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 By D.-E. LIEBSCHER Zentralinstitut fur Astrophyik, Potsdam-Babelsberg W. YOURGRAU 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 mechanics. 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 ) ) (3) 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. 21 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 . 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 (6) The special Lagrangian introduced by WEBER, viz., a . 9=z- GmAmB ( 1 A.B TAB +& - +?) 9 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 . BARBOUR 151 and BARBOUR et BERTOTTI 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 2m 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. D.-E. LIEBSCHER and W. YOURQRAU 22 if only the particles A belong t o the cloud. We get for small velocities in the cloud where QB = QmA 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 \o, 0, 0, 0 / Spontaneous Breakdown of Symmetry and Induction of Inertia 23 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, (15) 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 r 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 . 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= withr=]r-r*I Jq andna=- 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 I D.-E. LIEBSCHER and W. YOURGRAU 24 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 . 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. References [l] H.-J. TREDER, Die Relativitlt der Triigheit, Akademie-Verlag, Berlin 1972. [?I B. RIEMANN, Schwere, Elektrizitiit und Magnetismus, 2. Aufl., Hannover 1880.  W. WEBER,Gesammelte Werke Bd. 3 und 4, Leipzig 1890 und 1892.  D.-E. LIEBSCHER, Gerlands Beitr. Geophys. 8&,3 (1973). Kuovo Cimento 26 B, 16 (1975).  J. B. BARBOUR, [GI J. B. BARBOUR and B. BERTOTTI, Nuovo Cimento 58 B, 1 (1977).  E. KREISELand D.-E. LIEBSCHER, 1973, unpublished.  R. H. DICKE,The Theoretical Significance of Experimenteal Relativity, Gordon and Breach, Kew York 1964.  C. M. WILL, Astrophys. J. 163, 611 (1971). [lo] U. KASPERand D.-E. LIEBSCHER, Astron. Nachr. 2!)6,11 (1974). [ll] K;NORDTVEDT and C. M. WILL,Astrophys. J. lii, 775 (1972).  D.-E. LIEBSCHER, to be published in Ann. Physik Leipz.  B. E. EICHINOER, J. Math. Phys. 18, 1417 (1977). \Bei der Redaktion eingegangen am 15. Juli 1978. Anschr. d. Verf.: Dr.sc. D.-E. LIEBSCHER Zentralinstitut fur Astrophysik DDR-1502 Potsdam-Babelsberg Rosa-Luxemburg-Str. 1 7 a Prof. Dr. W. YOURORAU 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 to EICHINGER . But this approach is different from ours, because he considers only projective space, nrhereas we focus on projective space-time.