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Classical Spontaneous Breakdown of Symmetry and Induction of Inertia.

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Annalen der Physik. 7. Folge, Band 36, Heft 1, 1979, S. 20-24
J. A. Barth, Leipzig
Classical Spontaneous Breakdown of Symmetry
and Induction of Inertia
Zentralinstitut fur Astrophyik, Potsdam-Babelsberg
Foundations of Physics, University of Denver, Denver, Colorado (U.S.A.)
Abstract. We consider the existing forms of inertia-free mechanics as attempts to reduce the
a priori properties of space-time and to search for the mechanism inducing the actual properties of
local space-time regions by the matter-distribution in the universe. We explicate the problem of the
induction of the local Lorentz structure of space-time and establish a first solution.
Klassische spontane Symmetriebrechung und Induktion der Trlgheit
Inhaltsubersicht. Wir betrachten die existierenden Ansiitze einer tragheitsfreien Mechanik
als Versuche, die a-priori-Eigemchaften der Raum-Zeit zu reduzieren und nach Mechanismen zu
suchen, durch die die tatsachlichen Eigenschaften der lokalen Raum-Zeit-Gebiete yon der MaterieVerteilung im Kosmos induziert werden. Wir arbeiten das Problem der Induktion der lokalen LorentzStruktur der Raum-Zeit heraus und begrunden eine erste Losung.
Newtonian mechanics contains absoluteness of rotation and acceleration. The inertia
of the bodies enters a priori just as their gravitational charge. The equivalence of both
masses is solved in Einstein’s theory by describing t,he gravitation by the inertial field
gik.Dual to GRT are the attempts t o formulate inertia as the a posteriori consequence of
gravitation, i.e. t o formulate inertia-free mechanics. To this end, the Galilei group of
classical mechanics is kinematically extended in the existing propositions of inertia-free
The Lagrangian of classical mechanics,
is invariant with respect to the group of Galilei transformations, viz.,
where v represents the relative velocity of the two systems of reference and the matrix A
their relative orientation1). I n the invariance transformations of the Lagrangian of
Mach-PoincarB gravodynamics [ 11
9 = - v (l r A - r B l ,
1 ;A
- ;BI, (?A
- kB) ( r A - r B ) )
1) The indices A, B enumerate the particles, rA,rBare the radius vector, and mA,mB the inertial
masses of the particles A and B.
Spontaneous Breakdown of Symmetry and Iinduction of Inertia
the parameters u and ro are kinematically extended such that
r,(t) may absorb the term u(t) t. TREDERconsiders especially the Lagrangian
where is a numerical constant, G the gravitational constant, and mA the active gravitational mass2). Eq. (5) leads to the isotropy of Induced ineztial masses. The expression
for the potential (5) was introduced by RIEMANN
.[2] in the context of non-relativistic
electrodynamics. I n contrast, a Lagraagian of the form
9 = - v ( l r ~ - rB\, (+A - i.g) (rA - rg))
is invariant with respect to the full kinematical group of Euclidean space
The special Lagrangian introduced by WEBER[3], viz.,
a .
GmAmB ( 1
where T A B = I rA - rBl and (x is a numerical constant, leads t o anisotropy of induced
inertial masses, but eliminates the bucket paradax of NEWTON
151 and
et BERTOTTI[6] treat Lagrangians homogeneous in +A, so that the invariance
is extended to general transformations of time :
Their special choice for the Lagrangian is
All these transformations, eqs. (2), (4), (7), and
(lo), treat
the time separately. Simul-
taneity of events is absolutely defined. Inertial masses, i.e. the term
A is, do not
A 2
exist a priori in eqs. (3), (5), (6), (8), (9), and (11).One may therefore cafthese equations
special representatives of an inertia-free mechanics. .
Inertial masses enter u posteriori by reducing the Lagrangian t o small subsystems.
Summation of the particles in the surrounding universe, that is, in the so-called Machian
cloud, induces effective inertial masses. To show the method of exact deduction, we
select a particularly simple case, viz., the Lagrangian eq. (5). We separate the described
particle-system into a small subsystem and the surrounding Machian cloud. Corresponding to the state of the actual universe, we suppose the Machian cloud to be in a n
isotropic state of motion. Looking for the Lagrangian of the subsystem, we sum now
the particles of the cloud. Summation in eq. (5) results then in a constant, if A and B
are both particles of the cloud, and in the effective masses
*) c is a normalizing velocity. If c is chosen as the velocity of light, /?is of the order of unity.
if only the particles A belong t o the cloud. We get for small velocities in the cloud
QB =
2is the gravitational potential of the Machian cloud a t the point of
A C ~ ~ A B
particle B of the subsystem.
The Lagrangian of the subsystem eq. (12) is no longer invariant with respect t o
eq. (4)) but only with respect t o the Galilei group, eq. (2). This restriction of the original
invariance group is due to the special state of motion of the Machian cloud. A t first
glance the assumed state of motion of the cloud does not allow any invariance group
of the space at all, by reason of the fact that isotropy of a finite cloud defines a center
and therefore also absolute position and velocity with respect to this fixed center. The
Mach-Poincarb symmetry, eq. (4))is however not broken down t o the time translation
group, but only to the Galilei group. Relative positions are not defined, because in the
interior ( r < R) of a isotropic cloud ( r > R)the Newt,onian potential of the cloud is
constant, and only this potential enters the Lagrangian of the subsystem. The same
holds for the velocity-dependent terms of eq. (5))such that absolute velocity is also not
defined. I n more general Lagrangians than eq. ( 5 ) ,the Mach-Poincarh symmetry would
be broken down to the time-translation group.
The spontaneous breakdown of the symmetry of the total system by the special
state of motion of the Machian cloud produces the inertial masses of the particles of the
subsystem, which were not present in the original Lagrangian eq. (5). The same procedure shown here for the special Lagrangian eq. ( 5 )may be applied t o the Lagrangians
(3), (6), and (9). Similarly, it generates effective inertial masses by the spontaneous breakdown of the symmetry of the whole system in the reduction t o the subsystem. This
breakdown always takes place, because the Machian cloud has to be in some special
state of motion. I n general, symmetry breaks down t o time translations. Only if one uses
rTi in the Lagrangian and a n isotropic cloud, is Galilei symmetry left unbroken. I n
t h s context, the Galilei symmetry indicates a n isotropic state of motion of the universe.
We have hitherto considered small velocities alone. Galilei invariance can therefore
be applied without restriction. We need however Lorentz invariance t o deal a t least
with non-gravitational fields. One arrives a t Lorentz invariance by rewriting the oneparticle equation of motion as an approximation of a geodesic equation in a formally
Riemannian space. The corresponding metric may serve now as the equivalence metric
t o formulate special relativity in the framework of the equivalence principle. Vice versa,
the energy-momentum tensor has now t o be substituted for the particle masses, which
produced the equivalence metric. By these considerations we are led to the following
unsutz [ 71 : The a priori space-time has the Galilean metric
and the Galilean Beltrami operator
0, 0, 0 /
Spontaneous Breakdown of Symmetry and Induction of Inertia
The invariance group of the Space-time is the kinematical group eq. (7). The components
of the equivalence metric are potentials satisfying equations invariant with respect t o
this group. We get the following general form of the equivalence metric, namely,
where p2 is a numerical constant and yik dxi d g the velocit,y-dependent kinematically
invariant term
Evidently, eq. (15) represents the only velocity-independent and kinematically invariant construction of an equivalence metric. We replace now the summation of mA
by an integration of Tik. To this end we use the fact that Tik transforms as exiXk. The
substitution in eq. (16) furnishes
xz - xa
The potential part cikf dxi dxk of the equivalence
metric may contain a general functional f of the density Q = T i k g i k .
Let us now consider a small test region of the space. I n this region the equivalence
metric in zeroth order is generated by the surrounding universe. The anisotropy of mass
is then the anisotropy of the space part of the equivalence metric in the a priori coordinates and may be transformed away by a new choice of local inertial coordinates [a].
The energy-momentum tensor in this local region only produces a first-order perturbation, which may be developed in theqost-Newtonian framework [9]. Since the functional
f is not yet determined, one obtains a large variety for the values of the post-Newtonian
parameters. However, one of them is fixed: It is the coefficient of the potential Wi=
Tiknkarising in gio, resp. the corresponding post-Galilean invariant combination
m2 of parameters. This combination describes the quotient of the propagation velocities
of light and gravitation [lo]:
Because of the instantaneity of gravitational interaction in our ansatz, oi2 = -1 always
holds. According t o NORDTVEDT
and W n L [ 111, this parameter may be estimated by earth
tide analysis to be 1 m2 < .03 in agreement with Einstein’s theory, where m2 = 0.
Our aim was t o search for possible reductions of a priori properties of space-time,
i.e. for a larger invariance group, which reduces t o the actual space-time invariance group
b y regarding the universe as a “symmetry-breaking vacuum state” for the local subsystems. It seems that the main problem is t o direct this symmetry-breaking to the
Lorentz group instead of to the Galilei group. As long as the latter takes place, the resulting bi-metric ansatz terminates in m2 = -1. The kinematical extensions of the Galilei
group considered in eqs. (4),(7), and (lo), do not contain the Lorentz group and therefore
cannot be broken down t o this group. We have to search for other invariance groups
containing the Lorentz group to construct Machian models of space-time.
The conformal group seems to be too small an extension, because rotation is absolute
in conformally invariant theories just as in Newtonian mechanics. The general-relativistic
group of coordinate transformations is too large, since it leaves no structure to write a
mechanical Lagrangian before specifying a metric. But this metric should be the (‘output” of the theory instead of the “input”.
The most promising candidate is the projective group [12]. It does not define a priori
any distance (in contrast to eqs. (4),(7),and (lo)),nor orientation, and no time interval
either (just as cq. (10)). The actual motion of the universe determines a real projectively
infinite hypersphere - in the four-dimensional projective space-time - to which the
universe expands in the mean. This sphere remains invariant, when for the local subsystems all other syminetr ies break down. The subgroup d projective transformations,
leaving invariant an infinite hypersphere, is in fact just the Lorentz group, because scale
is defined by the mass of the universe. It is for this very reason that we believe the projective group to be the most promising candidate for a Machian group3). The mean of
the expansion velocity, weighted by the gravitational potential, would determine the
local light velocity. If the Machian cloud would not expand, it would define an infinite
point, and the resulting restricted group would be another group. I n this way the expansion of the universe is a necessary condition for the Lorentz invariance.
[l] H.-J. TREDER,
Die Relativitlt der Triigheit, Akademie-Verlag, Berlin 1972.
Schwere, Elektrizitiit und Magnetismus, 2. Aufl., Hannover 1880.
[3] W. WEBER,Gesammelte Werke Bd. 3 und 4, Leipzig 1890 und 1892.
Gerlands Beitr. Geophys. 8&,3 (1973).
Kuovo Cimento 26 B, 16 (1975).
[5] J. B. BARBOUR,
Nuovo Cimento 58 B, 1 (1977).
1973, unpublished.
[8] R. H. DICKE,The Theoretical Significance of Experimenteal Relativity, Gordon and Breach,
Kew York 1964.
[9] C. M. WILL, Astrophys. J. 163, 611 (1971).
Astron. Nachr. 2!)6,11 (1974).
and C. M. WILL,Astrophys. J. lii, 775 (1972).
to be published in Ann. Physik Leipz.
J. Math. Phys. 18, 1417 (1977).
\Bei der Redaktion eingegangen am 15. Juli 1978.
Anschr. d. Verf.: D.-E. LIEBSCHER
Zentralinstitut fur Astrophysik
DDR-1502 Potsdam-Babelsberg
Rosa-Luxemburg-Str. 1 7 a
Foundetions of Physics
University of Denver
Denver, Colorado 80210, U.S.A.
3) Another attempt to describe Lorentz invariance as a special outcome of projective space is due
[13]. But this approach is different from ours, because he considers only projective
space, nrhereas we focus on projective space-time.
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