8 O.V. Babourova, in:Abstracts of contrib. papers of the Cornelius Lanczos Int. conf. (NC. State Univ., USA, 1993) p. 100. 9 O.V.Babourova, M. Yu. Kor...

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arXiv:gr-qc/9708009v2 26 Jan 1998

O. V. Babourova∗ and B. N. Frolov† Department of Mathematics, Moscow State Pedagogical University, Krasnoprudnaya 14, Moscow 107140, Russia

Abstract The equation of perfect dilaton-spin fluid motion in the form of generalized hydrodynamic Euler-type equation in a Weyl–Cartan space is derived. The equation of motion of a test particle with spin and dilatonic charge in the Weyl–Cartan geometry background is obtained. The peculiarities of test particle motion in a Weyl–Cartan space are discussed. PACS Nos: 04.20.Fy, 04.40.+c, 04.50.+h

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∗ E-mail:[email protected] † E-mail:[email protected]

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Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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1. INTRODUCTION

In a previous work1 (which we shall refer as I) the variational theory of the perfect fluid with intrinsic spin and dilatonic charge (dilaton-spin fluid) was developed and the equations of motion of this fluid, the Weyssenhoff-type evolution equation of the spin tensor and the conservation law of the dilatonic charge were derived. The purpose of the present work is to investigate the equations of motion of such type of fluid and their consequences, one of which leads to the equation of motion of a test particle with spin and dilatonic charge in the Weyl–Cartan background. It is well known that the equations of charge particle motion in an electromagnetic theory are the consequence of the covariant energy-momentum conservation law of the system ‘particles–field’ and the electromagnetic field equations.2 In General Relativity the equations of matter motion are the consequence of the gravitational field equations. The reason consists in the fact that the Einstein equations lead to the covariant energy-momentum conservation law of matter. In the Einstein–Cartan theory3,4 the same situation occurs, citeHe:let but the conservation laws have more complicated form established in Ref. 4. In Refs. 6, 7 it was proved that in the generalized theories of gravity with torsion in a Riemann–Cartan space U4 based on non-linear Lagrangians the equations of the matter motion are also the consequence of the gravitational field equations. The similar result was established in a metric-affine space with curvature, torsion and nonmetricity.8 −10 In Sec. 2 we shall use this method for deriving the equation of dilaton-spin fluid motion in the form of generalized hydrodynamic Euler-type equation in a Weyl–Cartan space. In Sec. 3 this equation will be applied for obtaining the equation of motion of a test particle with spin and dilatonic charge in the Weyl–Cartan geometry background.

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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2. THE HYDRODYNAMIC EQUATION OF MOTION OF THE PERFECT DILATON-SPIN FLUID

In a Weyl–Cartan space Y4 the matter Lagrangian obeys the diffeomorfism invariance, the local Lorentz invariance and the local scale invariance that leads to the corresponding Noether identities which can be obtained as the particular case of the corresponding identities stated in a general metric-affine space11 (see Appendix on the notations used), 1 eσ ⌋Q)σ αα , DΣσ = (¯ eσ ⌋T α ) ∧ Σα − (¯ eσ ⌋Rαβ ) ∧ J βα − (¯ 8 1 D + Q ∧ Sαβ = θ[α ∧ Σβ] , 4 DJ = θα ∧ Σα − σ αα .

(2.1) (2.2) (2.3)

Here Σσ is the canonical energy-momentum 3-form, σαβ is the metric stress-energy 4-form, Sαβ is the spin momentum 3-form and J is the dilaton current 3-form. In case of the perfect dilaton-spin fluid the corresponding expressions for the quantities Σσ , σαβ , Sαβ and J were derived in I (see (I.5.3), (I.5.5), (I.5.6)). These expressions are compatible in the sense that they satisfy to the Noether identities (2.1), (2.2) and (2.3). The identities (2.2) and (2.3) can be verified with the help of the spin tensor evolution equation (I.4.4) and the dilatonic charge conservation law (I.4.2). The Noether identity (2.1) represents the quasiconservation law for the canonical matter energy-momentum 3-form. This identity is fulfilled, if the equations of matter motion are valid, and therefore represents in its essence another form of the matter motion equations. Let us introduce with the help of (I.5.4) a specific (per particle) dynamical momentum of a fluid element, πσ η := −

1 ∗u ∧ Σσ , nc2

πσ =

ε 1 uσ − 2 Sσρ u¯⌋Duρ . 2 nc c

(2.4)

Then the canonical energy-momentum 3-form (I.5.4) reads, p Σσ = pησ + n πσ + 2 uσ u . nc

(2.5)

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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Substituting (2.5), (I.5.5) and (I.5.6) into (2.1), one obtains after some algebra the equation of motion of the perfect dilaton-spin fluid in the form of the generalized hydrodynamic Euler-type equation of the perfect fluid, 1 1 p eσ ⌋Dp − η(ε + p)Qσ u ∧ D πσ + 2 uσ = η¯ nc n 8n 1 1 p eσ ⌋Rαβ ) ∧ Sαβ u + (¯ eσ ⌋Rαα ) ∧ Ju . −(¯ eσ ⌋T α ) ∧ πα + 2 uα u − (¯ nc 2 8

(2.6)

Let us evaluate the component of the equation (2.6) along the 4-velocity by contracting one with uσ . After some algebra we get the energy conservation law along a streamline of the fluid, dε =

ε+p dn . n

(2.7)

Comparing this equation with the first thermodynamic principle (I.2.14), one can conclude that along a streamline of the fluid the conditions ds = 0 ,

∂ε dS pq = 0 , ∂S pq

∂ε dJ = 0 . ∂J

(2.8)

are valid. The first of these equalities means that the entropy conservation law is fulfilled along a streamline of the fluid. This fact corresponds to the basic postulates of the theory.

3. THE EQUATION OF TEST PARTICLE MOTION IN A WEYL–CARTAN SPACE

Let us consider the limiting case when the pressure p vanishes, then the equation (2.6) will describe the motion of one fluid particle with the mass m0 = ε/(nc2 ) = const, with the spin tensor Sαβ and the dilatonic charge J, 1 1 1 eσ ⌋T α ) ∧ πα u − (¯ eσ ⌋Rαβ ) ∧ Sαβ u + (¯ eσ ⌋Rαα ) ∧ Ju . (3.1) u ∧ Dπσ = − ηm0 c2 Qσ − (¯ 8 2 8 The third term on the right-hand side of (3.1) represents the well-known Mathisson force, the second term represents the translational force that appears in spaces with torsion. The

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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forth term appears only in a Weyl–Cartan space. It has the Lorentz-like form with the Weyl’s homothetic curvature tensor Rαα as dilatonic field strength. The following Theorem is valid. Theorem. In a Weyl–Cartan space Y4 the motion of a test particle with spin and dilatonic charge obeys the equation, C C 1 eσ ⌋T α ∗ uβ D (Sαβ u) m0 u∧ D uσ = 2 u ∧ δσα D −¯ c C 1 1 eσ ⌋ R αβ ) ∧ Sαβ u + (¯ eσ ⌋dQ) ∧ Ju , − (¯ 2 16

R

C

R

(3.2)

C

where R αβ is a Riemann–Cartan curvature 2-form, D and D are the exterior R

covariant differentials with respect to a Riemann connection 1-form Γ αβ and a C

Riemann–Cartan connection 1-form Γ αβ , respectively. Proof. Using the decomposition (A.5) (see Appendix) the specific dynamical momentum of a fluid element (2.4) can be written in the form, πσ = m0 uσ −

C 1 1 Sσρ u¯⌋ D uρ − Sσρ Qρ . 2 c 8

(3.3)

With the help of the decomposition (A.5) one can prove that the evolution equation of the C

spin tensor (I.4.4) is also valid with respect to the Riemann–Cartan connection Γ αβ and reads, C

Πασ Πρβ u¯⌋ D S σρ = 0 ,

(3.4)

where Πασ := δσα + c−2 uα uσ is the projection tensor. Using (2.4) and (3.4) the left-hand side of the equation (3.1) can be represented as follows, C C C 1 1 u ∧ Dπσ = m0 u∧ D uσ − ηm0 c2 Qσ − 2 u∧ D (Sσρ u¯⌋ D uρ) 8 c C 1 1 ρ ρ λ − Sσρ u∧ D Q − ηSσρ Q Qλ u . 8 64

(3.5)

With the help of the decomposition (A.6) the third term on the right-hand side of (3.1) takes the form,

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space C 1 1 1 (¯ eσ ⌋Rαβ ) ∧ Sαβ u = (¯ eσ ⌋ R αβ ) ∧ Sαβ u + ηT ασρ uρ Sαβ Qβ 2 2 8 C 1 1 ρ ρ λ − ηSσρ u¯⌋ D Q + ηSσρ Q Qλ u . 8 64

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(3.6)

Substituting (3.6) and (3.3) in the right-hand side of (3.1), comparing the result with (3.5) and taking into account the equality (A.12), one obtains (3.2), as was to be proved. The Theorem proved has the important consequences. Corollary 1. The motion of a test particle without spin and dilatonic charge in a Weyl– Cartan space Y4 coincides with the motion of this particle in the Riemann space, the metric tensor of which coincides with the metric tensor of Y4 . Corollary 2. The motion of a test particle with spin and dilatonic charge in a Weyl–Cartan space Y4 coincides with the motion of this particle in the Riemann–Cartan space, the metric tensor and the torsion tensor of which coincide with the metric tensor and the torsion tensor of Y4 , if one of the conditions is fulfilled: i) the dilatonic field is a closed form,

dQ = 0;

ii) the dilatonic charge of the particle vanishes,

J = 0.

Corollary 3. The manifestations of the non-trivial Weyl space structure (when the dilatonic field Q is not a closed form) can be detected only with the help of the test particle endowed with dilatonic charge. The result of the Corollary 1 can be cosidered as a particular case of the Theorem stated in Refs. 8 – 10 for the matter motion in a general metric-affine space.

4. CONCLUSIONS

The perfect dilaton-spin fluid model represents the medium with spin and dilatonic charge which generates the spacetime Weyl–Cartan geometrical structure and interacts with it. The influence of the Weyl–Cartan geometry on dilaton-spin fluid motion is described by the Euler-type hydrodynamic equation. This hydrodynamic equation leads to the equation of motion of a test particle with spin and dilatonic charge in the Weyl–Cartan geometry

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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background, the special form of which is stated by the Theorem of Sec. 3. The important consequences of this Theorem mean that bodies and mediums without dilatonic charge are not subjected to the influence of the possible Weyl structure of spacetime (in contrast to the generally accepted opinion) and therefore can not be the tools for the detection of the Weyl properties of spacetime. For ‘usual’ matter without dilatonic charge the Weyl structure of spacetime is unobservable. In order to investigate the different manifestations of the possible Weyl structure of spacetime one needs to use the bodies and mediums endowed with dilatonic charge.

APPENDIX:

Let us consider a connected 4-dimensional oriented differentiable manifold M equipped with a linear connection Γ and a metric g of index 1, which are not compatible in general in the sense that the covariant exterior differential of the metric does not vanish, Dgαβ = dgαβ − Γγα gγβ − Γγβ gαγ =: −Qαβ ,

(A.1)

where Γαβ is a connection 1-form and Qαβ is a nonmetricity 1-form, Qαβ = Qαβγ θγ . A curvature 2-form Rαβ and a torsion 2-form T α , 1 Rαβ = Rαβγλ θγ ∧ θλ , 2

1 T α = T αβγ θβ ∧ θγ , 2

(A.2)

are defined by virtue of the Cartan’s structure equations, Rαβ = dΓαβ + Γαγ ∧ Γγβ ,

T α = Dθα = dθα + Γαβ ∧ θβ .

(A.3)

A Weyl–Cartan space Y4 is a space with a metric, curvature, torsion and nonmetricity which obeys the constraint (Q is a Weyl 1-form), 1 Qαβ = gαβ Q , 4

Q := g αβ Qαβ = Qα θα .

In a Weyl–Cartan space the following decomposition of the connection is valid,

(A.4)

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space 1 ∆αβ = (2θ[α Qβ] + δβα Q) , 8

C

Γαβ =Γ αβ + ∆αβ , C

where Γ

α β

8 (A.5)

denotes a connection 1-form of a Riemann–Cartan space U4 with curvature,

torsion and a metric compatible with a connection. The decomposition (A.5) of the connection induces corresponding decomposition of the curvature, C C C 1 Rαβ =R αβ + D ∆αβ + ∆αγ ∧ ∆γβ =R αβ + P αβ + δβα dQ , P αβ = P [αβ] , 8 C 1 1 1 T [α Qβ] − θ[α ∧ D Qβ] + θ[α Qβ] ∧ Q − θα ∧ θβ Qγ Qγ , P αβ = 4 8 16

(A.6) (A.7)

C

where D is the exterior covariant differential with respect to the Riemann–Cartan connection C

C

1-form Γ αβ and R αβ is the Riemann–Cartan curvature 2-form. In (A.6) the last term contains the Weyl homothetic curvature 2-form, 1 1 1 1 eα ⌋DQβ )θα ∧ θβ + Qα T α = dQ . Rαα = DQ = (¯ 2 2 2 2

(A.8)

The Riemann–Cartan connection 1-form can be decomposed as follows, C

Γ

α β

R

=Γ αβ + Kαβ ,

T α =: Kαβ ∧ θβ ,

1 Kαβ = 2¯ e[α ⌋Tβ] − e¯α ⌋¯ eβ ⌋(Tγ ∧ θγ ) , 2

(A.9) (A.10)

R

where Γ αβ is a Riemann (Levi–Civita) connection 1-form and Kαβ is a kontorsion 1-form.11 In a Riemann–Cartan space the covariant differentiation with respect to the transport connection12 is useful, tr

C

δσρ D:= δσρ D −¯ eσ ⌋T ρ .

(A.11)

In particular, the following equality is valid, tr

u∧ D uσ = u ∧

δσρ

C

eσ ⌋T D −¯

ρ

R

uρ = u∧ D uσ .

(A.12)

Perfect fluid and test particle with spin and dilatonic charge in a Weyl–Cartan space

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REFERENCES 1

O.V. Babourova and B.N. Frolov, “The variational theory of the perfect dilaton-spin fluid in a Weyl–Cartan space”, submitted toMod. Phys. Letters A.

2

L.D. Landau, E.M. Lifshitz, Classical Theory of Field (Pergamon, Oxford, 1962).

3

T.W.B. Kibble, J. Math. Phys. 2, 212 (1961).

4

A. Trautman, Symp. Math. 12, 139 (1973).

5

F.W. Hehl, Phys Letters 36A, 225 (1971).

6

O.V. Babourova, V.N. Ponomariov, B.N. Frolov, in: Gravitaciya i electromagnetizm (Universitetskoe, Minsk, 1988) p. 6 (in Russian).

7

O.V. Babourova, The variational theory of perfect fluid with intrinsic degrees of freedom in generalized spaces of the modern theory of gravitation, Dr. thesis (VNICPV, Moscow, 1989) (in Russian).

8

O.V. Babourova, in:Abstracts of contrib. papers of the Cornelius Lanczos Int. conf. (NC State Univ., USA, 1993) p. 100.

9

O.V.Babourova, M. Yu. Koroliov and B.N. Frolov, Izv. Vyssh. Uchebn. Zaved. (Fiz.) (Russian Physics Letters) No 1, 76 (1994) (in Russian).

10

O. V. Babourova, B. N. Frolov and M.Yu. Koroliov, “Peculiarities of matter motion in metric-affine gravitational theory” (LANL e-archive gr-qc/9502012, 1995).

11

F. W. Hehl, J. L. McCrea, E. W. Mielke and Yu. Ne´eman, Phys. Reports 258, 1 (1995).

12

A. Trautman, Bul. Acad. Pol. Sci. (Ser. sci. math., astr., phys.) 20, 895 (1972).