Nonholonomic Mapping Principle for Classical Mechanics in Spaces with Curvature and Torsion. New Covariant Conservation Law for Energy-Momentum Tensor

14 Pages·0.16 MB·English

by  Hagen Kleinert

Nonholonomic Mapping Principle for Classical
Mechanics in Spactimes with Curvature and
8
9 Torsion. New Covariant Conservation Law
9
for Energy-Momentum Tensor
1 ∗

n
a Hagen KLEINERT
J Institute for Theoretical Physics, FU-Berlin, Arnimallee 14,

5 D-14195, Berlin, Germany
1

3 (Received February 6, 2008)
v
3 The lecture explains the geometric basis for the recently-discovered
0 nonholonomic mapping principle which specifies certain laws of nature in
0 spacetimes with curvature and torsion from those in flat spacetime, thus
1
replacing and extending Einstein’s equivalence principle. An important
0
consequence is a new action principle for determining the equation of mo-
8
tion of a free spinless point particle in such spacetimes. Surprisingly, this
9
equation contains a torsion force, although the action involves only the
/
c metric. This force changes geodesic into autoparallel trajectories, which
q
are a direct manifestation of inertia. The geometric origin of the torsion
-
r force is a closure failure of parallelograms. The torsion force changes the
g
covariantconservationlawoftheenergy-momentumtensorwhosenewform
:
v is derived.
i
X
r
a 1. Introduction
According to Einstein’s equivalence principle, gravitational forces in a
small region of spacetime labeled by coordinates qµ (µ = 0,1,2,3) can be
removed by going into locally accelerated coordinates xa (a = 0,1,2,3) by
means of a general coordinate transformation xa = xa(q), such as a freely
falling elevator. A mass point lying at the center of mass of the elevator
does not feel any force. It undergoes no acceleration satisfying the Newton-
Einsteinequationofmotionx¨a = 0(a= 0,1,2,3), wherethedotdenotesthe
derivative with respect to the proper time σ. This observation has enabled
Einstein to find the physical laws in curved spacetimes knowing those in
flat spacetime. He simply transformed equations of motion from the locally
∗
[email protected], http://www.physik.fu-berlin.de/˜kleinert
(1)
2 jadwisin printed on February 6, 2008
Minkowskian coordinates xa to the original curvilinear coordinates qµ, and
postulated the resulting equations to describe correctly the motion in a
spacetime with gravitational forces.
In this procedure, the elimination of forces is not perfect: it holds only
at asingle point, the center of mass of the elevator, not in its neighborhood,
which is affected by tidal forces that cannot be removed in this way.
Mathematically, however, it is possible to remove the tidal forces with
the help of a coordinate transformations xa = xa(q) whose derivatives
∂ xa(q) do not satisfy the Schwarz integrability criterion, i.e., they do not
µ
possess commuting derivatives. Such transformations cannot be performed
in the laboratory, since the corresponding falling elevator would not exist.
The spacetime formed by the image coordinates xa would have defects, as
we shall see below, but it could be chosen to be free of tidal forces in the
neighborhood of the center of mass. If we admit such mathematical trans-
formations, Einstein’sequivalence principlecanbeformulated as amapping
principle: The correct equations of motion in the presence of gravitational
forces can be obtained by findingin an entire small neighborhood of a point
qµ a local set of coordinates xa with the above non-Schwarzian property
and a complete absence of forces, and by simply mapping the force-free
trajectories in these coordinates back into the original spacetime qµ.
This mathematical procedure opens up the possibility for discovering
the form of physical laws in more general spacetimes. For instance, we may
assume the existence of gravitational forces, which can only be removed
by transformations xa = xa(q) whose derivatives do not commute. Such
forces are outside of Einstein’s theory. In fact, such transformations can
be used to remove forces in an equation of motion, which appear in an
Einstein-Cartan spacetime with torsion. By postulating that the images of
the force-free trajectories in xa-spacetime are the correct trajectories in qµ-
spacetime we obtain the equations for the straightest lines or autoparrallels
in q-spacetime, thus contradicting present theories which find shortest lines
or geodesics for the particle trajectories.
Physically, autoparallel trajectories may be interpreted as a manifesta-
tion of inertia, which makes particles run along the straightest lines rather
than the shortest ones as generally believed [1, 2, 3, 4, 5]. In the absence of
torsion, the two lines happen to be the same, but in the presence of torsion
it is hard to conceive, how a particle should know where to go to make the
trajectory to a distant point the shortest curve. This seems to contradict
our concepts of locality.
Nonholonomic mappings have been of great use in the physics of vor-
tices and defects in superfluidsand crystals [6, 7, 8, 9, 10, 11, 12]. They are
also essential for solving the path integral of the hydrogen atom [13] via the
Kustaanheimo-Stiefel transformation [14]. In my lecture, I have first shown
jadwisin printed on February 6, 2008 3
how multivalued gauge transformations can be used to generate magnetic
fields and their minimal coupling to charged particles [15, 16, 17]. This
part is omitted in these printed notes to comply with page limitations [18].
These multivalued gauge transformations carry over to geometry by intro-
ducingmultivaluedinfinitesimallocalcoordinatetransformationsproducing
infinitesimal curvature and torsion in a flat background spacetime.
2. Nonholonomic Mapping Principle
Let dqµ be a small increment of coordinates in the physical spacetime.
This is mapped into a coordinate increment dxa via a transformation [13,
19, 20, 21]
dxa = ea (q)dqλ, (1)
λ
whose matrix elements ea (q) are multivalued tetrads. The transformation
λ
can be chosen such that the length of dxa is measured by the Minkowski
metric η , so that the metric g (q) in the spacetime qµ is given by
ab µν
g (q) = ea (q)eb (q)η , ea (q) ∂xa(q)/∂qλ. (2)
λµ λ µ ab λ
≡
Parallel transport in q-spacetime is performed with an affine connection
Γ λ(q) e λ(q)∂ ea (q) = ea (q)∂ e λ(q), (3)
µν a µ ν ν µ a
≡ −
where e λ(q) are reciprocal multivalued tetrads. Its antisysmmetric part is
a
the torsion tensor [22]
1
S λ(q) = Γ λ(q) Γ λ(q) , (4)
µν µν νµ
2h − i
and its covariant curl the curvature tensor
R κ(q) = e κ(q)(∂ ∂ ∂ ∂ )ea (q)
µνλ a µ ν ν µ λ
−
= ∂ Γ κ ∂ Γ κ Γ σΓ κ+Γ σΓ κ. (5)
µ νλ ν µλ µλ σν νλ σµ
− −
Note that if we were to live in x-spacetime, we would register S λ(q) as
µν
an object of anholonomity. In the coordinate system q, however, S λ(q) is
µν
observable as a torsion.
Recall the way in which the affine connection Γ κ serves to define a
µν
covariant derivative of vector fields v (q), vµ(q):
µ
D v (q) = ∂ v (q) Γ λ(q)v (q), D vλ(q)= ∂ vλ(q)+Γ λ(q)vν(q).
µ ν µ ν µν λ µ µ µν
−
(6)
4 jadwisin printed on February 6, 2008
If we lower the last index of the affine connection by a contraction, Γ
µνλ
g Γ κ, there exists the decomposition ≡
λκ µν
Γ = Γ¯ +K , (7)
µνκ µνκ µνκ
where Γ¯ is the Riemann connection, symmetric in µν,
µνκ
1
Γ¯ µν,λ = (∂ g +∂ g ∂ g ), (8)
µνλ µ νλ ν λµ λ µν
≡ { } 2 −
and
K = S S +S (9)
µνκ µνκ νκµ κµν
−
is an antisymmetric tensor in νκ, called the contortion tensor [22], formed
fromthetorsiontensorbyloweringthelastindexS = g S λ. Withthe
µνκ κλ µν
help of the Riemann connection, we may defineanother covariant derivative
D¯ v (q) = ∂ v (q) Γ¯ λ(q)v (q), D¯ vλ(q)= ∂ vλ(q)+Γ¯ λ(q)vν(q).
µ ν µ ν µν λ µ µ µν
−
(10)
3. Application to Particle Trajectories
As an illustration for nonholonomic mappings to spacetimes with curva-
ture and torsion we use the analogy with defect physics [6, 7, 8, 9, 10, 11].
We consider two types of special plastic deformations in crystals, by which
one produces a single topological defect called a disclination (a defect of
rotations) and a dislocation (a defect of translations) (see Fig. 1). Just as
crystals with dislocations, spacetimes with torsion have a closure failure,
implying that parallelograms do not close. As a consequence, variations
δqµ(τ) of particle trajectories parametrized arbitrarily by τ which in the
absence of torsion form closed paths, cannot be zero at both the initial and
the final point, but must be open at the endpoint of the trajectory [23, 24],
as illustrated in Fig. 2. As a consequence, the Euler-Lagrange equation
of spinless point particles receives a torsion force. This is quite surprising
since the Lagrangian of a trajectory qµ(τ),
L = M g (q(τ))q˙µ(τ)q˙ν(τ), (11)
µν
− q
(we use natural units with light velocity c = 1) contains only the metric
[24]. The new Euler-Lagrange equation reads
∂L d ∂L ∂L
= 2S λq˙ν , (12)
∂qµ − dτ ∂q˙µ µν ∂q˙λ
jadwisin printed on February 6, 2008 5
Fig. 1. Crystal with dislocation and disclination generated by nonholonomic coor-
dinate transformations from an ideal crystal. Geometrically, the former transfor-
mation introduces torsion and no curvature, the latter curvature and no torsion.
differing from the standard equation by the extra force on the right-hand
side involving the torsion tensor S λ. This extra force changes geodesic
µν
trajectories into autoparallel ones, whose equation of motion is
D
q˙ν(σ) q¨ν(σ)+Γ ν(q(σ))q˙λ(σ)q˙κ(σ) = 0, (13)
λκ
dσ ≡
where σ is the proper time defined by dσ = g dqµdqν. The geodesic
µν
equation without the extra force would containponly the Riemann part of
the connection and reads
D¯
q˙ν(σ) q¨ν(σ)+Γ¯ ν(q(σ))q˙λ(σ)q˙κ(σ) = 0. (14)
λκ
dσ ≡
A simple variational principle to derive the equation of motion (12) is
based on the introduction of an auxiliary nonholonomic variation δ-qµ(τ) of
a particle trajectory qµ(τ) which has the novel property of not commuting
with the τ-derivative d d/dτ [24]:
τ
≡
δ-d qµ(τ) d δ-qµ(τ) = 2S µq˙ν(τ)δ-qλ(τ). (15)
τ τ νλ
−
6 jadwisin printed on February 6, 2008
Fig. 2. Images under a holonomic and a nonholonomic mapping of a fundamental
path variation. In the holonomic case, the paths xa(σ) and xa(σ)+δxa(σ) in (a)
turn into the paths qµ(σ) and qµ(σ)+δqµ(σ) in (b). In the nonholonomic case
with S λ = 0, they go over into qµ(σ) and qµ(σ)+δSqµ(σ) shown in (c) with a
µν
closure failu6 re δSq2 = bµ at t2 analogous to the Burgers vector bµ in a solid with
dislocations.
Then (12) is a direct consequence of the new action principle [23, 24]
-
δ = 0. (16)
A
An important consistency check for the correct equations of motion is
based on their rederivation from the covariant conservation law for the
energy-momentum tensor which, in turn, is a general property of any field
theory invariant under arbitrary (singlevalued) coordinate transformations.
4. New Covariant Conservation Law for Energy-Momentum
Tensor
To derive this law, we study the behavior of the relativistic action
τ2
= M dτ g (q(τ))q˙µ(τ)q˙ν(τ) (17)
A − Zτ1 q µν
jadwisin printed on February 6, 2008 7
under the infinitesimal versions of general coordinate transformations
∂q′µ
dqµ dq′µ = αµ dqν; αµ , (18)
→ ν ν ≡ ∂qν
′
∂q
dq dq′ = α νdq ; α ν µ. (19)
µ µ µ ν µ
→ ≡ ∂q
ν
We shall write them as local translations
qµ q′µ(q) qµ ξµ(q), (20)
→ ≡ −
considering from now on only linear terms in the small quantities ξµ.
Inserting (20) into (18) and (19), we have
αλ (q) δλ ∂ ξλ(q), α ν(q) δ ν +∂ ξν(q), (21)
ν ν ν µ µ µ
≈ − ≈
and find from
∂qµ ∂q′µ ∂q′µ ∂qν
e µ(q) = e′ µ(q′) = = αµ (q)e ν(q), (22)
a ∂xa → a ≡ ∂xa ∂qν ∂xa ν a
∂xa ∂xa ∂qν ∂xa
ea (q) = e′a (q′) = = α ν(q)ea (q)
µ ∂qµ → µ ≡ ∂q′µ ∂q′µ ∂qν µ ν
the infinitesimal changes of the multivalued tetrads e µ(q) and ea (q):
a µ
δ e µ(q) e′ µ(q) e λ(q) = ξλ(q)∂ e µ(q) ∂ ξµ(q)e µ(q), (23)
E a a a λ a λ a
≡ − −
δ ea (q) e′a (q) ea (q) = ξλ(q)∂ ea (q)+∂ ξλ(q)ea (q). (24)
E µ µ µ λ µ µ λ
≡ −
The subscriptof δ indicates that these changes are caused by the infinites-
E
imal versions of the general coordinate transformations introduced by Ein-
stein.
To save parentheses, differential operators are supposed to act only on
the expression after it. Inserting (24) into (2), we obtain the corresponding
transformation law for the metric tensor
δ g (q) = ξλ(q)∂ g (q)+∂ ξλ(q)g (q)+∂ ξλ(q)g (q). (25)
E µν λ µν µ λν ν µλ
With the help of the covariant derivative (10), this can be rewritten as
δ g (q)= D¯ ξ (q)+D¯ ξ (q). (26)
E µν µ ν ν µ
For the coordinates qµ themselves, the infinitesimal transformation is
δ qµ = ξµ(q), (27)
E
−
8 jadwisin printed on February 6, 2008
which is just the initial transformation (20) in this notation.
We now calculate the change of the action (17) under infinitesimal Ein-
stein transformations:
δ δ
δ = d4q A δ g (q)+ dτ A δ qµ(τ). (28)
EA Z δg (q) E µν Z δqµ(τ) E
µν
Thefunctional derivative δ /δg (q)is thegeneral definition of the energy-
µν
A
momentum tensor of a system:
δ 1
A g(q)Tµν(q), (29)
δg (q) ≡ −2q−
µν
where g is the determinant of g . For the spinless particle at hand, the
µν
− −
energy-momentum tensor becomes
1
Tµν(q) = M dσq˙µ(σ)q˙ν(σ)δ(4)(q q(σ)), (30)
√ g Z −
−
where σ is the proper time. Equation (30) and the explicit variations (26)
and (27), bring (28) to the form
1 δ
δ = d4q√ gTµν(q)[D¯ ξ (q)+D¯ ξ (q)] dτ A ξµ(q(τ)).
EA −2 Z − µ ν ν µ −Z δqµ(τ)
(31)
A partial integration of the derivatives yields (neglecting boundary terms
at infinity and using the symmetry of Tµν)
δ = d4q ∂ [√ gTµν(q)]+√ gΓ¯ µ(q)Tλν(q) ξ (q)
EA Z n ν − − νλ o µ
δ
dτ A ξµ(τ). (32)
− Z δqµ(τ)
Because of the manifest invariance of the action under general coordinate
transformations,theleft-handsidehastovanishforarbitrary(infinitesimal)
functions ξµ(τ). We therefore obtain
∂ [√ gTµν(q)]+√ gΓ¯ µTλν(q) ξ (q)
ν νλ µ
n − − o
δ
dτ A δ(4)(q q(τ))ξµ(τ) = 0. (33)
−Z δqµ(τ) −
Tofindthephysicalcontentofthisequationweconsiderfirstaspacewithout
torsion. On a particle trajectory, the action is extremal, so that the second
term vanishes, and we obtain the covariant conservation law:
∂ [√ gTµν(q)]+√ gΓ¯ µ(q)Tλν(q) = 0. (34)
ν νλ
− −
jadwisin printed on February 6, 2008 9
Inserting (30), this becomes
M ds[q˙µ(σ)q˙ν(σ)∂ δ(4)(q q(σ))+Γ¯ µ(q)q˙ν(σ)q˙λ(σ)δ(4)(q q(σ))] = 0.
Z ν − νλ −
(35)
A partial integration turns this into
M dσ[q¨µ(σ)+Γ¯ µ(q)q˙ν(σ)q˙λ(σ)]δ(4)(q q(σ)) = 0. (36)
Z νλ −
Integrating this over a small volume around any trajectory point qµ(σ), we
obtain Eq. (14) for a geodesic trajectory.
A similar calculation was used by Hehl [5] in his derivation of particle
trajectories in the presence of torsion. Since torsion does not appear in the
action, he found that the trajectories to be geodesic.
The conservation law (34) can be written more covariantly as
D¯ Tµν(q) = 0. (37)
ν
This follow directly from the identity
1 1
∂ √ g = gλκ∂ g = Γ¯ λ, (38)
ν ν λκ νλ
√ g − 2
−
and is a consequence of the rule of partial integration applied to (31), ac-
cording to which a covariant derivative can be treated in a volume integral
d4q√ gf(q)D¯g(q) just like an ordinary derivative in an euclidean inte-
−
gRral d4xf(x)∂ g(x) After a partial integration neglecting surface terms,
a
Eq. (R31) goes over into
1 δ
δ = d4q√ g D¯ Tµν(q)ξ (q)+D¯ Tµν(q)ξ (q) dτ A ξµ(q(τ)).
EA 2Z − ν µ µ ν −Z δqµ(τ)
(cid:2) (cid:3)
(39)
whose vanishing for all ξµ(q) yields directly (37), if the action is extremal
under ordinary variations of the orbit.
Ourtheorydoesnotleadtothisconservationlaw. Inthepresenceoftor-
sion, the particle trajectory does not satisfy δ /δqµ(τ) = 0, but according
A
to (12):
δ ∂L d ∂L ∂L
A = = 2S λq˙ν , (40)
δqµ(τ) ∂qµ − dτ ∂q˙µ µν ∂q˙λ
theright-hand sidebeingequal to M2S q˙ν(σ)q˙λ(σ) if wechoose τ tobe
µνλ
−
the proper time σ. Inserting this into (33), equation (36) receives an extra
10 jadwisin printed on February 6, 2008
term and becomes
M dσ q¨µ(σ)+[Γ¯ µ(q)+2Sµ (q)]q˙ν(σ)q˙λ(σ) δ(4)(q q(σ)) = 0,
Z n νλ νλ o −
(41)
thus yielding the correct autoparallel trajectories (13) for spinless point
particles.
Observe that the extra term in (39) can be expressed via (40) in terms
of the energy-momentum tensor (30) as
d4q√ g2Sµ (q)Tλν(q)ξ (q). (42)
Z − νλ µ
We may therefore rewrite the change of the action (31) as
1
δ = d4q√ gTµν(q)[D¯ ξ (q)+D¯ ξ (q) 4Sλ ξ (q)]. (43)
EA −2Z − µ ν ν µ − µν λ
-
The quantity in brackets will be denoted by δ g (q), and is equal to
E µν
-
δ g (q) = D ξ (q)+D ξ (q), (44)
E µν µ ν ν µ
where D is the covariant derivative (6) involving the full affine connection.
µ
Thus we have
δ = d4q√ gTµν(q)D ξ (q). (45)
EA −Z − ν µ
Integrals over invariant expressions containing the covariant derivative D
µ
can be integrated by parts according to a rule very similar to that for the
Riemann covariant derivative D¯ (which is derived in Appendix A of [18]).
µ
After neglecting surface terms we find
δ = d4q√ gD∗Tµν(q)ξ (q), (46)
EA Z − ν µ
where D∗ = D + 2S λ. Thus, due to the closure failure in spacetimes
ν ν νλ
with torsion, the energy-momentum tensor of a free spinless point particle
satisfies the conservation law
D∗Tµν(q) = 0. (47)
ν
This is to be contrasted with the conservation law (37). The difference
between the two laws can best be seen by rewriting (37) as
D∗Tµν(q)+2S µ (q)Tκλ(q) = 0. (48)
ν κ λ