) and trace
part ($s=s^{c}{}_{c}$), we receive its associated component counteracting
shape, rotational and volume changes respectively, due to gravity (see
appendix 3.E). Of the aforementioned components we focus here on the last one.
Hence, considering the trace of (3.5) and the double projection of the Gauss-
Codacci formula (e.g. see eq. (1.3.39) in [19] and eq (92) in [10]) along the
magnetic direction $n^{a}$, we deduce that

$s=s^{c}{}_{c}=\mathcal{B}^{2}\mathcal{R}_{bd}n^{b}n^{d}=\mathcal{B}^{2}\left[\frac{2}{3}\rho+\mathcal{E}+\frac{\Pi}{2}+\left(\lambda^{2}-\frac{\lambda}{3}-\frac{2}{9}\right)\Theta^{2}+\alpha_{a}\alpha^{a}\right]\,,$
(3.6)

where $\rho$ denotes the energy density of matter;
$\Pi\equiv\pi_{ab}n^{a}n^{b}$ and $\mathcal{E}\equiv E_{ab}n^{a}n^{b}$ are the
anisotropic stress $\pi_{ab}$ and the tidal (or electric Weyl) tensor
$E_{ab}$, twice projected along the magnetic direction, respectively. In
deriving the above we have employed $\Sigma=\lambda\Theta$ and eqs (3.60) and
(3.63) from the appendix 3.E (see eq. (3.14) as well). Also, assuming an ideal
fluid model, the anisotropic stress terms in the above vanish. Then, of
particular interest is that the deformation due to gravitational
compression/expansion in (3.6) is determined by the density of matter and the
tidal tensor projected along the magnetic fieldlines. Note that there is no
magnetic input in (3.6). In fact, recalling that
$\rho_{\text{(magn)}}=\mathcal{B}^{2}/2$,
$\pi^{(\text{magn})}_{ab}=-B_{\langle a}B_{b\rangle}$ and therefore
$\Pi^{\text{(magn)}}\equiv\pi^{\text{(magn)}}_{ab}n^{a}n^{b}=-(2/3)\mathcal{B}^{2}$,
it turns out that the magnetic anisotropic stress exactly cancels the magnetic
energy density contribution (see also [9]). The aforementioned conclusion
becomes straightforward if we assume the simple case of a magnetic field as
the sole energy source of spatial curvature. Then, it is clear that (see also
[9])

$\mathcal{R}_{ab}=\frac{1}{3}\mathcal{B}^{2}h_{ab}+\frac{1}{2}\pi^{\text{(B)}}_{ab}\hskip
17.07164pt\text{with}\hskip 17.07164pt\mathcal{R}_{ab}n^{a}n^{b}=0\,,$ (3.7)

irrespective of the field’s strength. In other words, magnetic fields do not
self-gravitate or do not ‘feel’ their own gravity, no matter how strong the
latter may be.

####  3.4 Gravitational collapse of a magnetised fluid

The gravitational collapse of compact stellar objects, like white dwarfs,
neutron stars, black holes, as well as that of protogalactic clouds usually
involves (weak or strong) magnetic fields. In the context of general
relativity, independent studies have pointed out the unconventional tendency
of the $B$-fields to resist their own gravitational implosion. The same works
have also raised the question as to whether the magnetic presence and the
resulting Lorentz forces could actually halt the contraction of the
surrounding collapsing matter [7]-[12]. In addition, alternative studies of
charged collapse, this time employing the repulsive (electrostatic) Coulomb
forces, have found that the latter could also prevent the formation of
spacetime singularities [21]–[23]. The present section probes the
gravitational collapse of a highly conductive charged medium by means of the
Raychaudhuri equation and along the lines of [7]-[9]. Making a step further,
we take advantage of a $1+2$ spatial splitting and arrive at a simple
criterion which could decide the ultimate fate of homogeneously contracting
magnetised media.

#####  3.4.1 Using the Raychaudhuri equation

Traditionally, theoretical studies of gravitational collapse make use of the
Raychaudhuri equation which has been made famous as a keystone of singularity
theorems. Besides, in general terms, the formula in question covariantly
describes the volume evolution of a self–gravitating fluid element. In this
first subsection, we revisit the problem of gravitational implosion of a
highly conducting (magnetised) fluid with the aid of the Raychaudhuri
equation444Apart from its conventional application to timelike worldlines of
real (or hypothetical) observers, the aforementioned equation has been applied
to spacelike and null curves as well (e.g. see [5, 28, 29])., and in light of
our new knowledge regarding the behaviour of the associated magnetic field
(more specifically of relation (3.71)), as well as of our new developments in
the context of the 1+1+2 covariant formalism. Unlike previous independent
works, our study builds upon past research (see [9]-[16]) and leads to a
simple and clear criterion determining the fate of homogeneous and magnetised
gravitational collapse.

Before proceeding to the analysis, let us have in mind two crucial points.
Firstly, magnetic-line deformations are usually caused by electrically charged
particles, however relativistic spacetime curvature (gravity) also potentially
behaves as a deforming agent [7, 9]. Secondly, the magnetic tension reflects
the elasticity of the field lines and their tendency to react against any
agent that distorts them from equilibrium [9, 16, 18].

Let us start with the Raychaudhuri equation, which we have already written in
the form of (3.73). To proceed, we need to calculate the 3-divergence of the
acceleration vector (i.e. ${\rm D}^{a}\dot{u}_{a}$) which gives rise to
magneto-curvature tension terms, of crucial importance for our relativistic
study. In order to facilitate the analytic calculations, we assume that the
contracting fluid has nearly homogeneous matter555Note that the homogeneity of
the matter fields is a rather common approximation. In fact, spatial
homogeneity is a standard assumption in all typical singularity theorems [30,
31]. Besides, the assumption of homogeneous matter distribution does not
essentially affect the validity of our argument, since gradients in the fluid
and in the magnetic density distribution tend to inhibit gravitational
contraction, even within Newtonian physics. and magnetic energy density
distributions (${\rm D}_{a}\rho\simeq 0\simeq{\rm D}_{a}P\simeq{\rm
D}_{a}B^{2}$, where an equation of state of the form $P=P(\rho)$ has been
considered). However, we allow for $B^{b}{\rm D}_{b}B_{a}\neq 0$, so that we
can study effects caused by distortions of the magnetic forcelines (see the
following discussion). Subsequently, taking the 3-divergence of (2.22) in
combination with the 3-Ricci identities (eq. (3.3)) and Maxwell’s equations
(eq. (1.16)) we arrive at:

${\rm
D}^{a}\dot{u}_{a}=c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}+2(\sigma^{2}_{B}-\omega^{2}_{B})\,,$
(3.8)

where the scalars $\sigma^{2}_{B}={\rm D}_{\langle b}B_{a\rangle}{\rm
D}^{\langle b}B^{a\rangle}/2(\rho+P+B^{2})$ and $\omega^{2}_{B}={\rm
D}_{[b}B_{a]}{\rm D}^{[b}B^{a]}/2(\rho+P+B^{2})$ represent the magnetic
analogues of the shear and the vorticity respectively. Of special interest is
the purely relativistic (magneto-geometric) term $\mathcal{R}_{ab}n^{a}n^{b}$
which describes 3-D distortions of the magnetic forcelines due to the
curvature of the host spacetime. Note that all the terms on the right-hand
side of (3.8) are tension stresses triggered by the deformation of the
magnetic field lines. Each of these terms acts against the agent that caused
the deformation in the first place (e.g. the magneto-vorticity
$\omega^{2}_{B}$ is caused by rotational effects, $\omega^{2}$, and it tends
to counterbalance them. Observe the opposite signs of the pairs $\omega^{2}$,
$\omega^{2}_{B}$ and $\sigma^{2}$, $\sigma^{2}_{B}$ in (3.9)). Substituting
expression (3.8) into the Raychaudhuri equation (3.73), describing the
convergence/divergence of the timelike worldlines tangent to the $u^{a}$
field, our equation reads:

$\dot{\Theta}+\frac{1}{3}\Theta^{2}=-R_{ab}u^{a}u^{b}+c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}-2(\sigma^{2}-\sigma^{2}_{B})+2(\omega^{2}-\omega^{2}_{B})+\dot{u}^{a}\dot{u}_{a}\,,$
(3.9)

where $R_{ab}u^{a}u^{b}=(\rho+3P+B^{2})/2>0$ represents the total
(gravitational) energy density of the system. Note that if
$\dot{\Theta}+\frac{1}{3}\Theta^{2}<0$, the above equation implies that an
initially contracting congruence of worldlines will focus at a point
($\Theta\rightarrow-\infty$) within finite proper time. Hence, positive terms
on the right-hand side of the Raychaudhuri formula act against gravitational
collapse whilst negative ones in the inverse way.
Concerning the convergence/divergence of spacelike curves tangent to the
$n^{a}$ field (i.e. the magnetic forcelines), it can be described by the
following version of the Raychaudhuri equation (e.g. see [5] or [9]):

$\tilde{\Theta}^{\prime}=-\frac{1}{2}\tilde{\Theta}^{2}-\mathcal{R}_{ab}n^{a}n^{b}-2(\tilde{\sigma}^{2}-\tilde{\omega}^{2})+\tilde{D}^{a}n^{\prime}_{a}-n^{\prime
a}n^{\prime}_{a}\,.$ (3.10)

In the above, $\tilde{\sigma}^{2}=\tilde{D}_{\langle
b}n_{a\rangle}\tilde{D}^{\langle b}n^{a\rangle}/2$ and
$\tilde{\omega}^{2}=\tilde{D}_{[b}n_{a]}\tilde{D}^{[b}n^{a]}/2$ refer to the
magnitudes of the $2$-dimensional counterparts of the shear and the vorticity
tensor respectively. The prime denotes spatial differentiation along the
direction $n^{a}$. It worth considering the pure magnetic field case (i.e.
only magnetic field filling spacetime). Recalling that the magnetic field does
not self-gravitate (i.e. $\mathcal{R}_{ab}n^{a}n^{b}=0$), eq. (3.10) recasts
into:

$\tilde{\Theta}^{\prime}=-\frac{1}{2}\tilde{\Theta}^{2}\hskip
11.38109pt\text{accepting the solution}\hskip
11.38109pt\tilde{\Theta}(l)=\frac{\tilde{\Theta}_{0}}{2+\tilde{\Theta}_{0}l}\,,$
(3.11)

where $l$ denotes the proper length along the magnetic direction. Therefore,
we consider the following cases. First, if $\tilde{\Theta}_{0}>0$, the
magnetic forcelines keep diverging. Second and opposite, if
$\tilde{\Theta}_{0}<0$, the magnetic forcelines converge within finite proper
length (i.e. $\tilde{\Theta}_{0}\rightarrow-\infty$ as
$l\rightarrow-2/\tilde{\Theta}_{0}$). Last but perhaps most interestingly, if
the magnetic fieldlines happen to be at natural length (i.e.
$\tilde{\Theta}_{0}=0$), will remain so indefinitely unless an external agent
interferes. Note that the last case profoundly reveals, once again, the
elasticity of magnetic field lines, which fully counterbalance their gravity
via their tension.

#####  3.4.2 A non–collapse criterion

Having in mind the strong gravity conditions which characterise collapsing
compact stellar objects (and the counterbalancing relation of the paired terms
in (3.9)), we choose to focus our attention on the purely relativistic-
curvature terms666Note that $\dot{u}^{a}\dot{u}_{a}>0$ always, and therefore
it resists contraction in any case. (i.e.
$c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}$ which is positive in all cases
of realistic gravitational collapse and thus tends to inhibit the
gravitational pull of local matter, as encoded by the expression
$R_{ab}u^{a}u^{b}$). Besides, we expect that gravity ultimately dominates over
kinematics at the final stages of the collapse.
Regarding the magneto-geometric tension stress
$c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}$, it grows strong with
increasing curvature distortion during the collapse, in analogy with the
resisting power of a compressed elastic medium. In particular, if at some time
during implosion the magneto-curvature tension overwhelms gravity, namely if
the following condition holds,

$c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}>R_{ab}u^{a}u^{b}\,,$ (3.12)

we expect that contraction will be halted. In addition, if
$\dot{\Theta}+\Theta^{2}/3>0$ overall, it will turn into expansion. Making use
of the Gauss-Codacci formula (e.g. see expression (1.3.39) in [19]), the above
condition transforms into:

$2c^{2}_{\mathcal{A}}(\rho-\frac{1}{3}\Theta^{2})+3\alpha_{a}\alpha^{a}+3c^{2}_{\mathcal{A}}(E_{ab}-\frac{1}{3}\Theta\sigma_{ab}+\sigma_{ca}\sigma^{c}{}_{b}-\omega_{ca}\omega^{c}{}_{b})n^{a}n^{b}>\frac{3}{2}(\rho+3w\rho+\mathcal{B}^{2})\,,$
(3.13)

where the first of the two parentheses in the left-hand side represents the
isotropic part of the tension stress whilst the second one, the anisotropic.
It turns out that the latter must be nonzero which implies that the
gravitational collapse has to be anisotropic, if the tension stress is to
outbalance the gravitational pull of matter.
Subsequently, we will simplify our non-collapse condition by using a $1+2$
split of the spatial components and calculating the double projection of the
various quantities along $n^{a}$: $\mathcal{E}\equiv E_{ab}n^{a}n^{b}$,
$\Sigma=\lambda\Theta$, $\Sigma_{a}=\epsilon_{ab}\Omega^{b}+\alpha_{a}$,
$\sigma_{ca}\sigma^{c}{}_{b}n^{a}n^{b}=\Sigma^{2}+\Sigma^{a}\Sigma_{a}=\lambda^{2}\Theta^{2}+\Omega^{a}\Omega_{a}+\alpha_{a}\alpha^{a}$
and $\omega_{ca}\omega^{c}{}_{b}n^{a}n^{b}=\Omega^{a}\Omega_{a}$. Overall,
spatial distortions along the magnetic direction are encoded by the associated
projection of the Gauss-Codacci formula (recall (1.3.39) in [15]):

$\mathcal{R}_{ab}n^{a}n^{b}=\frac{2}{3}\rho+\mathcal{E}+\left(\lambda^{2}-\frac{\lambda}{3}-\frac{2}{9}\right)\Theta^{2}+\alpha_{a}\alpha^{a}\,.$
(3.14)

Recall that the anisotropic matter pressure (i.e.
$\Pi/2\equiv(\pi_{ab}n^{a}n^{b})/2$) has been set equal to zero on assuming a
perfect fluid model. It is worth noting that the effects of rotation, as
expressed by the vorticity vector, and included in the term
$\mathcal{R}_{ab}n^{a}n^{b}$, exactly cancel out. Also, the last term in the
right-hand side of the above is always positive (i.e.
$\alpha_{a}\alpha^{a}>0$), acting thus against contraction in all cases.
Taking then into account that:

$\frac{1}{2c^{2}_{\mathcal{A}}}\left(\rho+3P+\mathcal{B}^{2}\right)=\frac{1}{2}\left(1+4w+3w^{2}\right)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+(2w+1)\rho+\frac{1}{2}\mathcal{B}^{2}\,,$
(3.15)

and substituting (3.14) into the non-collapse criterion (3.12), the latter
becomes:

$\mathcal{E}+\left(\lambda^{2}-\frac{\lambda}{3}-\frac{2}{9}\right)\Theta^{2}+\alpha_{a}\alpha^{a}>\frac{1}{2}\left(1+4w+3w^{2}\right)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+\frac{1}{3}\left(1+6w\right)\rho+\frac{1}{2}\mathcal{B}^{2}\,.$
(3.16)

Subsequently, we consider, as application, some specific values of $\lambda$
parameter within the interval $(-1/3\,,~{}2/3)$ (determining the rate of
change for $\mathcal{B}$-recall the introduction to Part II). Given the rate
of change of matter density (i.e. $\rho\propto a^{-3(1+w)}$), and allowing
sufficient time for the collapse to evolve, we write down the dominant terms
composing the non-collapse criterion (3.16):

$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>\frac{1}{2}\mathcal{B}^{2}+\frac{26}{225}\Theta^{2}\hskip
22.76219pt\text{for}\hskip 28.45274pt\lambda=-\frac{1}{5}:\hskip
5.69054pt\mathcal{B}\propto a^{-2.6}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>\frac{1}{2}\mathcal{B}^{2}+\frac{5}{36}\Theta^{2}\hskip
28.45274pt\text{for}\hskip 28.45274pt\lambda=-\frac{1}{6}:\hskip
5.69054pt\mathcal{B}\propto a^{-2.5}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>\frac{1}{2}\mathcal{B}^{2}+0.21\Theta^{2}\hskip
22.76219pt\text{for}\hskip 28.45274pt\lambda=-\frac{1}{30}:\hskip
5.69054pt\mathcal{B}\propto a^{-2.1}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>\frac{1}{2}\mathcal{B}^{2}+\frac{2}{9}\Theta^{2}\hskip
34.1433pt\text{for}\hskip 28.45274pt\lambda=0:\hskip
5.69054pt\mathcal{B}\propto a^{-2}\hskip 5.69054pt\text{and}\hskip
8.53581ptw<\frac{1}{3}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>f(w)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+\frac{2}{9}\Theta^{2}\hskip
28.45274pt\text{for}\hskip 28.45274pt\lambda=0:\hskip
5.69054pt\mathcal{B}\propto a^{-2}\hskip 5.69054pt\text{and}\hskip
8.53581ptw>\frac{1}{3}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>f(w)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+\frac{1}{9}\Theta^{2}\hskip
28.45274pt\text{for}\hskip 28.45274pt\lambda=\frac{1}{6}:\hskip
5.69054pt\mathcal{B}\propto a^{-1.5}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>f(w)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+\frac{2}{9}\Theta^{2}\hskip
28.45274pt\text{for}\hskip 28.45274pt\lambda=\frac{1}{3}:\hskip
5.69054pt\mathcal{B}\propto a^{-1}\,;$
$\displaystyle\mathcal{E}+\alpha_{a}\alpha^{a}>f(w)\left(\frac{\rho}{\mathcal{B}}\right)^{2}+\frac{5}{36}\Theta^{2}\hskip
22.76219pt\text{for}\hskip 25.60747pt\lambda=\frac{1}{2}:\hskip
5.69054pt\mathcal{B}\propto a^{-0.5}\,,$

where $f(w)\equiv\left(1+4w+3w^{2}\right)/2$. The crucial quantities involved
in magnetised contraction are, according to our relativistic criterion: the
tidal stress tensor along the magnetic direction, i.e. $\mathcal{E}$, the
magnetic energy density, the ratio of (squared) matter to magnetic energy
density and the squared volume scalar. At this point let us recall that
$\mathcal{E}$ comes from $\mathcal{R}_{ab}n^{a}n^{b}$, and therefore describes
tidal distortions of the magnetic forcelines. Similarly,
$c^{2}_{\mathcal{A}}\mathcal{E}$, originating from
$c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}$, represents magneto-tidal
tension stresses, triggered by the increasing tidal deformation of the
magnetic fieldlines during contraction, and acting against the aforementioned
magnetic bending. On the other hand, the various density terms, appearing in
(3.4.2), contribute to the total gravitational energy (i.e.
$R_{ab}u^{a}u^{b})$, reinforcing thus the implosion process. It is worth
noting that the case $\lambda=-1/3$, for which the criterion becomes
particularly simple, refers to two dimensional contraction (see appendix 3.C
for further details).
To illustrate further the meaning of $\mathcal{E}$, let us recall that in
terms of Newtonian gravity, $E_{ab}$ is associated with the second-order
derivative of the gravitational potential $\Phi$ (precisely the Newtonian
tidal tensor) or equivalently with the first-order derivative of the tidal
forces $F^{a}$, in accordance with (e.g. see [1]):777In the context of
Newtonian theory, studying tidal forces presupposes the consideration of at
least two distinctive massive bodies. However, from a relativistic point of
view, we can envisage tidal forces as a result of the different curvature
effects (caused by the fluid’s spacetime energy distribution) experienced by
distinctive particles of the magnetised fluid.

$E^{\text{(Newt)}}_{ab}=\partial_{a}\partial_{b}\Phi-\frac{1}{3}(\partial^{c}\partial_{c}\Phi)h_{ab}\hskip
42.67912pt\text{and}\hskip
42.67912pt\mathcal{E}^{\text{(Newt)}}=\mathcal{F}^{\prime}-F^{a}n^{\prime}_{a}\,,$
(3.17)

where the latter relation comes from the double projection of the former along
$n^{a}$, and $\mathcal{F}\equiv F^{a}n_{a}$, $F^{a}n^{\prime}_{a}$ correspond
to tidal stresses acting along and normal to the magnetic forcelines
respectively.

Predicting actually the fate of nearly homogeneous gravitational collapse of a
highly conducting (magnetised) fluid remains an open question. Our results
indicate that the latter question reduces to whether the combination of the
tidal tensor along the magnetic field lines, and the squared volume scalar
${\Theta}^{2}$, increases faster than the magnetic or the matter to magnetic
energy density ratio. The answer seems to depend on the geometric background
in hand, and potentially on the problem’s initial conditions. Moreover, our
non-collapse criterion assumes the availability of sufficient time for the
contraction to evolve. Therefore, the possibility that the magnetic fieldlines
are broken meanwhile (before a potential halt of the process), is bypassed. We
address in detail the aforementioned crucial issue within the following
sections.

####  3.5 The law of magnetic elasticity under (volume) gravitational
distortions

With the present section we move on to the essential part of our work. In
detail, we discuss the law of magnetic elasticity, describe an enlightening
analogy to magnetised collapse, calculate the magnetic fracture limit and
apply it to the problem of gravitational collapse of compact stellar objects.
The present section should be studied along with appendix 3.C.

In reference to equation (3.9) we particularly observe that the magneto-
curvature tension stress:

$s^{*}=-c^{2}_{\mathcal{A}}u\,,$ (3.18)

counteracts the magnetised fluid’s gravity,
$R_{ab}u^{a}u^{b}=(1/2)(\rho+3P+\mathcal{B}^{2})>0$, namely the cause of
magnetic (volume) distortion in the form of
$u\equiv-\mathcal{R}_{ab}n^{a}n^{b}$ (so that $u<0$ is associated with closed
spatial sections and compression). We plausibly require that
$\mathcal{R}_{ab}n^{a}n^{b}>0$ ($u<0$) at all times during gravitational
contraction. In complete analogy with (3.9), the symmetric-trace-free and the
antisymmetric counterparts of (3.18) can be obtained via the shear and the
vorticity propagation formulae respectively, along with Euler’s equation of
motion. However, here we examine the magnetic elasticity against
gravitationally induced volume distortions (i.e. gravitational contraction).

#####  3.5.1 Insight into the law of magnetic elasticity

The meaning of (3.18)888The expression in question has appeared several times
in past works (e.g. see [7]-[10]) but it was not recognised or envisaged as an
expression of Hooke’s law of elasticity and therefore was not given its full
interpretation presented here. is that the tension stress $s^{*}$, tending to
restore the magnetic field into its initial (undeformed) state, is
proportional to the (gravitationally induced) volume distortion of the
magnetic forcelines (see also the appendices 3.C and 3.E)999Written here for
an ideal (magnetised) fluid.. According to eq. (3.14) (on assuming that the
magnetic direction remains constant during contraction, i.e. $\alpha_{a}=0$),
the latter reads:

$u\equiv-\mathcal{R}_{ab}n^{a}n^{b}=-\frac{2}{3}\rho-\mathcal{E}+g(\lambda)\Theta^{2}\,,$
(3.19)

where
$g(\lambda)\equiv-(\lambda^{2}-\lambda/3-2/9)=-(\lambda-2/3)(\lambda+1/3)$.
The minus sign in the right-hand side of (3.18) implies that $s^{*}$ acts
against the increasing magnetic distortion $u$. Note that in deriving the
above, we have taken into account eq. (1.3.41) from [19], as well as relations
(3.60), (3.63) from the appendix 3.C. It is straightforward to see that
condition $\mathcal{E}=-(2/3)\rho+g(\lambda)\Theta^{2}$ corresponds to the
natural (undeformed) ‘volume’ state of the magnetic field, where $s^{*}=0$.
The proportionality factor $0<c^{2}_{\mathcal{A}}<1$ (note the difference to
(3.5)) is always positive and its definition implies that the greater the
magnetic density contribution to the total fluid’s density, the more rigid the
magnetic fieldlines are (or the more they resist to their deformation). In
other words, eq (3.18) is a relativistic expression of Hooke’s law of
elasticity for a gravitationally distorted magnetic field, frozen into a
highly conducting fluid. Nevertheless, in contrast to an elastic spring, the
proportionality factor $c^{2}_{\mathcal{A}}$ is not a constant but a variable
quantity (a function of the ratio $\rho/\mathcal{B}^{2}$). Moreover, although
Hooke’s law is an approximate relation valid for sufficiently small
deformations, eq (3.18) seems to be valid for any deformation, given that the
Ricci identities (3.3) hold. Therefore, from our point of view, magnetic
fields appear to keep their elastic behaviour as well as to satisfy Hooke’s
law of elasticity no matter how big their deformation is.

Even if magnetic forcelines do not present an elastic limit101010The elastic
limit refers to that value of distortion beyond which the elastic medium is
unable to return to its initial state. Mathematically speaking, on setting the
external forces equal to zero, the deformation becomes zero as well. Of course
we do not know any such example of material in nature. under their
gravitational bending, one expects that they can support a finite amount of
distortion. Thus, we expect that there must be at least a fracture limit of
the magnetic fieldlines, predicted by the exact formula (3.18)111111In
contrast, Hooke’s law for elasticity, being a linear approximation-valid for
small values of deformation-, does not and could not contain such
information.. Beyond that limit, the magnetic field should stop being frozen
into the fluid, in the sense of magnetohydrodynamics (recall Alfvén’s
theorem). The significance of such a limit becomes clear on considering for
instance the astrophysical/cosmological phenomenon of magnetised gravitational
collapse. In particular, magnetic fields are known not to self-gravitate as
well as to have the potential to impede gravitational implosion from reaching
a spacetime singularity. Before proceeding to a definition and theoretical
calculation of the magnetic fracture limit under gravitational distortions, we
present our approach to magnetised contraction through an analogy.
We study the gravitational collapse of a magnetised fluid. In other words, we
study the contraction of a medium which behaves elastically along a specific
direction $n^{a}\parallel B^{a}$. The above statement points out the crucial
difference between ‘ordinary’ and magnetised gravitational collapse. We liken
the latter to the contraction of a waterballoon filled with water and
surrounded by water. Through this perspective, water stands for the
astrophysical/cosmic fluid whilst the elastic medium (balloon containing the
water) stands for the magnetic field. Both the balloon and the magnetic field,
force, in a sense, the water/fluid to behave elastically. In other words, the
water/fluid resists its contraction along the waterballoon/magnetic direction
via stresses increasing linearly in proportion to its deformation (Hooke’s law
for elasticity). Finally, we expect that magnetic forcelines, like the
waterballoon, can support a finite amount of contraction, which we estimate in
the following subsections.

#####  3.5.2 Magnetic fracture limit and gravitational contraction

In the first place, we claim that the fracture limit must correspond to a
maximum of the magnetic deformation with respect to proper time. However,
given that as the deformation of an elastic medium increases, so do the
internal tension stresses acting against it; it often happens that the maximum
deformation coincides with the maximum resisting tension stress (see the
following subsection 3.5.3). In reference to our case, because the law of
magnetic elasticity, eq. (3.18), is valid at all times during the collapse
(for any deformation), it is necessary that the fracture limit corresponds to
a double maximum, of the magnetic deformation and the magneto-curvature
tension as well (i.e.
$\dot{u}_{\text{fr}}=0=\dot{s}^{*}_{\text{fr}}$)121212The monotonic increase
of $u$ and $s^{*}$ during contraction is given by the problem’s nature..
Besides, a maximum of spatial deformation along a direction $n^{a}$, i.e.
$u=-\mathcal{R}_{ab}n^{a}n^{b}$, does not make physical sense by itself within
the problem of magnetised gravitational collapse. The quantity $u$ should
increase monotonically without ever reaching a maximum value. It is only
through the magnetic field presence (with its elastic behaviour encoded by
(3.18)), appearing in $s^{*}$ but not directly in $u$, that a maximum of the
latter acquires physical beingness together with a parallel maximum of the
former. Beyond that maximum value, the fieldlines of the magnetised fluid are
expected to be broken, namely to lose their coherence and stop connecting the
same fluid particles. Within the present subsection, we comment on our
gravitational collapse criterion. In the following subsection, we proceed to
the determination of the magnetic fracture limit.
Now in reference to the problem of magnetised gravitational collapse, we face
the following question. Do magnetic fieldlines affect the fate of
gravitational collapse? In particular, will they manage to impede contraction
towards a singularity or will they be inevitably broken beforehand? Let us
envisage magnetic elasticity under increasing gravity through the following
causal perspective. The magneto-curvature tension
$s^{*}=c^{2}_{\mathcal{A}}\mathcal{R}_{ab}n^{a}n^{b}$ is considered an
exclusive result of the system’s gravity $R_{ab}u^{a}u^{b}$ (recall eq
(3.10))131313Besides, during the principally relativistic phenomenon of
collapse, gravity is plausibly pointed out as the dominant term, cause of
magnetic deformation.. Hence, our argument can be stated this way: If $s^{*}$
turns out to be smaller than the fluid’s gravity at the fracture limit, then
it should have also been smaller earlier during the collapse. In such a case,
magnetic fieldlines are not able to impede contraction towards a singularity
(refer also to [10]). Oppositely, if $s^{*}$ overwhelms the fluid’s gravity at
the fracture limit, then magnetic forcelines halt the collapse. In other
words, our collapse criterion reads:

$s^{*}=-(c^{2}_{\mathcal{A}}u)_{\text{fr}}<(R_{ab}u^{a}u^{b})_{\text{fr}}\,.$
(3.20)

Note that for (magnetised) astrophysical objects in equilibrium, the total
gravitational energy density is generally by many orders of magnitude greater
than the magneto-curvature tension. This basically happens because the ratio
of the magnetic over the total energy density is for most (if not all)
applications very small, so that the Alfvén speed is many orders of magnitude
smaller than unity.
Whether the criterion in question is satisfied or not, seems to depend on the
precise value of $\lambda$ parameter. In other words, the question whether the
magnetic fieldlines manage to prevent contraction or not, reduces to how fast
the magnetic density increases.

#####  3.5.3 Specification of the magnetic fracture limit–Applications to
neutron stars and white dwarfs

In the present subsection we determine the magnetic fracture limit as a double
maximum, of the magnetic deformation $u$ and the magneto-curvature tension
stress $s^{*}$ as well. In practice, taking the dot derivative of (3.18) under
the condition $\dot{u}_{\text{fr}}=0$, we get:

$\dot{\left(s^{*}\right)}_{\text{fr}}=\dot{\left(c^{2}_{\mathcal{A}}\right)}_{\text{fr}}u_{\text{fr}}=\frac{1}{\mathcal{B}^{2}(1+\beta)^{2}}\left[\dot{\rho}+\dot{P}-\frac{2}{\mathcal{B}}\dot{\mathcal{B}}(\rho+P)\right]_{\text{fr}}u_{\text{fr}}\,,$
(3.21)

where $\beta\equiv(\rho+P)/\mathcal{B}^{2}$ (see appendix 3.G for details).
Assuming a polytropic equation of state (i.e. $P=k\rho^{\gamma}$, where $k$
and $\gamma$ are constant parameters), and setting (3.21) equal to zero, we
arrive at the condition:

$\left(\frac{\gamma}{\rho}\right)P^{2}+\left(2\lambda+\gamma-\frac{1}{3}\right)P+\left(2\lambda-\frac{1}{3}\right)\rho=0\,.$
(3.22)

Note that in deriving the above, we have taken into account the propagation
equation for the magnetic field at the magnetohydrodynamic limit, namely
$\dot{\mathcal{B}}=(\lambda-2/3)\Theta\mathcal{B}$ (see the introduction to
Part II). Moreover, it is simply verified that expression (3.21) is positive
for $P<(1/3-2\lambda)\rho/\gamma$, and negative in the opposite case. The real
solutions of the above quadratic are:141414It is easy to check that
$c^{2}_{\mathcal{A}}$ is an increasing function of proper time ($\Theta<0$ is
always assumed) for $P<\rho/\gamma$ whilst decreasing for $P>\rho/\gamma$.

$P_{\text{fr}}=k\rho^{\gamma}_{\text{fr}}=\left(\frac{\frac{1}{3}-2\lambda}{\gamma}\right)\rho_{\text{fr}}\Leftrightarrow\rho_{\text{fr}}=\left(\frac{\frac{1}{3}-2\lambda}{k\gamma}\right)^{\frac{1}{\gamma-1}}\hskip
42.67912pt\text{and}\hskip 42.67912ptP_{\text{fr}}=-\rho_{\text{fr}}\,,$
(3.23)

of which we accept the former and reject the latter, under the consideration
of ordinary collapsing matter. Note that eq (3.23a) provides the values (in
geometrised units) of matter density and pressure at the magnetic fracture
limit. Overall, we have associated the magnetic fracture limit with the unique
(double) maximum of $u$ and $s^{*}$ with respect to proper time. The
aforementioned point (a theoretical approach to the magnetic fracture limit)
is certainly expected to slightly differ, in practice, from the actual instant
in which the magnetic fieldlines are broken (i.e. $s^{*}=0$).
We can make use of the information provided by (3.23a), together with an
initial setting, in order to predict how much has the volume of a given
magnetised (collapsing) star changed until reaching its magnetic fracture
limit. In particular, with the aid of (3.80) (see appendix 3.F), we deduce
that the relation between matter density/scale factor at the fracture limit
(i.e. $\rho_{\text{fr}}$ and $a_{\text{fr}}$) and their counterparts at an
initial (hydrostatic equilibrium) state of the star (i.e. $\rho_{0}$ and
$a_{0}$), is

$\rho_{\text{fr}}=\rho_{0}\left(\frac{\frac{C}{k}a^{-3(1-\gamma)}_{0}-1}{\frac{C}{k}a^{-3(1-\gamma)}_{\text{fr}}-1}\right)^{\frac{1}{\gamma-1}}\hskip
28.45274pt\text{or}\hskip
28.45274pta^{-3(1-\gamma)}_{\text{fr}}=\frac{k}{C}\left[\left(\frac{\rho_{0}}{\rho_{\text{fr}}}\right)^{\gamma-1}\left(\frac{C}{k}a^{-3(1-\gamma)}_{0}-1\right)+1\right]\,.$
(3.24)

The above formula points out the sensible conclusion that the more dense a
star is, the earlier it reaches its fracture limit. In the following we
consider, as an application, two illustrative examples concerning a neutron
star and a white dwarf.

###### Neutron star of mass $M=1.5M_{\odot}$, radius $R=10$ km and average
magnetic field $\mathcal{B}\sim 10^{12}$ G

A neutron star with the above mentioned characteristics has matter and
magnetic densities $\rho\sim 10^{-14}~{}\text{cm}^{-2}$ and
$\mathcal{B}^{2}\sim 10^{-27}~{}\text{cm}^{-2}$, in geometrised units (i.e.
$c=1=G$). For conversion between cgs and geometrised units see e.g. the
appendix of [32]). Assuming that the matter of the star consists in average of
non-relativistic neutrons, the parameters of its polytropic equation of state
are $\gamma=5/3$ and $k=5.3802\cdot 10^{9}~{}$cgs$~{}\sim
10^{6.7}~{}\text{cm}^{4/3}$ (e.g. refer to [33] or [34] for details regarding
the parameters’ values). Finally, from (3.80) written in the initial
conditions, we determine the value of the integration constant,
$C=(\rho^{2/3}_{0}+k)/a^{2}_{0}\sim 10^{9.3}a^{-2}_{0}$. Taking into account
that $\mathcal{B}^{2}\propto a^{6\lambda-4}$ (refer to the introduction to
Part II), and substituting all the aforementioned values in (3.24b), we find
out that

$\lambda$ $a_{\text{fr}}/a_{0}$ $\hskip 11.38109pt\rho_{\text{fr}}/\rho_{0}$
$\rho_{\text{fr}}~{}(\text{cm}^{-2})$ $\hskip
11.38109pt\mathcal{B}^{2}_{\text{fr}}/\mathcal{B}^{2}_{0}$
$\mathcal{B}^{2}_{\text{fr}}~{}(\text{cm}^{-2})$ $\hskip
11.38109pt(\mathcal{B}^{2}/\rho)_{\text{fr}}$ $-1/30$ $1.0\cdot 10^{-1.3}$
$2.4\cdot 10^{6.3}$ $2.4\cdot 10^{-7.7}$ $2.9\cdot 10^{5}$ $2.9\cdot 10^{-22}$
$6.1\cdot 10^{-15}$ $-1/6$ $1.8\cdot 10^{-1.3}$ $2.3\cdot 10^{3}$ $2.3\cdot
10^{-11}$ $1.7\cdot 10^{5}$ $1.7\cdot 10^{-22}$ $7.4\cdot 10^{-11}$ $0$
$1.2\cdot 10^{-1}$ $7.7\cdot 10^{2}$ $8.0\cdot 10^{-12}$ $4.8\cdot 10^{3}$
$4.8\cdot 10^{-24}$ $6.0\cdot 10^{-13}$ $+1/10$ $3.6\cdot 10^{-1.3}$ $2.0\cdot
10^{2}$ $2.0\cdot 10^{-12}$ $3.4\cdot 10^{2}$ $3.4\cdot 10^{-25}$ $1.7\cdot
10^{-13}$

Therefore, we expect that the magnetic forcelines will be broken when the
neutron star’s radius becomes a hundred to ten times smaller (that is $1$ km)
than its initial (equilibrium) value. In parallel, the magnetic/matter density
ratio will have grown by zero to five orders of magnitude (recall that
$(\mathcal{B}^{2}/\rho)_{0}\sim 10^{-13}$).

###### White dwarf of mass $M=0.6M_{\odot}$, radius $R=1.4\cdot 10^{-2}$
$R_{\odot}$ and average magnetic field $\mathcal{B}\sim 10^{6}$ G

The white dwarf in question has matter and magnetic densities $\rho\sim
10^{-23}~{}\text{cm}^{-2}$ and $\mathcal{B}^{2}\sim
10^{-39}~{}\text{cm}^{-2}$, in geometrised units. Assuming that the stellar
fluid mainly consists of ultra relativistic electrons, the parameters of its
polytropic equation of state read $\gamma=4/3$ and $k\sim 10^{15}$ cgs$\sim
10^{4.3}~{}\text{cm}^{2/3}$. Under the above mentioned initial conditions, the
integration constant of (3.80) gives $C\sim 10^{7.6}~{}a^{-1}_{0}$. Overall,
(recalling once again that $\mathcal{B}^{2}\propto a^{6\lambda-4}$) we find
out that:

$\lambda$ $a_{\text{fr}}/a_{0}$ $\hskip 11.38109pt\rho_{\text{fr}}/\rho_{0}$
$\rho_{\text{fr}}~{}(\text{cm}^{-2})$ $\hskip
11.38109pt\mathcal{B}^{2}_{\text{fr}}/\mathcal{B}^{2}_{0}$
$\mathcal{B}^{2}_{\text{fr}}~{}(\text{cm}^{-2})$ $\hskip
11.38109pt(\mathcal{B}^{2}/\rho)_{\text{fr}}$ $-1/30$ $1.9\cdot 10^{-3}$
$3.4\cdot 10^{8}$ $3.4\cdot 10^{-15}$ $2.7\cdot 10^{11}$ $2.7\cdot 10^{-28}$
$7.9\cdot 10^{-14}$ $-1/6$ $1.4\cdot 10^{-3}$ $1.6\cdot 10^{9}$ $1.6\cdot
10^{-14}$ $1.9\cdot 10^{14}$ $1.9\cdot 10^{-25}$ $1.2\cdot 10^{-11}$ $0$
$2.2\cdot 10^{-3}$ $2.0\cdot 10^{8}$ $2.0\cdot 10^{-15}$ $4.2\cdot 10^{10}$
$4.2\cdot 10^{-29}$ $2.1\cdot 10^{-14}$ $+1/10$ $4.8\cdot 10^{-3}$ $1.3\cdot
10^{7}$ $1.3\cdot 10^{-16}$ $7.7\cdot 10^{7}$ $7.7\cdot 10^{-32}$ $5.9\cdot
10^{-16}$

We deduce that the stellar fluid reaches its magnetic fracture limit when its
radius shrinks to approximately one hundred thousand times its initial value
(that is the fracture radius is some hundreds of meters). As for the
magnetic/matter density ratio, it increases by about an order of magnitude,
i.e. $(\mathcal{B}^{2}/\rho)_{\text{fr}}\sim
10^{0.8}~{}(\mathcal{B}^{2}/\rho)_{0}\sim 10^{-15}$.

####  3.6 Discussion-Concluding remarks

On projecting Faraday’s equation at the MHD limit along the magnetic direction
$n^{a}$, and using a $1+2$ (covariant) spatial decomposition, we have shown
that it recasts into a solvable form. Its general solution for the magnetic
magnitude, i.e. $\mathcal{B}\propto a^{3\lambda-2}$, depends on a real
parameter $\lambda$, coming from the introduced physically senseful relation
$\Sigma\equiv\sigma_{ab}n^{a}n^{b}=\lambda\Theta$, between the shear and the
volume scalar. For a magnetic field decreasing/increasing during
expansion/contraction, the aforementioned real parameter obeys the constraint
$\lambda<2/3$. For orthogonal (magnetised) Bianchi cosmologies, the constraint
in question strictens to $-2/3\leq\lambda<2/3$.

Applying our magnetic evolution formula into the study of contracting
worldlines in nearly homogeneous magnetised gravitational collapse, we suggest
a non-collapse criterion. According to that, the contraction’s fate at
advanced stages is essentially determined by: the tidal scalar $\mathcal{E}$
(the tidal tensor projected twice along the magnetic direction), the magnetic
density $\mathcal{B}^{2}$, the ratio $(\rho/\mathcal{B})^{2}$ ($\rho$ :
density of matter) and the squared volume scalar $\Theta^{2}$. The actual form
of the criterion depends of course on the particular magnetic evolution rate
(i.e. on $\lambda$ parameter). Moreover, the precise evolution formulae for
$\mathcal{E}$ and $\Theta^{2}$ depend on the geometric background in hand.

After distinguishing and describing the kinematically and gravitationally
induced magnetic tension stresses, we focus our attention to a specific
magneto-curvature tension stress component, the one triggered by volume
distortions due to gravity. We point out the aforementioned tension as the
dominant stress counteracting (magnetised) gravitational collapse. The essence
of our reasoning lies in that gravitational deformation of magnetic forcelines
is governed by a particular form of Hooke’s law of elasticity, originating
from the magneto-curvature coupling monitored by the Ricci identities.
However, there are two basic features distinguishing gravitational
(relativistic) distortions of magnetic forcelines from mechanical distortions
of elastic materials. Firstly, unlike mechanical distortions of elastic
materials, Hooke’s law in the form of (3.18) is not an approximate expression
only valid for small magnetic deformations (thus magnetic forcelines do not
seem to have an elastic limit). In contrast, as long as Ricci identities are
an appropriate definition of spatial curvature for large values of the latter
(advanced stages of gravitational collapse), the law in question consists of
an exact expression, valid for any size of distortion. Secondly, the
proportionality factor in the elasticity law (3.18) is a variable instead of a
constant quantity.

Based on the aforementioned expression for gravito-magnetic elasticity, we
have suggested a calculation of the magnetic fracture limit under
gravitational volume distortions. Two explicit cases have been considered as
application, of a neutron star and a white dwarf. The results have been
summarised in two tables for some representative values of the $\lambda$
parameter. Addressing also the question whether magnetic forcelines manage to
impede contraction before being broken, our answer depends on the magnetic
density’s rate of change (i.e. on the precise value of $\lambda$ parameter).
Considering our results as new, we raise the fundamental problem regarding the
observational-experimental (and further theoretical) verification of the
gravito-magnetic elasticity law. Although magnetic elasticity under great
gravitational distortions is practically not a subject offered for study in
earthly laboratories, progress towards the experimental path could
alternatively and in the first place be achieved by examining magnetic
distortions under progressively increasing rotations.

Overall, our general law of variation for the magnetic field of an expanding
or contracting highly conducting fluid, is expected to provide
magnetohydrodynamics with a valuable theoretical tool. Interestingly, knowing
how the magnetic fieldlines explicitly behave during gravitational implosion,
could hopefully shed light on the various evolution phases of astrophysical
objects. The motivation for the above sentence essentially comes from
considering that many stars or protogalactic clouds are associated with (even
small) magnetic fields which are rapidly increasing during their collapse.
Concerning the potential applications in cosmology, our exact (not
approximate) evolution formula for the magnetic field could fortunately
refresh the question regarding the contribution of magnetic fields to the
kinematics of the early universe.

###  Appendix 3.A Propagation equations and constraints for the kinematically
induced magnetic tension stresses

In order to arrive at the propagation equation for $\sigma^{(B)}_{ab}$, we
make the following steps. First, project eq. (3.2) along the timelike
4-velocity $u^{a}$; second, project orthogonal to $u^{a}$ with the aid of
$h_{ab}$ and with respect to both indices (removing thus timelike terms);
third, take the symmetric and trace-free part of the resulting relation. The
equation in question finally reads:151515An index with bar denotes that the
associated component has been projected orthogonal to $u^{a}$.

$\displaystyle\dot{\sigma}^{(B)}_{\bar{a}\bar{b}}$ $\displaystyle=$
$\displaystyle-\Theta\dot{u}_{\langle a}B_{b\rangle}+2\dot{u}_{\langle
a}\omega_{b\rangle}{}^{c}B_{c}+{\rm D}_{\langle
a}\dot{B}_{\bar{b}\rangle}-\left(\sigma_{c\langle a}+\omega_{c\langle
a}+\frac{1}{3}\Theta h_{c\langle a}\right)\sigma^{c(B)}{}_{b\rangle}$ (3.25)
$\displaystyle-\left(\sigma_{c\langle a}+\omega_{c\langle
a}\right)\omega^{c(B)}{}_{b\rangle}-\frac{1}{2}B_{\langle a}q_{b\rangle}\,,$

where $\dot{u}_{a}=u^{b}\nabla_{b}u_{a}$ is the fluid’s acceleration and
$q_{a}$ its flux vector. On deriving the above we have taken into account eq.
1.3.1 of [15], as well as that

$\nabla_{a}B_{b}={\rm
D}_{a}B_{b}-u_{a}\dot{B}_{b}+u_{b}(\nabla_{a}u^{d})B_{d}+u_{a}u_{b}\dot{u}^{c}B_{c}\hskip
14.22636pt\text{and}\hskip 14.22636ptR_{a\langle bc\rangle
d}u^{a}B^{d}=\frac{1}{2}B_{\langle b}q_{c\rangle}\,,$ (3.26)

where eqs 1.2.6, 1.2.8 and 1.2.11 of [15] have been used on finding the latter
of the above. Following a similar procedure but taking the antisymmetric part
of (3.2) (via contraction with the 3-D Levi-Civita pseudotensor
$\epsilon_{abc}$) this time, we arrive at the propagation equation for the
magnetic tension induced by twisting effects:

$\displaystyle\dot{\omega}^{(B)}_{\bar{a}}$ $\displaystyle=$
$\displaystyle-3\epsilon_{abc}\dot{u}^{b}\sigma^{c}{}_{d}B^{d}-\epsilon_{abc}{\rm
D}^{b}\dot{B}^{\bar{c}}-\epsilon_{abc}\left(\sigma^{bd}+\omega^{db}+\frac{1}{3}\Theta
h^{bd}\right){\rm D}_{d}B^{c}$ (3.27)
$\displaystyle-(\dot{u}^{b}B_{b})\omega_{a}+(\dot{u}^{b}\omega_{b})B_{a}+H_{ab}B^{b}-\frac{1}{2}\epsilon_{abc}B^{b}q^{c}\,,$

where $H_{ab}$ is the magnetic Weyl component and we have taken into account
that:

$\dot{B}_{\bar{a}}=-\frac{2}{3}\Theta
B_{a}+(\sigma_{ab}+\epsilon_{abc}\omega^{c})B^{b}\hskip
14.22636pt\text{and}\hskip
14.22636pt\epsilon_{abc}R^{ebcd}u_{e}B_{d}=H_{ab}B^{b}-\frac{1}{2}\epsilon_{abc}B^{b}q^{c}\,.$
(3.28)

Note that eq (3.28a) is an expression of Faraday’s law at the MHD limit. On
the other hand, the spacelike part of (3.2) leads to the divergence conditions
for the aforementioned quantities. In detail, we start from the 3-D Ricci
identities (3.3). Subsequently, we take either its trace or its contraction
with $\epsilon_{abc}$. The former case leads to:

${\rm
D}^{b}\sigma^{(B)}_{ab}=\text{curl}\omega^{(B)}_{a}+2\omega_{ab}\left(-\frac{2}{3}\Theta
B^{b}+\sigma^{b}{}_{c}B^{c}\right)+\mu\omega_{a}-2\omega^{2}B_{a}+\mathcal{R}_{ba}B^{b}$
(3.29)

whilst the latter to

${\rm
D}^{a}\omega^{(B)}_{a}=\frac{1}{6}\Theta\mu-2\sigma_{ab}\omega^{a}B^{b}\,.$
(3.30)

In (3.29) $\mathcal{R}_{ab}$ represents the 3-D Ricci tensor. On deriving the
above we have made use of (3.28a) as well as of

$\omega^{a}B_{a}=\mu/2\hskip 14.22636pt\text{and}\hskip
14.22636pt\epsilon^{abc}\mathcal{R}_{dcba}B^{d}=-\frac{2}{3}\Theta\mu-4\sigma_{ab}\omega^{a}B^{b}\,,$
(3.31)

where $\mu$ is the charge density and (3.31a) is an expression of Gauss’s law
at the MHD limit. It is worth noting that for zero rotational distortions
(i.e. $\omega_{ab}=0$) of the magnetic field eqs (3.25) and (3.29)
significantly simplify to161616An ideal fluid (i.e. $q_{a}=0$) has been
assumed in the first equation.

$\dot{\sigma}^{(B)}_{\bar{a}\bar{b}}=-\Theta\dot{u}_{\langle
a}B_{b\rangle}+{\rm D}_{\langle
a}\dot{B}_{\bar{b}\rangle}-\left(\sigma_{c\langle a}+\frac{1}{3}\Theta
h_{c\langle a}\right)\sigma^{c(B)}{}_{b\rangle}\hskip
14.22636pt\text{and}\hskip 14.22636pt{\rm
D}^{b}\sigma^{(B)}_{ab}=\mathcal{R}_{ba}B^{b}\,.$ (3.32)

Overall, equations (3.25), (3.27), (3.29) and (3.30) determine the kinematics
of the magnetic tension stresses triggered by shear and vorticity effects.
Note that due to Gauss’ law at the MHD limit (i.e. ${\rm D}^{a}B_{a}=0$),
there are no magnetic tension stresses triggered by volume changes.

###  Appendix 3.B $1+2$ Covariant approach

In some cases, a further 1+2 decomposition of the 3-dimensional space (leading
to an overall 1+1+2 spacetime splitting–see [5], [25] and [26] for some
introductory information) in one specific spatial direction and a
2-dimensional surface orthogonal to it, may reveal additional useful
information about the problem in hand. This is more likely to happen when the
geometry, or the physics select a preferred spatial direction. For instance,
one could consider the radial component of a spherically symmetric spacetime,
or the rotation axis of a magnetised star, which may also happen to be
parallel to the direction of the magnetic forcelines. However, a split of the
spatial components may reveal valuable information about the problem in hand
even there are not any apparent, favorable geometric or physical conditions
(e.g. see the decomposition of Maxwell equations in the present piece of
work.).

#####  3.B.1 Spatial splitting

In what follows, we show how 3-D mathematical objects (vectors, tensors,
equations etc.) decompose into a component parallel to a spatial direction and
two components lying on a 2-D surface perpendicular to the aforementioned
direction (for a detailed presentation see also [25]). Let us introduce a
space-like unit vector $n^{a}$ orthogonal to $u^{a}$ ($n^{a}n_{a}=1$,
$n^{a}u_{a}=0$), which defines a specific spatial direction. Subsequently, we
can define the symmetric tensor $\tilde{h}_{ab}\equiv h_{ab}-n_{a}n_{b}$ which
projects vectors onto 2-D surfaces orthogonal to $n^{a}$
($\tilde{h}_{ab}n^{b}=0$, $\tilde{h}^{a}{}_{a}=2$,
$\tilde{h}_{a}{}^{c}\tilde{h}_{bc}=\tilde{h}_{ab}$). In analogy with the 1+3
formalism, 3-vectors and the corresponding second-rank, symmetric and trace-
free tensors are split in their irreducible components according to the
relations:

$v^{a}=Vn^{a}+V^{a},$ (3.33)

where $V\equiv v^{a}n_{a}$ and $V^{a}\equiv\tilde{h}^{a}{}_{b}v^{b}$ while

$v_{ab}=V(n_{a}n_{b}-\frac{1}{2}\tilde{h}_{ab})+2V_{(a}n_{b)}+V_{ab},$ (3.34)

where $V\equiv v_{ab}n^{a}n^{b}=-\tilde{h}^{ab}v_{ab}$,
$V_{a}\equiv\tilde{h}_{a}{}^{b}n^{c}v_{bc}$ and
$V_{ab}\equiv(\tilde{h}_{(a}{}^{c}\tilde{h}_{b)}{}^{d}-(1/2)\tilde{h}_{ab}\tilde{h}^{cd})v_{cd}$.
For instance, let us consider the 1+1+2 decomposition of the energy-momentum
tensor
$T_{ab}=g_{ac}g_{bd}T^{cd}=(\tilde{h}_{ac}-u_{a}u_{c}+n_{a}n_{c})(\tilde{h}_{bd}-u_{b}u_{d}+n_{b}n_{d})$,
which leads to:

$T_{ab}=\rho
u_{a}u_{b}+\tilde{\rho}n_{a}n_{b}+\tilde{P}\tilde{h}_{ab}+2u_{(a}q_{b)}+2n_{(a}\tilde{q}_{b)}+\Pi_{ab},$
(3.35)

where $\tilde{\rho}\equiv T_{ab}n^{a}n^{b}=P+\Pi$ and
$\tilde{P}\equiv(\tilde{h}^{ab}/2)T_{ab}=P-\Pi/2$ (therefore
$\Pi=(2/3)(\tilde{\rho}-\tilde{P}$)) are the analogues of relativistic energy
density and pressure defined in reference to spacelike curves with tangent
vector $n^{a}$. Regarding
$\tilde{q}_{a}\equiv\tilde{h}_{a}{}^{b}n^{c}T_{bc}=\Pi_{a}$ and
$\Pi_{ab}\equiv(\tilde{h}_{(a}{}^{c}\tilde{h}_{b)}{}^{d}-(1/2)\tilde{h}_{ab}\tilde{h}^{cd})T_{cd}$,
they represent the (2-D) surface (normal to $n^{a}$) counterparts of the
energy flux vector and the viscosity tensor respectively (refer to equation
(3.44) for the decomposition of the anisotropic stress tensor). We gather here
for reference all of the decomposition relations of vectors and tensors, which
we use throughout this thesis:171717Note that $\dot{n}_{a}n^{a}=0$ in eq.
(3.37) and therefore $\alpha_{a}n^{a}=0$.

$\dot{u}^{a}=\mathcal{A}n^{a}+\mathcal{A}^{a}$ (3.36)
$\dot{n}^{a}=\mathcal{A}u^{a}+\alpha^{a}$ (3.37) $\omega^{a}=\Omega
n^{a}+\Omega^{a}$ (3.38) $q^{a}=Qn^{a}+Q^{a}$ (3.39) $E^{a}=\epsilon
n^{a}+\epsilon^{a}$ (3.40) $B^{a}=\mathcal{B}n^{a}+\mathcal{B}^{a}$ (3.41)
$\mathcal{J}^{a}=jn^{a}+j^{a}$ (3.42)
$\sigma_{ab}=\Sigma(n_{a}n_{b}-\frac{1}{2}\tilde{h}_{ab})+2\Sigma_{(a}n_{b)}+\Sigma_{ab}$
(3.43)
$\pi_{ab}=\Pi(n_{a}n_{b}-\frac{1}{2}\tilde{h}_{ab})+2\Pi_{(a}n_{b)}+\Pi_{ab}$
(3.44)
$E_{ab}=\mathcal{E}(n_{a}n_{b}-\frac{1}{2}\tilde{h}_{ab})+2\mathcal{E}_{(a}n_{b)}+\mathcal{E}_{ab}\,.$
(3.45)

In the last equation, $E_{ab}$ is the electric component of the Weyl (long-
range) curvature tensor. There is also the magnetic tensor component $H_{ab}$.
Weyl curvature is associated with tidal forces and gravitational waves (e.g.
refer to [15]). The aforementioned decomposition relation will be used only
once when discussing the gravitational collapse of a magnetised fluid in
section 3.4. Finally, for some details concerning the meaning of the shear’s
scalar and vector components see the appendix section 3.C.

Regarding the derivatives of a general tensor field $T_{ab...}{}^{cd...}$, the
one along $n^{a}$ and the other projected on the 2-surface normal to $n^{a}$,
these are defined respectively as:

$T^{\prime}_{ab...}{}^{cd...}\equiv n^{e}{\rm D}_{e}T_{ab...}{}^{cd...}$
(3.46)

and

$\tilde{{\rm
D}}_{e}T_{ab...}{}^{cd...}\equiv\tilde{h}_{e}{}^{s}\tilde{h}_{a}{}^{f}\tilde{h}_{b}{}^{p}\tilde{h}_{q}{}^{c}\tilde{h}_{r}{}^{d}...{\rm
D}_{s}T_{fp...}{}^{qr...}\,.$ (3.47)

Finally, the 2-D Levi-Civita pseudotensor can be defined via the contraction
of its 3-D counterpart along the spatial direction of $n^{a}$,
$\epsilon_{ab}\equiv\epsilon_{abc}n^{c}$. It follows that:

$\epsilon_{ab}n^{b}=0\quad\text{and}\quad\epsilon_{ab}\epsilon^{cd}=2\tilde{h}_{[a}{}^{c}\tilde{h}_{b]}{}^{d}$
(3.48)

as well as that
$\epsilon_{abc}=n_{a}\epsilon_{bc}-n_{b}\epsilon_{ac}+n_{c}\epsilon_{ab}$.

#####  3.B.2 2-Dimensional kinematic quantities

In analogy with its 3-D counterpart, the motion on the 2-D surface orthogonal
to $n^{a}$ is characterised by a set of kinematic quantities which come from
the decomposition of the gradient of $n^{a}$. In other words, we have:

${\rm
D}_{b}n_{a}=\tilde{\sigma}_{ab}+\tilde{\omega}_{ab}+\frac{1}{2}\tilde{\Theta}\tilde{h}_{ab}+n_{b}n^{\prime}_{a},$
(3.49)

where $\tilde{\sigma}_{ab}\equiv{\rm D}_{\langle b}n_{a\rangle}$,
$\tilde{\omega}_{ab}\equiv{\rm D}_{[b}n_{a]}$ and $\tilde{\Theta}\equiv{\rm
D}^{a}n_{a}$ are respectively the shear and the vorticity tensors, the surface
expansion-contraction scalar and $n^{\prime}_{a}\equiv n^{b}{\rm D}_{b}n_{a}$
the spatial derivative of $n^{a}$ along its own direction. The sum
$\tilde{{\rm
D}}_{b}n_{a}=\tilde{\sigma}_{ab}+\tilde{\omega}_{ab}+\frac{1}{2}\tilde{\Theta}\tilde{h}_{ab}$
describes the relative motion of neighbouring spacelike curves orthogonal to
the surface in question.

It is worth comparing the 2-D version of the shear
$\tilde{\sigma}_{ab}\equiv{\rm D}_{\langle b}n_{a\rangle}$ with those of the
individual 1+2 components of its 3-D version $\sigma_{ab}\equiv{\rm
D}_{\langle b}u_{a\rangle}$. Concerning the 2-D vorticity tensor, it has only
one independent component (i.e. it consists of a vector along the one of the
two independent directions defining the 2-D surface), so that it can be
written as $\tilde{\omega}_{ab}=\tilde{\omega}\epsilon_{ab}$, where
$\tilde{\omega}^{2}=(1/2)\tilde{\omega}^{ab}\tilde{\omega}_{ab}$. Finally, the
condition $n^{\prime a}=0$ implies that the $n^{a}$ field is tangent to a
congruence of spacelike geodesics.

#####  3.B.3 Deriving the $2$-dimensional form of the Ricci identities

We begin with writing the Ricci identity in its $3$-dimensional form for a
vector lying on the $2$-surface normal to $n^{a}$, e.g. for $k^{a}$
($k^{a}n_{a}=0)$:

$2{\rm D}_{[a}{\rm D}_{b]}k_{c}=-2\omega_{ab}\dot{k}_{\langle
c\rangle}+\mathcal{R}_{dcba}k^{d}\,.$ (3.50)

Projecting the above orthogonal to the $n^{a}$-field, yields:

$\tilde{h}_{d}{}^{a}\tilde{h}_{e}{}^{b}\tilde{h}_{f}{}^{c}~{}{\rm D}_{a}{\rm
D}_{b}k_{c}-\tilde{h}_{d}{}^{a}\tilde{h}_{e}{}^{b}\tilde{h}_{f}{}^{c}~{}{\rm
D}_{b}{\rm
D}_{a}k_{c}=-2\tilde{h}_{d}{}^{a}\tilde{h}_{e}{}^{b}\tilde{h}_{f}{}^{c}~{}\omega_{ab}\dot{k}_{\langle
c\rangle}+\tilde{h}_{d}{}^{a}\tilde{h}_{e}{}^{b}\tilde{h}_{f}{}^{c}\tilde{h}_{g}{}^{i}~{}\mathcal{R}_{icba}k^{g}\,,$
(3.51)

where $\tilde{h}_{d}{}^{a}\tilde{h}_{e}{}^{b}\tilde{h}_{f}{}^{c}~{}{\rm
D}_{a}{\rm D}_{b}k_{c}=\tilde{{\rm D}}_{d}{\rm D}_{e}k_{f}$. Subsequently, we
take into account that $\tilde{{\rm
D}}_{a}k_{c}=\tilde{h}_{a}{}^{b}\tilde{h}_{c}{}^{d}~{}{\rm D}_{b}k_{d}$ which,
after expansion of the projection tensors, leads to:

${\rm D}_{a}k_{c}=\tilde{{\rm D}}_{a}k_{c}+n_{a}k^{\prime}_{c}-n_{c}({\rm
D}_{a}n_{d})k^{d}-n_{a}n_{c}(n^{d}k^{\prime}_{d})\,.$ (3.52)

Using the aforementioned expressions, eq (3.51) becomes:

$\tilde{{\rm D}}_{d}{\rm D}_{e}k_{f}-\tilde{{\rm D}}_{e}\tilde{{\rm
D}}_{d}k_{f}-(\tilde{{\rm D}}_{e}n_{d})k^{\prime}_{f}+(\tilde{{\rm
D}}_{e}n_{f})(\tilde{{\rm
D}}_{d}n_{g})k^{g}=-2\Omega\epsilon_{de}\dot{k}_{\bar{f}}+\tilde{h}_{g}{}^{i}\tilde{h}_{f}{}^{c}\tilde{h}_{e}{}^{b}\tilde{h}_{d}{}^{a}~{}\mathcal{R}_{icba}k^{g}\,,$
(3.53)

where we have made use of $\omega_{ab}=\epsilon_{abc}\omega^{c}$ and
$\epsilon_{abc}=n_{a}\epsilon_{bc}-n_{b}\epsilon_{ac}+n_{c}\epsilon_{ab}$, and
considered that the sequence according to which the projection tensors act on
$\mathcal{R}_{ifed}k^{g}$, does not matter. Expanding now the first term in
the left-hand side of (3.53) by deploying eq (3.52), and projecting once again
the resulting equation with
$\tilde{h}_{a}{}^{d}\tilde{h}_{b}{}^{e}\tilde{h}_{c}{}^{f}$ , we arrive at:

$2\tilde{{\rm D}}_{[a}\tilde{{\rm D}}_{b]}k_{c}=2\tilde{{\rm
D}}_{[b}n_{a]}k^{\prime}_{c}-2\Omega\epsilon_{ab}\dot{k}_{\bar{c}}+\left(\tilde{v}_{gb}\tilde{v}_{ca}-\tilde{v}_{ga}\tilde{v}_{cb}\right)k^{g}+\tilde{h}_{g}{}^{i}\tilde{h}_{c}{}^{f}\tilde{h}_{b}{}^{e}\tilde{h}_{a}{}^{d}~{}\mathcal{R}_{ifed}k^{g}\,,$
(3.54)

where $\tilde{v}_{ab}\equiv\tilde{{\rm
D}}_{b}n_{a}=\tilde{\sigma}_{ab}+\tilde{\omega}_{ab}+(\tilde{\Theta}/2)\tilde{h}_{ab}$.
Finally, employing our definition for the $2$-D Riemann tensor
$\tilde{\mathcal{R}}_{abcd}$ (see eq (3.56) in the following subsection), the
Ricci identity for a vector lying on the 2-surface normal to $n^{a}$, reads:

$2\tilde{{\rm D}}_{[a}\tilde{{\rm
D}}_{b]}k_{c}=2\tilde{\omega}_{ab}k^{\prime}_{c}-2\Omega\epsilon_{ab}\dot{k}_{\bar{c}}+\tilde{\mathcal{R}}_{dcba}k^{d}\,.$
(3.55)

We observe that the $2$-D form of the Ricci identity includes two vorticity
terms, describing rotation of the 2-surface normal to $n^{a}$; these are
$\tilde{\omega}_{ab}\equiv\tilde{\omega}\epsilon_{ab}\equiv\tilde{D}_{[b}n_{a]}$
and $\Omega\epsilon_{ab}$ with
$\Omega\equiv\omega_{a}n^{a}\equiv-n^{a}\text{curl}u_{a}=-(1/2)\epsilon_{ab}\tilde{{\rm
D}}^{a}u^{b}$, where $\Omega$ denotes the norm of the vorticity vector along
direction $n^{a}$. As far as we know, a $2$-dimensional form of the Ricci
identities has not appeared before in the literature.

#####  3.B.4 Finding the 2-D Ricci tensor and scalar in terms of the fluid
dynamic quantities

We define the Riemann tensor $\tilde{\mathcal{R}}_{abcd}$ of a 2-surface
orthogonal to a vector $n^{a}$, as follows:

$\tilde{\mathcal{R}}_{abcg}\equiv\tilde{h}_{a}{}^{d}\tilde{h}_{b}{}^{e}\tilde{h}_{c}{}^{f}\tilde{h}_{g}{}^{j}\mathcal{R}_{defj}-\tilde{v}_{ac}\tilde{v}_{bg}+\tilde{v}_{ag}\tilde{v}_{bc},$
(3.56)

where $\mathcal{R}_{defj}$ is the 3-D Riemann tensor and
$\tilde{v}_{ab}=\tilde{{\rm
D}}_{b}n_{a}=\tilde{\sigma}_{ab}+\tilde{\omega}_{ab}+(\tilde{\Theta}/2)\tilde{h}_{ab}$.
The above definition is analogous to its counterpart for the $3$-Riemann
tensor (e.g. see (1.3.34) in [19]). The 2-D Ricci tensor is derived from
relation (3.56) by contracting the indices $a$ and $c$, namely:

$\tilde{\mathcal{R}}_{bg}=\tilde{h}^{df}\tilde{h}_{b}{}^{e}\tilde{h}_{g}{}^{j}~{}\mathcal{R}_{defj}-\tilde{\Theta}\tilde{v}_{bg}+\tilde{v}^{c}{}_{g}\tilde{v}_{bc}\,.$
(3.57)

Projecting the $3$-D Riemann tensor (e.g. see eq. 1.3.35 in [15]) in
accordance with (3.57), we find out that:

$\tilde{\mathcal{R}}_{ab}=\frac{1}{2}\tilde{\mathcal{R}}\tilde{h}_{ab}+\Sigma\Sigma_{ab}+\tilde{\sigma}_{c\langle
a}\tilde{\sigma}^{c}{}_{b\rangle}+2\tilde{\sigma}_{c[a}\tilde{\omega}^{c}{}_{b]}+\Sigma_{c\langle
a}\Sigma^{c}{}_{b\rangle}+2\Omega\Sigma_{c[a}\epsilon^{c}{}_{b]}\,,$ (3.58)

where

$\tilde{\mathcal{R}}=-2\mathcal{E}+\frac{2}{3}\rho-\Pi-\frac{2}{9}\Theta^{2}-\tilde{\Theta}^{2}+\frac{2}{3}\Sigma\Theta-\frac{1}{2}\Sigma^{2}-2\Omega^{2}+\frac{4}{3}\tilde{\sigma}^{2}-2\tilde{\omega}^{2}+\frac{2}{3}\Sigma_{ab}\Sigma^{ab}$
(3.59)

is the scalar curvature of a 2-surface.181818Note that $\rho$ and $\Pi$ in
(3.59) include contributions from various forms of matter. In the above we
have used the definitions:
$2\tilde{\sigma}^{2}\equiv\tilde{\sigma}_{ab}\tilde{\sigma}^{ab}$ and
$\tilde{\omega}_{ab}\equiv\tilde{\omega}\epsilon_{ab}$.

###  Appendix 3.C The scalar and vector components of the shear tensor

191919The present appendix section is envisaged as a correction of Appendix A
in [10]. In particular, we point out here that eqs (102) and (106) of the
aforementioned work are valid, not generally, but within a specific framework.

The present appendix unit provides a technical supplement to calculations
spanning the main text of Part II. Throughout this thesis we encounter several
times products of tensors with the spacelike (unit) vector field $n^{a}$ (such
that $n_{a}u^{a}=0$ and $n_{a}n^{a}=1$), which is taken parallel to the
magnetic forcelines (i.e. $B^{a}=\mathcal{B}n^{a}$). In particular, our
calculations often involve the quantities $\Sigma=\sigma_{ab}n^{a}n^{b}$ and
$\Sigma_{a}=\tilde{h}_{a}{}^{b}\sigma_{bc}n^{c}$ (with
$\tilde{h}_{ab}=g_{ab}+u_{a}u_{b}-n_{a}n_{b}=h_{ab}-n_{a}n_{b}$, satisfying
$\tilde{h}_{ab}n^{b}=0$). Concerning the former, it can be written as

$\Sigma\equiv\sigma_{ab}n^{a}n^{b}\equiv{\rm D}_{\langle
b}u_{a\rangle}n^{a}n^{b}={\rm D}_{(b}u_{a)}n^{a}n^{b}-\frac{1}{3}\Theta
h_{ab}n^{a}n^{b}=u^{\prime}_{a}n^{a}-\frac{1}{3}\Theta\,,$ (3.60)

where $u^{\prime}_{a}=n^{b}{\rm D}_{b}u_{a}$. Note that the term
$u^{\prime}_{a}n^{a}$ does not generally vanish because the prime (′) is not
an actual spatial derivative operator. This means that ′ does not generally
satisfy the product (Leibniz) associative rule between (the timelike) $u^{a}$
and (the spacelike) $n^{a}$. To illustrate the meaning of the term
$u^{\prime}_{a}n^{a}$, let us consider the double projection of the volume
expansion/contraction tensor, $\Theta_{ab}=\sigma_{ab}+(\Theta/3)h_{ab}$,
along a spatial direction $n^{a}$, which reads:

$\Theta_{ab}n^{a}n^{b}=u^{\prime}_{a}n^{a}\hskip 28.45274pt\text{with}\hskip
28.45274pt\Sigma=\Theta_{ab}n^{a}n^{b}-\frac{\Theta}{3}\,.$ (3.61)

Within the main text we work under the consideration $\Sigma=\lambda\Theta$
($\lambda$: a real number), so that
$\Theta_{ab}n^{a}n^{b}=u^{\prime}_{a}n^{a}$ is a multiple of $\Theta$.
On the other hand, in reference to the vector shear component, namely
$\Sigma_{a}$, we have

$\Sigma_{a}\equiv\tilde{h}_{a}{}^{b}\sigma_{bc}n^{c}\equiv\tilde{h}_{a}{}^{b}n^{c}{\rm
D}_{\langle c}u_{b\rangle}=\tilde{h}_{a}{}^{b}n^{c}{\rm
D}_{(c}u_{b)}=\frac{1}{2}\Sigma_{a}+\frac{1}{2}\epsilon_{ab}\Omega^{b}+\frac{1}{2}\tilde{h}_{a}{}^{b}u^{\prime}_{b}\,,$
(3.62)

which subsequently leads to

$\Sigma_{a}=\epsilon_{ab}\Omega^{b}+\tilde{h}_{a}{}^{b}u^{\prime}_{b}\hskip
22.76219pt\text{or}\hskip
22.76219pt\Sigma_{a}=\epsilon_{ab}\Omega^{b}+\alpha_{a}\,.$ (3.63)

where in deriving (3.63b), we have taken into account the projection of
Faraday’s law normal to the magnetic direction $n^{a}$ (implying
$\tilde{h}_{a}{}^{b}\dot{n}_{b}\equiv\alpha_{a}=\tilde{h}_{a}{}^{b}u^{\prime}_{b}$).
Before closing the present section, let us examine the particular case
corresponding to $\Sigma=-\Theta/3$. The condition in question implies that
$u^{\prime}_{a}=0$, which means that the fluid velocity is homogeneous along
the magnetic forcelines (for $B^{a}=\mathcal{B}n^{a}$). In other words, the
magnetic fieldlines are envisaged as streamlines of the fluid. Moreover,
$\Sigma=-\Theta/3$ translates to $\Theta_{ab}n^{a}n^{b}=0$. Under the
aforementioned constraint, only accelerated (non-geodesic) motion or motion
due to vorticity is allowed along direction $n^{a}$. In reference to the
problem of nearly homogeneous magnetised gravitational collapse, recall that
there is no acceleration along $n^{a}$ (due to ${\rm D}_{a}\rho=0$). Also, if
the net charge density is zero (i.e. $\mu=0$), there is no vortex motion along
the magnetic direction, as follows from Gauss’s law. Therefore, for the
problem in question, condition $\Sigma=-\Theta/3$ or $\Theta_{ab}n^{a}n^{b}=0$
requires the total absence of motion parallel to the magnetic field. In other
words, it consists of a special case of $2$-dimensional gravitational
contraction.

###  Appendix 3.D Spatial decomposition of the magnetohydrodynamic constraints
and detailed solution of the equation
$\dot{\mathcal{B}}=\left(\lambda-\frac{2}{3}\right)\Theta\mathcal{B}$.

To begin with, let us consider the decomposition (orthogonal and along
$n^{a}$) of the constraints (2.17) and (2.18). In particular, the former
splits into:

$-\mathcal{B}^{2}\mathcal{A}_{a}-2\mathcal{B}\tilde{\rm{D}}_{a}\mathcal{B}+\mathcal{B}^{2}n^{\prime}_{a}=\mathcal{B}\epsilon_{ab}j^{b}\hskip
42.67912pt\text{and}\hskip
42.67912pt\tilde{\omega}\mathcal{B}=-\frac{j}{2}\,;$ (3.64)

As for the scalar equations (2.18), they are written as

$\Omega\mathcal{B}=\frac{\mu}{2}\hskip 42.67912pt\text{and}\hskip
42.67912pt\mathcal{B}^{\prime}+\tilde{\Theta}\mathcal{B}=0\,.$ (3.65)

In case of neutral total charge density (i.e. $\mu=0$), eq (3.65a) implies
that there is no angular velocity along the magnetic direction. Concerning
(3.65b), we observe that it is a covariant, linear, partial differential
equation of first order. Given that $\tilde{\Theta}\equiv 2l^{\prime}/l$ (with
$l\equiv\alpha\beta$ being202020The total scale factor we write as
$a\equiv\alpha\beta\gamma$, where $\alpha$, $\beta$, $\gamma$ represent the
anisotropy factors along the three spatial directions. the scale factor of the
surface orthogonal to $n^{a}$), the equation in question accepts the general
solution:

$\mathcal{B}\propto l^{-2}\,.$ (3.66)

Subsequently, we provide a detailed solution of Faraday’s equation in the form
of (2.16). In detail, as $\mathcal{B}$ is a scalar quantity, its covariant
differentiation is equivalent to its ordinary differentiation, so that

$\dot{\mathcal{B}}=u^{a}\nabla_{a}\mathcal{B}=u^{a}\partial_{a}\mathcal{B}=(u^{0}\partial_{0}+u^{1}\partial_{1}+u^{2}\partial_{2}+u^{3}\partial_{3})\mathcal{B}=\left(\lambda-\frac{2}{3}\right)\Theta\mathcal{B}\,.$
(3.67)

Now by defining new space-time variables $\tilde{x}^{a}$ such that212121Note
that here the repeated index $i$ does not imply summation of components.

$\tilde{x}^{i}=\int\frac{dx^{i}}{u^{i}}\,,$ (3.68)

expression (3.67) becomes:

$(\tilde{\partial}_{0}+\tilde{\partial}_{1}+\tilde{\partial}_{2}+\tilde{\partial}_{3})\mathcal{B}=\left(\lambda-\frac{2}{3}\right)\Theta\mathcal{B}\,,$
(3.69)

where $\tilde{\partial}_{i}$ are the new derivative operators with respect to
the variables $\tilde{x}^{i}$. Let us try to solve the latter equation by
assuming variables separation :
$\mathcal{B}=\mathcal{T}(\tilde{x}^{0})U(\tilde{x}^{1})V(\tilde{x}^{2})W(\tilde{x}^{3})$,
where $\tilde{x}^{0}$ is the new temporal variable and
$\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3}$ are the new spatial variables.
Relation (3.69) takes thus the form:

$\frac{\tilde{\partial}_{0}\mathcal{T}}{\mathcal{T}}+\frac{\tilde{\partial}_{1}U}{U}+\frac{\tilde{\partial}_{2}V}{V}+\frac{\tilde{\partial}_{3}W}{W}=\left(\lambda-\frac{2}{3}\right)\Theta(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})\,,$
(3.70)

We observe that each of the fractions in the above equation depends only on
one of the variables $\tilde{x}^{i}$. Subsequently, equation (3.70) holds if
and only if
$\Theta(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})=\Theta_{0}(\tilde{x}^{0})+\Theta(\tilde{x}^{1})+\Theta_{2}(\tilde{x}^{2})+\Theta_{3}(\tilde{x}^{3})$.
Therefore, the original partial differential equation reduces to four ordinary
differential equations of the form
$(\tilde{\partial}_{1}U/U)=\left(\lambda-\frac{2}{3}\right)\Theta_{1}(\tilde{x}^{1})$,
which are integrated directly to give
$U=c_{1}e^{\left(\lambda-\frac{2}{3}\right)\int{\Theta_{1}}d\tilde{x}^{1}}$.
Hence, it is overall clear to see that the solution for $\mathcal{B}$ can be
written as

$\mathcal{B}=\mathcal{C}e^{\left(\lambda-\frac{2}{3}\right)\left(\int{\Theta_{0}}d\tilde{x}^{0}+\int{\Theta_{1}}d\tilde{x}^{1}+\int{\Theta_{2}}d\tilde{x}^{2}+\int{\Theta_{3}}d\tilde{x}^{3}\right)}=\mathcal{C}e^{\left(\lambda-\frac{2}{3}\right)\left(\int{\frac{\Theta_{0}}{u^{0}}}dx^{0}+\int{\frac{\Theta_{1}}{u^{1}}}dx^{1}+\int{\frac{\Theta_{2}}{u^{2}}}dx^{2}+\int{\frac{\Theta_{3}}{u^{3}}}dx^{3}\right)}\,,$
(3.71)

where $\mathcal{C}$ is an arbitrary constant and we have found out that our
variables separation assumption turns out to be true222222Recall that the
original equation (3.67) is a partial differential one. However, we have shown
that it reduces four ordinary equations (see (3.70)). As a consequence, the
general solution we have found, eq. (3.71) is actually the only solution of
the original equation.. Equation (3.71), which is a solution232323As far as we
know, it is the first time that the solution in question appears in the
literature. of Faraday’s law at the MHD limit, tells us that if
$\Theta_{i}(\tilde{x}^{i})$ are continuous functions in a specific closed
interval $[\alpha_{1},\alpha_{2}]$ of their domain and they preserve constant
sign (e.g. $\Theta_{i}(\tilde{x}^{i})\leq 0$–implying continuous gravitational
contraction) for every value of their variable belonging in the interval, then
$\int_{\alpha_{1}}^{\alpha_{2}}{\Theta_{i}(\tilde{x}^{i})}d\tilde{x}^{i}<0$
and the magnetic field generally obeys an exponential type of increase with
respect to the spacetime variables. In fact, the aforementioned exponential
type behavior seems to be outward because on defining a scale factor
$a(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})$, such that
$\Theta=3\dot{a}/a$ (also $\Theta_{0}=3{\rm d}a_{0}/(a_{0}{\rm
d}\tilde{x}^{0})$ and $\Theta_{i}=3{\rm d}a_{i}/(a_{i}{\rm d}\tilde{x}^{i}))$,
equation (3.71) reduces to:

$\mathcal{B}\propto
a^{3\lambda-2}=\left(a_{0}(\tilde{x}^{0})a_{1}(\tilde{x}^{1})a_{2}(\tilde{x}^{2})a_{3}(\tilde{x}^{3})\right)^{3\lambda-2}\,.$
(3.72)

Finally, we shall keep in mind the following remarks. Firstly, on deriving
relations (3.71), (3.72) we have not adopted a specific coordinate reference
frame. Secondly, the evolution of $\mathcal{B}$ in each spacetime direction is
independent of its evolution in the other directions with respect to the
tilted variables only, where
$\mathcal{B}=\mathcal{T}(\tilde{x}^{0})U(\tilde{x}^{1})V(\tilde{x}^{2})W(\tilde{x}^{3})$.
The crucial equation (3.71), or (3.72), provides us the keystone for studying
magnetic fields in cosmological and astrophysical problems (refer to the
following sections).
In order to specify the constant $\mathcal{C}$, we observe that the key fluid
dynamic quantity related to the magnetic field, is the volume scalar $\Theta$.
Therefore, we turn our attention to the relation which describes its
evolution, the so-called Raychaudhuri equation (see the chapter’s main text or
e.g. [1]),

$\dot{\Theta}=-\frac{1}{3}\Theta^{2}-\frac{1}{2}(\rho+3P+\mathcal{B}^{2})-2(\sigma^{2}-\omega^{2})+D^{a}\dot{u}_{a}+\dot{u}^{a}\dot{u}_{a}\,.$
(3.73)

Considering an instant during which the fluid is found in its equilibrium
(static) state242424Such an instant could have been either the initial
instant-just before the collapse starts-or a transitional instant, during
which the collapse stops and the fluid starts expanding. (setting
$\Theta=0=\sigma^{2}$ and $\dot{u}_{a}=0=\omega^{2}$), we have
$\mathcal{B}=\mathcal{C}$, and (3.73) leads to (the star index refers to
equilibrium values in the following)

$\mathcal{C}^{2}=-(2\dot{\Theta}_{*}+\rho_{*}+3P_{*}),$ (3.74)

which means that $\mathcal{C}$ is a real constant if

$\dot{\Theta}_{*}<-\frac{1}{2}(\rho_{*}+3P_{*})<0.$ (3.75)

In other words, the rate of change of the volume scalar in the equilibrium has
to be negative and smaller than the gravitational mass of the system due to
conventional matter ($\frac{1}{2}(\rho_{*}+3P_{*})>0$).

###  Appendix 3.E Magneto-curvature tension stresses

Following the discussion in subsection 3.3.1, the magneto-curvature tension
stresses associated with shear, rotational and volume curvature distortions
are: 252525On deriving eqs (3.76)-(3.6) we make use of the so-called Gauss-
Codacci formula (e.g. see eq. 1.3.39 in [15]).

$\displaystyle s_{\langle ac\rangle}$ $\displaystyle=$
$\displaystyle\mathcal{B}^{2}\mathcal{R}_{d\langle ac\rangle b}n^{b}n^{d}=$
(3.76) $\displaystyle\mathcal{B}^{2}\left[\epsilon_{\langle
a|q|}\epsilon_{c\rangle
s}\left(\mathcal{E}^{qs}+\Omega^{q}\Omega^{s}\right)n_{\langle
a}n_{c\rangle}+\frac{1}{3}\left(\rho-\frac{1}{3}\Theta^{2}+3\Pi-2\Theta\Sigma-3\Sigma^{2}\right)+\Pi_{\langle
a}n_{c\rangle}\right]$
$\displaystyle+\mathcal{B}^{2}\left[-\pi_{ac}-2\left(\Sigma+\frac{\Theta}{3}\right)\Sigma_{\langle
a}n_{c\rangle}+\frac{\Theta}{3}\sigma_{ac}-\Sigma_{\langle
a}\Sigma_{c\rangle}\right]\,,$
$s_{[ac]}=\mathcal{B}^{2}\mathcal{R}_{d[ac]b}n^{b}n^{d}=\mathcal{B}^{2}\left[2\left(\frac{\Theta}{3}n_{[c}+\Sigma
n_{[c}+\Sigma_{[c}\right)\epsilon_{b]d}\Omega^{d}+\frac{\Theta}{3}\omega_{bc}\right]$
(3.77)

and

$s=s^{c}{}_{c}=\mathcal{B}^{2}\mathcal{R}_{bd}n^{b}n^{d}=\mathcal{B}^{2}\left[\frac{2}{3}\rho+\mathcal{E}+\frac{\Pi}{2}+\left(\lambda^{2}-\frac{\lambda}{3}-\frac{2}{9}\right)\Theta^{2}\right]\,,$
(3.78)

where $\pi_{ab}$ and $E_{ab}$ are the anisotropic stress and the tidal (or
electric Weyl) tensors respectively. Moreover, we have
$\Pi\equiv\pi_{ab}n^{a}n^{b}$,
$\Pi_{a}\equiv\tilde{h}_{a}{}^{b}n^{c}\pi_{bc}$, $\mathcal{E}\equiv
E_{ab}n^{a}n^{b}$,
$\mathcal{E}_{ab}\equiv(\tilde{h}_{(a}{}^{c}\tilde{h}_{b)}{}^{d}-(1/2)\tilde{h}_{ab}\tilde{h}^{cd})E_{cd}$,
$\Sigma\equiv\sigma_{ab}n^{a}n^{b}=\lambda\Theta$ and
$\Sigma_{a}\equiv\tilde{h}_{a}{}^{b}n^{c}\sigma_{bc}=\epsilon_{ab}\Omega^{b}+\alpha_{a}$
(we use the former of the last two expressions only in writing eq (3.78) from
the above), with $\epsilon_{ab}\equiv\epsilon_{abc}n^{c}$ being the 2-D
counterpart of the Levi-Civita pseudotensor,
$\Omega_{a}\equiv\tilde{h}_{a}{}^{b}\omega_{b}$ and $\tilde{h}_{ab}\equiv
h_{ab}-n_{a}n_{b}$ an operator projecting orthogonal to the magnetic field
direction $n^{a}$. We observe that tidal effects (electric Weyl components)
are associated with shape and volume magnetic distortions only. Assuming an
ideal fluid model, the anisotropic stress terms in the above vanish. Then, of
particular interest is that the deformation due to gravitational
compression/expansion in (3.78) is determined by the density of matter and the
tidal tensor projected along the magnetic fieldlines.

###  Appendix 3.F Temporal evolution of the matter density under a polytropic
equation of state

Considering an ideal, polytropic (i.e. $P=k\rho^{\gamma}$, with $k$ and
$\gamma$ constants) fluid at the MHD limit, the continuity equation,
$\dot{\rho}=-\Theta(\rho+P)$ ($\Theta=3\dot{a}/a$, $a$ denoting the scale-
factor of the fluid’s volume), reads the following explicit Bernoulli form

$\frac{d\rho}{da}+\frac{3}{a}\rho+\frac{3k}{a}\rho^{\gamma}=0\,.$ (3.79)

The equation in question accepts the general solution

$\rho=\left[Ca^{-3(1-\gamma)}-k\right]^{\frac{1}{1-\gamma}}\,,$ (3.80)

with $C$ (note that $C>0$ for $k>0$) being the integration constant. In the
cases of non-relativistic neutrons ($\gamma=5/3$) and ultra-relativistic
electrons ($\gamma=4/3$) the above equation recasts into

$\rho=\left(Ca^{2}-k\right)^{-3/2}\hskip 42.67912pt\text{and}\hskip
42.67912pt\rho=\left(Ca-k\right)^{-3}$ (3.81)

respectively. Obviously, the pressure of matter, $P=k\rho^{\gamma}$, increases
faster than its density for $\gamma>1$ (i.e. the cases we consider). We employ
the above equations in determining the magnetic fracture limit of a neutron
star and a white dwarf in the main text (see subsection 3.5.3).

###  Appendix 3.G Some auxiliary calculations

In reference to eq (3.21) (i.e. temporal derivative of the
Alfv$\acute{\text{e}}$n speed) in subsection 3.5.3, we employ the continuity
equation and the equation of state ($P=k\rho^{\gamma}$) to calculate the dot
derivative of matter density and pressure, i.e. $\dot{\rho}=-\Theta(\rho+P)$
and $\dot{P}=\gamma(P/\rho)\dot{\rho}$. Furthermore, we make use of the law of
magnetic contraction, $\dot{\mathcal{B}}=(\lambda-2/3)\Theta\mathcal{B}$
(under the ideal MHD approximation of a magnetised fluid, and condition
$\Sigma=\lambda\Theta$). Therefore, eq (3.21) recasts into:

$\dot{\left(s^{*}\right)}_{\text{fr}}=-\dot{\left(c^{2}_{\mathcal{A}}\right)}_{\text{fr}}u_{\text{fr}}=\frac{1}{\mathcal{B}^{2}(1+\beta)^{2}}\left[\dot{\rho}+\dot{P}-\frac{2}{\mathcal{B}}\dot{\mathcal{B}}(\rho+P)\right]u_{\text{fr}}=\frac{-\Theta(\rho+P)}{\mathcal{B}^{2}(1+\beta)^{2}}\left(\gamma\frac{P}{\rho}+2\lambda-\frac{1}{3}\right)u_{\text{fr}}=0\,,$
(3.82)

which clearly leads to (3.22). The fracture limit condition is mentioned
within the main text. Alternatively, assuming a barotropic equation of state
(i.e. $P=w\rho$ and $\dot{P}=w\dot{\rho}=(P/\rho)\dot{\rho}$ with
$w=\text{constant}$), eq (3.21) reduces to:

$\dot{\left(c^{2}_{\mathcal{A}}\right)}_{\text{fr}}=-\frac{1}{\mathcal{B}^{2}(1+\beta)^{2}}\left[\dot{\rho}+\dot{P}-\frac{2}{\mathcal{B}}\dot{\mathcal{B}}(\rho+P)\right]=\frac{\Theta(\rho+P)}{\mathcal{B}^{2}(1+\beta)^{2}}\left(\frac{P}{\rho}+2\lambda-\frac{1}{3}\right)=0\,,$
(3.83)

from which we deduce that:

$P_{\text{fr}}=\left(\frac{1}{3}-2\lambda\right)\rho_{\text{fr}}\,,\hskip
17.07164pt\text{with}\hskip
17.07164pt0\leq\left(w=\frac{1}{3}-2\lambda\right)\leq 1\hskip
5.69054pt\rightarrow\hskip 5.69054pt-\frac{1}{3}\leq\lambda\leq\frac{1}{6}\,,$
(3.84)

or that $P_{\text{fr}}=-\rho_{\text{fr}}$ (i.e. $w=-1$). The last solution is
directly rejected because ordinary collapsing matter is assumed.

### Bibliography

  * [1] G.F.R. Ellis, R. Maartens and M.A.H. MacCallum, Relativistic Cosmology (Cambridge University Press, Cambridge, 2012).
  * [2] C.B. Collins and G.F.R. Ellis, Singularities in Bianchi cosmologies, Phys. Rep. 56, 105 (1979).
  * [3] E.N. Parker, Cosmical Magnetic Fields (Clarendon Press, Oxford, 1979).
  * [4] L. Mestel, Stellar Magnetism (Oxford University Press, Oxford, 1999).
  * [5] V. Giantsos and C.G. Tsagas, Gravitational convergence, shear deformation and rotation of magnetic forcelines, Mon. Not. R. Astron. Soc 472, 869 (2017).
  * [6] C.G. Tsagas, Magnetic tension and the geometry of the universe, Phys. Rev. Lett. 86, 5421 (2001).
  * [7] C.G. Tsagas, Magnetic tension and gravitational collapse, Class. Quantum Grav. 23, 4323 (2006).
  * [8] A.P. Kouretsis and C.G. Tsagas, Raychaudhuri’s equation and aspects of relativistic charged collapse, Phys. Rev. D 82, 124053 (2010).
  * [9] C.G. Tsagas and P. Mavrogiannis, Melvin’s“magnetic universe”, the role of the magnetic tension and the implications for gravitational collapse, Class. Quantum Grav. 38, 195020 (2021).
  * [10] P. Mavrogiannis and C.G. Tsagas, How the magnetic field behaves during the motion of a highly conducting fluid under its own gravity–A new theoretical, relativistic approach, Phys. Rev. D 104, 124053 (2021).
  * [11] M.A. Melvin, Dynamics of Cylindrical Electromagnetic Universes, Phys. Rev. B 139, 225 (1965).
  * [12] K.S. Thorne, Absolute stability of Melvin’s magnetic universe, Phys. Rev. 139, B244 (1965).
  * [13] K.S. Thorne and D. MacDonald, Electrodynamics in Curved Spacetime - 3+1 Formulation, MNRAS 198, 339 (1982).
  * [14] S.G. Choudhury, Role of a magnetic field in the context of inhomogeneous gravitational collapse, Eur. Phys. J. Plus 137, 971 (2022).
  * [15] P. Mavrogiannis, Law of elasticity and fracture limit of magnetic forcelines under their gravitational deformation, Proc. R. Soc. A. 479, 20230048 (2023).
  * [16] C.G. Tsagas, Electromagnetic fields in curved spacetimes, Class. Quantum Grav. 22, 393 (2005).
  * [17] L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, Amsterdam, 1986).
  * [18] B. Carter and H. Quintana, Foundations of general relativistic high-pressure elasticity theory, Proc. R. Soc. London Ser. A 331, 83 (1972).
  * [19] C.G. Tsagas, A. Challinor and R. Maartens, Relativistic cosmology and large-scale structure, Phys. Rep. 465, 61 (2008).
  * [20] P. Mavrogiannis and C.G. Tsagas, Electromagnetic potentials in curved spacetimes, Class. Quantum. Grav. 38, 235002 (2021).
  * [21] I.D. Novikov, The replacement of relativistic gravitational contraction by expansion, and the physical singularities during contraction, Sov. Astron. 10, 731 (1967).
  * [22] V. de la Cruz and W. Israel, Gravitational bounce, NCimA 51, 744 (1967).
  * [23] A.K. Raychaudhuri, Spherically symmetric charged dust distributions in general relativity, Ann. Inst. Henri Poincaré 22, 229 (1975).
  * [24] D. Pugliese, J.A.V. Kroon, On the evolution equations for ideal magnetohydrodynamics in curved spacetime, Gen. Rel. Grav. 44, 2785 (2012).
  * [25] C.A. Clarkson and R.K. Barrett, Covariant perturbations of Schwarzschild black holes, Class. Quantum. Grav. 20, 3855 (2003).
  * [26] C.A. Clarkson, M. Marklund, G. Betschart and P.K.S. Dunsby, The electromagnetic signature of black hole ring–down, ApJ 613, 492 (2004).
  * [27] L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, Amsterdam, 1984).
  * [28] S. Kar and S. Sengupta, The Raychaudhuri equations: a brief review, Pramana 69, 49 (2007).
  * [29] G. Abreu and M. Visser, Some generalizations of the Raychaudhuri equation, Phys. Rev. D 83, 104016 (2011).
  * [30] R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  * [31] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge University Press, Cambridge, 1973).
  * [32] J.B. Hartle, Gravity: an introduction to Einstein’s general relativity (Addison Wesley, 2003).
  * [33] S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs, and neutron stars : the physics of compact objects (Wiley-VCH, Weinheim, 2004).
  * [34] J.S. Bielich, Compact star physics (Cambridge University Press, Cambridge, 2020).

###  Chapter 4 Magnetised Bianchi I cosmology

####  4.1 General introductory remarks on cosmic magnetic fields

Within the context of the standard cosmological model, large-scale
gravitational as well as electromagnetic perturbations are causally produced
via the inflationary mechanism. In particular, spacetime distortions initially
appear in the form of quantum fluctuations during the so-called Planck epoch.
Subsequently, due to the exponential expansion of the inflation era, these
quantum fluctuations are forced to pass out of the Hubble horizon, where they
freeze out in the form of classical perturbations. After inflation, during
reheating and the following radiation era, the electrical conductivity of the
initially poorly conducting cosmic medium increases rapidly [1]. As a
consequence, the electric fields gradually vanish and the currents freeze the
magnetic fields in with the cosmic fluid. In other words, the post-
inflationary universe can be causally described by the ideal
magnetohydrodynamical model, within the Hubble scale. Besides, the adoption of
the MHD approximation in the standard cosmological framework is in accordance
with the fact that only large-scale magnetic (not electric) fields have been
observed. In the following, our interest focuses on the evolution of large-
scale magnetic fields lying within the Hubble horizon.
Let us recall that in the MHD framework, magnetised ideal barotropic fluids
obey the continuity equation in the form: $\dot{\rho}=-\Theta(1+w)\rho$, where
$0\leq w<1$ is the barotropic index. Therefore, radiation ($w=1/3$) and dust
($w=0$) evolve according to $\rho_{\text{rad}}\propto a^{-4}$ and
$\rho_{\text{dust}}\propto a^{-3}$ respectively, where $a$ denotes the average
scale factor. Using the magnetic density evolution formula $\rho_{B}\propto
a^{6\lambda-4}$ (with $-2/3\leq\lambda<2/3$ -recall eq. (2.28)), the ratios of
the magnetic over the radiation and dust density read:

$\frac{\rho_{B}}{\rho_{\text{rad}}}=\left(\frac{\rho_{B}}{\rho_{\text{rad}}}\right)_{p}\left(\frac{a}{a_{p}}\right)^{6\lambda}\hskip
42.67912pt\text{and}\hskip
42.67912pt\frac{\rho_{B}}{\rho_{m}}=\left(\frac{\rho_{B}}{\rho_{m}}\right)_{p}\left(\frac{a}{a_{p}}\right)^{6\lambda-1}\,,$
(4.1)

where the suffix $p$ indicates the values of the involved quantities at the
present, and $a_{p}/a=1+z$, with $z$ being the redshift. When the two forms of
energy acquire equal densities, the associated scale factors,
$a_{\text{eq}~{}(B-\text{rad})}$ and $a_{\text{eq}~{}(B-m)}$, are:

$a_{\text{eq}~{}(B-\text{rad})}=\left(\frac{\rho_{\text{rad}}}{\rho_{B}}\right)^{\frac{1}{6\lambda}}_{p}a_{p}\sim
10^{-3/\lambda}a_{p}\hskip 28.45274pt\text{and}\hskip
28.45274pta_{\text{eq}~{}(B-m)}=\left(\frac{\rho_{m}}{\rho_{B}}\right)^{\frac{1}{6\lambda-1}}_{p}a_{p}\sim
10^{-\frac{22}{6\lambda-1}}a_{p}\,.$ (4.2)

In the above calculation we have taken into account that the present value of
intergalactic magnetic fields amounts to the order of $10^{-15}$ Gauss (e.g.
refer to [2]-[4]). Making use of natural units ($c=\hbar=k_{B}=1$) the
intergalactic magnetic energy density today is expressed in terms of GeV’s as
$\rho_{B}\sim 4\times 10^{-70}~{}\text{GeV}^{4}$, in accordance with the
equivalence: $1~{}(\text{Gauss})^{2}/(8\pi)\simeq 2\times
10^{-40}~{}\text{GeV}^{4}$ (e.g. see the appendix of [5]). Moreover, the
density of matter today is $\rho_{m}\sim
10^{-30}~{}\text{gr}/\text{cm}^{3}\sim 4\times 10^{-48}~{}\text{GeV}^{4}$
($\rho_{m}=\Omega_{m}h^{2}\rho_{\text{crit}}$ with $\rho_{\text{crit}}\sim
10^{-29}~{}\text{gr}/\text{cm}^{3}$ and $\Omega_{m}h^{2}\simeq 0.14$ today
[6]) whilst its radiation counterpart is
$\rho_{\text{rad}}=10^{-34}~{}\text{gr}/\text{cm}^{3}\sim 4\times
10^{-52}~{}\text{GeV}^{4}$ ($1~{}\text{GeV}^{4}\simeq 2\times
10^{17}~{}\text{gr}/\text{cm}^{3}$).
Subsequently, it is worth raising an issue related to the constraint that
cosmic nucleosynthesis imposes on the magnitude of the magnetic energy
density. In particular, magnetic fields are known to increase nuclear
reaction/transformation rates111Besides, magnetic fields contribute to the
expansion rate of the universe and thus indirectly affect the rate of nuclear
interactions., so that a potential domination of the magnetic energy density
over radiation, during the early stages of cosmic evolution, may be
incompatible with current predictions/observations regarding the nuclei
abundance in the universe. We attempt here a first approach to the question by