close

Вход

Забыли?

вход по аккаунту

?

Cosmology and Description of Local Space-Time Properties by Einstein's Equations.

код для вставкиСкачать
Annalen der Physik, 8. Folge. Band 36, Heft 1, 1078, S. 50-60
J. A. Barth, 1,eiuzig
Cosmology and Description of Local Space-Time Properties by
Einstein's Equations
By
u. KasrErr
Zentralinqtitnt fiir Astropliysik der Akadrniie dcr 'il'issenschaften der DDR, Potsdam-Babclsberg
A b s t r a c t . Wc consider a theory in which the global andlocalspace-timepropertiesaredcscribed
hy different laws. Onc consequence of such a theory is that the only time-dependent cosmological
models are such that their honiogenroos and isotropic three-spaces are closed. In the framework of
this theory the local space-time propcrtics are approxiniately described bei Einstein's oquations, hut
with Einstein's gravitational coupling nnniber now being a function of the niatter density filling
the universe.
Kosinologie und Beschreibung lokaler Itanin-Zeit-Eigenschaftm
durch Einsteiiis Gleicliringeii
T n h a l t s i i h e r s i c ht. Wir betrachtcn einc Theorie, in dcr die lokalen und globltleii Eigcnschaften
der liaum-Zeit durch verscliicdenr Geset.ze bescliriebcn werden. Eine IConseqnenz der Tlieorie iut,
daU nur solchc zeitabhangigen Irosniologischen Modclle oxistieron, dcren homogene nnd isotropc
Raunischnitte gcsclilossen sind. Kin Rahnien dcr hier betracliteten Theorie werden die lokalen RaumZeit-Eigenschaften angcniihert dnrch die Einsteinschen Gleichungen beschrieben. Jedoch ist die
Einxteinsche gravitat.ive Iiopplicngsz~iIiIeinc Fnnktion dcr Dichte der hlaterie, die die Struktur dcs
Tiniversuins bcstininit..
Iiitroductiori arid Suirirnary
Mainly t ~ v opictures have )wen d e ~ e l o l ~ eahout
d
the role of space-tiiur in pliysicu [I].
According to the first point. of view s p a c t 4 1 n e is something ahsolute wit.11 it,s own geometrical properties not infliicnced by t~rattcras we know it. Such a space-time lias been
thought, to he a vaciiutii or filled with a certain substance (called pleniiin by llescartes
and ether iti the
CeTitLirr). According t.o the second point of view space-tinir is
nothing absolute hut. determined by niattcr filling t.he universe.
Hert. we wniit to start fro111the first point of view, but we do not i n t m d to take i t
too literally. On the coiitrat,y, we woitld like t o foriiiulate onr point of view in t.he following way: F ~ > o i iai c e r t a i n p o i n t of c o s m o l o g i c a l d e v e l o p n i e r i t o n w a r d s i t
s e e m s t o b c p o s s i b l e t o descri1)e t h e g l o b a l a n d l o c a l s p a c e - t i m e p r o p e r t i c s
b y apparently d i f f e r e n t laws. Froiii t h i s c u t b e t w e e n global s t r u c t u r t : of
t h e u n i v e r s e a n d i t s l o c a l p r o p e r t i c s ina.y a r i s e t h e i t u p r c s s i o n t h a t . t ticre
is g i v e n a b a s i c geoiiic'trical st r u c t lire e s s e n t i a l l y i n d e p e n d e n t of loc.iLl
pliysics. 13ut t h i s b a s i c s t r u c t u r e w i l l h e i n f l u e n c e d h y l o c p l n i a t t e r d i s t r i b u t i o n s . 3 l o r e o v r r , w e will s u p p o s e t h a t t h e l a r g e s c a l e s t r u c t i i r c of t h e
u n i v e r s e i s Iiothilig al)solrite b u t d e t c r i i i i n e d b y a c e r t a i n s u b s t a n c e ( p l e n u I N , e t her).
Cosmology by Einstein’s Equations
51
The question, whether the kind of matter we know has developed itself froni this
substance, is not considered here. Our point of view is that if this process has really
taken place, it was unimportant for the global structure of the universe. That means we
suppose that the large scale structure of the universe is described in good approximation
if we start from a homogeneous and isotropic distribution of that substance and take
into account the possibility that its difitribution could have becn different in the early
phase of cosmological devclopenient.
If we consider the substancc which determines the global structure of the universe
to be completely different from the matter we know, then we can regard the theory
discussed in the following as a phenomenological description of a univcrsc with n matter
field wc do not know. Jn connection with the problem of variability of the gravitational
coupling number such theories have been discussed in the past 1.21.
In the following we shall call the substance which determines the global structure
of the universe plenum t o ease thc way of spcnking. Wc shall discuss two possibilities
for its energy momentum tensor. l n one case it is that of an incoherent fluid in the other
case that of radiation.
Once again, we want to point out that the plenuni shall deterniine the global structure of the universe, by which we mean that it shall not he possible t o say anything
ahout tht. metricul structure of the universe in case the mcrgy inonicntum tensor of the
plenuiii is equal to zero. As it is well-known, this is not the case in Einstein’s theory of
gravitation. It seems the reason for that is that Einstein’s theory is constructed like a
special-relativistic theory : The components of the metrical tensor arc introduced as new
dyniuiiical variables and one has to add to the Lagrange density a term describing the
free gravitational field. Therefore, one cannot be surprised. for instancc, that Einstein’s
theory gives cosmological models without any matter.
,\ceording to thc theory considered here it shall not be possible to make any statements about the largc scale structure of the universe, if the energy momentum tensor
of the plenum is allowed a t all to equal zero. The simple8t variational principle satisfying this requirement is
6 J @(L)R f = d 4 x
= 0.
(1)
Here. L is the J,agrange function of the plenum (incoherent fluid or radiation) and @
shall be such a functional of L that
d2@
@(L= 0) = 0, (d@/dh)L=o finite, - finite.
(2)
d L2
J n the following sonic consequences of the field eqnations obtained froiii (1) shall be
cnuiiierated for the casc that thc metrical propertics of the universe can be described b y
the Kohrrtson Walker metric
We have considered the ansatd)
@ ( L ):
L.
The field equations (22) together with ( 3 ) and (4) givc the following results in
(4)
case of a n
incolierent fluid plenum :
1.1. There exists a solution K =- const, 1,== const but thc three-spaces must be
fiat ( E = 0).
1.3. If we suppose that K and L arc not constant, we obtain the remarlrable result
that the three-spaces are spherically curved ( E == 4- I ) .
2)
4‘
A more general an8at.z is considered in the appendix.
52
U. KASPEK
1.3. Up to a constant of integration, the expansion factor K ( t ) is given by
(5)
This means the plenum can not he considered an incoherent fluid denunl. if it is extremely condensed (cosmologicalsingularity). From ( 5 ) we get the Hihble parameter If:
and the acceleration parameter y
q: = -
KK"
= 0.
K'2
(7)
This seems to be in the range of possible values [:3].
I n case we describe the plenum by radiation we can take over analogously what we
have said in 1.1. But also in the case of radiation only spherically curved three-spaces
( E = + I ) are possible. The expansion factor K ( t ) follows a similar law:
1
K(1) = v t .
v3
Even if we knew t.he exact value of the Huhble parameter and the acceleration parameter
we could not, decide in this way whether t,he plenum is an incoherent fluid 01'a rsdiat.ion
plenum.
NOW,I would like t.0remark on the singularity problem. As has been mentioned, cosmologicd models considcrcd up to now have a singularity at t = 0 (lim e = 00). It. is
t -to
not clear, whether cosmological models with more general three-spaces must, also have
such a singularity. The proof t.hat t,tiis is the case in Einstein's t.he?ry is mainly t m e d on
the fact that the expression
R,at$m19L
(9)
is a definite one for niat.ter with the usual properties [4]. But it seems that the ficld
equations following from (1) do not allow easily t o decide whether expression (9) is ct
definite one in this case, t.oo. Furthermore, we must refer to another point. Singularity
theorems we know from the study of Einstein's theory tell us soniet.hingabout geodesical
conipleteness of the considcred space-time. This kind of coniplct,enessis iniportant for
Einstein's theory because incoherent matter moves on geodesics. But, according to the
theory considered here, this is not the case if the curvctture scalar R is space-dependent
(three-spaces which are not homogeneous). Then it is possible t.hat geodesical completeness
looses its physical meaning.
The local geometrical propert,ies of spice-time shall be determined by the following
variational principle
6 J vG (L,R 4-AL,) d% = 0 .
(10)
LAfis the Lagrange function of the local matter distribution. The Lagrange function
of the plenum is denoted by Llc.r l shall be a constant the dimension of which is that of
an inverse squared length. This is one possible ansatz and is perhaps connected with
the assumption that certain microscopic quantities are not cosmologically constant (See
below!). The local metrical tensor gInn shall be the sum of the cosmological metrical
tensor gmn and a tensor h,,, which is determined hy the local inatt,er distribution:
K
Ymn
Ynw
R
+ hnm*
(1')
Cosmology by Einstein’s Equations
53
We obtain Z
, from the respective special-relativistic Lagrange function by introducing
gmn and its derivatives and asking that L, is a scalar with respect to general coordinate
transformations. The curvature scalar R in (1 0) shall also be formed using gmn a
q given
by (1 1).
Local geometrical properties arc characterized by taking the cosmological metric gnm
E
and the density L K of the plenum as constant. The tensor h,, describes the deviation
from thc locally flat background metric gmn. The field equations describing the field h,,
E
follow from (10) by variation of h,,,,t. They are Einstein’s equations up to the fact that
Einstein’s gravitational coupling number x is determined by the Lagrange function
(rest mass density) of the plenum and the inverse squared length A :
x=
A
LK
--,
(12)
If we put LK m
g/cm3, then A : =
must be -10-” cni-2 or I m 10% cm.
That is the length belonging to the Hulnble time. In connection with the discussion of
Mach’s principle a relation similar to (12) always arises [5].
The numerical value of L, used above is suggested by astronomical observations
of the matter which can more or less directly act upon our instruments. But if there is
not any connection between the plenum and the matter concentrated in galaxies or
background radiation we could start froni a completely different value of ZK or A .
Choosing a certain value of L K or A means giving a measure for what is to be considered
as a local phenomenon. For instance, if we put L, m
g/cm3 Einstein’s equations
are valid in regions of length much smaller than lo2*cin and in time intervals much smaller
than 1018sec. Starting from a density value of the plenum much smaller than
q/cni3
the “age of the universe” would be very much greater than the Hubble time and we
could arrange it that the so-called metagalaxy is to be considered a local phenomenon.
I n particular, we point out that Einstein’s gravitational coupling number x is a
rising function for decreasing values of the plenum rest mass density or rising age of
universe. There holds the relation
x‘
=3
x
t-l
:(- =
3ni-1 if we start from
@ = L”
or, if we put K ’ / K equal to the present value of the Hubble parameter II,,
Reccnt analysis of planetary radar-ranging data has put a limit on such a cosmological
variation: x ’ / x < 4 10-10 (years)-l [ 6 ] . The result that x is a rising function of the age
of universe and thc expansion factor K a linear function of t is in accordance with that
obtained by TREDERin connection with his considerations of Mach’s principle 171. On
the other hand, the BRANSDICKEJORDAN
theory [2] predicts that x is a decreasing
function of the age of universe.
To finish this section we would like to remark on the question, whether masses
very much greater than the mas8 of thc sun have to collaps. As i t is well-known, according to Einstein’s theory there do not exist any spherically symmetric, static, and stabil
matter distributions with “usual” properties and inasses niuch greater than thc mass of
the sun. Such matter distributions would have to collaps. This is a result of Einstein’s
theory. But if masses greater than that of the sun were to be considered as cosmological
objects we could not niake any statements about their cosmological development in the
framework of Einstein’s equations bccause these equations would only give a correct
description of local phenomena.
a
64
LT. KASPER
Field Equations Describing tho Large Scale Structure of tho Kniverso
First let us collect the conventions used throughout the following. The curvature
tcnsor is defined by
r;m,l
- I;:,,,, + rtrnr3- r,lr:m.
R ; ~=
~~:
(14)
w e put
Rk.: = RZ,,,.
(15)
The T,agrange function L of the plenum shall be that of a fluid the pressure of which is
equal to zero in one case (incoherent fluid plenum) and ,oj3 in the other case (radiation
plenum). We have to put
L = -Q
(1 6)
where 2 is the rest inass density of thc plcnuni. The energy morncnt,um tensor of thc
plenum is defined by
and reads in our case
Tmn
=
-(e -+ p ) umun - wmn.
Units are used such t,hat c2 =
have
u,&m = - 3..
(1.8)
of
the
iiietric
tensor
is
-+2,
so
that.
we
1. The signature
(I!))
The method to obtain the energy momentum tensor (18) from (17) is prescnted in [S].
Here, we point out that the conserved density Q* used in [8] is identical to the rest
mass density Q of the incoherent flujd plenum. Therefore, the relation
(eu");, = 0
holds.
The Euler Lagrange equations
following from the variational principle (1) can be written as
These equations do not tcll us anything about the metrical structure of the universe
in case that the density of the plenum equals zero and the fiinction @ has the propertiesz)
@ ( L 0 ) = 0,
=;;
ELo
fjnite,
d2@
(m)L=o
finite.
There is a certain similarity between the field equations ( 2 2 ) and those ones of the
BRANS
I ~ I C K E theory. The diffwence comes from the fact that we do not suppose @
to be an independent scalar field but it function of the plenum rest mass density. Therefore, we obtain a term from varying @ by gmn. Moreover, there is no term describing R
free scalar field. The third expression in (22) is obtained from the first one of eq. (21) like
that of the Brans 1)icke theory, too.
2)
Only then we can put L = 0.
_-
Cosmology by Einstein’s Equations
;)3
If W F suppose the field equations ( 2 2 ) are satisfied. their comriant divergence gives,
if we put SP = L ,
y;t;jl
4- yl:tlR,ll
+ LR,,,, = 0
(23)
*
.\c~or.clingto (18), the energy nionicntuiu tmsor of
a11
incmhercnt fluid plenuin is
rnl= - - - n ~ , , , t ~ ~ ~
I
(24)
1)
and t l w e hold the relations
R(pL”’);sa
=; 0 ,
(25)
+
IZQU,,;,U” - ?(&:,
u,,~u”)
R,,,
== 0.
We recognize that the world lines deteriiiiticd by
(26)
itre only then gcodejics, if the projcc.tion of R,),on tlic 1oc:il t hree-spece orthogonal to u,,,
is equal to zero.
Cosmological Models
Wc base our considerations on the Robertson Walker metric
(28)
which inca,ns we suppose that the three-spaces arc lic~iiiog~~neous
and isotropic. The nonvanishing components of t hc lticci tensor are then
(29)
(30)
Taking into accoiint the rclation
oK3 = const
following from cq. (20) in case that we
tions reduce to
K4
- = 0,
K
(31)
have an incolierent
fliiid p l c n ~ n i the
, field equa(32)
if we put
@(I,) -- L .
Onc solntion3) of (32), (33) is
K = const, g = const’,
c: := 0.
--
(34)
(35)
(36)
3) Can one be sure that this is not the solution of the cosmological problem? How strongly may
wo believe that a gravitational field determines the large scale structure of the unircrse?
U. KASPER
66
This is an understandable rcsiilt. The distrihution of the plenuni sliould not depend on
tinre if the plenuni is not thc sourve of a gravitational field.
If K is n function of t with K' $ 0 eqs. (32), (38) are only identical for
&=I
and the expansion factor K ( t ) is given. up to a constant of integration, by
(37)
The most remarkable featurc of this result iu that the curvature of the three-spaces 8
can only he positive, which mcans the three-spaces are closed. Such a result has oftcn
heen required in connection with the discussion of Mach's principle "31.
I,et us now consider thc case of a radiation plenuni. Froin t,he continuity equation
satisfied by the conserved matter density c* [S] the relation
Q K=
~ const
(39;
follows now. The field equnt ions reduce to
Kt
= 0,
K
if we put
@ ( L )= L .
Again, one possible solution is
K
&
= copst,
=;
Q
= const',
0.
For a frinction K(1) with K'
& =
1
(42)
(43)
+ 0, the two eqs. (40), (41) are identical, if
(44)
and the expansion factor is, up to a constant of integration.
1
K ( t ) = -t .
1/5
(46)
Even if the plenuni is supposed to be a radiat.ion plenum. only closed three-spaces are
possible.
ltemorks on thc Singiiltarity Problem
An essential supposition of the proofs given in [4] that certain space-times are
desically incomplete, is that
ice-
lla,VaV" 5 0
(46)
for all non-spacelike vectors V a (Here. the <_ sign is a consequence of our definition of
the curvature tensor and the chosen signaturc). In Einstein's theory, (46) is identical
to the requirenicnt
@
+ 3 p 2 0.
(47)
Cosmology by Einstoin’s Equations
63
if we take the four velocity un as non-spacelike vector in (46). But, starting from the field
equations (22) we obtain in the case of an incoherent fluid plenum, for inst,ance,
for a normalized ( V m V m = -1) tirnclike vector VTP.
As we have already pointed out above, for the theory considered here the yuestion
whrither a space-time is coniplete with respect to non-geodesics seems to be niore important than the question whether a space-time is geodesically complete, because the
“particles” of an incoherent fluid plenum can move on non-geodesical world lines in
cosinological models with inore general geoirictrical structures of the three-spaces. (See
qs. (25). (as)!)
Einstein’s Equations
In the first section of this paper we started from the variational principle (1)
I/G
6$
RLK d4x = 0.
(In the following we shall write LK instead of L for the Lagrange function of the plenum.) This principle determines the global metrical structure of the universe and the
distribution of the plenum. The next assumption was that the local metiiaal properties
of space-time and the physical processes are described by the variational principle (10)
I/G
+
6 j(EL,
AL,) a42 = 0 .
All the symbols used here have been explained above.
Perhaps, it would have becn more suggestive starting from the variational principle
(10). We outline this way here. The reason why we have chosen the other way above,
is that one of the main points of this paper is the opinion that there is a certain cut
between the global and local structure of space-time and we wanted to stress this point
from the beginning. The largc scale structure of the universe and local space-time properties are described by different field equations. It is not possible to obtain the field
equations ( 2 2 ) for the global structure of the universe by averaging the field equations
for local spacctime properties following from (10).
Einstein’s theory starts from the variational principle
and was applied originally to weak gravitational fields, that means to fields which can
be considered small perturbations of the flat Minkowski metric
gmn = Vinn
+ hmn*
Such weak gravitational fields can also naturally incorporated into the existing formalism of quantum field theory.
But shortly after he had published his equations Einstein himself applied these equations to describe the large scale structure of the universe. With this extension to strong
gravitational fields arose a lot of well-known problems and there remains the question,
whethcr ths extension is correct or not. Against the point of view that one can apply
Einstcin’s equations to strong gravitational fields, one can argue in the following way:
The dimension of x-
.
[x-l] = g/cm = (g/cm3) cm2 = mass
. (1ength)z
(50)
(density)
is equal to the dimension of the product of t i mass density and a squared length. If wo
put this mass density equal to
g/cm3 suggested by astronomical observations as
the mean density of the universe and use as a length that which one obtains 1)yv niultiplying the Hubble time by c (velocity of light), that means 10%cm, we get thc value of
Einstein’s gravitational coupling number ic. This suggests to write
6 J / f S ( L x R4-AL,,{) d4x =: 0 ([A1 = (lcngth)-2)
(51)
instead of (49) which inturn suggests that (51) can only describe local phenoniena. for
which the Lagrange function of the matter filling the universe and A are to be considered
constant. Here, the question arises of how to describe local phenomena on the cosniic
time scale. One possibility were, as mentioned by TREDER,
to choose the Rosen interpretation of Einstein’s theory but with it Robertson Wallter metric as background
instead of the flat Minkowski metric. Above we have assunred that A i s a constmt on
the eosniological time scale. This is only one possibility. Its meaning seems to be thc
assuniption that a t least one of the quantities: the masses of elementary pitrticlcs,
Planck’s “constant” h, and the “vacuum” velocity of light should depend on the age o f
the universe! As we know since Eddington’s and DTRAC’S
[lo] discussion of the connection between cosmological and quantum numbers the relation
x-1 M
cSm:K
(m:= mass of the proton)
h2
(52)
holds arid we infer from (12) and (31) for the case of an incoherent fluid plenum
&I
de--
K4
h2
c3mg
(33)
so that A = const means that at least one of the quantities h, c, tnP should dcpcnd on the
age of universe. On the other hand. if no one of them has this property, then A has to be
a function of the age of universe.
The step to the: starting point of this paper is the assumption that L, can be neglectcd
if one wants to describe the large scale structure of the universe, which then follows
from
6 $ 1 / q L x I l d 4 x = 0.
(54)
Several authors, among them JORDAN,BRAXS
and DICKE,and BERGMANS
121, have
considered the possibility that the gravitational coupling number is variable and supposed it to be a function of a certain scalar field. If we assume that the substance (here callcd
plenum) which determines the global structure of the universe is completely differcnt
from the matter we know, one could interpret LK as the Lagrange function of it phcnomenological description of a scalar field. But the field theoretical description of this
scalar field would be different from that in [a).
A discussion on the subject of this paper with Prof. TREDER
is acknowledged hcn.
Appendix
Up to now, we have only considcred the ensatz
( A I)
@ = L.
From the point of view taken in the last section, this secnis to be the most natural one.
Here we start from the assuniptiori
@ = P.
(A 2)
Thcn, the field equations (22) reduce to
Kf2 E
(I.-3n)T+-==0,
( A 3)
K
K2
( 1 - 3 % ) 2 - +K‘2
(1
-:3?2)-+2(1
F
-:+n )-=o
K” ,
K2
K2
K
Cosmology by Einstein’s Equations
59
(incoherent fluid plenuni),
(1 - 4n)
(1 -
+ z2E
K2
K12
K’2
4n)2K2
=
( A a)
0,
-j- (1 - 4n)
8
K”
K
- -k 2 ( 1 - 112) -=
K2
0
(radiation plenum).
If we put n = 1/3 (n = 1/4) and, as a consequence, E = 0, in the case of an incoherent
fluid plenum (radiation plenum) there does not follow anything concerning the large
scale structure of the universe and distribution of plenum. For 1 - 3n 0 (1 - 4n ;t. 0)
we get the rcsult that the expansion factor K ( t )is a linear function oft. Jts derivative is
given by
+
or
respectively. We see the three-spaces have positive curvature for
1/3 < n (incoherent fluid plenum)
(A 9)
or
I /4< n (radiation plenuni) ,
(A 10)
respectively.
The three-spaces have negative curvature for
1/3 > n (incoherent fluid plenum)
or
1/4 > n (radiation plenum),
respectively.
References
[l] R. H. DICKEin H.-Y. CHIUand W. F. HOFFMAXNN
(ed.): Gravitation and Relativity, Chapter 7 ,
W. A. Benjamin, Inc., New York-Amsterdam 1964.
[2] P. JORDAN,
Schwerkraft und Weltall, Vieweg u. Sohn, Braunschweig 1955;
C. H. BHAKSand €2. H. DICKE,
Phys. Rev. 124,925 (1961);
P. G. BERCIXAXK,
Int. J. Theor. Phys. 1, 25 (1968).
[3] S. WEINBERG,Gravitation and CoBmology, John Wiley and Sons, Inc., Xew York-LondonSydney-- Toronto 1972.
[4] S. W. HAWKIKCI
and G. P. R. ELLIS,
The b r g e Scale Structure of Space-Time, Cumbridgo
University Prose, Cambridge 1973.
[5] See, for oxample, [l].
[(;I I. I. Siurmo, W. B. SmTiI, M. E. Asn, R. P. ISGALLS
and G. H. PETTENGILL,
Phys. licv. Lett.
26, 2 i (1971).
60
U. KASPER
[7] H.-J. TREDER,Die Relativitiit der Triigheit, Akademie-Verlag, Berlin 1972.
[8] V. FOCK,
Theorie von Raurn, Zeit und Gravitation, Akademie-Verlag, Berlin 1960.
[9] H. HONLin H.-J. TREDER
(ed.): Entstehung, Entwicklung und Perepektiven der Einsteinschen
Gravitationstheorie, Akademie-Verlag, Berlin 1966.
[lo] P. A. M. DIRAC,
Proc. R. SOC. (Landon) 166 A, 199 (1938).
Bei der Redaktion eingegangen am 8. Juni 19iS.
Anschr. d. Verf.: Dr. U. KASPER
Zentralinst. f. Astrophysik d. AdW der DDR,
DDR-1502 Potsdam-Babelsberg,Rosa-Luxemburg-Str.17a
Документ
Категория
Без категории
Просмотров
1
Размер файла
557 Кб
Теги
local, properties, times, space, description, equations, einstein, cosmology
1/--страниц
Пожаловаться на содержимое документа