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Consistency of the Averaging Procedure in the Dualistic Approach to Gravitation.

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A N N A L E N D E R PHYSIK
7. Folge. Band 45.1988. Heft 6, S. 393-456
Consistency of the Averaging Procedure in the Dualistic
Approach to Gravitation
By M. N. MAHANTAt
Indian Institute of Advanced Study, Shimla, India
A b s t r a c t . The Einstein approximation on which the averaging procedure in the recent dualistic
approach is based is further elaborated and used t o derive important results. This lends further credibility t o the consistency of the procedure.
Die Konsistenz der Mitteilung im dualistischen Ansatz fur die Gravitation
I n h a l t s u b e r s i c h t . Die Einsteinsche Naherung, auf der die Mittelung im dualistischen Ansatz
basiert, wird weiter ausgebaut und wichtige Ergebnisse werden hergeleitet.
1. Introduction
Over the years 1979-83 a novel approach to gravitation called the dualistic approach
[l]has been developed by the author. The approach has been called dualistic because i t
is an attempt to deal with the microscopic and macroscopic aspects of gravitation within
a single framework, the macroscopic aspect being an averaged out effect of the microscopic aspect not unlike the case of Lorentz and Maxwell forms of electrodynamical
equations.
The basic assumption in the new approach is that hadrons (the ultimate particles
with composite structures) are microuniverses with a conformally flat space-time.
Mathernatically this is realizing by adopting a variational principle of the form
6 f I(-g)1’2
d4X =
0,
(1)
+
+
+
with I = Piikl(Riikl
- (gikdil
gild, - gikdjl - gir di,)) agiidii
KL. (2)
Here Piikl has the symmetry properties of Riikl,dii is symmetric and a , k constants.
The field equations are
R..
- d.
(3)
zikl - g i k 21
gil
- gi&l - g i l d i k ?
pii
1f 4 agii
If 4 a(dik - d g i k )
(4)
gklpikli =
pi;v - +,k
-
=~
~
i
k
where the constant K is redefined and
,ik
d d p W arid d giidii.
The identities are
( P y - Zjk);$
=
0,
(dik - dgik).kE I f 2(Rik - 1f 2 Rgik);k= 0 ,
,
(5)
394
Ann. Physik Leipzig 45 (1988) 6
Derivation of (6), (7) is based on (3) which is the condition for conformally flat spacetime. A contraction of eq. ( 5 ) gives
12K
R ~ 6 d = -a T ,
6
or, since R = - 0 H , if we take the conformally flat space-time in the form
H3
ds2 = H2q.dXi
27
dd-
H2[(dXo)2- ( d d ) 2- (dz2)’ - ( d ~ 3 ) ~ ] ,
(9)
we have the Nordstrom equation
6 3 n ~ = H
12K T .
a
The system (3)-(9) is supposed to be appropriate for the microscopic world. The
transition to the macroscopic or Einstein system is brought about by rewriting eq. (5)
as
and averaging locally (at each point of space-time) over hadronic regions to smooth out
the high frequency effects. As a result the tensor Fk is supposed to manifest as a macroscopic energy momentum tensor in the mixed form which we call Ti(,,,ac)k
and then the
corresponding macroscopic metric tensor which we call g(mac)ii
is to be obtained by
solving the Einstein field equations
-.
-.
.8K .
Ch= Rk - I f 2 6bR = - -T3
a (rnax)k’
As shown in [ l ] in
, bringing about this transition a certain approximate first used
by Einstein [ 2 ] in his critique of the Norsdtrom theory seems to be necessary. It is to a
further clarification and justification of this procedure that this paper is chiefly devoted.
2. The Einstein Approximation
This approximation was used by Einstein to furnish a very direct proof of the equality of inertial and gravitational masses of an isolated material system i n Nordstrom’s
theory of gravitation [ 2 ] . If the conformally flat metric in this theory is assumed in the
form (9) then the approximation is equivalent to neglecting the space-time variations of
(In
inside the volume of the particle (in contrast for example with the variations in
T in the same volume).
We shall extend it further to assume its validity in the space-time regions of hadronic
dimensions in which the averaging is performed locally. Thus we shall generally treat
(In H),$approximately as constants in each such region surrounding a space-time point.
As shown in [ I ] this approximation results in the Ricci tensor Ri for the metric (9)
having a triple space-like eigenvalue el and a simple time-like eigenvalue eo given by
(el M
2
5 A l H , where AIH
= gifH,iH,i
395
M. N. MAHANTA,Dualistic Approach to Gravitation
the normalized eigenvector corresponding to
eo being
AO,i” (A1 H)-1‘2 H,i
(13)
(for more details see appendix I).
Using now a result due to Eisenhart [3] we may write down the following representation for the Einstein tensor of the metric (9) in the same approximation.
GL = Ric - 1/2 SLR = (eo - el) ( L/H2dlH) qizH,kH,z- 1/2(@, el) S:,
+
(14)
when averaged over hadronic regions we get
-.
-,
. GL = Rf,- i / 2 Sf,R = (Go - &) ( l / H 2All?) $‘H,kH,Z
- 1/2(@0
+
Gl)
SL,
(15)
which is the same as
-.
-_
_ -
+
G,?= Rf, - 1/2 SLR = (& - &) A,/&,/ - 1/2(G0 &) S.
(16)
Eq. (16) again is an Eisenhart representat’ion since 2.0/k,2.bLcan also be regarded as the
covariant and contravariant forms of the eigenvectors of Rf, for the eigenvalue (eo)the
corresponding metric tensor being g(mitx)ii.
This can be seen easily from the fact that
we can obtain the equations
-.
(RL Bo,j = 0 ,
(17)
Sico)
(I?; - &j0)A;/ = 0 ,
from the corresponding eq. s.
(18)
(R;- dieo) A O / i = 0 ,
(R:A;, = 0 ,
(19)
(20)
by averaging over hadronic neighbourhoods and using the Einstein approximation.
But (18) can also be obtained from (17) by using g(mac)iiand gflAac)for lowering and
raising suffixes. T h i s means that we must have
$Lnac)&/i
= gii&/io ; gii(moc)%, = gi&/
(21)
= giiH3i;
gii(mac)H’i= g,jjH3i,
(22)
or equivalently
g&,H,i
which in -turn requires
det(g$,,, - gii) % 0 ; det(gif(mac)
- gii) w 0
(23)
to the approximation used.
Taking the partial derivatives of both sides of the first eq. of (22) w.r.t. x” and
using
(1/H H,i),k M 0 ,
and ( 2 2 ) we can easily show that in the same approximation
gfAac),k ~
,
i 9;iH.i-
(24)
A similar result g(mac)ii;,kH’i
m gii,kH,i follows from 2nd eq. of (22).
The Einstein approximation can be given a more compact form. We may also regard
it as the condition of approximate constancy of [h~(-g)],~
where g- det gii, in hadronic
regions surrounding any space-time point. The equivalence is seen from the relation
[In(-g)],; = 8(ln H),i, since g = - H E .
396
Ann. Physik Leipzig 45 (1988) G
Since [In(-g)],i is a slowly varying quantity in hadronic neighbourhoods of spacetime points we shall assume that even after the averaging process it retains its form (in
the approximation used). We thus write
[W-g)l,i
M
(25)
w-g(rnac)),i>
or
3. An Important Application
We shall now make use of the results developed in the foregoing section in the derivation of the important identities
-.
(27)
T{mac)k;i
= 9-;;i= 0 .
This result is needed to fully justify the reasonings leading to the Einstein field equak the macroscopic energy momentum tensor T(mac)k:
tions (11)and calling y
We start from the microscopic eqs.
F k ; i = 0, which can be written equivalently as
(28)
We multiply (28) by the element d4x = dxo dxl dx2 dx3 and integrate over the hadronic
surrounding the point xi. Since gLmgim,k
are of the type (1/H H,i) which
region (,Z0)
are treated as approximate constants in such regions, we have
( Ti(- g ) i / 2 ) , j - 1/2 g l m g j m , k T ; (-g)1/2 = 0 .
(ASi(--g)ll2 d4x
- l/2
glmgjm,k(
1
Ti(-g)1/2 d42 = 0 .
),i
(29)
Now by assumption
and so
J Ti(-g)l/2 d4x = (ZO)
T;mac)k.
(31)
4
We may replace (,Yo)by (-g(mac))112 d4x, where d44x is now to be taken in a macroscopic
sense. Thus we get from eq. (29)
(T[mac)k(-g(mac))),i
= 1/2 gZmgginz,k(T~mac)l(-g~moc))li2.
(32)
Now from eqs. (ll),(16)
-
-.
aa (RL
- 1/2 IYJ)
= -
8K [(Go - el) Ai/&,k
T[mac)k =
a
- 1/2(?0
+ e,, &I.
(33)
Substituting for Tlmac)k
on the r.h.s. of eq. (32) we see that in the first (of the two terms)
and a
term there will be a factor gzmAoll which by (21) can be replaced by g~~a,ac)AO/l
factor gim,& can be replaced by g(,ac)im,k A/,/from the counterpart of eq. (24). Thus the
first term is unaffected if we replace the g's by the g&,ac)'s. As for the second term the
factor gZmgLm,kcan be replaced by g$ac) g('(mac)lm,kfrom (26). Thus we finally have
397
M. N. MAEANTA,Dualistic Approach t o Gravitation
in the approximation used
(T{mac)k (-g(mac))1'2),i
x (-g(mac))'/')
==
-
1/2 g f z a c ) g(mac)jm,k(Tfmac)l
0,
(34)
or
T[mac)k;j = 0 ,
(35)
where the tensor g(mac)ii
is used in the covariant differentiation. Thus the transition to
as the new metric tensor
the macroscopic or Einstein system is complete with g(mac)ij
the macroscopic energy momentum tensor.
and Ti(ma,:)k
Appendix I
The fact that under the Einstein assumption the Ricci tensor for the conformally flat
metric (9:i has a triple space-like eigenvalue Po can be surmised in the following way:
From the expressions of the Ricci tensor for the metric (9) it can be seen that the
eigenvalues can be functions of H and the following differential invariants only :
R
(ii) A,H
(iii) A,H
G
giiHiij= 0 H
+ "€1A
H
'
= giigklH,iH,kH;il.
Now iin AIH and A,H the space derivatives appear symmetrically and there are no
mixed derivatives, this cannot be said of d,H. But we can show that under the Einstein
approximation A H , m 0 showing that there is a triple space-like root el and a simple
root eo which is time-like since its eigenvector H,i is assumed to be time-like.
Proof:
1
w -H,jH,i
H
-
in our approximation.
Also
+w
1
2
1
( L ~ , H-) ~ (LI,H)~
(ZlH)2= 0 .
gifgkzH,iH,LH,,L H
H
Of course this proof is not straight forward. But it can be verified directly from the
expression of the Ricci tensor [ 3 ] that indeed under the Einstein approximation the
representation (14) is valid. This amply justifies the underlying assumptions for the
same.
Thus
Appendix 1.1
The r.h.s. of eq. (14) already represents a. perfect fluid distribution in an approximately conformally flat S,and so the class of the metric has to be one in the same approximation according to a result by Gupta [4] .We may prove it as follows:
398
Ann. Physik Leipzig 45 (1988) 6
I n our case, we have
giving
p2= X d l H
e
(A)
+
and ec2 3p = 0,
8nG
and p being the density and pressure respectively.
Thus in Gupta’s notation
b27. . - p(-~o/Jo/j
Sii)
+
7
with
With a little calculation i t can now be shown that in the Einstein approximation
bii;k - b i k ; i
= 0,
which is the required class one condition. Also
u.2 =
.- 201%
- (d1H)--1/2 H ,i 7
and
showing U iM
~ I Q , where
~ ,
y~ is a scalar, proving the second part of Gupta’s result
A c k n o w l e d g e m e n t s . The author would like to express his acknowledgement
and thanks to the Indian Institute of Advanced Study, Shimla, for the grant of a fellowship and facilities for the work reported.
References
[l] MAHANTA,M. K . : Ann. Physik (Leipz.) 41 (1984) 357 and the references therein.
[‘L] EINSTEIN,A.: Phys. Z. 14 (1913) 1249.
[3] EISENIIART,
L. P. : Riemannian Geometry. Princeton N. J. 1966.
[4] GIJPTA,
Y. K.: Ind. J. Pure and App. Math. 7 (1976) 190.
Bei der Redaktion eingegangen am 31. Marz 1986.
Korrespondenz an: Dr. Y. K. GUPTA
Dept. of Mathematics
University of Roorkee
Roorkee-24 7 66 7, India
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