VORTEX and
space-time
distortion
Z.Yoshida
1and S.M.Mahajan
21Graduate School
of
FrontierSciences, TheUniversityof
Tokyo2Institute for
FusionStudies, The Universityof
Texas atAustin AbstractThe Universe is filled with $s$
‘vortexes” (such
as
galaxies, accretion disks, starsand planetary systems, etc.) that clump and wind-up with magnetic fields.
Strik-ingly absent in this rich narrative of growth and evolution of the cosmic systems,
isa satisfactory ”universal‘’ mechanismthat could havegenerated the original seed
magnetic field. Because the explosive expansion of the universe must immensely
dilute themagneticfieldstrength, verystrongfields must haveoriginated inthe early
universe. Exploiting thespace-time distortion inherent in relativistic dynamics, we
have unearthedjustthe mechanism that, by breaking the topological constraint
for-bidding theemergence of magnetic fields(vortexes), allows“generalvorticities” –
naturally coupled vortexes of matter motion and magnetic fields–to be created in
anidealfluid. Thenewly postulated relativistic mechanism,arising fromthe
interac-tion betweenthe inhomogeneous flowfieldsandinhomogeneousentropy, maybean
attractiveuniversal solutiontothe origin problem.
1
Introduction
Vortexisthe most
common
appearance
ofexisting, sustaining,and evolving“heterogene-ity,” at
every
scale hierarchy, in the Universe. Oncea
vortex is created, it behavesas
ifit
were
alive; vortex is basicallya
coherent, stable “object,’‘ while its motion isconsid-erably complex; its interactionswith othervortexes
are
not like those ofparticles, sinceinteractions, in general,
may
penetrate into the identityof each vortex–quantization ofvortexis not
a
strait-forward notionbecauseoftheessential nonlinearityofthe kinematicdescription (while
many
different approacheshave been proposed, and have madesome
interestingprogresses).
Among rich narrative ofvarious aspects ofvortexes, the “origin problem” is
one
ofthe mostchallenging. The fact that the circulationmustvanish for
every
ideal force(in-cluding the thermodynamic force
as
longas
entropy isconserved)forbids theemergence
ofvorticity (or
an
axial vector field) in any ideal leading order model. Thisfundamen-tal obstacle (aconservationlaw known
as
Kelvin’s circulationtheorem), anchoredon
thegeneralHamiltonianstructureofideal kinematics[1,2, 3],seemingly inhibits thecreation
of the
very
fistvorticity intheUniverse; sincethe vorticity of fluidmotionis unified withmagnetic field is
simultaneously
questioned[4]. Invoking “non-ideal effects” has beenthe only known
recourse
to change the vortical state ofa
fluid; A typical example is thebaroclinic mechanism[5],
or
Biermann$battel\gamma[6]$, involvingnon-ideal thermodynamicsin which the gradients of
pressure
and temperature havedifferent
directions[7, 8]. Avelocity-space non-equilibriumdistribution also provides
a
source
ofmagnetic field viathe so-called Weibel instability[9]. In early cosmology, inflation[10, 11], QCD phase
transition[12, 13],
or
radiationeffect[14]couldcreatea
source.
While these mechanismsmay,
and likely will, play important roles in magnetic-field generation atsome
scales,none
of these could beconsidered
a
universal mechanismoperating
atall scales[4].In thepresent work,
we
demonstrate thata
purelyideal mechanism,originating inthespace-timedistortion(shearing)causedbythedemands of special relativity,
can
breaktheoffendingtopologicalconstraint. Vorticity,then,
may
be generated throughan
interactionbetween the inhomogeneous flow fields and inhomogeneous entropy. The
new
mecha-nism is universal, andisstrong enough to
overcome
dissipationeven
for relatively weakflows[15, 16].
2 Generalized
vorticity
and
circulation
theorem
The mathematical and dynamical similarity between magnetic fields and fluid
vorticity
imparts
both elegance andusefulness
tothe concept of generalizedvorticity.
Unlessex-plicitly stated,generalizedvorticity,denoted by $\Omega$, will symbolizeall
physical quantities
of this nature.
The “origin problem” has its genesis in the fact that the circulation associated with
$\Omega$
mustvanish for$evel\gamma$ “ideal force” includingthe entropy conserving thermodynamic
force. The
reasons
lie deep in the Hamiltonian structure governingthe dynamics ofan
idealfluid; the constrained dynamics implies theconservation of
a
“topological charge”that
measures
the generalized vorticity of the fluid –the invariance of the generalizedhelicity,which,for
a
non-relativisticchargedflow,takes thefamiliarform$K= \int P\cdot\Omega dx$,where $P=mV+(q/c)A$ isthe canonical momentum and $\Omega=\nabla\cross P$ isthe generalized
vorticity
or
generalized magnetic field $(m$:
mass
ofa
particle, $q$:
charge ofa
particle,$V$
:
fluid velocity, $A$:
vectorpotential, $B$:
magnetic field). Consequently, inany
“ideal”leading ordermodel, $\Omega$
(consisting of both magnetic andkinematic components) cannot
emerge
froma
zero
initial value.The problem of unearthing
a
primary generation mechanism for the magnetic field,found to be important in
every
scale hierarchy ofuniverse, has defieda
satisfactoryso-lutionto date[4]. Since the topological constraint
on
the ideal fluidforbids the vorticityto
emerge, one
resorts to “non-ideal dynamics” to affecta
change. However,a
satisfac-torily strong anduniversal mechanism, operating atall scales, is not known. The search
for such
a
universal mechanism provided the stimulus for thispaper
in whichwe
makea
clean break with the standard practice: Instead of relyingon
non-ideal effectswe
willshow that $\Omega$
can
be generated in strictly ideal dynamics,as
long
as
the dynamics isex-plicitly embedded in the space-time dictated by the demands of special relativity. The
generalized $vollicity$ is, then, generated through
a
source
term bom out oftheinhomoge-Figure 1: Transport of
a
loop and circulation[15]. Givena
loop$L$inspace,
thecirculationof
a
vector field $P$is
the integralsll
$P\cdot dx$.
Two loops $L(\tau)$ and $L(\tau’)$, connected bythe “flow” $dx/d\tau=U$ (the parameter $\tau$
may
be regardedas
time),are
shownin
thefigure. A circulation theorem pertains to
a
“movement” of loops; the rate of changeofcirculation is calculated
as
(1). Ona
loop $L(\tau)$ carried by the fluid (i.e., $\tau=t$ and$U=V)$,thecirculation is conservedbecause $\ _{(l)}\nabla \mathcal{E}\cdot dx\equiv 0$(Kelvin’scirculation law).
Togeneralize the argumenttothe relativistic regime,
we
have to immersethe loop in the4-d space-time and transportitby the 4-velocity $dx_{\mu}/d\tau=U_{\mu}$;
see
Fig. 2. The explicitlyLorentz
covariant
equality,written
in terms oftheproper
time $\tau=s=ct/\gamma$ is found tobe$d(\phi_{(s)^{P^{\mu}dx_{\mu})/ds}}=f_{(.’)}(\theta^{A}P^{\nu}-\partial^{v}\mu)U_{\nu}dx_{\mu}$
.
Thus if the fluid equation couldbe castin the form $(\theta^{A}P^{\nu}-\partial^{v}P^{\mu})U_{\gamma}=\theta^{l}\varphi$, the circulation would, indeed, be conserved. The
relativistic space-timecirculation
conserves
in ideal fluids;see
(2).neous
entropy. To set the stage fora proper
relativistic calculationwe
begin withsome
non-relativistic preliminaries and
see
howan
“ideal‘’ mechanicsrestricts the topology offields.
The circulation
&
$\delta$Q, associated witha
physical quantity $\delta Q$, calculated along theloop $L$,
may
bezero or
finite dependingon
whether$\delta Q$equalsan
exactdifferential $d\varphi(\varphi$being
a
statevariable)or
not. Forexample,if$\delta Q=Td\sigma$($T$:
temperature,$\sigma$:
entropy),thecirculation isgenerally finiteand
measures
the heat gainedina
quasistatic thermodynamiccycle.
An ideal
fluid
can
be viewedas
a
realization ofan
infinite number ofideal isolatedcycles covering
space.
Alongthetimedependent loop$IXt$),convected by the fluidmotion(see Fig. 1), the rate of change ofcirculation associated with the canonical momentum
$\ _{(()}P\cdot dx$is identically
zero:
connecting
twoloops$L(t)$and$L(l)$by$a$ flow”$dx/dt=U$ ,therate of change ofcirculationiscalculated
as
$\frac{d}{dt}\oint_{L(l)}P\cdot dx=\oint_{L(\tau)}[\partial_{\tau}P+(\nabla\cross P)\cross U]\cdot dx$
.
(1)The ideal equation of motion for the momentum $P=mV$
can
be written in the form$\partial,P+(\nabla\cross P)\cross V=-\nabla 8$with the
energy
density$8=P^{2}/2m+\phi+H(\phi$:
potentialenergy,
$H$
:
enthalpy). Hence, therate ofchange ofcirculation equals the circulation ofan
exactfluid-dynamic force derived from the
energy
density, i.e., $\phi_{(,)^{\nabla\epsilon\cdot dx}}=$Sl
$(l)^{d\epsilon}=0$.
Inthe standard non-relativistic description of
an
ideal fluid, therefore, if the $inl$tial statehasno
circulation (vorticity), the laterstate will also be vorticity-free (Kelvin’s circulationtheorem). For the vorticity to becreated, the “force”
on
the fluid must not bean
exactdifferential.
(a) (b)
$t$
Figure 2: Transport of
a
surface (and its boundary) in space-time. Two figurescompare
the evolution of
a
surface and its boundary (loop) moved, respectively, by (A) thenon-relativistic velocity $(dx_{j}/dt=V_{j}$
: 3-vector
$)$ and (B) the relativistic 4-velocity $(dx_{\mu}/ds=$$U_{\mu})$
.
The figuresare
drawn in the space-time x-y-twith $V/c=(\tanh x,O,O)$ (thus $\gamma=$ $sech^{-1}x)$.
In the Lorentz-covarianttheory, the circulation theorem applies toa
loop$L(s)$thatismoved by the4-velocity$U_{\mu}$in the4-dimensionalspace-time. However,thevorticity
(or
magnetic
field)is
a
reference-dependent quantity
definedon
the synchroniccycle$L(t)$,requiring
a
mapping from the naturally (relativistically) distorted $L(s)$to $L(t)$; this mapmultiplies the thermodynamic force by
a
Jacobian weight$\gamma^{-1}$ breaking the exactness ofthe
differential
form.In a plasma, the momentum must be generalized to the canonical momentum that
includes the EMpart, i.e., $P=mV+(q/c)A$
.
Thegeneration ofa
canonical circulation(orvorticity),then,implies the
emergence
ofmagnetic field.3
Relativistic circulation
theorem
in
space-time
Interestingly enough, the space-time unity imposed by special relativityprovides
a
path-way
to create vorticity. This purely kinematic relativistic effectactsbyimposinga
Jaco-bian weight $\gamma^{-1}=\sqrt{1-(V/c)^{2}}$ that destroys the exactnessof the ideal thermodynamic
$Zero.Thusvor\ddagger icitycouldbecreatedwithinpureforce;relativitytransfo-stheloopintegra]g_{(t}i_{yidea1dynamics}^{dH,to\oint_{L(t)}\gamma^{-1}dH}$which is
no
longerFor
a
geometricvisualizationofthenew
creationmechanism,letus see
how relativitybrings about
a
fundamental reconstruction ofthe notion ofcirculation. In the relativisticspace-time, the loop $L(t)$ pertaining to
a
“synchronic space” $(t=$ constantcross
sectionofspace-time in
a
reference frame)ceases
to be the appropriate geometric object alongwhich thecirculation mustbe evaluated (seeFig. 1). The loop
moves
in space-time witha
4-velocity $U^{\mu}=(\gamma, \gamma V^{j}/c)$ ($V^{j}$: the reference-frame velocity) and therelativistic
cir-culation must be described
as
a
function of theproper
time $s$. In Fig. 2, the respectiveevolutions of the “synchronic loop” $L(t)$ and the “relativistic loop” $L(s)$
are
compared.The synchronicity of the loop $L(s)$ is broken by the nonuniformity of the
proper
time.4-vector$\mu$ along$L(s)$ obeys
$\frac{d}{ds}(\oint_{L(s)}\phi dx_{\mu})=\oint_{L(s)}(\theta^{!}\wp^{v}-\partial^{v}\wp^{\mu})U_{v}dx_{\mu}$
.
(2)If$\mu$ is
an
appropriatemomentum,the relativistic equation ofmotionrelatesthe integrand$(\theta^{l}\wp^{v}-\partial^{v}\phi)U_{v}$ with
an
effective
force. If the force is exact, the relativistic circulationwill beconserved;the
ideal
fluid, indeed,obeys thisrelativistic
circulation theorem.How-ever, vorticity (or magnetic field) is defined
on
synchronicspace
(hence, it isreference-dependent); its circulation still
pertains
to the synchronic loop $L(t)$.
The field must bemapped from the naturally distorted $L(s)$ back to $L(t)$–this reciprocal distortion,
repre-sented by
a
Jacobian$\gamma^{-1}$, impartsa
sheartothethermodynamic force(i.e,changes$dH$to.
$\gamma^{-1}dH)$destroying itsexactness.
Theseformalconsiderationswill,now, betranslated into
an
explicit calculationshow-inghowrelativityhelps
us
tocircumventthe“no-circulation”theorem. Acovarianttheoryof
vorticity
generationfollowsfrom the recentlyformulated unifiedtheory ofrelativistic,hotmagneto-fluids[17]. The central constructionof thistheory istherelativistic
general-ized
4-momentum
$\wp^{\mu}=mcfU^{\mu}+(q/c)A^{\mu}$ ($A^{\mu}$: 4-vector
potential) and theanti-symmetric
tensor
$M^{\mu v}=\theta^{l}\wp^{v}-\partial^{v}\emptyset^{I}=mcS^{\mu v}+(q/c)F^{\mu v}$, (3)
where $S^{\mu v}=\theta^{l}(fU^{v})-\partial^{v}(fU^{\mu})$ is the flow-field tensor representing both the inertial
and thermal forces, and $F^{\mu v}=\theta^{l}A\nu-\partial^{v}A^{\mu}$ is the electromagnetic tensor. The factor$f$
represents the thermally induced increase in effective
mass
$(h\equiv fmc^{2}$ relates $f$ to themolarenthalpy$h;h$ is
an
increasing function of temperature $T,$ $f\approx 1$ fornon-relativisticrisingto$f\approx 6.66$for$T=1MeV$)$[1,18]$
.
Instandardtextbooks andpapers
$h=(p+p)/n$with$\rho$ and$p$being the
proper energy
density andpressure,
respectively. The generalizedvorticity $\hat{\Omega}$
(orthe generalized magnetic field $\hat{B}$
) is defined by $\nabla\cross\wp$ $(or (c/q)\nabla\cross\wp)$,
where $\wp$is thevectorpart of$\mu$
.
It mustbeemphasized that theflow-EMfield
tensor$M^{\mu v}$containsboththe inertial and thethermal forces.
Following
an
explicitlycovariant
procedure, theequation
ofmotion
$\partial_{\mu}T^{\mu v}+qnU_{\mu}F^{\mu v}=0$, (4)
where $T^{\mu\nu}=nhU^{\mu}U^{v}-Pg^{\mu v}$is perfect fluidenergy momentumtensorand theright hand
sideis theLorentzforce,
can
bedisplayedas
[17]$cU_{\mu}M^{\mu v}=T\partial^{\nu}\sigma$, (5)
where
we
havewrittenas
$\partial^{v}h-n^{-1}\partial^{\gamma}P=T\partial^{v}\sigma$ invoking a“thermodynamicrelation;’witha
temperature$T$ anda
molar entropy$\sigma$,torepresent thenon-exactresidual ofthe left-hadside($T$ and$\sigma$
are
assumedtobe numbers independent ofthechoiceofcoordinate).Thevectorpartof(5)
$q( \hat{E}+\frac{V}{c}\cross\hat{B})=\frac{T\nabla\sigma}{\gamma}$, (6)
with thegeneralizedelectricand magnetic
fields
given by$\hat{E}$
$=$ $E-(1lq)[(\partial_{l}(\gamma fmV^{j})+\nabla(\gamma fmc^{2})]$, (7)
$\hat{B}$
is
a
conciseway
inwhich the equation ofmotionofa
hot relativistic fluid isexpressedina
form
reminiscent
of thenon-relativistic version. Byconstruction $(S^{\mu}V$was
definedtohavetheexactformof$F^{\mu v}$), the generalizedfields
satisfy Faraday’s law$\partial_{l}\hat{B}=-c\nabla\cross\hat{E}[19]$
.
The
appearance
of
$\gamma^{-1}$on
the right-hand side of(6) is dueto the
mapping
backof therelativistic space-time onto the synchronic
space
in which the conventional circulationandthe vorticity
are
to be calculated. Toevaluatethe rateofchangeof$\hat{B}$(withrespect to
the reference
time
$t$),we
mustgo
back to (6) whose curl reveals thesource
for magneticfield generation:
$\partial_{l}B-\nabla\cross(V\cross\tilde{B})=-\nabla(\frac{cT}{q\gamma})\cross\nabla\sigma\equiv \mathfrak{S}$, (9)
where the right-hand-side generationtermis brokenintothefamiliar baroclinicterm$\mathfrak{S}_{B}=$
$-(c/q\gamma)\nabla T\cross\nabla\sigma$andtherelativistically induced
new
term$\mathfrak{S}_{R}=-(\frac{cT}{q})\nabla\gamma^{-1}\cross\nabla\sigma=-(\frac{c\gamma}{2qn})\nabla(\frac{V}{c})^{2}\cross\nabla p$
.
(10)4 Relativistic
source
of magnetic field
Thediscoveryof$\mathfrak{S}_{R}$is the principalresultof this
paper.
Following conclusions
are
readilydeducible:
1$)$ Forhomogeneousentropy, thereis
no
vorticity
drive–either baroclinic
or
relativis-tic.
2$)$As long
as
the kineticenergy
is inhomogeneous, itsinteractionwith inhomogeneous entropykeeps $\mathfrak{S}_{R}$ non-zero,
even
ina
barotropic fluid.3$)$ When baroclinic driveisnonzero, and, in addition,
the kinematicandthermal
gra-dients
are
comparable,we can
estimate
$\frac{|\tilde{(}0_{R}|}{|\mathfrak{S}_{8}|}\approx\frac{(V/c)^{2}}{1-(V/c)^{2}}$
.
(11)
For highly relativistic flows(cosmicparticle-antiparticle plasmas, electron-positron
plas-mas
in the magnetosphere of neutron stars, relativisticjets, etc.), $\mathfrak{S}_{R}$ will be evidentlydominant,and
can
be far larger than the conventional estimatesfor the baroclinicmecha-nism. Onemustalso remember thatmostlong lived plasmas will tendtohave$\nabla T\cross\nabla\sigma=0$
becauseofthethermodynamiccouplingof temperatureand entropy. In this largemajority
ofphysical situations, $\mathfrak{S}_{R}$
may
be the only vorticitygeneration mechanism;no
physical
constraints will force the alignment of thegradientsof kinematic$\gamma$and statistical $\sigma$
.
Thus,therelativistic drive is trulyuniversal.
5
Separation of kinetic vorticity
and
magnetic field
After having shown that thenew drive$(\tilde{o}_{R}$will always dominatethetraditional baroclinic
drive $\mathfrak{S}_{B}$ for relativistic plasmas,
we
willnow
attemptto estimate its strength in
a
fewrepresentative
cases.
Since the basic $theo1\gamma$ pertains to thegeneration of the generalizedvorticity$\Omega$, the eventual
apportioningof$\hat{\Omega}$
part will be
a
difficult system-dependent exercise–for example, ifthe plasmaconsistsofrelativistic electrons in
a
neutralizing ionbackgroundor
itisan
electron-positron pairplasmawhereboth species
are
dynamic.Here
we
considera
typical pair plasma, neutral in its rest frame, with density $n_{+}=$$n_{-}=n=$ constant[16]. The
suffix
$+$ $(-)$ labels thepositive
(negative)particles. Wealso
assume
that the particles have thesame
homogeneous temperatureso
thattheirtempera-ture modified effective
masses
$m_{\pm}^{*}\equiv f_{\pm}m$ ($m$:
rest mass)are
also thesame
$(m_{+}^{*}=m_{-}^{*}=$$m”=$ const.). The generalizedcanonical momenta
are
$\wp_{\pm}^{j}=m^{*}cU_{\pm}^{j}\pm(e/c)A^{j}$,
and the associated generalized vorticities
are
$\frac{1}{m}*\nabla\cross\wp_{\pm}$ $=$ V$\cross(cU_{\pm})\pm\frac{e}{mc}*B$
$\equiv\omega_{\pm}\pm\omega_{c}$, (12)
in
termsofwhich, theinductionequation
(9) takes the form$\partial_{l}(\omega_{f}\pm\omega_{c})-\nabla\cross[V_{\pm}\cross(\omega_{\pm}\pm\omega_{c})]=-\nabla\cross(\frac{cT\nabla\sigma_{\pm}}{\gamma_{\pm}m^{*}})$
.
(13)To close the system,
we
needa
determiningequation for$\omega_{c}=eB/(m^{*}c)$(thenormalizedmagnetic field). When the large-scale slowly evolving EM isdecoupled from the photons,
the displacementcurrent
may
be neglected[20] and theresulting Ampere’slaw$\nabla\cross B=\frac{4\pi}{c}J=4\pi en(U_{+}-U_{-})$ (14)
may be written
as
$\delta^{2}\nabla\cross\omega_{c}=c\hat{n}(U_{+}-U_{-})$
.
(15)Here $\delta=c/\omega_{pe}$ (electron inertia length) with$\omega_{pe}^{2}=4\pi e^{2}\overline{n}/m^{*}$ (plasma frequency), $\overline{n}$ is the
average
density,and$\hat{n}=n/\overline{n}$isthenormalized density. The curl of(15):$\delta^{2}\nabla\cross(\hat{n}^{-1}\nabla\cross\omega_{c})=\omega_{+}-\omega_{-}$ (16)
shows thatthe magnetic field is related to the difference in the normal vorticities of the
two fluids.
We denote the
generation
drivesas
$G_{\pm}=- \nabla\cross[\frac{cT\nabla\sigma_{\pm}}{\gamma_{\pm}m}*]$
.
Assuming $V_{+}\approx V_{-}\approx\overline{V}$, and defining$\overline{\omega}=(\omega_{+}+\omega_{-})/2$,
we may
rewrite (13)as
$\partial_{l}\overline{\omega}-\nabla\cross(\overline{V}\cross\overline{\omega})=\frac{G_{+}+G_{-}}{2}$, (17)
$\partial_{l}\omega_{c}-\nabla\cross(\overline{V}\cross\omega_{c})=\frac{G_{+}-G_{-}}{2}$
.
(18)Here
we
haveapproximated, using(16)andassuming alarge scale $(\gg\delta)$,$\omega_{c}+(\omega_{+}-\omega_{-})/2=\omega_{c}+(\delta^{2}/2)\nabla\cross(\hat{n}^{-1}\nabla\cross\omega_{c})$
6
Cosmological
application
One of theprimary motivationsto lookfor
an
idealdrivewas
toinvestigateif sucha
drivecould generate
a
magnetic field in early universe (when the plasma is in strict thermalequilibrium)thatis strong enoughtoleave itsmarkin
an
expandinguniverse. Wepresenthere
a
possiblescenariothat couldemerge
inthe light of thecurrentrelativistic drive. Thescenario is intertwinedwith the thermal history of theuniverse. Although there
are
earlierhottereras, let
us
beginour
considerations around 100$MeV$.
(i) $100MeV(10^{12}K)$
age
$(- 10^{-4}s)$:
At this time the muon-antimuonare
begin-ning to annihilate, and primary constituents of the universe
are
electron-positron pairs,neutrionos-antineutrinos
and photons, allin thermalequilibrium, witha
$vel\gamma$smallamountof nucleons (protonsandneutrons).
(ii) $10MeV$
age
$(\sim 10^{-2}s)$:
The mainconstituents
are
electron-positron pairs,neu-trinos and photons, and
a
very
small amount of nucleons (protons getting considerablymore
thanneutrons). Atthisstageneutrinosare
decoupled andare
freely expanding. Theelectron-positron pairs and photons
are
coupled and in thermal equilibrium.(iii)0.5$MeV$
age
$(\sim 4s)$:
The neutrinosare
in free expansion. The electron-positronpairs
are
beginningto annihilate.(iv) 0.$1MeV$
age
$(\sim 180s)$:
Almost all pairsare
gone,
and whatwe
havenow
isan
electron-proton plasma contaminated by lots ofneutrons andgammas (gammas
are
stillelectromagneticallycoupled).
(v)Thisplasmacontinuesfor
a
longtime,butit isbarelyrelativistic(protonsare
not).Nucleosynthesisconverts
some
protons andneutrons toHe (about25% of the mass). Butwe
continue withan
electron-ion mildly relativistic and then essentially nonrelativisticelectrons ti114000$K$ when atomic hydrogen forms by the absorption of electrons in the
protons. At thistimethe universeis about
400000 years
old.Notice that till there is plasma whether electron-positron
or
electron-proton (He), theradiation keeps theparticles in thermalequilibrium,
so
there isno
baroclinicterm. Hencethe onlything that could generate seed vorticity is the relativistic
source.
Nowwe
couldenvisage the “magnetic field” generation in several stages:
1$)$ The universal “ideal” relativistic drive creates seed vorticity in the $MeV$
era
oftheearly universewhen the electron-positron$(e_{-}, e_{+})$ plasma isthe dominantcomponent
decoupled ffom neutrinos. In the context of this
paper,
this is the crucial element of thetotal scenario–the restis cobbling together pieces of highly investigatedphenomena.
2$)$ Below 1 $MeV$,
as
the $(e_{-}, e_{+})$pairs beginto annihilate, the electron-proton plasmatendstobe the dominantcomponent. This
era
lastsfor400,000years
till the temperaturefalls to4000$K$when the plasma disappears and the radiation decouples frommatter.
Dur-ingthis relativelylong era, the seed field, created in the$MeV$era, isvastly magnified by
what
we
call the Early Universe Dynamo (EUD).3$)$Atthehydrogen formationtime,thismagnetic field(howeverbigitis)isdecoupled
frommatter–thereis
no
plasmaleft,and like the photons, the macroscopic magnetic fieldbecomes
a
relic and red shifts (goes down in intensity) conserving flux. The relic fieldmanifests in later
eras
appropriatelydiluted (conserving flux)by thecosmic expansion.4$)$ This diluted field would, then, provide the seed for
an
intergalacticor a
galactic7
Concluding
remarks
We have found that
a
recourse
to special relativityuncovers
an
ideal, ubiquitous,fun-damental vorticity generation mechanism. The exploration ofthis mechanism is likely
to help
us
understand, inter alia, the origin ofthemagnetic
fields in astrophysical andcosmic settings.
Weend this
paper
by makinga
fewcomments aboutthefiner points concemingvor-ticity,
the generalized vorticity, and the relativistic generalized vorticity. As the physicalsystem becomes
more
andmore
complicated (froman
uncharged fluidtoa
charged fluidto
a
relativistic charged fluid),one
must inventmore
andmore
sophisticated physicalvariables
so
thatthe fundamental dynamical structure (vortical form), epitomizedin (5)is maintained. We do this because the
very
beautiful vortical structure isso
thoroughlystudied that reducing
a
more
complicated
system to thisform
immediately advancesour
understanding of
new
largerphysical systems or, possibly, ofmore
advancedspace-time
geometnes.
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(2000).[9] E. S. Weibel,Phys. Rev. Lett. 2, 83 (1959).
[10] M. S. Tumer,L. M. Widrow,Phys. Rev. D37,
2743
(1988).[11] E. A. Calzetta, A. Kandus,F. D.Mazzitelli, Phys. Rev. D57,
7139
(1998).[12] C. J. Hogan,Phys. Rev.Lett. 51, 1488 (1983).
[13] J.M. Quashnock, A. Loeb,D. N. Spergel, Astrophys. J. 344,L49 (1989).
[14] E. R. Harrison,Phys. Rev. Lett. 30, 188 (1973).
[16] S. M. Mahajan and Z. Yoshida,Phys. Plasmas 18,
055701
(2011).[17] S.M. Mahajan, Phys. Rev. Lett. 90,
035001
(2003).[18] In the relativistic regime, the definition ofvorticity must be appropriately
general-ized: Based
on
the Lorentz covariantmagneto-fluid unified theory[17], thecanon-ical momentum P $=mf\gamma V+(q/c)A$ modifies through the relativistic factors $\gamma$
and
f.
The latterrepresents the increase ofthe effectivemass
by large randommo-tions;
we
may
writef
$=f(T)=K_{3}(mc^{2}/T)/K_{2}(mc^{2}/T)$ fora
Maxwellian fluid,where $K_{j}$isthemodified Besselfunction.Thefactor
f
isrelatedtoenthalpythroughH $=mn_{R}f$where$n_{R}$ isthe fluid density in therest frame;
see
D. I. Dzhavakhrishvili,N. L. Tsintsadze, Sov. Phys.JETP37,
666
(1973); V I. Berezhiani, S. M. Mahajan,Phys. RevE52,
1968
(1995).[19] Inthedefinitionof the generalized electricfield,noticethe terms$\nabla(\gamma fmc^{2})\equiv\nabla(\gamma h)$,
and $(\partial_{l}(\gamma fmV^{j})$ that contain manifestations of the thermal effects. Combining the
first of these with the right-hand sideentropy termyields
an
effectivepressure
forcewhich reproduces the standard
pressure
term in the nonrelativistic limit $(\gammaarrow 1)$:
$-\nabla(\gamma h)+\gamma^{-1}T\nabla\sigmaarrow-n^{-1}\nabla P$
.
Thesecondterm$(\partial_{l}(\gamma fmV^{j})$containstime derivativesof
f.
This termwitha
little manipulation, andhelp from thezero
componentof(5),can
beconverted,ifone
so
wished,intoa
termproportional to$\partial_{t}p$. Forex
ample,eq.
(9) of T. Katsoulous and W. Mori, Phys. Rev. Lett. 61, 90 (1988) will be the
one
dimensional limitof suchanequationderivedfrom (5).
[20] Thedisplacement-current term$\partial_{l}E/c$in Ampere’slawwill
appear
as
$-\omega_{pe}^{-2}\partial_{1}^{2}\omega_{c}$on
the left-hand side of(16). In the time scale $\tau\gg f/c(\ell$ is the length scale of the
structures), this term
may
be neglected with respect to$\delta^{2}\nabla\cross(\hat{n}^{-1}\nabla\cross\omega_{c})$, and theD’Alembert operator collapses to the elliptic operator, eliminating the EM
waves.
The displacement-current termmaynotbe neglected when