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Non-trivial Solutions of the Bach Equation Exist.

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Annalen der Physik. 7. Folge, Band 41, Heft 6, 1984, S. 436-436
J. A. Barth, Leipzig
Non-trivial Solutions of the Bach Equation Exist
Zentralinstitut fur Astrophysik der AdW der DDR, Potsdam-Babelsberg
I n connection with fourth order gravitational field equations, cf. e.g. [l, 21 where
the breaking of conformal invariance was discussed, the original BACHequation,
B$1. . = 0 ,
enjoys current interest. Eq. (1)stems from a Lagrangian
and variation gives, cf. BACH[ 3 ] ,
An Einstein space,
is always a solution of the BACHequation (1).But eq. (1)is conformally invariant, and
therefore, each metric, which is conformally related to an Einstein space, fulfilles eq. (l),
too. We call such solutions trivial ones.
Now the question arises whether non-trivial solutions of the BACHequation (1)do
or do not exist, and the present note will give an affirmative answer. As a by-product,
some conditions will be given under which only trivial solutions exist. Observe that
eq. (1)is conformally invariant whereas eq. (2) is not. Therefore, a simple counting of
degrees of freedom does not suffice.
Because the full set of solutions of eq. (1) is not easy to describe, let us consider
some homogeneous cosmological models. Of course, we have to consider anisotropic
ones, because all Robertson-Walker models are trivial solutions of eq. (1). Here, we
concentrate on the diagonal Bianchi type I models
= dt2
- a?
with Hubble parameters hi = a i l daifdt, h = Zhi and anisotropy parameters mi =
hi - h/3. The Einstein spaces of this kind are described in [4], for 1 = 0 it is just the
Kasner metric ai = t)%,L'pi = Zp? = 1. All these solutions have the property that the
quotient of two anisotropy parameters, milmi, (which equals (3pi - 1)/(3pi- 1) for
the Kasner metric) is independent of t, and this property is a conformally invariant
one. Furthermore, it holds: A solution of eqs. (l),( 3 ) i s a trivial one, if and only if the
quotients milmi are constants.
Ann. Physik Leipzig 41 (1984) 6
Restricting now to axially symmetric Bianchi type I models, i.e., metric ( 3 ) with
h, = h,, the identity ,Emi = 0 implies ml/m2 = 1, m,/m, = m$m, = - 2 , i.e., each
axially symmetric Bianchi type I solutiim of eq. (1)is wnformally related to an Einstein
space. (Analogously, all static spherically symmetric solutions of eq. (1)are trivial ones,
cf. [ 5 ] . )
Finally, the existence of a solution of eqs. (l),(3) with a non-constant m1/.m2will be
shown. For the sake of simplicity we use the gauge condition h = 0, which is possible
because of the conformal invariance of eq. (1).Then the 00 component and the 11component of eq. (1) are sufficient t o determine the unknown functions h, and h,; h, =
-hl - h, follows from the gauge condition. Defining r = (h: h,h, + hg)l/a and
p = hl/r, eq. (1) is equivalent to the system
c = const.,
p2 5 413,
9(dp/dt)2f i = [2r d2r/dt2 - (&/at), - 4711 (4r2 - 3p2r2).
3d2(pr)/dt2= 8p13
As one can see, solutions with a non-constant p exist, i.e., ml/m2 is not constant for
this case.
Result. Each solution of the system (3), (4),(5) with dpfdt 0 represents a non-trivial
solution of the BACH
equation (1).
The author thanks Dr. G. DAUTCOURT
and Prof. Dr. H. TREDER
for stimulat,ing
[l] WEYL,H.: Raum, Zeit, Materie, 4. Aufl. Berlin: Springer-Verlag 1921.
H.; TREDER,H.-J.; Y o r ; ~ c ~ aw.:
u , Ann. Phys. hipzig 3b (1978) 471.
[3] BaoH, R.: Math. Zeitschr. 9 (1921) 110.
141 K R A M JD.,
~ , et al.: Exact solutions of Einstein’s field equations. Berlin: Verlag der Wissenschaften 1980, eq. (11.52).
R.: Rep. math. phys. 17 (1980) 15.
Bei der Redaktion eingegangen am 16. November 1984.
Anschr. d. Verf.: Dr. H.-J. SCHMIDT
Zentralinstitut fur Astrophysik
der AdW der DDR
DDR-1502 Potsdam-Babelsberg
R.-LuxemburgStr. 1 7 a
Verlag Johann Ambrosius Barth, DDR-7010 Leipzig, SalomonstraBe 18b; Ruf 70131.
Verlagsdirektor: K. WIECKE
Chefredakteure : Prof. Dr. Dr. h.c. mult. H.-J. TREDER,DDR-1502 Potsdam-Babelaberg,
Rosa-Luxemburg-Str. 17a
Prof. Dr. G. RICSTBR, DDR-1199 Berlin-Adlershof, Rudower Chaussee 6
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