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Weyl integrable space-time: A model of our cosmos?
Volume 98A, number 1,2
PHYSICS LETTERS
3 October 1983
WEYL INTEGRABLE SPACE—TIME: A MODEL OF OUR COSMOS? M. NOVELLO and H. HEINTZMANN’ Can tro Brasilefro de Fesquisas Fisicas, CBPF/CNPq, Rua Xavier Sigaud, 150, 2290-Rio de Janeiro,
RJ,
Brasil
Received 26 April 1983 Revised manuscript received 29 July 1983
We introduce a Weyl integrable space—time as a consequence of a dynamical process. We study the particular case of a vector field non-minimally coupled to gravity. The consequences for cosmology are considered. We find that it is possible to connect one riemannian region of space—time with another riemannian region of space—time by a series of Weyl integrable domains.
It is well known that Einstein’s field equations are if one varies derivable from a lagrangian .12E the metricg~.However, it is also possible to start from an arbitraiy affine geometry and, using Palatini’s method, varying independently the metricg~and the affme object ~ we obtain ~
g,~~ ~ R
—
=
~
—
=
O~
— —
L
+ E
—
.
(3)
‘
(ib)
performed a la Palatini. In this case g~p;~=g~~ ~ where A2 =A A~
“
~
—[(A2)~/A2]g~, (4)
—
Here A~= (A
-‘
If LM contains any kind of fields coupled minimally to gravity variation ‘a la Palatini implies the same relation (la) and consequently we assume the riemannian structure of space—time by means of a prescribed dynamic process. Elsewhere [1] we have already pursued the idea of a modified are lagrangian, electromagnetism and gravitation coupledwhere non-minimally. The lagrangian 1
J?~~4A~ k1R]
(la)
L M
—
We have shown there that an interesting (eternal) universe without singularity is a possible solution of the equations which follow from the variation of ala Einstein. Here we pursue the idea that the variation is to be
repectively. The advantage of this procedure is to introduce the dependence of the structure of the geometry on dynamics. In general, for the full lagrangian of geometry and matter,we have that L is a simple sum L
we considered reads L = .~/T~[_ !f fPP
0,A 1 ,A2 ,A3) is an electromagnetic four-vector potential and in the riemannian case we found that “longitudinal” photons (A00,A~=0) are capable of curving space—time, unifying thereby electromagnetism and gravity [2]. In the Palatini case we obtain a Weyl integrable space—time (WIST), since eq. (4) reads now (5)
=
X ~2), which
On leave of absence from Institut tiir Theoretische Physik
is a necessary and sufficient with 0: = ln(A condition for a WIST. This WIST is conformally riemannian, i.e., we can write ds~IsT= ~22(x)~ (6)
der Universit~tK~1n,Cologne, Fed. Rep. Germany. Partially supported by IBM do Brasil.
Besides eq. (4) the remaining set of equations obtain-
10
~‘~“
?~
.
0.031-9163/83/0000—0000/S 03.00 © 1983 North-Holland
Volume 98A, number 1,2
PHYSICS LETTERS
3 October 1983
(7 f fly ~a in which double bar means covariant derivative in the riemannian part of the connection.
tional clocks are(7b) related in different places by A2.conLooking at eq. we see that the gravitational stant is renormalized in our new theory and this is exactly the reason why non-singular cosmological solutions arise in this theory.
Fk1 + A21.1 GP11
We note,regimes furthermore, thatthe it is possibleremains to obtain asymptotic in which normA2
ed from (3) are RAL — —
\
—
=
—E~1V
—
RA ~ A 11
—
‘
(7b)
in which E~
11—=f~f~1, + ~gf~f°~ and is the energy —momentum tensor for the matter (minimally coupled to gravity). We note that in order for2the = conspace—time to be riemannian we must have A stant. This case has been considered before in a completely different context (and independently of each other) by Dirac [3] and Nambu [4,5]. Dirac suggested a new classical theory of the electron. The main argument rests on the fact that breaking the gauge symmetry of electrodynamics enables us to use the superfluous variables to describe electric charges. Set A,~A!~A~= Defining Vj.L = f3~A~ we define a velocity field. Dirac argues then that its physical significance is nothing but the velocity which any charged particle flow has ifit is introduced in such an electromagnetic field, Nambu examined the quantum case and arrived at the conclusion that A2 = constant implies a spontaneous breakdown of Lorentz invariance, which then could help in the interpretation of the photon as a Goldstone boson, being the massless excitations associated to the break of symmetry. Both authors found that the requirement for consistency with experiments leads us to set j3 = m/e. Our equations of motion (which follow from T~11 = 0) modify Dirac—Nambu in only one essential point: it introduces a cosmological rest mass for the photon. Therefore our theory agrees completely with standard electrodynamics as far as laboratory experiments are concerned. We note further that our theory incorporates one essential aspect of ~me recent work of Canuto and co.workers: atomic clocks and gravita.
constant. In these regions the structure of space—time is riemannian. Thus it is possible that through a se quence of Weyl integrable regions of space—times riemannian domain of space—time becomes relateda asymptotically to a different riemannian domain. This is along the lines of some recent work of Zel’dovich [6] , Guth [7] and others who also tried to avoid the Big Bang singularity by introducing a sufficiently large cosmological “constant” A which however is not constant as it is thought to be zero or very small today (and initially represented the pseudo energy—momentum tensor of fluctuations of empty space—time).We arrive here at a somewhat different scheme by enlarging the mathematical structure of space—time. Already the most trivial solution of our set of field equations contains a free function, which can be used to make the “age” of our universe arbitrarily large and provides thereby possibly a viable model of a cosmos of biological age. Further study of the subject is in progress. -
References [1] M. Novello and
H. Heintzmann, Preprint CBPF-NF-101983, submitted for publication. [2] M. Novello andNature J.M. Salim, Phys. Rev. D20 (1979) [3] P.A.M. Dirac, 168 (1951) 906;Proc. R. Soc.377. A209 (1952) 292 [4] Y. Nanbu, Suppl. Progr. Theor. Phys. Extra (1968) p. 190. [5] R. Righi and G. Venturi, mt. J. Theor. Phys. 21(1982) 63.
[61 Ya.
Zeldovich, Soy. Phys. Usp. 24 (1981) 3,216. [7] A.H. Guth, Phys. Rev. D23 (1981) 347.
11
PHYSICS LETTERS
3 October 1983
WEYL INTEGRABLE SPACE—TIME: A MODEL OF OUR COSMOS? M. NOVELLO and H. HEINTZMANN’ Can tro Brasilefro de Fesquisas Fisicas, CBPF/CNPq, Rua Xavier Sigaud, 150, 2290-Rio de Janeiro,
RJ,
Brasil
Received 26 April 1983 Revised manuscript received 29 July 1983
We introduce a Weyl integrable space—time as a consequence of a dynamical process. We study the particular case of a vector field non-minimally coupled to gravity. The consequences for cosmology are considered. We find that it is possible to connect one riemannian region of space—time with another riemannian region of space—time by a series of Weyl integrable domains.
It is well known that Einstein’s field equations are if one varies derivable from a lagrangian .12E the metricg~.However, it is also possible to start from an arbitraiy affine geometry and, using Palatini’s method, varying independently the metricg~and the affme object ~ we obtain ~
g,~~ ~ R
—
=
~
—
=
O~
— —
L
+ E
—
.
(3)
‘
(ib)
performed a la Palatini. In this case g~p;~=g~~ ~ where A2 =A A~
“
~
—[(A2)~/A2]g~, (4)
—
Here A~= (A
-‘
If LM contains any kind of fields coupled minimally to gravity variation ‘a la Palatini implies the same relation (la) and consequently we assume the riemannian structure of space—time by means of a prescribed dynamic process. Elsewhere [1] we have already pursued the idea of a modified are lagrangian, electromagnetism and gravitation coupledwhere non-minimally. The lagrangian 1
J?~~4A~ k1R]
(la)
L M
—
We have shown there that an interesting (eternal) universe without singularity is a possible solution of the equations which follow from the variation of ala Einstein. Here we pursue the idea that the variation is to be
repectively. The advantage of this procedure is to introduce the dependence of the structure of the geometry on dynamics. In general, for the full lagrangian of geometry and matter,we have that L is a simple sum L
we considered reads L = .~/T~[_ !f fPP
0,A 1 ,A2 ,A3) is an electromagnetic four-vector potential and in the riemannian case we found that “longitudinal” photons (A00,A~=0) are capable of curving space—time, unifying thereby electromagnetism and gravity [2]. In the Palatini case we obtain a Weyl integrable space—time (WIST), since eq. (4) reads now (5)
=
X ~2), which
On leave of absence from Institut tiir Theoretische Physik
is a necessary and sufficient with 0: = ln(A condition for a WIST. This WIST is conformally riemannian, i.e., we can write ds~IsT= ~22(x)~ (6)
der Universit~tK~1n,Cologne, Fed. Rep. Germany. Partially supported by IBM do Brasil.
Besides eq. (4) the remaining set of equations obtain-
10
~‘~“
?~
.
0.031-9163/83/0000—0000/S 03.00 © 1983 North-Holland
Volume 98A, number 1,2
PHYSICS LETTERS
3 October 1983
(7 f fly ~a in which double bar means covariant derivative in the riemannian part of the connection.
tional clocks are(7b) related in different places by A2.conLooking at eq. we see that the gravitational stant is renormalized in our new theory and this is exactly the reason why non-singular cosmological solutions arise in this theory.
Fk1 + A21.1 GP11
We note,regimes furthermore, thatthe it is possibleremains to obtain asymptotic in which normA2
ed from (3) are RAL — —
\
—
=
—E~1V
—
RA ~ A 11
—
‘
(7b)
in which E~
11—=f~f~1, + ~gf~f°~ and is the energy —momentum tensor for the matter (minimally coupled to gravity). We note that in order for2the = conspace—time to be riemannian we must have A stant. This case has been considered before in a completely different context (and independently of each other) by Dirac [3] and Nambu [4,5]. Dirac suggested a new classical theory of the electron. The main argument rests on the fact that breaking the gauge symmetry of electrodynamics enables us to use the superfluous variables to describe electric charges. Set A,~A!~A~= Defining Vj.L = f3~A~ we define a velocity field. Dirac argues then that its physical significance is nothing but the velocity which any charged particle flow has ifit is introduced in such an electromagnetic field, Nambu examined the quantum case and arrived at the conclusion that A2 = constant implies a spontaneous breakdown of Lorentz invariance, which then could help in the interpretation of the photon as a Goldstone boson, being the massless excitations associated to the break of symmetry. Both authors found that the requirement for consistency with experiments leads us to set j3 = m/e. Our equations of motion (which follow from T~11 = 0) modify Dirac—Nambu in only one essential point: it introduces a cosmological rest mass for the photon. Therefore our theory agrees completely with standard electrodynamics as far as laboratory experiments are concerned. We note further that our theory incorporates one essential aspect of ~me recent work of Canuto and co.workers: atomic clocks and gravita.
constant. In these regions the structure of space—time is riemannian. Thus it is possible that through a se quence of Weyl integrable regions of space—times riemannian domain of space—time becomes relateda asymptotically to a different riemannian domain. This is along the lines of some recent work of Zel’dovich [6] , Guth [7] and others who also tried to avoid the Big Bang singularity by introducing a sufficiently large cosmological “constant” A which however is not constant as it is thought to be zero or very small today (and initially represented the pseudo energy—momentum tensor of fluctuations of empty space—time).We arrive here at a somewhat different scheme by enlarging the mathematical structure of space—time. Already the most trivial solution of our set of field equations contains a free function, which can be used to make the “age” of our universe arbitrarily large and provides thereby possibly a viable model of a cosmos of biological age. Further study of the subject is in progress. -
References [1] M. Novello and
H. Heintzmann, Preprint CBPF-NF-101983, submitted for publication. [2] M. Novello andNature J.M. Salim, Phys. Rev. D20 (1979) [3] P.A.M. Dirac, 168 (1951) 906;Proc. R. Soc.377. A209 (1952) 292 [4] Y. Nanbu, Suppl. Progr. Theor. Phys. Extra (1968) p. 190. [5] R. Righi and G. Venturi, mt. J. Theor. Phys. 21(1982) 63.
[61 Ya.
Zeldovich, Soy. Phys. Usp. 24 (1981) 3,216. [7] A.H. Guth, Phys. Rev. D23 (1981) 347.
11