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In 1922, Kottler put forward the program to remove the gravitational potential, the metric of spacetime, from the fundamental equations in physics as far as possible. He successfully applied this idea to Newton's gravitostatics and to Maxwell's electrodynamics, where Kottler recast the field equations in premetric form and specified a metric-dependent constitutive law. We will discuss the basics of the premetric approach and some of its beautiful consequences, like the division of universal constants into two classes. We show that classical electrodynamics can be developed without a metric quite straightforwardly: the Maxwell equations, together with a local and linear response law for electromagnetic media, admit a consistent premetric formulation. Kottler's program succeeds here without provisos. In Kottler's approach to gravity, making the theory relativistic, two premetric quasi-Maxwellian field equations arise, but their field variables, if interpreted in terms of general relativity, do depend on the metric. However, one can hope to bring the Kottler idea to work by using the teleparallelism equivalent of general relativity, where the gravitational potential, the coframe, can be chosen in a premetric way., Comment: 72 pages latex with 6 figures; based on an invited talk given at the Annual Meeting of the German Physical Society (DPG) in Berlin on 20 March 2015, Working Group on Philosophy of Physics (AGPhil); a short version will be submitted to IJMPD

Coupling fermions to gravity necessarily leads to a non-renormalizable, gravitational four-fermion contact interaction. In this essay, we argue that augmenting the Einstein-Cartan Lagrangian with suitable kinetic terms quadratic in the gravitational gauge field strengths (torsion and curvature) gives rise to new, massive propagating gravitational degrees of freedom. This is to be seen in close analogy to Fermi's effective four-fermion interaction and its emergent W and Z bosons., Comment: title changed, matches published version, 7 pages

In the recent decades, it became more and more popular for engineers, physicists, and mathematicians alike to put the Maxwell equations into a generally covariant form. This is particularly useful for understanding the fundamental structure of electrodynamics (conservation of electric charge and magnetic flux). Moreover, it is ideally suited for applying it to media with local (and mainly linear) response behavior. We try to collect the new knowledge that grew out of this development. We would like to ask the participants of EMTS 2016 to inform us of work that we may have overlooked in our review., Comment: 5 pages latex, 4 figures, submitted to EMTS 2016 (Int. Symposium on Electromagnetic Theory) on 14-16 Aug. 2016 in Espoo/Finland. We'd like to ask authors whose work we may have overlooked to let us know this

We take a quick look at the different possible universally coupled scalar fields in nature. Then, we discuss how the gauging of the group of scale transformations (dilations), together with the Poincare group, leads to a Weyl-Cartan spacetime structure. There the dilaton field finds a natural surrounding. Moreover, we describe shortly the phenomenology of the hypothetical axion field. --- In the second part of our essay, we consider a spacetime, the structure of which is exclusively specified by the premetric Maxwell equations and a fourth rank electromagnetic response tensor density $\chi^{ijkl}= -\chi^{jikl}= -\chi^{ijlk}$ with 36 independent components. This tensor density incorporates the permittivities, permeabilities, and the magneto-electric moduli of spacetime. No metric, no connection, no further property is prescribed. If we forbid birefringence (double-refraction) in this model of spacetime, we eventually end up with the fields of an axion, a dilaton, and the 10 components of a metric tensor with Lorentz signature. If the dilaton becomes a constant (the vacuum admittance) and the axion field vanishes, we recover the Riemannian spacetime of general relativity theory. Thus, the metric is encapsulated in $\chi^{ijkl}$, it can be derived from it. [file CarlBrans80_07.tex], Comment: 21 pages, 2 figures, invited contribution to the Festschrift for Carl Brans' 80th birthday, reference to P. Russer added, comments of Alan Kostelecky taken care of

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummer's (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research., Comment: 7 pages, 3 figures

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution., Comment: 96 pages, 17 figures, pdflatex. Invited review article. To appear in Wei-Tou Ni (editor) "One Hundred Years of General Relativity: Cosmology and Gravity," World Scientific, Singapore (2015)

We study properties of the fourth rank elasticity tensor C within linear elasticity theory. First C is irreducibly decomposed under the linear group into a "Cauchy piece" S (with 15 independent components) and a "non-Cauchy piece" A (with 6 independent components). Subsequently, we turn to the physically relevant orthogonal group, thereby using the metric. We find the finer decomposition of S into pieces with 9+5+1 and of A into those with 5+1 independent components. Some reducible decompositions, discussed earlier by numerous authors, are shown to be inconsistent. --- Several physical consequences are discussed. The Cauchy relations are shown to correspond to A=0. Longitudinal and transverse sound waves are basically related by S and A, respectively., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1208.1041

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\cal T}^{ijkl} = - {\cal T}^{jikl} = - {\cal T}^{ijlk}$. Thus, ${\cal K}\sim {\cal T}^3$, see Eq.(46). (i) If $\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\cal T$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\cal K}\sim R^3$ and, in the special case of general relativity, ${\cal K}$ reduces to the Kummer tensor of Zund (1969). This $\cal K$ is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose $\cal K$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$., Comment: 54 pages, 6 figures, written in LaTex; improved version in accordance with the referee report

In special relativity, quantum matter can be classified according to mass-energy and spin. The corresponding field-theoretical notions are the energy-momentum-stress tensor T and the spin angular momentum tensor S. Since each object in physics carries energy and, if fermionic, also spin, the notions of T and S can be spotted in all domains of physics. We discuss the T and S currents in Special Relativity (SR), in General Relativity (GR), and in the Einstein-Cartan theory of gravity (EC). We collect our results in 4 theses: (i) The quark energy-momentum and the quark spin are described correctly by the canonical (Noether) currents T and S, respectively. (ii) The gluon energy-momentum current is described correctly by the (symmetric and gauge invariant) Minkowski type current. Its (Lorentz) spin current vanishes, S = 0. However, it carries helicity of plus or minus one. (iii) GR contradicts thesis (i), but is compatible with thesis (ii). (iv) Within the viable EC-theory, our theses (i) and (ii) are fulfilled and, thus, we favor this gravitational theory., Comment: 10 pages latex. Invited talk delivered at the XV Workshop on High Energy Spin Physics `DSPIN-13' in Dubna, Russia, 08--12 October 2013

Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the temporal frequency $\omega$, the spatial wave (co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating $\omega$ and $k$. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is represented by two light-cones. Kummer, whilst analysing quadratic line complexes as models for light rays in an optical apparatus, discovered in the framework of projective geometry a quartic surface that is linked to the Fresnel one. Given an arbitrary dispersionless linear (meta)material or vacuum, we aim to establish whether the resulting Fresnel surface is equivalent to, or is more general than, a Kummer surface., Comment: Invited lecture at "Electromagnetic Spacetimes", Wolfgang Pauli Institute, Vienna, 19--23 Nov 2012