on defining the spin moment in the tetrad theory of gravitation
TRANSCRIPT
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.
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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
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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.
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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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