Generalized, or power-law, inflation is shown to necessarily exist for a simple, anisotropic (Bianchi Type I) cosmology in the Einstein-Cartan gravitational theory with the Ray-Smalley (RS) improved energy-momentum tensor with spin. Formal solution of the EC field equations with the fluid equations of motion explicitly shows inflation caused by the RS spin angular kinetic energy density

Bradas, James C.

Fennelly, A. J.

Smalley, Larry L.

English

NASA Technical Reports Server

INFLATION IN EINSTEIN-CARTAN THEORY WITH
ENERGY-MOMENTUM TENSOR WITH SPIN
by
A. J. Fennelly <a>
MS-47 Teledyne Brown Engineering
Cuomings Research Park
Huntsvilie, AL 35807
James C. Bradas (a)
AMSMI-RD-GC-T, Research Development and Engineering Center,
U.S. Army Missile Command,
Redstone Arsenal, AL 35898
and oo
Larry L. Smalley
ES-65, Space Science Laboratory ^
Marshall Flight Center, AL 35812
101 16(1) 1MI1AIICK IK N88-215il7
EIKSTEIN-CABTAN atEOfl l SIT£ IKIEGY-IOUEBTOB
TENSO£ WJ1H Sfli: ( N A E 8 ) 18 p CSCL 20C
Uuclas
G3/77 01 467 16
https://ntrs.nasa.gov/search.jsp?R=19880015163 2020-03-20T07:16:25+00:00Z
Abstract
Generalized, or "power-law," inflation is shown to necessarily
exist for a simple, anisotropic, (Bianchi Type-I) cosmology in the
Binsten-Cartan gravitational theory with the Ray-Smalley improved
energy-momentum tensor with spin. Formal solution of the EC field
equations with the fluid equations of motion explicity shows inflation
caused by the US spin angular kinetic energy density.
(a) Also with Department of Physics, University of Alabama in
Huntsville, AL 35899.
INFLATION IN EINSTEIN-CARTAN THEORY WITH
IMPROVED ENERGY-MOMENTUM TENSOR WITH SPIN
by
A. J. Fennelly, James C. Bradas and Larry L. SmaHey
Introduction. The early universe is expected to undergo a
period of inflation (exponential expansion) due to the presence of bare
quantum fields which behave like a cosmological constant term in the
field equations. That period of accelerated expansion can also exist as
a state of "power-law inflation" in which the expansion scale factor is
a power-law function of the time.^ In either case, R - e^ fc of R - t^,
inflation. is important because it may offer a simultaneous solution to
the cosmological problems of horizons, homogeneity, and flatness.^
Inflation may also provide a solution to the origin of the
presently observed large-scale isotropy of the university.3 Its
presence minus the acceleration which would be produced by a now zero
cosmological constant, which is quite effective in isotropizing the
universe.^  There is, however, the not fully resolved question of whether
inflation occurs, and is effective in producing isotropy, if the initial
shear of spatial curvature of the universe is sufficiently large.^ One
would therefore like to find models with inflation, even power-law
inflation, that possess inevitable inflation no matter how large the
shear.
Power—law inflation has been invoked Co explain the structure of
inhomogeneity in the universe.3>6 it has been discussed as the result
of particle-creation processes in the early vacuum dominated universe,?
the result of evolution and compactification of certain models of
Kaluza-Klein cosmology,8 the consequence of the antisymmetric part of
the affine connection in the nonsymmetric theory of gravity of Moffat,^
and the effect of a root-mean square spin energy densitylO in the
Einstein-Cartan theory with the Ray-Smalley improved energy-momentum
tensor with spin.H In this paper, we show that a generalized (or
power-law) inflationary (and super-inflationary) epoch arises naturally
and inevitably in the early universe in a cosmological model in the
Einstein-Cartan gravitation theory with the improved energy-momentum
tensor with spin of Ray and Smalley.
Torsion, Spin Energy Density, and the Ray-Smalley Improved
Energy-Momentum Tensor With Spin. Spin is a Lorentz-invariant quantum
mechanical phenomenon. This means that it exists in flat space and as
such represents rotation with respect to nothing. It must ,therefore,
properly be included in the definition of internal energy of a
fluid.12 Furthermore, rotation itself can have a variety of forms, even
if we are considering only rotating coordinate systems and references
frames and not rotating matter with or without internal spin.13 The
former requries a reformulation of the fluid lagrangian and the first
law of thermodynamics to include the angular kinetic energy density of
spin in determination of the equations of motion and field equations.^
The latter requires that torsion be included in any apacetime
formulation of physics that includes an affine connection.^
- In a further convincing argument, Penrose has shown that torsion
naturally arises when the general class of conformal transformations of
metrics is allowed to include complex transformations.*** This occurs if
one wishes to preserve the Lorentz-invariant two - component spinor
calculus in as natural a form as possible. With a conformal
transformation gao » ft ft gao for complex ft, the torsion is generated as
Tabc • i <Vd log ft - Vd log fl> Eabcd, where i = /~ and Eabc<* ij} the
four-dimensional alternating symbol.
Thus, we are directed to the Ray-Smalley formalism for the
Einstein-Cartan theory with improved energy-momentum tensor with spin.
The internal energy e of a fluid with spin is then given by
de = Tds - Pd(l/p) + j.
 w dsij
2 J
•here T,S,P, p, w^j and sLJ are the fluid's temperature, specific
entropy, pressure, density of inertia, spin angular velocity, and
• specific spin angular momentum. The spin angular velocity Wjj is
defined b y . . . . . .
where a^£, uk, and //^ are the tetrad vectors, four-velocity of the
fluid, and the covariant derivative. The spin density Sjj of the fluid
is given by
S.. = p< (a\ a2. - a1, a^) = ps. .
where < is a coupling constant. The two combine to form the spin
kinetic energy density T of the fluid:
T - 1 S.
2
One choice that can be made for the tetrad vectors is to identify one of
them with the fluid four velocity: U*- » a^ *-.
A lagrangian has been given for this model by Ray and Smalleyll
but is not necessary here. Appropriate variations lead to equations of
motion and field equations. First of these is the equation of motion
for the spin:
where D is the total convective derivative.. .The field equation is
G«
where the modified torsion tensor is
and the square brackets denote antisymmetrization. The perfect fluid
ikpart of the energy-momentum tensor T_ is:
- p(l + e + P/P) U1 0* + gikP (4)
and its intrinsic spin part T-1^ is:
..K)£ ,", ^ „_,
 r_¥Tuy,JJ .vC(i_K)-t + OUX~S"'^ U nJ (5)
where the last two terms arise from the inclusion of spin as a
thermodynamic variable in the first law of thermodynamics. The torsion
is detemined from the spin density tensor by
where Sjjk is the canonical spin-density tensor of Halbwachs:^-^
S. .K a c..nKxj siju
The antisymmetric field equation is
and is mainly an expression for the conservation of angular momentum.
The above equations constitute the Einstein-Car tan theory with the Ray-
Smalley energy-momentum tensor.
A notable success of the theorty has been the solution for a
static cylinder with perfectly aligned spins of its constituent
particles, found by Tsoubelis.l? It is a significant improvement, in
that consistent exterior-interior matching conditions are fulfilled, and
that the exterior gravitational field possess off-diagonal terms in the
metric which cannot be transformed away. This indicates that a true
magnetic-trype gravitational field is generated by spin in the improved
EC with RS energy-mometum tensor with spin.
Cosmological Models in Gravitation Theories with Torsion. A
large number of cosmological models have been exhibited in Einatain-
Car tan theories. However, we shall mention only a representative few of
them, of direct relevance here. The most recent study of homogeneous
and isotropic cosmological models in torsion gravity theories with the
standard energy-momentum tensor has been given by Goenner and Muller—
Hoissen for lagrangian densities containing terms in T2 and R2, a more
general gravitational lagrangicis than what we consider later.^
Contributions to the field equations arising from the T2 and R2 terms
will also drive an inflationary epoch, ^ beyond the contribution of the
spin kinetic energy density terms shown here. •
Anisotropic cosmologies for EC theory with the standard
lagrangian and standard energy - momentum tensor have been studied in an
interesting manner by Tsoubelis and by Lorenz.^ l~23 Those represent the
clearest treatment of anisotropic cosmologies with torsion for Bianchi
Types VI0 and VIIO,20 Type V,20*2* Kantowski-Sachs geometries,22 and a
general class of Bianchi types with magnetic field.2*
There have been some recent limited studies of cosmological
models in EC theory with the improved RS energy-momentum tensor with
spin. These have mostly concentrated on the Godel solution or Godel-
like cosmologies.2 '^2^ Their particular success has been in providing
reasonable perfect-fluid sources for the Godel models but with,
furthermore, causal behavior of the geodesies.
In expanding models, there have been very few investigations
published thus far using the RS energy-momentum tensor. One was a
consideration of the role of spin, with the energy-momentum tensor with
spin but not the improved energy-momentum tensor with spin. 26 While we
"do not completely agree with all the details of their calcuationa, we
believe their conclusion is accurate. Their computation shows that
internal spin drives expansion of a fluid positively.
Gasperini has demonstrated that the improved RS energy-momentum
tensor with spin may drive inflation. 10 It is proven approximately and
ad hoc for a flat Robertson-Walker (homogeneous and isotropic)
cosmological model. Much of the argument is unsatisfactory in that
lacks rigor, but the main conclusion has a certain validity. Our
'result, which confirms both that of Bedran and Vasconcellos-Vaidya^ and
that of Gasperini, 10 shows that such approximations are unnecessary to
demonstrate that inflation is inevitable if the universe is correctly
described by the EC theory with the improved RS energy-momentum tensor.
Inevitable Inflation in a Simple Anisotropic Einstein
Cartan/Ray-Smalley Cosmological Model. We will now examine a simple
anisotropic model to demonstrate our result. The spatial geometry is
Bianchi Type I, indicating we consider only the effects of shear
(dynamic anisotropy) and not of curvature (kinematic or static
anisotropy). The anisotropy is necessary so that global spin angular
kinetic energy effects can be examined without resorting to the
averaging arguments and approximations necessary in the Roberton-Walker
model.10
We choose Che metric Co have Che form first introduced by
Misner:27
d82 , _ dt2 + e2tx e2Bi dx1- dxJ, (6)
2a
where e • R (t) the scale factor, and BJJ is a 3 x 3 symmetric,
tracefree, matrix. We use the method of differential forms and so write
ds2 - ga8 W* W*
where we take g-jj to be the Lorentz metric and Wa to be a basis of
differential forms.28 We then have that W° = dt and W1 = ed
 e
B
. dx-J.
To show more clearly how torsion is inserted in the connection forms, we
rewrite the first Car tan equation in the form
wx -
where WU are the connexion forms and d indicates the exterior
derivative operation on a form. The dual to the basis are the tetrad
vectors themselves and they obey the usual orthonormality relations
*u ^av * ^ uv an(* ^ i a^^ = Hij wnere Hij signifies the usual Lorentz
metric. Using tetrads consistent with the choice of Bianchi Type I, we
take the nonzero components of Sj4 to be
S12 " K(X)
where <(X) is some general function. The tracefree proper torsion
•fm
Kij is related to the spin density by KJJ 3 psij 0K.
Proceeding with the usual calculations, including use of the
second Cartan equation, we find the usual relativistic quantities and
equations: The affine connexion forms are
and
K Tjk » where ajfc an<* Tjk are c^e symmetric and
antisymmetric parts of
Jt
and the overdot indicates the time derivative. Keeping maximum simplic-
ity, we choose a comoving fluid with four-velocity ua 3 nS >
ua * 5*. The Einstein-Cartan equations can be put into the form
G - 3a2 - 1 a,. a±j - p(l + e), (8a)
oo •=• ij
G* /* - 9a2 - 3 a.. aij - 3p + 2pr Sji,
K » 6a j ij iJ (8b)
and
-
 TiK° *j
(8c)
K
 - 1 T S1K
-j ij K -2 Kj
10
with GQ£ a To£ " o. It seems as though the Goo = Too equation has no
spin kinetic energy terms, but one should recall that the term e con-
tains spin kinetic energy terms via Eq. (1). Indeed we could approxi-
mately write : . . . >
e - TS - P/P + I W slj
2 J
under certain circumstances. But it is true that under most circum-
stances the term in p itself in the Too component will dominate.
Requirements of symmetry and consistency give that
° + 3° T° - 0
which solves to Ti2° * ^ 12° (°) e ~^ a» where Tj2 (°) " an integration
constant (initial value). We can absorb the (1 + e) term into p and
manipulate and combine the Einstein-Car tan equations to find
o - - 1 a., a J - i (P + P) + I PT s J
 (9)
1 ij 2 3 i3
• • * • • ry
Then substitution of a - R/R -. (R/R)
and a " R/R into Eq. (24) gives
- 1 P^-E-I+P-1!!3 (10)
3 CT 6 2 2 1J
We must now determine the way each term on the right hand side of Eq.
(10) scales with the volume expansion factor R. It is clear that while
the terms for shear energy density pa = % O^j axJ, inertia density a,
11
and pressure p are negative, the term for spin angular kinetic energy
density contributes positively to R. One must see if it ever dominates.
To examine that we need only establish the dominant scaling relation of
each energy density with R, formally and to leading order.
The contracted Bianchi identity and E-C field equations combine
to give the usual conservation equation for the inertia density, with
equation of state p » -yp : P/P * 3* (I4?) * o, with the solutions
p 3 P0 R~3(l*y). The shear evolution equations have the usual leading
order formal solution which gives the scaling law PQ * p<j R~6. The
tensor Tij is identically the rotation rate of the observer's Fermi-
transported reference triad2' and so equals v^j of Eq. (2). Dimensional
analysis^  shows that its evolution (from the EC equations) scales with
o —iproper time t as T£j • Tij t *.
The scaling relation between R and time t is given from the
G00 3 Too field equation. Depending on which form of matter, or the
shear, dominates; R ** t^ '3 for shear or stiff (scalar fields) matter
(q » 1), and R " t1/2 for radiation (y = -1/3), and R ~ t2/3 for dust or
low-pressure matter (y * 0). The scaling relation for the given density
comes from the equations of motion above. That gives Sjj t ~*. Com-
bining those results for density, spin density, and tetrad rotation, we
find that
Combining the above results into Eq. (11), we obtain the equations
P
'
2
 2 p °R- 5+ 1 ° °
I CT T
-
PoR'- + p T sCT
 °
 (12a)
12
for dust,
-3 o -5 ° c-° «R - -Po R - 2 P_° R + I P t.. S R
3~ 3 - 3 - -- -- (12b)
for radiation, and
0
 -5 o °R-12
3" ° ° T ° 13 (12c)
for stiff matter or scalar fields. In each of Eqs. (12), for
sufficiently small values of R, the spin density term will dominate and
thus an early epoch cosmology will have its expansion accelerated by the
spin kinetic energy density of the fluid. The shear cannot effectively
damp the acceleration for sufficiently early epochs.
Conclusion. Note that the important term in the equations is of
the form Tfj S1J and so it doesn't matter what components of Tjj and S*-J
are nonzero. For'computational simplicity, we could choose only T^2 >* 0
and S*2 / Q, and then (312 is the only off-diagonal component of CTjj. We
do find an * 022 from this (axial symmetry) and the true fluid
vorticity then turns out to be Q^2 * ~ ^12 °^ o- *n fcke future, we
will extend this work to other Bianchi types (for curvature anisotropy)
and magnetic and magnetohydrodynamic models (for matter anisotropies).
The computation presented here at least shows that the EC theory with RS
EMT provides us with cosmological models with inevitable inflation in
the early universe, exhibiting a simple and natural explanation for the
apparent inflationary epoch which may have occurred.
13
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17


Inflation in Einstein-Cartan theory with energy-momentum tensor with spin

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