on defining the spin moment in the tetrad theory of gravitation

4
ON DEFINING THE SPIN MOMENT IN THE TETRAD THEORY OF GRAVITATION V. N. Tunyak UDC 530.12 : 531.51 A general definition of the spin moment is presented in the tetrad formulation of the relativistic theory of gravitation; it is based on the conditions for the invariance of the corresponding action integral relative to infinitesimal tetrad transformations (the so-called tetrad spin moment) and infinitesimal coordinate transformations (the so-called coordinate spin moment). It is shown that the tetrad formulation of the general theory of relativity (TFGTR) and the tetrad theory of gravitation (TTG) in a space of absolute parallelism lead to fundamentally different definitions of spin, since in the Riemannian geometry of the TFGTR only the coordinate spin moment is physically meaningful, whereas in the space of absolute parallelism of the TTG only the tetrad spin moment has essential significance. It is also indicated that the Pellegrini-Plebanski theory (PI~I ") leads to an unsatisfactory hybrid definition of spin in the form of the coordinate spin moment of the gravitational and boson fields and the tetrad spin moment of the gravitational and fermion fields, the gravitational field entering into these spin moments of the PVI~ with op- posite signs. A general formulation of the weak law of conservation was presented earlier [1, 2] within the framework of the tetrad theory of gravitation (TTG) in a space of absolute parallelism Wt; this theory takes the tetrad components h(~) as the 16 dynamic potentials of the gravitational field. A consideration of this weak law of con- servation in the important particular case of arbitrary infinitesimal coordinate transformations x"'= X~+-:~ (1) leads [1, 2] to a fundamentally interesting definition of the sources of the tetrad g-ravitational field, in the form of the total canonical ener~,-momentum tensor of nongravitational matter; this of course plays a fundamental part in the special-relativistic theory of energy-momentum localization [3]. The aim of the present investiga- tion lies in making a further study of the weak law of conservation of the TTG [1, 2] for infinitesimal coordi- nate transformations of a special type x ~' = x ~ q- g~.~ x: a~, a~ : a[~: i , c~ a~ ==0 (2) and infinitesimal rigorous tetrad transformations ~h~(~) = b(~)(~ h(~ "), b(~)(:~ + b(~)(~) = 0, 0~ b(~)(~) = O, (3) leading to two fundamental definitions of the spin moment in the TTG in the form of the so-called coordinate and tetrad spin moments respectively. In solving this problem we shall start from the single variational principle of the TTG [1, 2] + = o (4) for a certain self-consistent set of gravitational and boson fields described by tensor potentials QA (A, B, C, ... are the collective tensor indices), and fermion fields represented by Dirac bispinors ~I, and ~. The expres- sions L = L m + Lh, Lh, L m respectively represent the total Lagrangian, the gravitational Lagrangian of the TTG [1, 2], and the Lagrangian of nongravitational matter t m = Lm (h(~ ~), QA, V'~ QA, W, Oa~', ~, O~ ~"), (5) V. I. Lenin Belorussian State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 68-71, May, 1976. Original article submitted December 1, 1975. This material is protected by copyright registered in tile name of Plenum Publishing Corporation, 227 [Vest 1 7th Street, New York, N Y 10011 No part ] of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic mechanical, photocopying, I microfilm ing, recording or otherwise, withou t written pernzission of t)~e publisher. A copy of this article is available from the publisher for $ 7. 50. 599

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ON DEFINING THE SPIN MOMENT

IN THE TETRAD THEORY

OF GRAVITATION

V. N. T u n y a k UDC 530.12 : 531.51

A general definition of the spin moment is presented in the tetrad formulation of the re la t ivis t ic theory of gravitation; it is based on the conditions for the invariance of the corresponding action integral relat ive to infinitesimal te t rad t ransformat ions (the so-cal led te t rad spin moment) and infinitesimal coordinate t ransformat ions (the so-cal led coordinate spin moment). It is shown that the tetrad formulation of the general theory of re la t ivi ty (TFGTR) and the tetrad theory of gravitation (TTG) in a space of absolute para l le l i sm lead to fundamentally different definitions of spin, since in the Riemannian geometry of the TFGTR only the coordinate spin moment is physically meaningful, whereas in the space of absolute para l le l i sm of the TTG only the tetrad spin moment has essential significance. It is also indicated that the Pe l l eg r in i -P l ebansk i theory (PI~I ") leads to an unsat isfactory hybrid definition of spin in the form of the coordinate spin moment of the gravitational and boson fields and the te t rad spin moment of the gravitational and fermion fields, the gravitational field entering into these spin moments of the PVI ~ with op- posite signs.

A general formulation of the weak law of conservat ion was presented ear l ie r [1, 2] within the f ramework of the tetrad theory of gravitat ion (TTG) in a space of absolute para l le l i sm Wt; this theory takes the tetrad components h(~) as the 16 dynamic potentials of the gravitational field. A considerat ion of this weak law of con- servat ion in the important par t icular case of a rb i t r a ry infinitesimal coordinate t ransformat ions

x " ' = X~+-: ~ (1)

leads [1, 2] to a fundamentally interesting definition of the sources of the tetrad g-ravitational field, in the form of the total canonical ene r~ , -momen tum tensor of nongravitational matter ; this of course plays a fundamental part in the specia l - re la t iv is t ic theory of ene rgy-momentum localization [3]. The aim of the present investiga- tion lies in making a further study of the weak law of conservat ion of the TTG [1, 2] for infinitesimal coordi- nate t ransformat ions of a special type

x ~' = x ~ q- g~.~ x: a~, a~ : a[~: i , c~ a~ == 0 (2)

and infinitesimal r igorous tetrad t ransformat ions

~h~(~) = b(~)(~ h(~ "), b(~)(:~ + b(~)(~) = 0, 0~ b(~)(~) = O, (3)

leading to two fundamental definitions of the spin moment in the TTG in the form of the so-cal led coordinate and tetrad spin moments respect ively.

In solving this problem we shall s tar t f rom the single variational principle of the TTG [1, 2]

+ = o (4)

for a certain se l f -consis tent set of gravitational and boson fields described by tensor potentials QA (A, B, C, ... are the collective tensor indices), and fermion fields represented by Dirac bispinors ~I, and ~ . The expres- sions L = L m + Lh, Lh, L m respect ively represen t the total Lagrangian, the gravitational Lagrangian of the TTG [1, 2], and the Lagrangian of nongravitational matter

tm= Lm (h(~ ~), QA, V'~ QA, W, Oa ~', ~, O~ ~"), (5)

V. I. Lenin Belorussian State University. Translated f rom Izvest iya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 68-71, May, 1976. Original ar t ic le submitted December 1, 1975.

This material is protected by copyright registered in tile name o f Plenum Publishing Corporation, 227 [Vest 1 7th Street, New York, N Y 10011 No part ] o f this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic mechanical, photocopying, I microfilm ing, recording or otherwise, withou t written pernzission o f t)~e publisher. A copy o f this article is available f r o m the publisher for $ 7. 50.

599

constituting a di rect covar iant general izat ion of the spec ia l - re la t iv i s t i c express ion L m to the space W 4 [1, 2]. The covar iant der ivat ive of the tensor potential QA in W 4 has the fo rm

w, QA = o~ QA-- ~ Q ~ , ( 6 )

where 7~x ---- h (~) 0r h(~ ~) is the connectechaess (compendeney) of the absolute para l l e l i sm; fA---fl~ a r e ce r t a in com- binations of Kronecker symbols 5 ~ depending on the tensor rank of QA. The general formulat ion of the weak law of conservat ion corresponding to this variat ional pr inciple has the fo rm �9

* { OL., ~QA-b c)Lm ~ F + OLin .~-.t-~oh~ ~') vo, ,~L+ O(Vo, QA) O(w,~') 0(v,,~----) •

OL~ • [U~(,) - 0(wQA) Q ~ h ~ ) f ~ l } = O , (7)

where 6 is the fo rm variat ion of the function; 6 ~ is the covariant der ivat ive re la t ive to the Cris toffel connec- tion der ived f rom the Riemannian me t r i c g~ = h~ :) h~)~(~)(~), W(=)o)=diag (1, -- 1, - - ], -- 1) is the local Minkowski me t r i c

U'V'(~,) --___ OLh/O (Or h~: )) (8)

the so-ca l led superpotent ial [1, 2].

If we substi tute the potential var ia t ions due to the infinitesimal coordinate t ransformat ion (2)

[h(~ ") = a.., x'- (gP" 0 e h(~ ") + ..p r g' , -- ', -r~, (9)

~-qa = a~.. QB/] ~" g ' : + a= x" {#= 0.~ QA + QB 1~,~ O.~ g": }, (10) ~'tF=aop x:apcg ,, [WF =a=. x : O ~ , (ii)

in the weak law of conservat ion (7) we obtain the following law of conservat ion of the total (orbital PP (rw and coordinate spin VP (rw) moment:

0 ~) {V g [PP=~ + VP'~]} = 0, (12)

where V~- = 2U ~ I,~1. (13)

In o rde r to de te rmine the t e t rad spin moment W(#)(v);% we substi tute the potential var ia t ions due to the infinitesimal t e t r ad t rans format ion (3)

~ Q a = 0 , ~ : --~ b(r)(~) %)(r)~F, ~ _ / b r W:(r)(~) ' (14) 4 4

into Eq. (7) where (r#v =i T [# y v ] is the so-ca l led ma t r ix spin tensor y # = h ( a ) y ( ~ ) j T(~) a r e cer ta in spec ia l - re la t iv i s t ic Dirac ma t r i ces . We accordingly a r r i v e at the t e t rad spin moment , conserved in accordance with the equation

V, W(~)(~) x = 0 (15)

of the gravitat ional field and nongravitat ional mat te r

(~)(~, = 2 UI(~)(~)I -}- S(v,)(,,) , (16)

where

I ~ OL., qt OL,,, ~ (17) O (w qa) 0 (w ~) !

is the spin moment of nongravitat ional mat ter . Comparison between the resul tant express ions for the coord i - nate and te t rad spin moments shows that, within the f r amework of the TTG, only the definition of the t e t rad spin moment general iz ing the spec ia l - re l a t iv i s t i c definition of spin (and t r ans fo rming into the la t ter when the gravitat ional field is excluded) remains physical ly consis tent in W 4 space, while the coordinate spin moment (13), which does not contain the cor responding contribution of nongravitat ional ma t t e r , and f u r t h e r m o r e pos- ses ses the opposite s ign to the te t rad spin moment of the gravitat ion field obtained f rom (16), has no physical meaning. It is also an important fact that, according to (12) and (15), the principle of cor respondence with the spec ia l - re la t iv i s t i c law of conservat ion of the total moment

0~ {P~~ + S~ ~} = 0 (18) 0 0

600

is sa t i s f ied in the TTG by v i r tue of the law of conserva t ion of the total t e t rad spin moment of the gravi ta t ional field and nongravi ta t ional ma t t e r .

It is a lso of fundamental in te res t to compare this r e su l t of the TTG with the analogous r e su l t of the TFGTR [4-7]. The Lagrangians of the TTG L h and L m a re in this case r e spec t ive ly rep laced by the te t rad

,

Lagrangian of MSller, L h [8] and the g e n e r a l - r e l a t i v i s t i c Lagrangian of nongravi ta t ional ma t t e r

L~ =L~(h(~ ~), QA, V,~QA, ~P, V~ ~', ~ , v~ iF), (19)

where ~Tp ,I, is the F o k - I v a n e n k o covar iant der iva t ive [9] (in genera l the a s t e r i s k denotes quantit ies in Vr con- st i tuting analogs of the cor responding quantit ies in W4). The cor responding genera l formulat ion of the weak law of conservat ion in the TFGTR has the f o r m

"{ w ~'~ L + OLin 0 (V~ QA)

0 (V ~ ~')

~QA+ OLin g~,+ 0(Wo'7)

0 (v~ QA) (20)

where h~, . = _l ( ~ q_ ~ q_ S ~ ) . (21) 2

If we subst i tute the potential var ia t ions cor responding to (9)-(11) and (3), (14) in Eq. (20), we find the coordi - nate and te t rad [4-7] spin moments of the TFGTR:

- 2 U '~ I~pl _}- , ( 2 2 )

2 "~ W(~)(~) x = __ Vo {h ~ [(~) h(~)l x} (23)

Since Eq. (22) r e p r e s e n t s the total spin moment of the gravi ta t ional field and nongravi ta t ional ma t t e r , which in the spec i a l - r e l a t i v i s t i c l imit ing case t r a n s f o r m s into the cor responding spin moment of nongravi ta t ional m a t - t e r , while Eq. (23) contains no contribution of nongravi ta t ional ma t t e r and sa t i s f i e s the conserva t ion law iden- t ica l ly , independently of the sa t i s fac t ion of the field equations [6, 7], only the definition of the coordinate spin moment may be r e g a r d e d as phys ica l ly consis tent within the f r a m e w o r k of the TFGTR. We thus a r r i v e at a fundamental ly in teres t ing conclusion, namely , the essen t ia l ly different or igin of the spin in the non-Eucl idean theor ies of gravi ta t ion in Riemannian space and the space of absolute pa r a l l e l i sm , r e spec t ive ly leading to the coordinate spin moment in the TFGTR and the t e t rad spin moment in the TTG. The r e a s o n for this difference, as may read i ly be seen on compar ing Eqs. (7) and (20), is the di f ference in the f o r m of the weak conservat ion laws in the space of absolute p a r a l l e l i s m W 4 and the Riemannian space V 4. Thus, for example , in considering the r igorous t e t r ad t r ans fo rma t ions (3) and (14), we find that p r e c i s e l y because of the absence of N - t e r m s of the (21) type f r o m (7) the spin moment of nongravi ta t ional ma t t e r is re ta ined in the te t rad spin moment of the TTG (16), whereas , on the contrary , the p re sence of these N - t e r m s in (20) leads to the exclusion of the contr i - bution of nongravi ta t ional ma t t e r f rom the t e t r ad spin moment of the TFGTR (23).

In conclusion, let us also consider the definition of spin in the Pe l legr in i - P lebanski theory (PPT) [10]. Cer ta in intr insic shor tcomings of this theory assoc ia ted with the hybrid definition of the sources of g rav i t a - t ional field were noted in [1, 2]. Another c h a r a c t e r i s t i c failing of the PI~T l ies in the hybrid definition of spin in the f o r m of the coordinate spin moment of the gravi ta t ional and boson fields and the t e t r ad spin moment of the gravi ta t ional and fe rmion fields, the gravi ta t ional field enter ing into these spin moments of the PPT with opposite signs.

LITERATURE CITED

1. V.N. Tunyak, Izv. Akad. Nauk BelorussSSR, Set. F iz . -Mat . Nauk, No. 5, 98 (1973). 2. V .N. Tunyak, Izv. Vyssh. Uchebn. Zaved. , Fiz . , No. 1, 91-97 (1975). 3. D .D . Ivanenko and A. A. Sokolov, C l a s s i c a l Field Theory [in Russian] , GITTL, Moscow (1951). 4. V . I . Rodichev, Izv. Vyssh. Uchebn. Zaved. , Fiz . , No. 1, 142 (1965). 5. V . I . Rodichev, Theory of Gravi ty in an Orthogonal Refe rence System [in Russian] , Nauka, Moscow (1974). 6. B . N . Fro lov , in Modern l:Yoblems of Gravi ta t ion [in Russian] , Izd. TGU, Tbt l is i (1967). 7. B . N . Fro lov , Summar ies of Contributions to the Fifth Internat ional Conference on Gravi ta t ion and the

Theory of Relat iv i ty [in Russian] , Izd. TGU, Tbi l is i (1968), p. 71.

601

8. C. M~ller , Ann. of Phys. , 12, 118 (1961). 9. V .A. Fok and D. D. Ivanenko, Compt. Rend., 188, 1470 (1929).

10. C. Pel legr ini and J . Plebanski , Mat. Fys. b-1~r. Dan. Vid. Selsk., 2, No. 4 (1963).

S O M E F E A T U R E S O F T H E E L E C T R I C A L

F O R M I N G O F M D M S Y S T E M S B A S E D

ON S I L I C O N O X Y N I T R I D E F I L M S

V. A. B u r d o v i t s i n , B . L . G a l a n s k i i , K. I . S m i r n o v a , a n d Y u . B. Y a n k e l e v i c h

UDC 621.382.2

Resul ts a r e given in this paper of a study of the e l ec t r i ca l fo rming p r o c e s s in th in - f i lm MDM s y s t e m s based on f i lms of s i l icon oxyni t r ide of vary ing composi t ions . It was obse rved that the forming voltage and forming r a t e of a th in - f i lm A1 - SixNyO z -A1 s y s t e m depends essen t ia l ly on the composi t ion of the d ie lec t r ic f i lm while the c h a r a c t e r i s t i c s of the fo rmed s t r u c t u r e s depend weakly on the o x y g e n / n i t r o g e n ra t io in the fi lm. The r e su l t s obtained a r e in good a g r e e m e n t with Shnnrov's model of forming.

The phenomenon of e l ec t r i c fo rming is obse rved in a wide range of th in - f i lm m e t a l - d i e l e c t r i c - m e t a l (MDM) s y s t e m s . However , the m e c h a n i s m of this phenomenon has not yet been elucidated. The r e su l t s of nu- me rous studies [1] indicate that the pr inc ip les of e l ec t r i ca l fo rming depend fundamental ly on the p rope r t i e s of the d ie lec t r ic f i lm. The re fo re , it is pa r t i cu l a r ly impor tan t to compare the pr inc ip les of this phenomenon in MDM s y s t e m s with var ious insulat ing l aye r s .

This paper r e p o r t s the r e s u l t s of a s tudy of the fo rming p r o c e s s in MDM s t ruc tu r e s . The s t r u c t u r e s included f i lms of var ious composi t ions of s i l icon oxyni t r ide which a r e used as co ld -e l ec t ron e m i t t e r s .

M E T H O D

We chose oxynl t r ide f i lms produced by sput te r ing a s i l icon t a rge t with a shaped b e a m of ions of an act ive gas [2]. A mix tu re of oxygen and ni t rogen was used as the ac t ive gas. By vary ing the ra t io of these gases in the mix tu re we were able to produce s i l icon oxynitr ide f i lms of different composi t ions . The d ie lec t r ic f i lms were applied in a vacuum chamber ; the p r e l i m i n a r y vacuum (4.10 -6 m m Hg) was produced by mechanica l and oil diffusion pumps; and the working p r e s s u r e (2 .10 -4 m m Hg) was es tab l i shed by leaking in the act ive gas. The subs t r a t e was at r o o m t e m p e r a t u r e during sput ter ing. In o rde r to moni tor the p rope r t i e s of the f i lms , we r eco rded the absorpt ion s p e c t r a in the UV, vis ible , and IR reg ions and we m e a s u r e d the r e f r a c t i v e index at a wavelength of ~t =6328 .~ by an e l l i p somet r i c method. Theo e l ec t r i ca l studies w e r e c a r r i e d out on A1-SixNyO z - A1 MDM s y s t e m s with a s i l icon oxynitr ide f i lm of 300 A. The lower and upper a luminum e lec t rodes , r e s p e c - t ively, 2000 and 100 A thick, we re applied by vacuum evaporat ion. We used glass for the s u b s t r a t e s of the MDM s y s t e m s . All the e l ec t r i ca l m e a s u r e m e n t s w e r e made at constant voltage in a vacuum of 1 . 1 0 -6 m m Hg. The v o l t - a m p e r e c h a r a c t e r i s t i c s were r e c o r d e d using a fou r -p robe method [3] on an xy r e c o r d e r with a vol - tage r a m p of 0.1 V / s e c .

R E S U L T S

The effect of the composi t ion of the gas mix tu re on the p rope r t i e s of the d ie lec t r ic f i lm is i l lus t ra ted in Table 1, where 3.ma x is the wavelength cor responding to the m a x i m u m absorpt ion in the IR spec t rum; n is the r e f r a c t i v e index; and E 0 is the optical band gap. The value of E 0 was de te rmined f r o m the re la t ionsh ip for in- d i rec t t rans i t ions [4]

T o m s k Insti tute of Automated Control Sys tems and Radioe lec t ron ics . T rans la t ed f r o m Izves t iya Vys- shikh Uchebnykh Zavedenii , Fizika, No. 5~ pp. 71-74, May, 1976. Original a r t i c l e submi t ted December 9, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N .Y . 10011. No part [ o f this publication may be reproduced, stored in a .retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, I imlerofilmlng, recording or otherwise , wi thout written permission o f the publisher. A copy o f this article is available f rom the publisher for $Z50 .

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