odd old paper ideas

arXiv:gr-qc/0011064v1 20 Nov 2000What can Physics learn from Continuum Mechanics ? Alexander Unzicker Institute for Meteorology University of Munich Theresienstr. 37, D-80799 M¨ unchen, Germany e-mail: [email protected] November 20th, 2000 Dedicated to Prof. Ekkehart Kr¨ oner Abstract This paper is mostly a collection of ideas already published by various authors, some of them even a long time ago. Its intention is to bring the reader to know some rat her unknown papers of different fields that merit interest and to show some relations between them the au thor claims to have observed. In the first section, some comments on old unresolved problems in theore tical physics are collected. In the following, I shall explain what relation exists between Feynman graphs and the teleparallel theory of Einstein and Cartan in the late 1920s, and the relation of both to the theor ies of the incompressible aether around 1840. Reviewing these developments, we will have a look at the cont inuum theory of dislocations developed by Kr¨ oner in the 1950s and some techniques of differential geom etry and topology relevant for a modern description of defects in continous media. I will then illus trate some basic concepts of nonlinear continuum mechanics and discuss applications to the above theories. B y doing so, I hope to attract attention to the possible relevance of these facts for ‘fundamental’ physic s. Contents 1 Old unresolved problems in theoretical physics 2 2 Lorentz symmetries in elastic solids 3 2.1 Frank’s discovery . . . . . . . . . . . 3 2.2 MacCullagh’s theory . . . . . . . . . 4 3 Einsteins teleparallel theory 5 3.1 The early papers 1928 . . . . . . . . 5 3.2 Torsion in Riemannian geometry . . 5 3.3 The electromagnetic field . . . . . . 6 3.4 Dislocations - a tool to understand torsion . . . . . . . . . . . . . . . . . 7 3.5 Cartan and topology . . . . . . . . . 74 Topological defects 8 4.1 Quantum behaviour . . . . . . . . . 8 4.2 Homotopic classification of defects . 9 5 Nonlinear continuum mechanics and spacetime analogies 10 5.1 Basic concepts . . . . . . . . . . . . 10 5.2 Nonlinear extension of MacCullagh’s proposal . . . . . . . . . . . . . . . . 11 5.3 Topological defects as charges in MacCullagh’s theory . . . . . . . . . 11 5.4 Energy density in electromagnetism and continuum mechanics . . . . . . 14 5.5 Nonlinear theory of incompressibility 15 5.6 Waves . . . . . . . . . . . . . . . . . 16 6 Conclusions 16 1 1 Old unresolved problems in theoretical physics After the foundation of modern physics with its cornerstones quantum mechanics and general rel- ativity up to 1930, theoretical physics has devel- oped in a less revolutionary manner in the past decades. Richard Feynman mentioned in his No- bel lecture (1965) that he was driven by the hope tocalculate the rest energy of an electron - which is an experimentally well-known quantity of 0.511 MeV. Not much has happened yet towards the solu- tion of this problem. Another example where physi- cists seem to have surrendered regards the mass ratios mp/me= 1836 .15...,mn/me= 1838 .68..., mµ/me= 206 .768...of protons, neutrons, myons and others with respect to the electron. It seems like an evidence of incapacity for present day physics that the only attempts to calculate these numbers are some playing with powers of e′sandπ′swith- out any physical background.1The same develop- ment has taken place with the fine structure con- stant α= 137 .03597 .... Feynman commented: ‘It is one of the greatest damn myster- ies of physics. We know what kind of a dance to do experimentally to mea- sure this number very accurately, but we don’t know what kind of a dance to do on a computer to make this num- ber come out - without putting it in se- cretely ! ... all good theoretical physicists put this number on the wall and worry about it’.2 Today’s physicists seem to prefer announcing some great unification now and then instead. Even more remote to a solution is the problem3of the ratio of the electromagnetic and gravitational force, which is around 1040. In this sense, physics hasn’t moved closer towards a great unification since the specu- lations of Eddington (1929) and Dirac (1939). But should physicists disregard these problems forever ? Is there a reason why nature does not permit us to resolve these puzzles as a matter of principle, like the quadrature of the circle ? If this should be the 1How to play at least efficiently, one can read in Bailey and Ferguson (1989). 2Feynman 1985, chap. 4. 3This problem, which was already mentioned by Dirac (1939), was recently put in evidence by Weinberg (1999).case, physicists havn’t done their homework yet by proving their ‘ π’ to be transcendent. Quantum electrodynamics had a great success after having calculated the the magnetic moment 1.00115965 µBof the electron and the Lamb shift of about 1040 MHz. However, in view of the above un- solved questions4, does this justify to build a general theory of physics on renormalization ? Even Feyn- man himself was never convinced of the correctness of that theory: ‘It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.’ (Feynman 1985, p. 128). Dirac expressed himself more drastically: ‘This is just not sensible mathematics. Sensible mathe- matics involves neglecting a quantity when it turns out to be small - not neglecting it because it is in- finitely great and you do not want it.’5However, since the times of QED a kind of monoculture of physical theories have been developed on its concep- tual basis. Even Feynman commented self-ironically on the theories based on QED: ‘so when some fool physicist gives a lec- ture at UCLA in 1983 and says: ”This is the way it works, and look how won- derfully similar the theories are,” it’s not because nature is really similar; it’s be- cause the physicists have only been able to think of the same damn thing, over and over again.’ (Feynman 1985, chap. 4, p.149). Is it therefore not astonishing that general rela- tivity remained ‘off side’ from the ‘rest’ of theoreti- cal physics.6 In Ryder’s (1985) book on quantum field theory we can read: ‘the quantisation of the theory is beset by great problems.’... ‘in electrodynamics the field is 4As Feynman pointed out in his famous lectures (Feyn- man, R.B., and Sands 1963), the quantization of electrody- namics did not resolve the basic inconsistency of electrody - namics that predicts (with Coulomb’s law and the energy density of the field) an infinite energy for the electron. 5cited by Kaku (1993), p. 12. 6Interestingly, experimenters face an embarrassing uncer- tainty (1 ,5 10−3) arising from discrepant measurements of the value of the gravitational constant G. Recently, Vargas and Torr (1999) suspected a theoretical reason for this. 2 an actor on the spacetime stage, whereas in gravity the actor becomes the spacetime stage itself.’ Then the comment follows: ..‘In view of this, the particle physicist is justified in ignoring gravity - and be- cause of the difficulties mentioned above is happy to!’ This kind of reasoning seems to be searching a key in the shine of a lamppost because one can see there, even if the key had been lost elsewhere in the dark. In view of the unresolved problems physicists are rather justified to dig out and reconsider old ideas of excellent scientists - and because of the pleasure of reading their papers they should be happy to! 2 Lorentz symmetries in elas- tic solids 2.1 Frank’s discovery This section should guide the readers attention to a paper ‘On the Equations of Motion of Crystal Dis- locations’ of Frank (1949). The abstract follows: ‘It is shown that when a Burgers screw dislocation moves with velocity vit suf- fers a longitudinal contraction by the factor/radicalBig 1−v2 c2, where cis the veloc- ity of transverse sound. The total en- ergy of the moving dislocation is given by th formula E=E0/(1−v2 c2)1 2, where E0is the potential energy of the disloca- tion at rest. Taylor dislocations behave in a qualitatively similar manner, com- plicated by the fact that both longitu- dinal and transverse displacements and sound velocities are involved.’ A visualization of dislocations in crystals is given in fig. 2 in section 3.4. As has been pointed out by Frank, dislocations appear to have a particle- like behaviour. Their motion in a crystal close to the velocity of transverse sound is analogous to the mo- tion of a particle close to the speed of light. Is this just a coincidence ? At the first look, dislocations are a very special kind of defect. For deriving the above result, Frank considers the deformation field of a screw dislocation ux= 0, u y= 0 uz= (b/2π)arctany x,(1)which is, since∂2 ∂x2uz+∂2 ∂y2uz= 0 and divu= d dzuz= 0, a statical solution of pure shear type of the Navier equation −(λ+2µ)grad div u+µ curl curl u=ρ∂2u ∂t2(2) the displacement components u= (ux, uy, uz) have to satisfy. For the following, it is sufficient to con- sider only the last two terms. A dislocation propagating in x-direction with ve- locity vmust be represented by a time-independent function of x′,yandz, where x′=x−vt. With this substitution,∂2 ∂t2becomes v2∂2 ∂x′2, and the remaining terms of eqn. 2 read: µ(∂2 ∂y2+∂2 ∂z2) + (µ−v2ρ)∂2 ∂x′2= 0 (3) With the further substitution x′′=x′/radicalBigg 1−v2ρ µ= (x−vt)(1−v2 c2)−1 2,(4) where c=/radicalbig µ/ρis the velocity of transverse sound, the solution of a propagating dislocation has a form identical to the solution (1) of the dislocation at rest, apart from the substitution x→x′, a ‘Lorentz contraction’. As Weertman and Weertman (1979), p. 8, com- mented: ‘This distorsion is analogous to the contrac- tion and expansion of the electric field surrounding an electron.’ Frank went ahead and showed that the elastic energy of the moving screw dislocation in- creases with the factor (1 −v2 c2)−1 2. The question arises, whether this relativistic be- haviour7is a consequence of the special solution (1) or a more general effect. However, things would get more complicated only if the dilatational part divu 7Radiation damping. Interestingly, a phenomenon of radiation damping seems to occur in the dynamics of dislo- cations (Kosevic 1962; Kosevic 1979). That means, a part of the energy used in order to accelerate a dislocation is dissi - pated by the production of transversal sound waves. It shoul d be noted that radiation damping in classical physics is ever y- thing but well understood. As e.g. (Dirac 1938), eq. 24, Lan- dau and Lifshitz (1972), par. 75 or Feynman et al. (1963), chap. 28 point out, the Lorentz force /vectorFL=e/vector vx/vectorBis just an approximation for small values of dv/dt, and a general formula for the radiation emitted by an accelerated electro n does not exist. The quantization of electrodynamics didn’t resolve this problem either. 3 of the displacement in eqn. 2 does notvanish. If it vanishes instead, the above transformation x→x′′ can obviously be applied to every solution of (2). The condition of the vanishing dilatation can be for- mally realized by letting go λto infinity, which does physically mean that the medium is incompress- ible.8The reader who is interested in details may look how other authors like Eshelby (1949), Weert- man and Weertman (1979), G¨ unther (1988), and in a somewhat redundant way, G¨ unther (1996) have developed these analogies further, obtaining all fea- tures of special relativity including time dilatation etc. Thus, propagating solutions in an incompress- ible elastic continuum behave exactly as relativistic particles, if the speed of light is identified with the velocity of transverse sound. 2.2 MacCullagh’s theory Can these relativistic effects, apart from being a cu- riosity of elasticity theory, have a deeper meaning ? It does not seem so, because all attempts of describ- ing fundamental physics with continuum mechanics in the 19th century have been falsified by the fa- mous experiments by Michelson and Morley that seem to have disproved the concept of an aether. What’s wrong here ? The point is, the physicists of the 19th century imagined particles as made of an external substance distinct from the ‘aether’, which for some reason can pass through it without (or with infinitely little) friction even with infinitely great ve- locities. The aether theorists Young and Fresnel, Stokes, Navier, Cauchy, Lord Kelvin and Green never thought of particles as being topological defects cre- ating a displacement field - which is not astonishing, since the first examples of such defects, dislocations in solids, were discovered in 1934 by Taylor, almost 30 years after aether theories had disappeared from the stage of theoretical physics. In view of the results of Frank (1949) and others, however, one must say thatthe wrong concept was not describing spacetime as an elastic continuum but a wrong or missing pic- ture of particles moving in it . Therefore, theories of an incompressible aether like for example the one of MacCullagh (1839) do not contradict the the exper- iments of Michelson and Morley, if moving particles 8Of course, we are dealing with linear elasticity and have tacitly assumed small displacements. The nonlinear issues are discussed below.are assumed to be propagating solutions.9 Let’s have a look how MacCullagh10identified the elecromagnetic quantities with those of an in- compressible elastic aether: According to this theory, one may identify the electric11field/vectorEwith the curl of the displacement field∇xuand the magnetic field strength /vectorHwith its time derivativedu dt. Then, the Navier equation (2) reduces to ρ∂2u ∂t2=µ curl curl u (5) which is ( µis the shear modulus) equivalent to Maxwell’s µ0∂/vectorH ∂t=1 ǫ0curl/vectorE, (6) whereas div/vectorH= 0 follows directly from the in- compressibility condition divu= 0 which implies divdu dt= 0. By definition div curl u= 0 and curld dtu=d dtcurlu(7) holds, which correspond to Maxwell’s second pair of equations in vacuo . Whittaker (1951), p. 143, com- ments: It is evident from this equation (eqn. 5) that if divuis initially zero it will be always zero; we shall suppose this to be the case, so that no longitudianal waves exist at any time in the medium. One of the greatest difficulties which beset elastic-solid theories is thus completely removed.12 Before I discuss in section 5.3 the topological is- sues how charged particles may enter in this model, 9As has been pointed out by Dirac (1951), aether theories do not contradict quantum mechanics either, as long as the absolute velocity of the aether material appears as a nonmea - surable quantity. 10This was the result presented in 1839 to the Royal Irish Academy by MacCullagh and published in 1848 in Trans.Roy. Irish Acad. xxi, p.17. The interested reader is referred to the excellent review on aether theories by Whittaker (1951) , p. 142. ff; p. 280 11To be precise, the electrostatic induction /vectorD, which has to be divided by the dielectricity constant ǫ0to obtain /vectorE. 12MacCullagh assumed furthermore the elastic energy to be a function of curlu. As we shall see later, this additional assumption is not necessary. 4 I would like to point out the connection with a paper of Einstein in 1930, where he states: ‘(46), (47) ... correspond to Maxwell’s equations of empty space.’ Einstein does not mention MacCullagh, although according to historians (Kostro 1992) he had given up his rejection to ether already around 1920. But it is better to tell the story from the beginning: 3 Einsteins teleparallel theory 3.1 The early papers 1928 There is an amazing contrast between the public admiration for Einstein for having developed gen- eral relativity and the importance that is given to his later work in differential geometry.13More than once I happened to hear the statement that after 1920 Einstein had published just nonsense. Symp- tomatically, his work on teleparallel geometries has not been translated in English yet.14Of course, there is a reason for the disregard of Einstein’s work of the years after 1920: the conflict he had with quantum mechanics. His continuous objections, for example at the Solvay conference in 1927, could not unsettle the success of the new theory. On the con- trary: people were realizing more and more that quantum mechanics was a good physical theory be- cause it described the experiments, and got tired of the philosophical attacks launched by Einstein. In plain words: Einstein was a nuisance in the 1920s, and it is quite understandable that physicists were annoyed of the work that he proposed as an alterna- tive to quantum mechanics and called unified field theory. Thus, why should we deal with his cumber- some tensor calculus developed in a series of pa- pers15and follow all his attempts that at the end were discarded by Einstein himself ? A closer look at these papers, however, reveals that there is not only no contradiction to quantum mechanics on a con- ceptual level16, but there even arise some surprising facts from that geometry that remind us from the 13To avoid confusion, it should be mentioned that this the- ory distinguishes substantially from the so-called Einste in- Cartan-Sciama-Kibble (ECSK) theory. See Hehl, D.Kerlick, v.d.Heyde, and Nester (1976) for a review of several theorie s including torsion. 14A translation of some of his papers is availible under www.lrz.de/˜ aunzicker/ae1930.html . 15Einstein (1928b); Einstein (1928a); Einstein (1930). 16Of course, the formalism is quite different from that of quantum mechanics, as that of GR is.quantum behaviour of particles. I will discuss that in section 4. But let’s listen to Einstein (1928b) now: ‘Riemannian Geometry has led to a physical description of the gravitational field in the theory of general relativity, but it did not provide concepts that can be assigned to the electromagnetic field. Therefore, theoreticians aim to find nat- ural generalizations or extensions of Rie- mannian geometry that are richer of con- cepts, hoping to get to a logical construc- tion that unifies all physical field con- cepts under one single leading point.’ 3.2 Torsion in Riemannian geometry Einstein was convinced that the geometric descrip- tion of physics does not stop at the rather special case of Riemannian geometry.17In a later paper (Einstein 1930), he says: ‘To take into account the facts (...) grav- itation, we assume the existence of Rie- mannian metrics. But in nature we also have electromagnetic fields, which can- not be described by Riemannian metrics. The question arises: How can we add to our Riemannian spaces in a logically natural way an additional structure that provides all this with a uniform charac- ter ?’ In the following Einstein refers to an idea that Cartan had pointed out to him already in 1922 - and Einstein did not understand at that time-, the ‘Columbus connection’.18For Columbus, navigating straight meant going westwards. In terms of differ- ential geometry: parallel transport of vectors means keeping a fixed angle to the lines of constant lati- tude, whereas usually the straight lines on a sphere are defined as the great circles19(fig. 1). 17It should be mentioned that the notion of Riemannian ge- ometry seems to have changed. Einstein intended a geometry in which the connection was determined by the metric only, with the absence of torsion (see also Schouten 1954,Bilby et al. 1955). Modern texts like Nakahara (1995) instead re- quire just the existence of a Riemannian metric. 18Connection is the differential geometric entity that gov- erns the law of parallel transport of vectors. 19Therefore, it is necessary to distinguish between autopar- allels, on which vectors remain parallel, and extremals , that maximize the covered distance . 5 Figure 1: Visualization of the connection (vector transport rule) proposed by Cartan to Einstein. While transporting a vector (along the dotted line) the angle with the meridians is kept fixed. Thus, di- rections may be compared globally (Whenever we are speaking of ‘west’, ‘east’, ‘north’ and ‘south’, we are comparing directions globally !). If this is pos- sible, a teleparallel connection can be given to the manifold and the curvature tensor vanishes. Surprisingly, with this new connection the sphere has zero curvature but nonzero torsion.20If one looks at Fig.1, it becomes clear what Einstein said: ‘In every point there is a ... orthogonal n-bein21. (...) The orientation of this n-beins is not important in a Rieman- nian manifold. We assume, that these (...) spaces are governed by still another direction law. We assume, (...) it makes sense to speak of a parallel orientation 20One can imagine best the difference between curvature and torsion with differential forms. Both are 2-forms, that means quantities that have to be integrated over a 2-surface . If one transports a vector along a closed curve that bounds this surface, in the case of curvature it comes back rotated , and in the case of torsion shifted . Because this shift is done by a vector, torsion is called a vector -valued form, whereas curvature could be called a ‘rotation-valued’ form. Correc tly speaking, it is a Lie−algebra - valued form. If the vector be- comes just rotated (in the so-called metric-compatible cas e) the curvature form takes values in so(3), the Lie algebra of orthogonal rotations in three-dimensional space. For an in tro- duction to differential forms, see Flanders (1963) or Nakaha ra (1995). 21n-‘leg’, from German ‘bein’, means northogonal unit vec- tors.of all n-beins together (...).’ That was the idea that Einstein applied to space- time- describe the same physics with another differ- ential geometric entity. Instead of nonzero curvature and vanishing torsion he proposed vanishing curva- ture and nonzero torsion - from the example fig. 1 it should be clear that this does not change the ge- ometry of space. The advantage is that torsion in four dimensions has more components that curva- ture - that means one can pay the bill for describing gravity and hope that electromagnetism comes out of the additional components. The problem was not that Einstein did not find tensor identities that were equivalent to Maxwell’s equations, he actually found too many of them - and nobody knows which identity is the right one that represents Maxwells’ equations - if there is any. For several reasons (see, e.g. Unzicker 1996, section 2.7), the proposed field equations (Einstein 1930, eqn. 29 and 30) must be wrong.22 This does not imply, however, that the quantities he considered cannot have a reasonable meaning. 3.3 The electromagnetic field We shall stop here as well for a moment and in- vestigate what differential geometric quantities Ein- stein proposed for the electromagnetic field. In first approximation, he defines the electromagnetic field aaµin (Einstein 1930, eqn. 45) as aaµ=¯haµ−¯hµa, (8) the antisymmetric part of the vielbeins ¯hµa. The vielbeins h, as we shall see below, are nothing other than a generalization of the deformation gradient in continuum mechanics. In the case of a compatible deformation, the antisymmetric part defined in (8) is just the curl of the displacement vector u- the same quantity23that had been proposed by MacCullagh ! Thus, this part of Einstein’s proposal was a kind of recycling MacCullagh’s old idea - I don’t know 22In a letter to Salzer (1938, published in 1974), Einstein named as a reason for the failure of his teleparallel theory i ts representation of the electromagnetic field in first approxi ma- tion, which does not transform as a tensor. We shall touch this problem in section 5.3. 23Actually, it is not clear from Einstein’s paper whether he considered the aaµas the tensor of the electromagnetic field or its dual ( /vectorEand/vectorBinterchanged) - for Maxwell’s equations in empty space it makes no difference. 6 if he was aware of that and if he had liked it, if he were. It seems that at that time Einstein had given up denying the existence of an aether (Kostro 1992), but probably not because he was aware of that relation to MacCullagh’s theory. Einsteins theory, however, is in a sense more gen- eral than MacCullagh’s - Einstein’s continuum can- not be described by a compatible deformation gen- erated by a displacement field; this is a consequence of the nonvanishing torsion. We will see that there remains a close relation between Einstein and MacCullagh as well. For this, a little excursion is needed to understand what tor- sion means. 3.4 Dislocations - a tool to understand torsion Figure 2: Examples of an edge (left) and a screw (right) dislocations in a crystal. In 1952 Kondo revealed in an article of his won- derful review ‘RAAG memoirs - the unifying study of basic problems in physics and engeneering by means of geometry’ the relation between disloca- tions and torsion. He discovered that torsion could be identified with a density of dislocations piercing through a surface element. The various components of torsion can be visualized in the example Fig. 2, an edge dislocation (left) and a screw dislocation (right) in a crystal. Suppose direction 1, 2, 3 point to the right, backwards and up as indicated. Then in the left picture, after surrounding a surface ele- ment in the 1-3-plane one gets shifted in direction 1, therefore this gives a contribution to the T1 13com- ponent of the torsion tensor. In the right picture, after surrounding a surface element in the 1-2 plane the shift is in direction 3, therefore this contributes to the T3 12component. Note that the singularity line of the dislocation in the left case goes in direction 2and is perpendicular to the shift (Burgers vector), and in the right case parallel to the shift (both in direction 3). Torsion is just a continuous version of dislocation density, that means one lets the lattice spacing go to zero while maintaining the quantity shift per surface element. Bilby, Bullough, and Smith (1955) have observed this equivalence independently and Kr¨ oner (1959; 1960) made a beautiful theory out of it.24 Now, what can we learn from that ? On the one hand, that differential geometry with torsion is a good tool for describing dislocation behaviour in crystals. On the other hand, we are able to give a phys- ical interpretation to abstract geometries like those proposed by Einstein. In particular, there is no need to stick to the notion of a continuous torsion field. Spacetime could as well be endowed with a discrete torsion on a microscopic level that appears as dislo- cation density on the large scale. In this case it could be described by a compatible displacement field u which has, however, singularities. Dislocations can be seen as singularities with Dirac-delta-valued tor- sion, but not all singularities in an elastic solid need to be dislocations. In section 5.3 I will discuss a topological defect that carries torsion without be- ing a dislocation. 3.5 Cartan and topology To be fair, one must say that Einstein did exclude that possibility and postulated a priori singularity- free solutions. Cartan,25however, told him that postulating singularity-free solutions may create topological complications: ‘As far as singularity-free solutions are concerned, it seems to me, the question is extremely difficult. (...) It is quite pos- sible that the existence of singularity- free solutions imposes purely topological conditions on the continuum. (...) The 24Kr¨ oner was fascinated by the similarities of this geom- etry and wrote: ” We have seen that Riemannian geometry was too narrow to describe dislocations in crystals. Is ther e a reason why space–time has to be described by a connection that is less general than the general metric–compatible affin e connection ?” (Kr¨ oner 1960, par. 18) 25The interesting discussion between Einstein and Cartan is cited in the book by Debever (1979). 7 space in which the group exists, there- fore depends from the topological point of view , on the constants Λk ij(the tor- sion tensor), and every choice of the con- stants gives a space (or family of spaces) which is topologically defined. In short, every singularity-free solution of system (1),26creates from the topological point of view the continuum in which it ex- ists’. (letter to Einstein dated Jan 3rd, 1930) Unfortunately, Einstein was not very interested27in the topological issues that arise in geometries with nonvanishing torsion: ‘I cannot tell anything about the connec- tivity properties of space, but it seems unavoidable to demand singularity-free solutions...’ (letter to Cartan dated Jan 30th, 1930) Einstein’s theory, however (or, in general, theo- ries with torsion), allows an interpretation as geome- try with a density of singularities on the microscopic level. What do we gain with speculating about a dis- crete version of torsion and the interpretation as topological defects ? I consider this interesting be- cause it establishes a connection to a theory of physics that has been considered to be in blatant contradiction to Einstein’s unified field theory - quantum mechanics. We shall see this in the fol- lowing. 4 Topological defects 4.1 Quantum behaviour Consider a pair of edge dislocations (as shown in fig. 2 a) in a two-dimensional view fig. 3. It is clear that to every such defect exists an antidefect (in this case, with the Burger’s vector pointing in the opposite direction). If the two dislocations of oppo- site sign in the left and the right part of the picture start propagating towards each other, there will be 26The equation Λγ αβ;µ= 0, whereby Λγ αβis the torsion tensor. 27For a discussion of these topics, in particular the Einstein - Cartan correspondence see also the papers by Vargas (1991; 1997; 1999).Figure 3: Two edge dislocations of opposite sign in a two-dimensional view. Their existence cannot be deduced by counting lattice points on the bound- ary as in the case of one single dislocation. If the two dislocations move towards each other, they will annihilate in the center. an annihilation in the center. No topological irreg- ularity of the lattice will be maesurable, even if the elastic energy stored before will give rise to some lattice waves. If two dislocations as those in Fig. 3 move towards each other with a given velocity, it is even conceivable that the annihilation energy cre- ates two other defects - not necessarily of the same structure. Thus, sticking to the particle picture, the encounter could be even seen as a scattering pro- cess, or it could appear as if the two dislocations pass through each other without interacting. This doesn’t seem extraordinary at all, but has some noteworthy consequences if we compare the motion of these defects with the motion of classical particles. Fig. 4 a) shows the motion of a single dislocation propagating in x-direction from point PtoQ. The slope in the x−tdiagram is a measure of its velocity. Analogously, fig. 4 can be interpreted as a Feynman Diagram for an electron propagating from PtoQ (Feynman 1985, p. 99 and p. 125). The two signs of the dislocations correspond to the two signs of an electron and a positron; the latter one may be seen as an electron travelling backwards in time. If one measures only the events PandQ, be- sides the ‘direct path’ Fig. 4 (a) the scenario (b) is possible as well: while propagating, the dislocation encounters its antidefect created by a spontaneous pair creation process and cancels out, whereas the other ‘half’ of the pair, identical to the original de- fect, continues propagating. Of course, there may be many other scenarios 8 P PQ Q AC t tx x a) b) Figure 4: The propagation of dislocations is analo- gous to propagating electrons. When measuring the events in PandQ, there is no method to detect whether a defect propagating from PtoQgoes a ‘di- rect’ path (a) or has a creation–annihilation process plugged in between (b). The ‘Feynman diagrams’ (a) and (b) are indistinguishable. with the same experimental outcome corresponding to the various Feynman graphs (see Feynman 1985, p. 125). Obviously, it doesn’t make sense to assign an ‘identity’ to this kind of ‘particles’. Once two defects have the same structure, they are identical. This kind of behaviour is well-known in quantum me- chanics. Since particles are indistinguishable, they have to be described by Bose-Einstein or Fermi- Dirac statistics rather than by the classical Maxwell distribution. Furthermore, it is clear that it makes no sense to speak about a ‘trajectory’ of the dislocation. This reminds us from the result of the double slit experi- ment that tells us that it makes no sense to say the electron passed through the one slit or the other. Hence, dislocations behave not only relativisti- cally, but also as quantum mechanical particles. Ein- stein, who introduced a geometry that describes dis- locations may have been closer to the discovery of the puzzling quantum behaviour as he liked. 4.2 Homotopic classification of de- fects In the above sections we have seen some examples of topological defects. For a precise definition of topological defects and for their classification, ho-motopy theory is needed. It has been applied first by Figure 5: Topological defect in a nematic liquid crys- tal corresponding to the nontrivial element of the first homotopy group of the order parameter group RP2. The circle represents a path in RP2that can- not be contracted. Toulouse and Kleman (1976) to defects in ordered media, good review articles are Rogula (1976), Mi- neev (1980), Michel (1980), Dzyaloshinskii (1980), Monastyrsky (1993) and Nakahara (1995), chap. 4.8. Ordered media have a so-called order parameter, in the example of fig. 5 the orientation of the bars in a nematic liquid crystal28. Since this orientation is described by a director , the corresponding order pa- rameter space is the projective plane RP2, which describes all possible orientations of these directors in space. In three dimensions, line defects (like the example fig. 5) can be detected by surrounding them by closed lines (homeomorphic to the circle S1) and point defects by surfaces (homeomorphic to the sphere S2). A topological defect occurs if the line or surface, mapped to the order parameter space, can- not be contracted to identity. Consequently, the first homotopy group consists of noncontractible loops in the order parameter space, and the second homo- topy group to noncontractible closed surfaces. Non- trivial examples of the third homotopy group are more difficult to visualize, a quite comprehensive ex- ample (the Shankar monopole) is given by Nakahara (1995). The dislocations discussed above are line de- fects. The order parameter is the position of the 3- bein. Since after surrounding a defect this 3-bein is never rotated, the homotopy group consists of the 28A fluid consisting of little rods. 9 possible shifts (by the discrete amount of the Burg- ers vector) in 3d-space, which is Z Z×Z Z×Z Z. Later I shall investigate the homotopy groups of SO(3), the rotations in three-dimensional space. 5 Nonlinear continuum me- chanics and spacetime analo- gies 5.1 Basic concepts It has been known for a long time that the governing equations (2) of elasticity theory in the incompress- ible case lead to Maxwell’s equations in empty space. As far as elasticity is considered, these equations are no more than a rather crude approximation - the classical linear theory that applies to small defor- mations. If one wants to push further the analogies between an elastic continuum and spacetime, there is no physical reason to assume the deformations to be small, in particular in the neighbourhood of topological defects which - in the view of section 2 - should take the part of elementary particles. There- fore, there is a quite natural option to generalize electrodynamics to a nonlinear theory29- just see how the real, nonlinear physics of elastic continua works. Deformation gradient. Nonlinear elasticity goes back to the work of Cauchy in (1827). One assumes an undeformed, euclidean continuum with Cartesian coordinates X= (X, Y, Z ) (the so-called ‘reference configuration’) and attaches in every point a displacement vector uthat points to the coordinates x= (x, y, z ) of the deformed state (‘configuration’): u=x−X.30 Fromuone deduces the basic quantity in contin- uum mechanics that transforms the coordinates X of the undeformed state to those xof the deformed state: the so-called deformation gradient F:=∂x ∂X, 29We should not forget the inconsistencies of classical linea r electrodynamics mentioned above. Various attempts to mod- ify electrodynamics are described in Feynman et al. (1963), chap. 28.5. 30I use the common notion, e.g. in Truesdell and Toupin (1960), par. 15 ff., Truesdell and Noll (1965), or, in a more modern and didactical introduction, Beatty (1987).or, in components, F:= 1 +uX uY uZ vX 1 +vY vZ wX vY 1 +wZ ,(9) where ( u, v, w ) denote the components of uand the subscripts differentiation. It was Cauchy’s merit to discover the importance of the symmetrical tensor B:=FFT, (10) which is now called rightCauchy-Green tensor.31 Polar decomposition. In tensor calculus it is a very common operation to split tensors in a sym- metric and skew-symmetric part. In continuum me- chanics, however, it would not make much sense to split the deformation gradient tensor in that way, because one could not assign the physical meaning of a deformation to the results any more. The above splitting would be a linear operation based on addi- tion of matrices. However, if a material undergoes a deformation F1and then a successive deformation F2, one has tomultiply the matrices F1andF2to obtain the result F12- which is a noncommuative operation. Only for the small deformations (a case which is frequently assumed in linear elasticity) one can add (F1−E) and ( F2−E) (Eis the unit matrix) and get an approximate result F12−E. Fortunately, mathematicians have developed the right way to split Fmultiplicatatively- it’s the polar decomposition32of F=RU=VR, where UandV33are positive, symmetric matrices andRis a rotation matrix, that means an element of the special orthogonal group SO(3). Of course, a product in the above equation means matrix mul- tiplication, and VandUare in general different because of their noncommutativity. 31C:=FTFis called the left Cauchy-Green tensor. 32Fhas to be nonsingular. 33This decomposition is unique. While one can find a proof of this theorem in many books, it is rarely told how to do it in practice. Surprisingly, there is no way to express the coe ffi- cients of RandUby the fijin general. Since the solution of this problem involves the zeros of the characteristic polyn om of the matrix, the general formula of Cardano makes the solution inhibitively complicated even for computer algeb ra systems. 10 Differential geometry. Even if there is very lit- tle overlap between the languages of books on elas- ticity and books on differential geometry34, one should be aware of the similarity of some concepts. The deformation gradient Fcorresponds to the basis 1-forms ϑi(cfr. Nakahara 1995, section 7.8). The dif- ference is just that the ϑimaintain their meaning as quantities that transform coordinates, even if they cannot be deduced from a displacement field uany more. Since gµν=ϑi µϑi ν,35is equivalent to eqn. (10), the Cauchy tensor Bacquires the meaning of a met- ric. The so-called compatibility conditions which are necessary for the existence of a displacement field u(see Vargas and Torr 1993 or Unzicker 1996), are expressed in differential geometry by the vanishing of both the curvature and the torsion tensor. Since the Cauchy tensor Band the stress tensor Tare both symmetric and coaxial, the meaning of a metric can be assigned to the latter tensor, too. An interesting comment on the duality of these tensors was given by Kr¨ oner (1986). Despite of the elegance of differential forms I do not know, however, how to express the above polar decomposition essential for nonlinear elasticity properly in that language. 5.2 Nonlinear extension of MacCul- lagh’s proposal Before going ahead, let’s keep track of the quantities MacCullagh and Einstein had proposed for the elec- tromagnetic field. The components of MacCullagh’s curluor ∂iuk−∂kui are just the skew-symmetric part of the deforma- tion gradient tensor F(see eqn. 9). The same holds for the the antisymmetric part of the vielbeins ¯hµa (eqn. 8) Einstein proposed for the electromagnetic field. Einstein was aware that this could be an approx- imation only, but he did not put in question the process of splitting the hµa’s in a symmetric and skewsymmetric part. From MacCullagh’s point of view, however, it would have been a rather natu- ral option to pass from linear elasticity to the more general nonlinear equations. Thus, if one wants to develop the analogy be- tween spacetime and an elastic continuum as con- 34An exception is Marsden and Hughes (1983). 35see also Einstein 1928b, eq. (3), Einstein 1930, eq. (7).sequent as possible, one should apply to the viel- beins hµaor the deformation gradient Fthe polar decomposition theorem and identify the electric field with the rotational part R, which takes values in SO(3).36 According to MacCullagh, the magnetic field corresponds to the velocity of the aether elements (Whittaker 1951, pp. 142, 280). Taking this into ac- count, in the proposed nonlinear extension, the elec- tromagnetic field takes values in SO0(3,1), the con- nected component of the Lorentz group37. Since this does not create further topological complications, I will sometimes restrict the discussion to SO(3) in the following. As mentioned above, MacCullagh’s theory did not allow charges because of the vector analysis rule div curl = 0, applied to the displacement field u. The nonlinear extension of MacCullagh’s idea will help to overcome this difficulty by introducing a topological defect that acts as a ‘source’ of the ro- tations. 5.3 Topological defects as charges in MacCullagh’s theory Circular edge disclination. If we assume space- time locally to be described by the deformation gradient F, by the polar decomposition theorem F=R Ufollows that a unique field Rcan be as- signed to every point of spacetime.38In the language of the homotopic description of topological defects, one may regard R, which takes values in SO(3), as an order parameter field and ask about possible de- fects. Since the first homotopy group π1(SO(3)) is Z Z2, the group with two elements, mathematics pre- dicts the existence of a line defect. The defect is then a closed singularity line, because defect lines cannot end inside the medium. In an elastic solid, this defect can be realized as follows: One cuts the continuum along a (circular) sur- face, twists the two faces against each other by the amount of 2 πand rejoins them again by gluing (Un- zicker 1996).39 36For small fields, this is equivalent to MacCullagh’s pro- posal curlu. 37The velocity of the aether material, according to special relativity, can be seen as a pseudorotation in the x−t-plane. 38If this is the case, a teleparallel connection can be defined, see also Truesdell and Noll (1965), par. 34. 39The deformation gradient takes then infinite values at 11 Figure 6: Scematic description how to produce the topological defect 1 ∈π1(SO(3)) = Z Z2in an elas- tic continuum. The solid torus is removed. Then the material is cut along the hatched surface. Af- ter twisting the cut faces by the amount of 2 π, the material is rejoined. The dotted line represents a closed path in SO(3) which is not contractible. To obtain a line defect, the solid torus can be shrunk to a singularity line. Note that after the cut, the same material elements are rejoined. An alternative way to describe the same process is (see fig. 6): Imagine R I3filled with elastic material and remove a solid torus centered at the origin and withzas symmety axis (fig. 6). Then the comple- ment40is double connected due to the ‘bridge’ along thez-axis. One cuts now the material along the sur- face bounded by the inner circle of the torus in the x−y-plane (hatched surface in fig. 6). Now the cut faces may be twisted, for example clockwise the face of the positive z-direction and counterclockwise the face of the negative one, and glued together again. If each of the twists amounts to π, the material ele- ments meet their old neighbours again, so to speak, since the total twisting angle is 2 π. To come back to the above description, now let the removed torus become infinitely thin. A defect of this kind has been investigated first by Huang and Mura (1970), who called it edge the boundary of this circular surface. 40For topologists, the complement is a solid torus as well, ifR I3is compactified by adding the infinitely distant point. Two solid tori, merged in the described way, form the three- dimensional sphere S3.disclination loop. Physically, it is more similar to a screw dislocation , even if closed loops of this kind do not exist (Unzicker 1996). Mura considered gen- eral twisting angles (Frank vector’s), whereas for topological reasons the twisting angle of 2 πis the most interesting.41To come back to homotopy the- ory, immagine now a closed path in the complement (the dotted line in fig. 6) that surrounds the re- moved torus , e.g. the singularity line. A twist of 2 π inSO(3) is equivalent to identity, thus the path is closed in SO(3) as well. Since it is not contractible, the defect corresponds to the nontrivial element 1 of the first homotopy group Z Z2. This topological description of the continuum defect has been given first by Rogula (1976). Gaussian surface integral. We shall see now that the defect described above can be seen as an electrical charge in the nonlinear extension of Mac- Cullagh’s theory. Except at the singularity line, a continous field of the deformation gradient Fis given everywhere. Consequently, also the field Rob- taind by polar decomposition is continous, since in the region of the cut-and-glue surface the clockwise and the counterclockwise twist by the amount of π are the same element of SO(3). As the electrical field /vectorE, one could regard an element of SO(3) as having a ‘direction’ ( ϑ, ϕ) (a point on the two-dimensional sphere S2which de- termines the rotation axis), and a ‘length’ r, the rotation angle42(0≤r≤πand a direction in 3D- space 0 ≤ϑ≤π; 0≤ϕ≤2π)43. However, even if this object has direction and length, it is not re- ally a vector, since ‘vectors’ with opposite directions and length πare the same object.44This detail of replacing curluby/vector rwill avoid the consequence/integraltext/integraltext curlud f= 0 and make charges possible. Consider a closed surface fthat contains just the circular singularity line, like the surface shown in fig. 8. Then, in analogy to Gauss’s theorem, consider 41Any twisting angle which is not a multiple of 2 πwould destroy the distant parallelism, i.e. the vielbeins could n ot be defined any more. Accordingly, disclination density is usua lly described by a nonvanishing curvature tensor. 42this is sometimes called an ‘axial vector’. 43This is called a representation of an abstract group like SO(3). Another representation uses Euler angles (Love 1927). 44It should further be noted that this ‘vector’ has no‘com- ponents’, since due to the noncommutativity of SO(3) the superposition principle does not hold. 12 Figure 7: Example of a closed surface (for a better visualization, a sector is removed) surrounding the singularity line (dotted) generated by the shrinking of the torus in fig. 6. The Gaussian integral over this surface fis considered in the following. the integral ρ=/integraldisplay/integraldisplay /vector r d f, (11) where /vector rdenotes the ‘vector’ in the above representa- tion of SO(3). If we shrink the surface fin fig. 7 to a minimal size in which it contains just the singular- ity line, /vector rpoints upwards for z >0 and downwards forz <0. If one integrates now as if/vector rwould be a vector, the integral value is 2 πtimes the area of the circle. Note that the field /vector rwould be discontinous if it were a real vector field, but is continous, if its val- ues are correctly interpreted as elements of SO(3). However, the nonzero value ρof the integral appears to be a charge if /vector ris as usual identified with the electrical field /vectorE. This charge is a consequence of the topological properties of the group of rotations in three-dimensional space. Hence, if MacCullagh’s idea is consequently extended by means of nonlin- ear continuum mechanics, electrically charged ‘par- ticles’, i.e. topological defects become possible. Quantization and screwsense. It is further- more remarkable that any closed surface (like fin fig. 7) may contain only integer values of these de- fects, in accordance with the observed quantization of the electrical charge. Figure 8: Visualization of the Gaussian integral ρ=/integraltext /integraltext /vector rd f(The surface fin fig. 7 is shrunk to a minimal size that includes the singularity). If ro- tations in 3D-space a represented by a ‘vector’ /vector r, the rotations in the neighbourhood of the defect in fig. 6 can be represented as the vectors attached at the surface that contains the singularity line. ρhas nonzero value due to he fact that the discontinous ( at z=0) ‘vector’ field /vector rrepresents a continous field of elements of SO(3). One may however raise the question how posi- tive and negative charges that occur in nature are distinguished in this model, since the above defect corresponds to the element 1 of the first homotopy group Z Z2. Abstractly, two defects of Z Z2would cancel out by the rule 1+1 = 0. It should be noted, however, that this defect is not described completely by the first homotopy group. Coiled line defects like the above one do influence also higher homotopy groups (as π2 is influenced by π1of the projective plane, see Naka- hara (1995) and Mineev (1980)). For example, the above considerations on the integral eqn. 11 would hold also for a defect called Shankar’s monopole (Shankar 1977), representing the the nontrivial el- ement 1 of the third homotopy group π3(SO(3)), which is Z Z. Apart from this it is important to note that the above defect can be realized in twodifferent ways. Twisting clockwise for z >0 and counterclockwise forz <0 defines a screwsense, and if the twisting is done vice versa, the result is not the same defect but its mirror image. It is clear that a defect can be annihilated completely only by its mirror image, not by an identical defect. If two identical defects merge, the line singularity will disappear, but the result will be a nontrivial element of the third homotopy group Z Z.45 45The second homotopy group is trivial. 13 5.4 Energy density in electromag- netism and continuum mechanics Since SO(3) is compact, the electrical field could take finite values46only. This becomes interesting if one assigns an energy density to the electromag- netic field. Then, integrating the energy density in the neighbourhood of an electron would not yield infinite energies as in classical electrodynamics. If one respects however the analogy to nonlin- ear elasticity theory, the elastic energy47should not depend on the rotation R. Rather it can be demon- strated that it depends on the eigenvalues of the Cauchy tensor only - this can be deduced by frame indifference and material symmetry considerations (Beatty 1987). I have pointed out how MacCullagh’s proposal of an incompressible elastic medium (Whittaker 1951, p.142) leads in linear approximation to Maxwell’s equations. MacCullagh derived this by postulating the energy density function to depend on the rota- tion of the volume elements of the aether. This ad- ditional assumption, however, is not necessary since Maxwell’s equations already follow from the incom- pressibility condition. The question arises if the analogy between space- time and an elastic continuum can be developed as close as possible or if are we forced - contrarily to the theory of continuum mechanics - to assign an energy density that depends on the electric field, i.e. to the rotation of the volume elements. But isn’t the energy density w=1 2ǫ0(E2+c2B2) of the electromagnetic field an experimental fact that compels us to choose the latter option ? Since this formula, together with Coulomb’s law, leads to the inconsistency of a infinite self-energy of the elec- tron, it is interesting to see the comment given by Feynman on w: ‘In practice, there are infinitely many possibilities for wandS(Sis the Poynt- ing vector) and up to now nobody has thought about an experimental method that allows to say which is the right one! People think that the most sim- ple possibility is the right one, but we 46Given an appropriate norm on SO(3), like the absolute value rof the ‘vector’ mentioned above. 47We do consider only the case in which an energy density can be defined, which is called the case of hyperelasticity . Truesdell and Noll (1965) advocate the more general case.must admit that we don’t know for sure which one describes the correct localiza- tion of the electromagnetic field energy in space.’ (Feynman, R.B., and Sands 1963, chap. 27.4) In view of this, one should note that in a deformed elastic solid it is obviously impossible that in a bounded region there is a nontrivial48fieldRwith a trivial Cauchy tensor B49. Thus, the energy density one ‘observes’ for the electromagnetic field could be hidden in the deformation field, as elasticity theory says. Strain-energy function. Let’s review briefly how nonlinear continuum mechanics describes the localization of energy: It is convenient to introduce the so-called princi- pal invariants (I1, I2, I3) of a tensor that are defined as follows: I1=λ1+λ2+λ3(trace) (12) I2=λ1λ2+λ2λ3+λ3λ1 (13) I3=λ1λ2λ3(det) (14) (theλiare eigenvalues). Then, in general, the energy density Wis afunction of the principal invariants W=W(I1, I2, I3) of the Cauchy tensor, and the stress tensor50Tis given by51 T=β1B+β0E+β−1B−1(15) with the βi(I1, I2, I3) being functions of the princi- pal invariants of B. For small deformations, i.e. for values of the I’s close to 1, the β’s have fixed values - that can be related to the known elastic constants in linear elasticity.52Of course, this complications 48‘nontrivial’ means also nonconstant here. 49One would seek a theorem of matrix analysis that relates the global properties of these two fields; to my knowledge, it does not exist yet. 50Tis defined as traction per surface element. There are media with so-called microstrucure where Tis not symmetric any more. The interested reader is referred to the papers of Mindlin (1964) and Toupin (1964). 51This is called the constitutive equation for isotropic ma- terials. Note that there are no higher powers of B. This is a consequence of the theorem of Cayley-Hamilton, see f.e. Beatty (1987). 52As we see, nonlinear continuum mechanics has invented scalar-tensor-theories a long time ago. In Brans-Dicke the ory (1963), the gravitational ‘constant’ Gis a function of space- time, too. 14 need not to be realized in nature, but one should keep in mind the possibility that constants of na- ture are just the weak-field limit of field-dependent functions. In conclusion, the nonlinear extension of the spacetime - elastic continuum analogy has interest- ing consequences for the localization of energy in space. This has to be developed and clarified fur- ther, but may become a possibility to overcome the inconsistencies classical electrodynamics still has to face. Stopping these considerations at a necessarily in- complete stage, I’d like to mention the theoretical peculiarities that another element of MacCullagh’s theory has as consequence - incompressibility. 5.5 Nonlinear theory of incompress- ibility Notwithstanding its beauty, nonlinear elasticity has suffered from the fact that, due to its complexity, the calculation of concrete problems was inhibitively difficult. A big progress towards this direction was done in the late 1940s by Rivlin (1948), who - start- ing from rather practical problems given to him by the British rubber producer’s association he worked for - developed many results of theoretical impor- tance (Rivlin 1948). The nonlinear condition of incompressibility is given by I3=λ1λ2λ3=detB= (detF)2= 1.(16) andnot, as many texts on linear elasticity state, divu=0. As a consequence, the elastic energy W depends on the principal invariants I1andI2ofB only. Rivlin considered an energy density of the form W(I1, I2) =C(I1−3) +D(I2−3) (17) which is the definition of a Mooney-Rivlin material53 with the constants CandD. One should note that this kind of material is in a certain sense the most simple among the isotropic incompressible ones,54 53From incompressibility follows I3= 1. 54Of course, one may further assume C=D, a case which is called Neo-Hookean.but has still twoelastic constants,55whereas in the corresponding linear case only oneconstant, f.e. the shear modulus µdescribes the elastic properties of an isotropic incompressible material.56 Rivlin’s cube. Rivlin (1948; 1974) considered a cube of incompressible elastic material loaded uni- formely by three identical pairs of equal and oppo- sitely directed forces acting normally on its faces. The surprising result was (see, f.e. Beatty (1987), p. 1719 for a derivation) that besides the trivial so- lution λ1=λ2=λ3= 1 there are six (!) others, from which three are stable and three inherently unstable. Hence, there might be the possibility of different stable solutions for the same topological defects, too. Ericksen’s problem. Determining the deforma- tions arising from a given distribution of body forces and surface tractions for a material with arbitrary response functions βiis possible for very few sim- plified cases. Therefore the attention of the theo- reticians concentrated on deformations which arise from surface tractions alone. These defromations are called controllable and are also described by the equivalent condition of the vanishing divergence57 55It is not difficult to understand why incompressible non- linear elasticity needs more than one ‘shear modulus’. The reason is that squeezing and stretching are qualitatively d if- ferent deformations. Consider a cube made of an incompress- ible elastic material under pressure on the x−yfaces. It will shorten in x-direction and, in order to preserve its vol- ume, elongate in the x−andy- directions. In linear elas- ticity, the elastic energy depends just on the ratio of the length change in z-direction, which is the same for elonga- tion and shortening. For large deformations, f.e. squeezin g to the half of the height or doubling it by stretching, the nonvanishing components of the deformation gradient would befxx=λ1=fyy=λ2=√ 2, fzz=1 2(squeezing) or fxx=λ1=fyy=λ2=√ 2 2, fzz= 2 (stretching). The prin- cipal invariant I1(trace) of the Cauchy tensor B=FFTis therefore 2 + 21 4= 41 4in the first case and1 2+1 2+ 4 = 5 in the latter (vice versa for I2). Thus, nonlinear elasticity distin- guishes between stretching and squeezing, or, in other word s, elongation in one or two dimensions. 56In linear elasticity, the incompressibility condition is σ= 1 2(Poisson’s ratio), k=∞(compression modulus), λ=∞ (Lame’s constant) or Y= 3µ(Young’s modulus). See Feyn- man, R.B., and Sands (1963), chap. 38; Love (1927), p. 103 for the definitions of the various constants. Only two of them (in the compressible case) are independent. 57In practice, this is still not easy to calculate, since the derivations have to be taken with respect to the deformed, curvilinear coordinate system. 15 of the stress tensor divT= 0. The reason why dealing with controllable defor- mations is still a ‘dirty’ work is that the material- dependent response functions βistill enter the cal- culations. However, in nonlinear elasticity exists an interesting class of solutions in which the βidrop out of the final result, and these solutions are called ‘universal’.58 Therefore, theoreticians were particulary inter- ested in controllable deformations which are inde- pendent of the βi(universal deformations), and Er- icksen (1954) was the first to ask the question: ‘which deformations are possible in every isotropic, perfectly elastic body ?’. Forcompressible materials the answer (Ericksen 1955) is that only homogeneous deformations (that means with a constant deformation gradient F) are possible. Surprisingly, for incompressible materials this is not the case. Ericksen gave examples of 4 different families of deformations and conjectured this classification to be complete. In the meantime, however, a fifth familiy has been detected, and it is still an open question if there are others or not (see Beatty (1987) and Saccomandi (2000) for a descrip- tion of the families). This is just to give an example why the incompressible case is of theoretical inter- est. 5.6 Waves Of course, the highly nonlinear condition eqn. 16 again complicates a lot the nice behaviour of eqn. (2) that led to linear electrodynamics. One important question is: May in the nonlinear case still waves exist that correspond to the electromagnetic waves we observe ? Almost nothing is known about waves in the general case. However, for the simpler case of eqn. (17 ) men- tioned above some remarkable results hold: ”‘That is, in a Mooney-Rivlin material subject to homogeneous strain, all dis- turbances parallel to a given transverse principal axis are propagated at a com- mon speed and in unchanged form. Per- haps this is a characterizing property 58The most common and simple example is the simple shear of a rectangular block that in nonlinear elasticity ca n- not be produced by shear stresses only. Rather the shear stress is determined by the difference of the normal stresses only, see Truesdell and Noll (1965).of the Mooney-Rivlin material; in any case, the possibility of waves of perma- nent form is certainly unexpected in a theory of finite deformation.”’ (Truesdell and Noll 1965, par. 95, p. 351 above). Quite recently, Boulanger and Hayes (1994) and Boulanger (2000) have discovered that these soliton- like solutions which maintain their wave form while propagating exist also in the case of arbitrary, finite deformations.59. That means that not only the classical behaviour of electromagnetic waves is recovered from nonlin- ear elasticity, but there are hints for a particle-like behaviour of propagating solutions coming from the nonlinear treatment. 6 Conclusions The main purpose of this paper was to attack two popular preconceptions among today’s physicists. The first one regards the compatibility of aether theories with the experimental facts of special rela- tivity. It has been given evidence that not the con- cept of the aether as such is wrong, but the idea of particles consisting of external material passing through the aether. Rather the aether is a concept that yields special relativity in a quite natural way, provided that topological defects are seen as parti- cles. Independently from this, topological defects ap- pear interesting, because they have been shown to behave as quantum mechanical particles under var- ious aspects. The second prejudice regards the compatibility of quantum mechanics with Einstein’ attempts of a unified field theory using teleparallelism. While there is no doubt that this theory presented in the stage around 1930 is wrong, I hope to have con- vinced the reader that it is worth to be studied as well. On the one hand, there is a very close relation -probably unknown to Einstein- to the theories of the incompressible aether, on the other hand Ein- stein’s theory anticipated the continuum theory of topological defects developed in the 1950s. There- fore, there is a clear possibility that quantum the- ory may emerge from the geometries Einstein con- 59The authors investigated even the more general case of a previously deformed material. 16 sidered, even if his verbal attacks at that time still support today’s common opinion of the incompati- bility of his unified theory and quantum mechanics. As a consequence, the author proposes to de- velop the far reaching analogies between spacetime and an elastic continuum in the most natural way- leaving the linear approximation and apply the non- linear theory of finite deformations wherever possi- ble. The serious shortcut of MacCullagh’s theory, the impossibility of electrical charges, can be overcome by applying the nonlinear theory. Other features of the general theory, like the localization of energy, appear promising or at least not contradictory to experimental facts. Even if some topics neccessarily have been dis- cussed in a qualitative manner, the paper should contribute to help mathematics play a stronger role in theoretical physics. In view of the open problems mentioned in sec- tion 1, the development of the theories that have come up in past decades seems to be rather ex- hausted. One should therefore ask the question if physics can gain further insight from discussing events on the Planck scale or from the mathematics of differ- ential geometry and homotopy groups. A reconsideraton of some old-fashioned physical theories with modern mathematics could be even more than a matter of historical interest that physi- cists and mathematicians of the 19th and the begin- ning 20th century merit. Acknowledgement. I wish to thank Daniel Gru- miller and Prof. Millard Beatty for commenting on the manuscript. 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