A NNALES DE L ’I. H. P., SECTION A
F RIEDRICH W. H EHL P AUL VON DER H EYDE
Spin and the structure of space-time
Annales de l’I. H. P., section A, tome 19, n
o2 (1973), p. 179-196
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179
Spin and the Structure of Space-Time (*)
Friedrich W. HEHL Paul von der HEYDE Institute for Theoretical Physics
of the Technische Universitat Clausthal, D-3392 Clausthal-Zellerfeld, West Germany
" Newton successfully wrote apple = moon, but you cannot write apple=neutron. "
J. L. SYNGE
"
... la torsion de l’Univers continue a etre nulle dans le vide. "
E. CARTAN Ann. Inst. Henri Poincaré,
Vol. XIX, no 2, 1973,
Section A :
Physique théorique.
ABSTRACT. - The
gravitational
field in GeneralRelativity (GR)
iscoupled
to theenergy-momentum
tensor ofmatter,
i. e. it is thedynamical
manifestation
ofenergy-momentum.
In Section 1 wetry
to collectall
arguments
which show that it is veryplausible
to look also for adyna-
mical
manifestation
ofspin-angular
momentum of matter.In Section 2 it turns out that this program can be fulfilled
by
genera-lizing
Riemanniangeometry
ofspace-time
to a Riemann-Cartangeometry.
The affine connection is now
asymmetric
and chosen in such a way that the covariant derivative of the metric still vanishes. Thenewly
intro-duced contortion tensor or,
equivalently,
Cartan’s torsion tensor describeindependent
rotationaldegrees
of freedom of thespace-time
continuum.Hence we
couple
contortion tospin-angular
momentum in a similar wayas metric to
energy-momentum.
In Section 3 we discuss certain situa- tions in 3-dimensional continuum mechanics where the Riemann-Cartangeometry
hasalready
been used. Weespecially get
a clear idea of thephysical interpretation
of torsion.(*)
This article is based on a Seminar given by the first named author (F.W.H.)at the Institut Henri Poincare, Paris on November 28, 1972. This author would like to thank for the invitation.
ANNALES DE L’INSTITUT HENRI POINCARE .
180 F. W. HEHL AND P. VON DER HEYDE
In Section
4,
with thehelp
of an actionprinciple,
GR isappropriately generalized.
The new fieldequation (4.9) generalizing
the Einsteinequation
is derived. In Section 5 we compare theemerging U4-theory
with GR. The
U4-theory
describes in a unified way usualgravitational
interaction and a very weak universal
spin-spin
contact interaction.For very
high
matterdensities, spin
becomes the dominant source of thegravitational
field. Hence in theneighborhood
ofsingularities,
the metric of
space-time
isexpected
to be determined to alarge
extendby
thespin
distribution of matter.1. A DYNAMICAL THEORY OF SPIN ?
The
only property
of matter which enters GeneralRelativity (GR)
is its
energy-momentum
distribution. This is sufficient and worksquite
well in
macroscopic physics.
If one wants topenetrate
into more micro-scopic domains, however,
not much isexperimentally
known and one hasto
extrapolate
in one or another direction.The
theory
we shallspeak
of is anattempt
to find agravitational
fieldtheory
formicrophysics. Nevertheless,
we will still work in the frame- work of classical fieldtheory
with a matterfield (xk) (k
=0, 1, 2, 3).
The process of field
quantization,
if it is necessary at all in the context ofGR,
is notyet applied
to thistheory.
In
macrophysics
we can describe matterby quantities
such as matterdensity, velocity,
pressure, etc. Inmicrophysics,
in the framework of classical fieldtheory,
we start fromSpecial Relativity (SR). Using
therepresentations
of the Poincare group it turns out thatfields,
or let us sayelementary particles,
are labelledby
m and s, mbeing
the mass,connected with the translational
part, s being
thespin,
connected with the rotationalpart
of the Poincare group. Hence m as well as s are kinema-tically
related to the Minkowskispace-time
continuum of SR.In the context of a field theoretical formalism m
corresponds
to the(canonical) energy-momentum
tensor and s to the(canonical) spin- angular
momentum tensor(=-T~i~).
This can berecognized,
as iswell
known,
in a field theoreticalLagrangian
formalism with thehelp
of Noether’s theorem :
(1.1 )
mass m ~ translation -~(canonical) energy-momentum
(1. 2) spin s
- rotation - ¿..(canonical) spin-angular
momentumTi l’‘ .
Macroscopically
we live in anunpolarized world,
if wedisregard
suchfairly untypical things
asferromagnets.
Inmacrophysics spin usually
averages out and the
dynamical properties
of matter arecorrectly
described
by energy-momentum
alone. It is due to thisfact,
as itVOLUME A-XIX - 1973 - N° 2
181
SPIN AND THE STRUCTURE OF SPACE-TIME
fact,
as it appears to us, that thedescription
ofspace-time by
theRiemannian
geometry
of GR remains valid.In conventional GR the
gravitational
field is thedynamical
manifes-tation of
energy-momentum [see
Sakurai(1960),
e.g.]. Consequently
the
gravitational
field iscoupled
to theenergy-momentum
tensor ofmatter. Let be
given
thepotential
of thegravitational field,
i. e. themetric
(Xk)
of the Riemannian spaceV,~
of GR.(~ ~xk), (Xk»)
is the
Lagrangian density
of matter, then the(metric) energy-momentum
tensor ~i~ is defined
according
to Hilbert(1915) by ( 1. 3)
Here e
# V -
det g~~ and V means the covariant derivative withrespect
to theV4.
The kinematical definition ofenergy-momentum (1.1)
is now
superseded by
thedynamical
definition(1.3). Weyl (1961)
hasput
this in thefollowing
sentence : ~ Thegeneral theory
ofrelativity
alone,
which allows the process of variation to beapplied
to the metrical structure of theworld,
leads to a true definition ofenergy ".
A variationof the metric leads to a variation of the mutual distances of the events of
space-time.
Hence such a " deformation " is of a translationaltype.
But if
spin
does not average out, i. e. if we look for thegravitational properties
ismicrophysics ?
Would it not betempting
to consider the motions " translation " and " rotation " on anequal footing ? Why
should translations and
energy-momentum
be of more fundamentalimportance
than rotations andspin-angular
momentum ? " There does not exist any knownproperty
ofparticles showing
thatspin
is lessimportant
than mass",
asLurcat (1964)
hasspelled
it out. If onestudies the notion of
Regge trajectories,
e. g., onerecognizes
thatspin
does not seem to have a lower
position
than mass.Accordingly
let us look for adynamical
manifestation ofspin
in ananalogous
manner, as it was done forenergy-momentum
in Einstein’s GR.[Technically speaking
we arelooking
for a local gaugetheory (Yang-
Mills
theory)
for the Poincare group. SeeUtiyama (1956),
Sciama(1962,1964)
and Kibble(1961)].
Thisdynamical theory
ofspin
should bea
general
relativistic fieldtheory,
which in the limit ofmacrophysics (and
normal matterdensities)
goes over intoGR,
i. e. GR should be valid in themacrophysical region,
whereas inmicrophysics
a more detailedtheory
should beappropriate.
2. THE AFFINE
CONNECTION i’~ ~
Experience
seems to teach us that thelength
ofmeasuring
rods andthe
angle
between two of them do notchange
underparallel
transfer.ANNALES DE L’INSTITUT HENRI POINCARE
182 F. W. HEHL AND P. VON DER HEYDE
In an affine and metric
space(-time)
this isfulfilled,
as soon as the metric iscovariantly
constant withrespect
to the affine connection Hence wepostulate
(2.1)
If we furthermore assume a
symmetric connection,
(2.1)
leads to the connection of a Riemannian spaceV~
(2.2)
i. e. to the Christoffel
symbol
of the 2ndkind,
whichrepresents
thegravitational
fieldstrength
in GR.If we
drop
theassumption =0, (2.1) yields
the connection ofa Riemann-Cartan space
U4 (1) : (2.3)
The contortion tensor
K**kij
can beexpressed through
Cartan’s torsion tensor(2.4)
according
to(2.5)
The Riemann-Christoffel curvature tensor of a
U.~
is defined in the usual way as " vector vortex ". Wefinally get
for it[see
Schouten(1954), e. g.]
.(2. 6)
There exist the
following arguments (not
all of thembeing independent
of each
other, however)
to believe that(2.3)
is the correctspace-time
connection :
a. If we look for a
dynamical
manifestation ofspin,
weexcept
thespin
to
correspond
to ageometrical quantity
of the samerank,
i. e. a 3rd(1) A connection of this type has already been used by Infeld in an attempt to formulate a unified field theory, c f. Tonnelat (1965), e. g. Nevertheles, the formalism and the interpretation of Infeld’s theory and the U4-theory presented here are completely different.
183
SPIN AND THE STRUCTURE OF SPACE-TIME
rank tensor
antisymmetric
in two indices(24 independent
compo-nents),
as is the case with the contortionKijk (or
thetorsion
Sij k
= -b. The contortion tensor
Kijk
describes rotationaldegrees
of freeedom ofspace-time,
as will berecognized
if oneparallelly
transfers tetrads from onepoint
to another. This fact will also be illustrated with thehelp
of a model in Section 3[compare (3.9)].
c. Thus in
analogy
with(1.3)
wepostulate
thedynamical
definitionof
spin-angular
momentumaccording
to(2. 7)
where a’ ==
(03C8, ~03C8) [compare (3.12)].
This definition whichcouples
contortion tospin,
works in the sense that thequantity
idefined
by (2.7)
later on turns out to be identical with the canonicalspin
tensor mentionedalready
in(1.2).
d. The kinematical relation between
spin-angular
momentum and rota-tions mentioned in Section 1 allows for an
interpretation
ofspin
asa real internal
rotation,
asused,
forinstance, by
deBroglie
and hiscoworkers in the
rotating liquid droplet
model ofelementary particles [see
deBroglie (1963)
and Halbwachs(1960),
e.g.].
This semi- classicalunderstanding
ofspin
is also inherent in our considerations.According
to an axiom due to Minkowski(1958),
it shouldalways
be
possible
to transformsubstance,
i. e. massiveparticles
to rest.If we
interpret spin
as an internalmotion,
this is nolonger possible
for
spinning
massiveparticles,
and the axiom breaks down. Then theproportionality
of 4-momentum and4-velocity
is nolonger valid,
a fact which
immediately
leads to anasymmetric energy-momentum
tensor as was
repeatedly
stressedby
deBeauregard (1942, 1943, 1959, 1963)
and others. Hence one shouldexpect
forspinning
massiveparticles
aslight
modification of conventional SR. Of course thisautomatically
would lead to acorresponding
modification ofGR,
too.Let us consider as an
example
aferromagnet
in afreely falling
and
non-rotating laboratory.
Across theferromagnet
there is a non-vanishing macroscopic spin density.
If we take the affine connection of
space-time (2.3),
in the caseof
vanishing
contortionKijk
we are able to choosegeodesic
coor-dinates for the
laboratory
andthereby
to transform away Christoffel’ssymbol { ~ ~.
Then SR should be valid in accordance with theequivalence principle.
But for anon-vanishing
contortiontensor,
r ~
cannot be transformed to zero anylonger,
and for that reason SRis modified.
ANNALES- DE L’INSTITUT HENRI POINCARE
184 F. W. HEHL AND P. VON DER HEYDE
This is the case of the
ferromagnet
with its intrinsicspin motion,
because herespin
andtherefore, according
to(2. 7),
also contortion cannot vanish. Then aslight
violation of theequivalence principle
is to be
expected. Morgan
and Peres(1962)
haveshown,
interalia,
that such a
possibility
is not ruled out asyet experimentally
for testbodies with
aligned spins.
Since forvanishing spin
conventional GR shouldresult, spin
shoulddepend algebraically (and
not via a diffe-rential
equation)
on the contortion ofspace-time.
e.
By
the theorems onholonomy [c{.
Lichnerowicz(1955),
e.g.]
curvatureand torsion of a
U.,~
arerelated, respectively,
to the groups of rotations and translations in thetangent
spaces of theU,~4,
as wasnicely
workedout
by
Trautman(1972
a,b) :
(2. 8)
rotation -~ curvature -~energy-momentum, (2. 9)
translation - torsion -spin-angular
momentum.Curvature
according
to GRcorresponds
toenergy-momentum,
andcomparing (2.8)
and(2.9)
with(1.1)
and(1.2), respectively,
we areled to the
correspondence
torsion Nspin-angular
momentum, as wasalready
to beexpected according
to(d).
Such
being
assured that there isgood
evidence for the affine connec-tion
(2. 3),
let us refer to some of the former work onThe differential
geometric
notion of torsion was introducedby
E. Cartan(1922, 1923, 1924, 1925).
He was aware of the fact that torsion should havesomething
to do with what wetoday
callspin.
But he did notpresent
agenuine theory.
After certaininvestigations mainly by
deBeauregard (1942, 1943, 1959), Papapetrou (1949),
andWeyl (1950),
Sciama
(1962, 1964)
and Kibble(1961)
gave adynamical theory
ofspin.
Kroner and one of the authors
[Hehl
and Kroner(1965),
Hehl(1966,1970)]
gave the
explicit
form of the affine connection and rederived thetheory
from a different
starting point using
another formalism. For morerecent work see for instance Clerc
(1971,
1972 a,b),
Datta(1971), Hayashi
andBregman (1973),
Hehl and Datta(1971),
Lenoir(1971), Kopczynski (1972),
and thehighly interesting investigations
of Trautman(1972
a,b, c).
Fore more detailed references see Hehl(1973).
3. DISLOCATIONS : A MODEL OF TORSION Strain and Force Stress
Dislocations are certain defects in
crystals. According
to Kondo(1952)
and
Bilby, Bullough,
and Smith(1955)
the continuumtheory
of dis-colations can be formulated in a 3-dimensional space with torsion. In
185
SPIN AND THE STRUCTURE OF SPACE-TIME
this context a
physical interpretation
of torsion was discovered for the first time.Compare
the review article of Kroner(1964),
e. g..Fig. 1. - Model of an elastic body : An ideal cubic crystal in an undeformed
state. The part shown is supposed to be a volume element dV in the continuum limit.
Let us first consider classical
elasticity theory.
In order to describethe deformation of the
crystal
infigure 1,
we use a cartesian coordinatesystem xa
fixed in space. The lattice constant of thecrystal
is assumedto be small in
comparison
withdx;x.
All deformations areperformed isothermally.
Fig. 2. - The homogeneous dilatation of the crystal in xl-direction is maintained by the force stress The mean distances of the lattice points have changed;
strain and force stress occur macroscopically.
If we
displace
thearbitrary
materialpoint
with the coordinates X(X to X(X + sa(x~),
thecrystal
ingeneral
will be deformed(fig. 2).
The mutual distance ds of the
points
xaand xa
+dxa after
the defor-mation process turns out to be
(3.1)
dS2 = + 2dx03B1 dx03B2 (03B1, (3,
... =1, 2, 3).
In classical
elasticity, by (3.1),
the strain tensor == can beexpressed
for small relativedisplacements
as ,(3.2)
=ANNALES DE L’INSTITUT HENRI POINCARE 13
186 F. W. HELH AND P. VON DER HEYDE
In
order to preserve thedeformation,
there acts a force dF~ on eacharbitrarily
oriented area element in the.crystal df# according
to(3.3)
The force stress is the static response of the
crystal
to the strain~3 . 4)
Here L is the free energy of the
crystal. Shortly :
Strainproduces
force stress and vice versa.
Now we make the
following interesting
observation :(3.1)
definesthe metric of a 3-dimensional Riemannian space
Vs.
This space can be constructedaccording
to thefollowing prescription :
a. Each material
point
of the deformed continuum is characterizedby
the cartesian coordinate xa which it has before deformation.
b.
Lengths
andangles
are measuredby comparsion
with the unde-formed continuum.
Thereby
each deformation ismapped
into a certain metric space.Two
parallel (lattice)
vectorsC~
ofequal length,
attached in the unde- formed state to thepoints
xaand xx
+dxex,
after deformation differby (3. 5)
The anine connection
raw
can beexpressed
in i. e. reducesin this case to the connection of a
V 3 :
(3.6)
The strain tensor
(3.2)
is of a veryspecial kind,
and the correlated space iscorrespondingly simple.
It is a Euclidean space, since de Saint- Venant’scompatibility
relations are fulfilled :(3.7)
= curvature
tensor).
But of which kind is the strain
belonging
to a non-Euclidean spaceV~ ? Imagine
thecrystal
is " blown up "irregularly by
amacroscopic
distri-bution of interstitials. This causes a
macroscopic
strain ~a~ and a corres-ponding
self stress Ingeneral,
the strainproduced by
these inter-stitials cannot be derived from a
displacement
field sa and thus0,
i. e. ~a~ now has six
independent
translational functionaldegrees
offreedom. If we cut the continuum into small
pieces dV,
it will relaxand the self stress will
vanish,
but the elements dV do not fittogether
any
longer.
Hence we conclude : self stress is causedby
the fact that the non-Euclidean continuum is forced into a Euclidean space.2
187
SPIN AND THE STRUCTURE OF SPACE~TIME
Contortion and Moment Stress
Let us look at
figure
3 in order to understand that a Riemannian spaceV3
is toospecial
to describe alltypes
of deformationsoccuring
Fig. 3. - Deformation of the crystal by edge dislocations of the 1 type alters
the relative orientation of the crystal structure. Thereby the vector in x2-direction, parallelly displaced along the xl-axis, will rotate : there occurs a closure failure of the infinitesimal parallelgram. The crystal’s deformation will be maintained by the moment
stress 03C4**121.
The mean distances of the lattice points have not changed, hence no macroscopic strain and stress are produced.in a
crystal.
Withrespect
to theedge
dislocations infigure
3 we observethe
following :
a. The infinitesimal
parallelogram
shown was forced openby
the dislo-cations,
which we canimagine
to haveimmigrated
from the outside of the volume element. A closure failuredby
wasproduced, by
which we are able to define the dislocation
density according
to(3.8)
= - = area
element).
b. The mean distances of the lattice
points
have notchanged.
Hencethere occurs no
macroscopic
stress andstrain : { ocj3, y }
= 0.c. The orientation of the lattice structure has
changed,
however. Conse-quently
theparallel displacement
is related to a rotation(3 . 9)
The deformation measure
Kocpy
=2014 is called contortion and contains nineindependent
rotational functionaldegrees
of freedom.According
toNye (1953),
it can beexpressed
in terms of the dislocationdensity (3.10)
d. In order to preserve the
deformation,
there has to act a momentANNALES DE L’INSTITUT HENRI POINCARE .
188 F. W. HEHL AND P. VON DER HEYDE on the area elements :
(3 .11)
The moment stress is the static response of the
crystal
to thecontortion
’).T
(3.12)
In short : Contortion
produces
moment stress and vice versa(~).
On the one hand infinitesimal
parallelograms
are broken upby
immi-grating dislocations,
on the other hand in a space withnon-vanishing
torsion closed infinitesimal
parallelograms
areimpossible
ingeneral. By
this and
(3.9)
it is evident thatmacroscopically
the deformation of acrystal containing
dislocations should bemapped
into a Riemann-Cartan space U,~ . Its connection reads(3.13)
Using (3.10)
we have(3 .14)
hence Cartan’s torsion and dislocation
density
are identical notions.Accordingly
torsion isdirectly
measurableby
the closure failure(3.8).
An Elastic
Space-Time
ContinuumContaining
Dislocations We will work out now the close relation between the 3-dimensional statics of a dislocatedcrystal
withpoint
defects and the 4-dimensional" statics " of the
space-time
continuum. Thefollowing
facts are well-known :
(mi)
The 4-dimensionalgeneralization
of the force stress tensor is the(metric) energy-momentum
tensor ail.(m2)
Thegeneralization
of the definition(3.4)
of the force stress tensorimmediately
leads to thedynamical
definition(1.3)
of energy- momentum. The connection seems clear : variation of the metric gijmeans a variation of the distances of the
space-time
continuum.As response to this
deformation, through
each 3-dimensionalhyper-
surface
element,
there acts an infinitesimal4-momentum, identifying space-time
as a sort of an elastic continuum.(2) A theory of a continuum with independent rotational degrees of freedom and moment stress was developped already at the beginning of our century by the
Cosserats (1909). Cartan was influenced by their work in introducing torsion in GR.
1973 2
189
SPIN AND THE STRUCTURE OF SPACE-TIME
(m;1)
The affine connection(3.6) corresponds
to the Christoffelsymbol
of GR
(2.2).
(si)
The 4-dimensionalgeneralization
of the moment stress tensor is thespin-angular
momentum tensor Atregions
wherespin- angular
momentum ispresent,
there acts an infinitesimalspin- angular
momentumthrough
anarbitrary
orientedhypersurface
element.
An
analogous
continuation seems now tosuggest
itself :(s)
Thegeneralization
of the definition(3.12)
of the moment stress tensor is thedynamical
definition(2.7)
ofspin-angular
momentum.Thus
spin
is the response ofspace-time
to a variation of the contor- tionKijk, identifying space-time
as a sort of a dislocated continuum.(~) Consistently (3.13)
leads to theasymmetric
connection(2.3).
Thisis an additional
argument
for the correctness of the affine connec-tion
(2.3).
4.
U.-THEORY
Let us start with the
special
relativistic material action function in cartesian coordinates(c
=velocity
oflight,
d~ = 4-volumeelement) : (4.1)
According
to ourhypothesis
all events should takeplace
in aU’f-
We go over to curvilinear coordinates and
apply
the minimal substitution(4.2)
We end up with the material action function
(4.3)
Let be k m 2 X 10-48
dyn-1
the relativisticgravitational
constant and11 the
density
of the curvature scalar of theU;.
Then weget
for thetotal action
function, using
the conventionalsimplicity arguments,
inanalogy
with GR :(4.4)
The
independent
variablesare ~, g,
and S.ANNALES DE L’INSTITUT HENRI POINCARE
190 F. W. HEHL AND P. VON DER HEYDE
Hamilton’s
principle (4 . 5)
leads
finally
to(4.6)
and,
after somecomputation (3),
to(4. 7)
(4.8)
(modified
torsion tensor = k Xspin).
(4 . 7)
is of thegeneral
form of Einstein’s fieldequation,
but it hasadditionally
anantisymmetric part. (4. 8)
states thatspace-time
possesses torsion at thosepoints
wherespin
is.present.
Asexpected,
weget
analgebraic
relation between torsion andspin. Apart
from a constant,modified torsion and
spin
are synonyms. Substitution of(4.8)
into(4.7)
leads to one
single
fieldequation [Hehl (1970)] : (4 . 9)
Here Gij
(~ ~
is the conventional Einstein tensor of the V4 and cij the metric(and symmetric) energy-momentum
tensor(1.3)
taken withrespect
to theLagrangian density entering
the action function(4.3).
(4 . 9)
issuggestive
in the sense that forvanishing spin
oneimmediately recognizes
that(4.9)
goes over to Einstein’s fieldequation
of GR. Theexplicit U~
terms in the bracket areclearly
exhibited as corrections to theoriginal V4-theory.
Using
Noether’s theorem one is able to derive identities for theLagrangian density
in the usual manner. This leads to the observation (3) In deriving(4.7) ~ is
de fined according to+ ~- (~ k
. +k ) here V
Vk + 2S~.
Later on E; turns out to be identical with the canonical energy-momentum tensor. The antisymmetric part of (4.7), expressing angularmomentum conservation, supplies no independent components to the field equation.
This is clear from the derivation, since we varied with respect to the symmetric
field gij. Sciama (1962) and Kibble (1961), combining the tetrad formalism with
a Palatini technique, were the first who derived the field equations (4.7) and (4.8).
VOLUME A-XIX - 1973 - NO 2
191
SPIN AND THE STRUCTURE OF SPACE-TIME
that and are identical with the canonical
energy-momentum
and
spin-angular
momentumtensors, respectively.
Furthermore onegets
the conservation theorems in theU4
as follows(4) :
Incidentally (4.10)
can beput
in another form. If we denote theright-hand
side of(4. 9) by
k we havefj
0.Integrating (4.10)
forspinning
dust on thebackground
of external curvature and contortionfields,
one is led to theequation
of motion[Hehl (1970, 1971),
see also Trautman(1972 c)].
Since therein contor- tion iscoupled
to theantisymmetric part
of theenergy-momentum distribution, basically according
to it ispossible
inprinciple
to measure contortion and torsion of
space-time, respectively.
Two remarks should be added :
a. For a
photon
thespin
is aquantity
not gauge invariant. A closer look on the fieldequations
shows that for thephoton
the substitu- tion(4.2)
is notvalid,
hencephotons
do notproduce
torsion(5).
b. The field
equations (4.7)
and(4.8)
are also valid for matter describedby spinor
fields. Oneonly
has to refer thespinors
to tetrads and tomodify
thedynamical
definition of theenergy-momentum
tensor.No new features do appear.
5. UNIVERSAL SPIN-SPIN CONTACT INTERACTION Let us compare our results with conventional GR.
Already
in(4.9)
we have noticed that
U4-theory,
among otherthings, supplies
correctionwhat is important for deriving the equation of motion.
(~) 1969 Imbert discovered a transvers shift in the total reflection of a circularly polarized light beam as proposed by de Beauregard [see Imbert (1972) and references given therein]. In the interpretation of this Imbert-shift due to de Beauregard, Imbert, and Ricard (1971) it is assumed, that the energy-momentum distribution of the photons within Fresnel’s evanescent wave, in spite of being in the vacuum, has to be described by the asymmetric energy-momentum tensor of de Broglie. If this interpretation is correct, photons in the vacuum would produce torsion in contrast to our conclusion above.
It is not clear to us, however, whether this interpretation is compulsory. Interes- tingly the Imbert-shift may be calculated by only using Maxwell’s theory and the appropriate boundary conditions, see Imbert, loco cit. Thus a non-MaxweIlian behaviour of photons seems not to be present.
ANNALES DE L’INSTITUT HENRI POINCAR~
192 F. W. HEHL AND P. VON DER HEYDE
terms
quadratic
in thespin.
If we trace this back to the total actionfunction,
it turns out that theU4-Lagrangian
differs from theV4-Lagrangian
of GR to 1 st order in the term(5.1)
Accordingly (5.1) represents
a very weak universalspin-spin
contactinteraction characteristic for a
U~
andproportional
to thegravitational
constant. In this
theory
there is no "spin
field " which is emitted andthereby
the carrier of a newinteraction ;
there is rather a very weak interaction as soon asspinning
matter is in contact with each other.In
particular
there results a universal self-interaction ofspinning
matter,which leads to non-linearities in an
analogous
manner as energy- momentum does viagravitational
interaction. Hence aU.;-theory supplies
a unifieddescription
of the universallong-range gravitational
interaction and a universal weak
spin-spin
interaction ofvanishing
range.
The deviation of the
U4-theory
fromV,.-theory
is very smallindeed,
as can be seen from
(4.9)
because the 2nd term carries a factor k2. In order to estimate the relative contribution of thespin
interaction to theright
hand side of(4.9),
let us use forspinning
dust the semi-classicalapproximation
~~~ = p c2 ui uf and = Sij cuk(p
= matterdensity,
s~~ ==
spin density,
ui =velocity
of thedust).
Weget (5.2)
For
particles
with mass mcarrying
aspin
of the order of ~, weexpect (5 . 3)
if the
spins
are’parallelly
oriented.(5.3)
substituted in(5.2)
leads to(5.4)
The
spin
terms are of the same order ofmagnitude
as the matterdensity
term as soon as(5.5)
For electrons this occurs at the
huge density
of about 10~8VOLUME A-XIX - 1973 - NU 2
193
SPIN AND THE STRUCTURE OF SPACE-TIME
This estimate shows that the contribution from
spin
togravitational
interaction can be
neglected
in the case of normal matter densities. In theregion
of densities of the order of(5.5)
orhigher, however, spin
becomes the dominant source of the
gravitational
field. Hence it is to beexpected
that the metric of aU~,
in theneighborhood
ofsingu- larities,
is determined to alarge
extendby
thespin
distribution of matter.There are indications
(")
that thegravitational
interaction ofparallel spins
is of arepulsive type.
If so, one couldspeculate
whetherthereby
the occurence of
singularities might
beprevented
ingravitational collapse
and
cosmology.
A firstattempt
in this direction has been undertakenby Kopczynski (1972) [see
also Trautman(1972 b)].
His verysimple cosmological
model of the Robertson-Walkertype
with aspatially
constant
spin density
is somewhat unrealistic in itsassumptions,
butit shows how
spin
couldprevent singularities
inprinciple.
In the mean-time we have
proved
that inKopczynski’s
model it is at densities of the order of(5.5)
thatspin
becomes effective.For a Dirac
particle,
as iswell-known,
the canonicalspin-angular
momentum tensor is
totally antisymmetric
and henceequivalent
toan axial vector. In this case
(5.1)
is the exact difference between theU4-
andV4-theory
and weget
an axial vector interaction(5.6)
matrices).
The axial vector interaction
(5.6)
can also be discovered in another way. One starts with the DiracLagrangian
andapplies
the formalism of Section 4. Then the matterequation,
after substitution of(4.8),
turns out to be a non-linear
spinor equation
of theHeisenberg-Pauli type (7), (i2
==20141,
= covariant derivative withrespect
to theV4) : (5.7)
(6) Spin at rest can be shown to produce time-like " screw dislocations " in space- time [Hehl (1970)]. Parallel screw dislocations, according to 3-dimensional dislocation
theory, repel each other.
Q In the original Heisenberg-Pauli equation there enters a fundamental length
L ~ 10-13 cm. We can arrive at a spinor equation with such a constant by a suitable
choice of the Lagrangian density of the field. If we choose in (4.4) instead of
~(F)-~~(; ~)-}-6T;~K~/o~
with a dimensionless constant a, we get (5 . 7)with I -~ a 1. In (4.9), for the 2nd term on the right-hand-side, we wouldha ve
k2 - (a k)2. Such a choice of the Lagrangian would seem artificial to us, however.
All this has already been remarked on by Peres (1962). For still more general Lagrangians, we refer to Hayashi and Bregman (1973), see also Hehl (1973).
ANNALES DE L’INSTITUT HENRI POINCARE
194 F. W. HEHL AND P. VON DER HEYDE
This
equation
isequivalent
toya Va
111 = im W. In(5.7)
it canbe seen
directly :
torsion leads to a self-interactionintroducing
additionalnon-linear terms in the Dirac
equation.
Thus in thisspecial
case theaffine connection of
space-time
is influencedby
an axial vector.Let us
just
note that it isappealing
tospeculate
whether this axial vector interaction is a classicalanalogue
of the weak interaction of elemen-tary particle physics.
There is another deviation from GR. The
right-hand-side
of(4.10) (5.8)
is
expected
to be a volume forceacting
on eachspinning particle
in thepresence of a
gravitational
field. Thispresumptive
force is of the Mathissontype [Mathisson (1937)]. Nevertheless,
it is different fromit,
because(5.8)
vanishes for ~z - 0.(5.8)
wouldcorrespond
to aslight
violation of the
equivalence principle,
a factbeing
mentioned in Section 2.With
today’s experimental techniques (5.8)
does not seem to bemeasurable.
6. SUMMARY
In one
respect, U4-theory
seems togive
a final answer : if space- time possessestorsion,
then torsion hassomething
to do withspin- angular
momentum of matter. Hence all theories whichtry
to connecttorsion,
say with theelectromagnetic field,
are obsolete.Consequently
we understand the
potential physical meaning
of torsion. Theutility
of an
asymmetric
connection has beenshown,
theIJ4
as aphysically
reasonable
generalization
of theV~
seems to be near at hand.On the other side we could make
plausible
the introduction of torsion in connection with materialspin,
i. e. adynamical
manifestation ofspin
seems to be a natural
thing.
We know asyet
noargument, however,
which could enforce a torsion upon
space-time.
Hence we
just
have to wait forexperiments
or astronomical observa- tionsrefering
toextremely high
matterdensities,
which will show the existence of the very weak universalspin
contact interaction characteristic for aspace-time
continuumobeying
the Riemann-Cartangeometry.
ACKNOWLEDGEMENTS
One author
(F.W.H.)
would like to thank Prof. O. Costade Beauregard,
Dr. R.-L.
Clerc,
and Dr. J. Madore forinteresting
discussions aboutquestions
on thepossible space-time geometry
and Dr. C. Imbert forVOLUME A-XIX - 1973 20142014 ? 2