On Kottler's Path: Origin and Evolution of the Premetric Program in Gravity

Home , Friedrich Kottler

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2 Contents

1 Introduction 4

2 Premetric 3-vector calculus 7

3 The Maxwell equations in premetric 3-vector form 11

4 Jump conditions derived from the premetric Maxwell equations 16

5 Constitutive law 19

6 A legacy of Kottler: premetric fundamental physical constants versus the speed of light 20 6.1 Vacuum admittance λ0, vacuum speed of light c ...... 20 6.2 Absolute dimension of a physical quantity ...... 22 6.3 Counting procedures, the Josephson constant KJ and the von Klitzing constant RK ...... 23 6.4 The vacuum speed of light at last ...... 26

7 Kottler’s program for Newton’s gravity 30

8 Gravitomagnetism and critical assessment of Kottler’s approach in gravity 32

9 Linear and local electrodynamics 35 9.1 A look at the history...... 36 9.2 Localandlinearresponse...... 38 9.3 Axion part of the magnetoelectric moduli found in the multiferroic Cr2O3 ...... 40 9.4 Premetric reciprocity analysis ...... 41

10 Waves, rays, generalized Fresnel equation 43 10.1 Hadamard’s method, Tamm-Rubilar tensor density ...... 43 10.2Eikonalmethod ...... 45 10.3 Kummer tensor density of electrodynamics ...... 47

3 11 Spacetime metric derived 49 11.1 Suppression of birefringence in vacuum: the light cone . . . . . 49 11.2 Dilaton, metric, axion ...... 50

12 The sign of the electromagnetic energy, the Lenz rule, and the emergence of the Lorentz signature 51

13 Concluding remarks 54

Since the notion of metric is a complicated one, which requires measurements with clocks and scales, generally with rigid bodies, which themselves are systems of great complexity, it seems undesirable to take metric as fundamental, particularly for phenomena which are simpler and actually independent of it. Edmund Whittaker (1953)

1 Introduction

The physical reality of the metric can, as noticed by van Dantzig [29], hardly be “...denied by anyone who has ever been pricked by a needle, i.e. who has felt its rigidity and the smallness of its curvature.” Yes, we felt this, too, repeatedly. That is, the reality of the metric is beyond doubt. Still, as Whittaker observed, there are phenomena in physics that are independent of the metric or for which the metric turns out to be irrelevant. If you count the number of electrons within a prescribed domain, for example, this number does not refer to a length measurement—and a domain, a container, can be deﬁned without the necessity to have a length measure available. Historically, the premetric program in physics was ﬁrst clearly spelled out by Friedrich Kottler in 1922 in two papers on “Newton’s law and metric” and on “Maxwell’s equations and metric” [86, 87]. A short history of the subject is provided in Whittaker’s book [174, Vol.2, pp. 192–196] to which we refer for more details. We will discuss Kottler’s two papers in a modern formalism and will show that the premetric nature of Newton’s (non-relativistic) grav-

4 itational attraction law is phenomenologically not as well grounded as that of the Maxwell equations. However, in relativistic theories of gravity, the premetric program leads to a better understanding of the interdependence between topological, affine, and metric concepts, that is, between those concepts related to the surgery of a manifold, to parallel transfer, and to spatio- temporal distances, respectively. In van Dantzig’s nomenclature [29, 26, 27, 28], the premetric Maxwell equations are called “fundamental equations” whereas the set of metric- dependent “linking equations” relates the excitations to the field strengths, and the potentials to the currents (typically by Ohm’s law). The linking equations depend on the properties of the vacuum or of the matter involved. They can be adapted to the structure of the “material” considered, whereas the fundamental equations are, on the classical level, unalterable. This qualitative difference between premetric and metric quantities is also reflected in the universal constants of physics; for an overview of the fundamental constants, see Flowers & Petley [46] and Mohr et al. [109, 110]. There are universal scalars, like the Planck constant ~, the elementary charge e, the magnetic flux quantum Φ0, which are invariant under all coordinate transformations (that is, under the diffeomorphism group). They all can be counted. Today single electrons, for example, can be manipulated by nanotechnological means. In contrast, the speed of light is only a P(oincaré)- scalar, cf. Fleischmann [44]. It is merely invariant under the rigid Poincaré transformation of the flat Minkowski space of special relativity. Since a speed requires for its measurement spatial and temporal distances, the speed of light c is metric-dependent and is, in this sense, less fundamental than ~, e, Φ0. We all know this since the deflection of light near the Sun has been experimentally confirmed. In other words, c is only “universal” provided gravity can be neglected. Since in our terrestrial laboratories gravity is relatively weak, it can be neglected if c is measured. In Secs. 2 to 5, we display, in a modern disguise, Kottler’s electrodynamic results of 1922. In Sec. 2, a premetric 3-vector calculus is set up. In Sec. 3, we start from a 4-dimensional premetric version of the Maxwell equations in the calculus of exterior differential forms and put the Maxwell equations into the formalism of Sec. 2; for textbook representations, one could compare with Schouten [151], Bamberg & Sternberg [9], and Scheck [149]. In Sec. 4—this section can be skipped at a first reading—the jump condition at the boundaries between different materials are directly derived from the premetric Maxwell equations. In Sec. 5, the Maxwell-Lorentz constitu-

5 tive law for vacuum electrodynamics is presented. Here, for the first time, the metric of spacetime enters electrodynamics. This concludes our review of Kottler’s results [87] of 1922 on electrodynamics. Sec. 5 immediately invites to a reflection about the physical dimensions and units in electrodynamics and on the fundamental universal constants involved. This is done in Sec. 6, in particular the status of the electric constant ε0 and the magnetic constant µ0 are elucidated and the Josephson and the von Klitzing constants put into a proper perspective. Next we turn to Kottler’s version of Newton’s gravity. In Sec. 7 the corresponding results of Kottler are presented in 3-vector calculus and in exterior differential forms. We are not aware of other literature on these thought- provoking considerations of Kottler. Subsequently, in Sec. 8, we can extend these gravitoelectric results of Kot- tler by introducing gravitomagnetism. We will follow ideas that were pro- nounced already by Heaviside [55]. This will us lead to quasi-Maxwellian gravitational field equations, which will be interpreted in terms (i) of general relativity and (ii) of the teleparallelism theory of gravity. In the former approach gravitomagnetism defies a proper premetric framework, in the latter approach, however, the gravitational field variable is premetric (namely the coframe) and the constitutive assumption seems to be the requirement of the coframe to be orthonormal. This appears to be in the sense of the Kottler idea. In Sec. 9 we will postulate a premetric local and linear response law for classical electrodynamics, starting from ideas developed in Sec. 6.4. This law will thoroughly be studied in Secs.10 and 11. In particular, we will find out that we can derive a conformal metric by insisting on the vanishing of birefringence in the medium under considerations. In this way, we found a new way to understand the emergence of the concept of a metric of spacetime. In Sec. 12 we will show that even the signature of the metric can be understood this way and that it is related to the sign of the electromagnetic energy and the Lenz rule in the context of Faraday’s induction law. In Sec. 13 we conclude with some remarks.

A short curriculum vitae of Friedrich Kottler (FK) (∗Vienna 1886, †Roche- ster, New York 1965): FK was born as a son of a Lutheran attorney-at-law and a Jewish mother. He was a theoretical physicist, who ﬁrst worked in relativity theory. He got a Ph.D. in 1912 on spacetime lines in Minkowski’s

6 world.1 From 1914 to 1918, FK served as an officer in the Austrian army during the first World War. Already in 1918, he wrote an extended paper on a detailed analysis of Einstein’s gravitational theory [85]. In 1922 he studied the role of the metric tensor in Newton’s gravitational theory and in electrodynamics. During the late 1920s, FK turned to optics, inter alia, and till 1938 was an academic teacher and Professor at the University of Vienna. In 1938, apparently because of political reasons and of his Jewish mother, he was fired from his position by the Nazis (Austria and Germany were united at this time), and he emigrated to the US. There, he worked in industry with the Eastman Kodak Company, once a dominant player in the photographic film sector.2 FK became an American citizen in 1945. Around 1955, FK wanted to retire at Kodak and tried to get his University position back. The University of Vienna made him an honorary professor of mathematical physics, but FK informed the University that he could not teach before the Winter term of 1956/57. The teaching of FK in Vienna, however, apparently never materialized. A short CV of FK up to 1938 is available [88], written by FK himself. It was sent to Einstein and Pauli in order to help FK to find a position in the US. Havas [54], in a monograph by Goenner, Renn, Ritter, and Sauer [48], gave a very insightful review of the development of the relativity theory in Austria from 1900 to 1938, Therein the role of Kottler is also clearly visible. An obituary of FK of the Vienna University can be found under [89].

2 Premetric 3-vector calculus3

Although one can hardly overestimate the importance of the metric in physics, it is remarkable how far one can go in the development of meaningful diﬀeren- tial-geometrical tools for direct applications in ﬁeld theory.

1Kottler applied the four-dimensional tensor analysis (of Ricci-Curbastro and Levi- Civita [138]) to relativity theory even before Einstein became aware (via Grossmann) of this tool. 2Around 2000, one of us (fwh) met an American physicist (HJZ), who worked in FK’s laboratory. Apparently, FK was the boss of some major laboratory in the Eastman Kodak Company. 3Our more philosophically inclined readers may not want to follow the erection of the vector calculus and the setting up of the premetric Maxwell equations, including the constitutive law, in detail. We collected for them the main results of Secs.2, 3, and 5 in Table I, Table II, Table III and in Figures 1, 2, which they may want to turn to directly.

7 We begin with the discussion of a 3-dimensional space M3. We will not assume any Riemannian metric to be defined on it. Here we will demonstrate that, despite a widely spread belief, one can construct, essentially, a 3-vector calculus without any metric. Let xa, with a =1, 2, 3, be local coordinates in some chart that we choose ∂ in such a way that the vectors ∂a := ∂xa are linearly independent at any point of the chart. Thus, they form a coordinate basis of the tangent space. Consequently, the three 1-forms dxa comprise a basis of the cotangent space b b at any point, being dual to the coordinate frame ∂a, that is, ∂a dx = δa, where denotes the interior product of a vector with a form. Since⌋ dxa is ⌋ a coframe, the 3-form ǫ := dx1 dx2 dx3 is defined in this local chart. One immediately notices, though,∧ that ǫ∧is not invariant under the change of a a b coordinates. Indeed, under a transformation x x (x′ ), we have → a 1 ∂x ǫ ǫ′ = J − ǫ, J := det . (1) −→ ∂x b ′ Thus, the 3-form ǫ is a density of the weight 1. − There are more densities of various (exterior) ranks. Namely, we define the set of three 2-forms by ǫa := ∂a ǫ, the set of three 1-forms ǫab := ∂b ǫa, ⌋ ⌋ and, finally, an object ǫabc := ∂c ǫab. From (1) it is clear that these forms ⌋ are all densities of weight 1. At the same time, ǫa is a covector-valued − 2-form, ǫab an antisymmetric 2-tensor-valued 1-form, and ǫabc is a totally antisymmetric 3-tensor density. From the definition ǫ = dx1 dx2 dx3, we have ǫ = 1. Other compo- ∧ ∧ 123 nents of ǫabc are equal 1 for even/odd permutations of the indices 1,2,3, or ± zero when any two indices coincide. Moreover, the components of this object (Levi-Civita symbol) have the same values in all local coordinates. Indeed, a a b under a transformation x x (x′ ) we find → d e f 1 ∂x ∂x ∂x ǫabc ǫabc′ = J − a b c ǫdef = ǫabc. (2) −→ ∂x′ ∂x′ ∂x′ We can uniquely define a similar object ǫabc as the solution of the equation abc [a b c] [] ǫ ǫmnk = δmδnδk . This is a density of weight +1, and the brackets denote antisymmetrization of the indices involved, see Schouten [151]. One may wonder of how to use these objects. With their help we can, to a certain extent, circumvent the absence of metric in M3 and construct a substitute of the full-fledged vector calculus. We call it the poor man’s vector

8 calculus, since at one’s disposal there are only the operations of the exterior and the interior products and of the exterior diﬀerential d. ∧ ⌋ To begin with, let us consider an arbitrary 1-form (or covector) ω. In local a coordinates it is represented by its components ωa according to ω = ωadx . It is clear that we can work equally well with the components ωa, keeping in mind their properties under coordinate transformations. Let us now calculate the exterior diﬀerential dω. In local coordinates it reads:

a 2 3 3 1 dω = dωa dx = (∂ ω ∂ ω )dx dx +(∂ ω ∂ ω )dx dx ∧ 2 3 − 3 2 ∧ 3 1 − 1 3 ∧ +(∂ ω ∂ ω )dx1 dx2. (3) 1 2 − 2 1 ∧

The vector valued 2-form density ǫa forms the basis in the space of 2-forms. It has the components

ǫ = dx2 dx3, ǫ = dx3 dx1, ǫ = dx1 dx2. (4) 1 ∧ 2 ∧ 3 ∧

Comparing (3) with (4), we can define a differential operator curlǫ which maps covectors into vector densities of the weight +1. In the exterior form language a it is defined implicitly by (curlǫ ω) ǫa := dω or explicitly in components by

1 (curlǫ ω) := ∂2ω3 ∂3ω2, 2 − (curlǫ ω) := ∂3ω1 ∂1ω3, (5) 3 − (curlǫ ω) := ∂ ω ∂ ω . 1 2 − 2 1 a One can check directly that (curlǫ ω) transforms as

a a a ∂x′ b (curlǫ ω) (curlǫ ω)′ = J (curlǫ ω) , (6) −→ ∂xb that is, like a vector density of the weight +1. 1 a b Likewise, given an arbitrary 2-form ϕ = 2 ϕab dx dx , we can consider ∧ a its expansion with respect to the basis of 2-forms ǫa, namely ϕ = ϕ ǫa. This deﬁnes a mapping of the antisymmetric tensor components ϕab into the components of a vector density of the weight +1. Explicitly, b 1 2 3 ϕ := ϕ23, ϕ := ϕ31, ϕ := ϕ12. (7)

In particular, for any two covectors va and ua, a “vector product” is defined b b ab which maps them into a vector density [v ǫ u] of weight +1, via the relation a × [v ǫ u] ǫa := v u. × ∧ 9 This mapping allows to define one more differential operator, this time acting on 2-forms. At first, one immediately sees that

a a dx ǫb = δ ǫ. (8) ∧ b Consider now the exterior differential of a 2-form dϕ. Using its expansion a with respect to ǫa and the property (8), one finds dϕ =(∂aϕ )ǫ =: (divǫ ϕ) ǫ, which defines the divǫ operator: it maps antisymmetric 2-tensors into scalar densities of weight +1. Explicitly, b

a divǫ ϕ := ∂1ϕ23 + ∂2ϕ31 + ∂3ϕ12 = ∂aϕ . (9)

One can straightforwardly verify the transformation law under the change of a a b b the local coordinates x x (x′ ): → 1 (divǫ ϕ) (divǫ ϕ)′ = J − (divǫ ϕ) . (10) −→ Equivalently, one can write the operators (5) and (9) with the help of the totally antisymmetric tensor density:

a abc abc (curlǫ ω) = ǫ ∂b ωc, divǫ ϕ = ǫ ∂a ϕbc. (11)

We collected our main results in Table I. In the next section, we extend this three-vector formalism to four spacetime dimensions.

Table I. Premetric diﬀerential operators (we use only partial derivatives!)

operator domain argument image components ∂1f grad = ∂ scalars f covectors ∂ f  2  ∂3f ω vector ∂ ω ∂ ω 1 2 3 − 3 2 curlǫ = ∂ covectors ω densities ∂ ω ∂ ω ×  2   3 1 − 1 3  ω3 weight + 1 ∂1ω2 ∂2ω1 1 − antisym. ϕ23 =ϕ ˆ scalar   2 1 2 3 divǫ= ∂ 2nd rank ϕ31 =ϕ ˆ densities ∂1ϕˆ + ∂2ϕˆ + ∂3ϕˆ ·  3  tensors ϕ12 =ϕ ˆ weight +1  

10 3 The Maxwell equations in premetric 3-vector form

In 1907, Minkowski fused the time coordinate t and the space coordinates xa into a fundamental structure, the 4-dimensional spacetime manifold (“world”). Basically, he achieved this by using Maxwell’s ﬁeld theory of electromagnetism. Let us denote the local spacetime coordinates by xi =(t, xa). The three building blocks of the classical electrodynamics are the electromagnetic ﬁeld strength 2-form F , the electromagnetic excitation 2-form H, and the electric current density 3-form J. The Maxwell equations are written in a generally covariant form, which is valid for all coordinates and for all reference frames:

dF = 0, (12) dH = J. (13)

The electromagnetic field strength can be decomposed into the electric and magnetic fields E and B. (14) a Their components, Ea and B , are identified with the components of the field 1 i j strength tensor, F = Fijdx dx , as follows: 2 ∧ 1 2 3 Ea = Fa0, B = F23, B = F31, B = F12. (15)

Accordingly, the electromagnetic ﬁeld strength 2-form reads

a 1 2 3 2 3 1 3 1 2 F = dt Eadx + B dx dx + B dx dx + B dx dx . (16) − ∧ ∧ ∧ ∧ It is worthwhile to note that the local spacetime coordinates xi =(t, xa) are absolutely arbitrary—not necessarily the Cartesian ones. If we change the local coordinates

a i i i j t = t(t′, x′ ) x x = x (x′ ) a a b , (17) −→ (x = x (t′, x′ ) the electric and magnetic ﬁelds transform into

b b Ea′ = L aEb PabB , (18) a ab − a b B′ = Q Eb + M bB , (19) − 11 where the transformation matrices read

b b c d b ∂x ∂t ∂x ∂t ∂x ∂x L a = a a , Pab = ǫbcd a , (20) ∂x′ ∂t′ − ∂t′ ∂x′ ∂t′ ∂x′ ∂t ∂xb ∂xc ∂x a Qab = ǫacd , M a = det ′ . (21) ∂x c ∂x d b ∂x d ∂xb ′ ′ ′ As we see, for a general spacetime transformation, the components of electric and magnetic fields are mixed up. In a special case of a pure spatial transformation, a a b t = t′, x = x (x′ ), (22) the above formulas reduce to ∂xb Ea′ = a Eb, (23) ∂x′ c a a ∂x ∂x′ b B′ = det B . (24) ∂x d ∂xb ′ In other words, the electric and magnetic fields transform contragrediently. From the 3-dimensional point of view, E is a 3-covector, whereas B is a 3- a vector. This explains the different position of indices Ea vs. B . Moreover, according to (24), the magnetic field is a vector density and not a true vector. In this table, two twisted electric charge and current forms ρ and j rep- resent the source of the electromagnetic field. The excitation of spacetime is represented by two twisted forms and . The force acting on the test particle is determined by two untwistedD formsH E and B. It is straightforward to find the homogeneous Maxwell equation in 3- components. Substituting (16) into (12), we find

∂ E + B˙ =0, ∂ B =0. (25) × ·

Here the dot denotes the time derivative, ˙ = ∂t, and the diﬀerential operator ∂ is, in components, represented by ∂a . In Cartesian coordinates, this { } operator coincides with the nabla ∇. The metric-free diﬀerential operators show up here: we used the alternative boldface notation ∂ = curlǫ and × ∂ = divǫ for the divergence (9) and the curl (5). · In a similar way, we introduce the magnetic and electric excitations

H and D, (26)

12 D H

E B

Figure 1: Faraday-Schouten pictograms of the electromagnetic ﬁeld, see Schouten [151].

a by identifying their components Ha and D with the components of the 1 i j excitation tensor, H = Hijdx dx , as follows: 2 ∧ 1 2 3 a = H a, = H , = H , = H . (27) H 0 D 23 D 31 D 12 Then the excitation 2-form can be rewritten as a 1 2 3 2 3 1 3 1 2 H = dt adx + dx dx + dx dx + dx dx . (28) ∧ H D ∧ D ∧ D ∧ Under the change of the spacetime coordinates (17), we ﬁnd the transformation law b b a′ = τ L a b + Pab , (29) Ha abH aD b ′ = τ Q b + M b . (30) D H D Here we took into account that H is a twisted 2-form, which is manifest in ∂xi ∂xi the presence of an extra sign factor τ = det ′ / det ′ . For orientation- | ∂x j | ∂x j preserving coordinate transformations τ = +1, and τ = 1 in case the orientation is changed. For the pure spatial transformations,− we ﬁnd ∂xc b det ∂x′d ∂x a′ = ∂xc a b, (31) H ′ ∂x H det ∂x d ′ c a a ∂x ∂x′ b ′ = det . (32) D ∂x d ∂xb D ′

13 Accordingly, we conclude that H is a twisted 3-covector, whereas D is a twisted 3-vector density with respect to the spatial transformations (22). Corresponding attributions can be found in, for instance, Schouten [151], Truesdell & Toupin [168], and in Russer [146], and they are collected in Table II.

Table II. The electric current and the electromagnetic ﬁeld

object name math. independ. related reflec- absolute object compon. to tion dimen. electric ρ charge twisted ρ volume ρ q = el. − density 3-form charge electric j current twisted ja area j q/t density 2-form − electric twisted 1, 2, 3 area q D excitation 2-form D D D −D magnetic twisted 1, 2, 3 line q/t H excitation 1-form H H H −H electric E field untwisted E1, E2, E3 line E φ/t strength 1-form magnetic B field untwisted B1, B2, B3 area B φ = mag. strength 2-form flux

Finally, we deﬁne the electric current density J and the electric charge density 1 i j k ρ by identifying the components of the current 3-form J = Jijkdx dx dx : 6 ∧ ∧ J 1 = J , J 2 = J , J 3 = J , ρ = J . (33) − 023 − 031 − 012 123 The current 3-form then reads

J = ρ dx1 dx2 dx3 J 1dt dx2 dx3 J 2dt dx3 dx1 J 3dt dx1 dx2. (34) ∧ ∧ − ∧ ∧ − ∧ ∧ − ∧ ∧

14 D

excitation

2−forms

electric H

1−forms E field strength magnetic B

Figure 2: Different aspects of the electromagnetic field. The four quantities , , E,B jointly constitute the electromagnetic field, see Raith’s edition of Bergmann-SchaeferD H [137]. These four fields are all of a microscopic nature.

It is straightforward to derive the corresponding transformation law, taking into account that J is the twisted 3-form: i ∂x ∂t′ ∂t′ b ρ′ = det ρ + J , (35) ∂x j ∂t ∂xb ′ i a a a ∂x ∂x′ ∂x′ b J ′ = det ρ + J . (36) ∂x j ∂t ∂xb ′ ∂xi Note that here we have the determinant of the 4 4 Jacobi matrix ′ of ∂x j the spacetime coordinate transformation, whereas× in (21) and (24) we have ∂xc the determinant of the 3 3 Jacobi matrix ′ of the spatial transformation. × ∂x d Under the pure spatial transformation (22), ∂xc ρ′ = det ρ, (37) ∂x d ′ c a a ∂x ∂x′ b J ′ = det J , (38) ∂x d ∂xb ′ hence ρ is a twisted 3-scalar density, and J is a twisted 3-vector density. The twisted property is manifest in the modulus of the determinant in the transformation law, cf. with (24).

15 Substituting (28) and (34) into (13), we can recast the inhomogeneous Maxwell equation into

∂ H D˙ = J, ∂ D = ρ. (39) × − · 4 Jump conditions derived from the premetric Maxwell equations4

In this section, we discuss the jump conditions in premetric electrodynamics. In the standard metric presentation of the Maxwell theory, these conditions appear in various situations: (i) On the timelike boundary between two different types of media; (ii) on the spacelike Cauchy initial surface; (iii) on the lightlike wave-front surface. The jump conditions cannot be treated as independent ingredients of the theory, as it is presented sometimes in the literature, see for instance Lakhtakia et al. [94, 95]. Quite to the contrary [157, 124], these conditions are straightforward consequences of the field equations. In the framework of premetric electrodynamics, we are able to derive unique covariant metric- free jump conditions in a unified way. When a metric is present, the three different aforementioned metric-dependent conditions arise as special cases. We present here a brief discussion, see [75] for the technical details and for explicit calculations. Let us consider an arbitrary surface S defined by the equation ϕ(xi) = 0. It is then convenient to derive the covariant boundary conditions directly from the 4-dimensional Maxwell equations by means of their singular solutions. Observe first that a 2-form F , which is discontinuous on the surface S, can be written with the help of the Heaviside step-function θ(τ) as

(+) i i i (+) i ( ) i F (x ), if ϕ(x ) > 0, − F (x )= F (x ) θ(ϕ)+ F (x ) θ( ϕ)= ( ) i i (40) − ( − F (x ), if ϕ(x ) < 0. Substituting this expression into the homogeneous Maxwell equation (12), we ﬁnd F q =0 . (41) ∧ 4This section is not needed for understanding the rationale of the premetric approach. Readers who want to concentrate on the premetric and epistemological aspects of our article can skip this section.

16 Here we denoted the jump 2-form by

(+) i ( ) i F := F (x ) − F (x ) . (42) − We introduced the wave covector (or 1-form)

i q := dϕ = qi dx , qi = ∂iϕ. (43)

Making use of the splitting (14)–(16) of the electromagnetic ﬁeld strength 2-form F into the electric E and magnetic B ﬁelds, we are able to recast (41) into a space-time decomposed form

B q =0 , E q B q =0 . (44) · × − 0 Here we use the obvious notation qi = q0, q ; the central dot in (44) denotes the contraction of a vector with a covector{ (}not the metric-dependent scalar product). In order to deal with inhomogeneous Maxwell equation (13), we need a description of a current that is localized on the surface S. For such a singular current, we use a representation

(S)J = δ(ϕ)j q , (45) ∧ 1 i j with the regular 2-form j = 2 jijdx dx . This deﬁnition guarantees explicitly the principal properties of the surface∧ current:

(i) On a 4-dimensional manifold, the surface current is given as a 3-form. Thus, it can be directly substituted into the inhomogeneous ﬁeld equation (13).

(ii) The current is localized on the surface S = xi ϕ(xi)=0 itself and vanishes outside of it. This property is provided{ | by the factor} δ(ϕ).

(iii) Due to the factor q, the current is tangential to the surface. Indeed, in view of (43), (S)J dϕ = 0. ∧ (iv) The 2-form j and the 3-form (S)J have the same absolute dimension of an electric charge.

(v) The surface current is preserved, even if we choose, for the same surface, a diﬀerent boundary function ϕ(xi).

17 Similarly to (40), we use the step-function representation for the electromagnetic excitation H and introduce the jump 2-form

(+) i ( ) i H := H(x ) − H(x ) . (46) − As a result, in the inhomogeneous Maxwell equation dH = J, the current J is the sum of an ordinary bulk current and the singular surface current (45). As a consequence, H j q =0 . (47) − ∧ In complete analogy with the decomposition of the excitation (26)–(28), 1 i j s we split the 2-form j = 2 jijdx dx into the two pieces j and σ. We identify the components as follows: ∧

s 1 2 3 ja = j0a, σ = j23, σ = j31, σ = j12 . (48)

Then the current reads,

j = dt jsdxa + σ1 dx2 dx3 + σ2 dx3 dx1 + σ3 dx1 dx2. (49) ∧ a ∧ ∧ ∧ The equation (47) can then be rewritten in the space-time decomposed form D q = σ , H q + D q = K . (50) · × 0 Here the surface charge and the surface current densities are deﬁned by

σ = σ q , K = σ q + js q. (51) · 0 × The system (44) and (50) expresses the jump conditions for an arbitrary 3- dimensional surface S in a 4-dimensional spacetime. Generally, it represents the condition for a moving surface, whereas the static case arises for q0 = 0. The same conditions are applicable for the wave-front surface. Then the surface charge and the current are zero, and we obtain the two premetric conditions for wave-fronts,

F q =0 , H q =0 . (52) ∧ ∧ These conditions are of central importance. They can be used as a starting point for deriving the dispersion relation for electromagnetic waves. Inciden- tally, one can equivalently write the decomposed jump conditions (44) and (50) in terms of components, if needed.

18 5 Constitutive law

The premetric Maxwell equations (25) and (39) are generally covariant and valid in all coordinates and reference systems. Moreover, the spacetime geometry and the metric is not specified. The geometrical structure of spacetime enters only the constitutive law that relates the excitation H with the field strength F . The constitutive relation depends on the specific dynamical contents of the electrodynamic theory. In Maxwell-Lorentz electrodynamics, the constitutive law in vacuum reads

ε H = λ ⋆F, λ = 0 . (53) 0 0 µ r 0

Here ε0 and µ0 are the electric and magnetic constants and λ0 determines the vacuum admittance of about 1/(377 ohm). Note that λ0, in contrast to ε0 and ⋆ µ0, is a 4-dimensional diffeomorphism invariant scalar. The star denotes the Hodge duality operator determined by the 4-dimensional spacetime metric. Let us split the local and linear constitutive law (53) into space and time. Then the excitation H, according to (28), decomposes into the vector density D with components a (upper index!) and a covector H with components D a (lower index); here a, b =1, 2, 3. Similarly, the field strength F , according H to (16), decomposes into a covector E with components Eb and a vector density B with components Bb, respectively. If the 3-dimensional positive definite metric has components gab, the Maxwell-Lorentz spacetime relation (53) becomes (g := det gcd)

a ab 1 1 b = ε0 √gg Eb , a = gab B . (54) D H µ0 √g

ab ∗ ab 1 ∗ In Cartesian coordinates, √gg = δ , √g gab = δab. If we introduce the g ab g 1 1 3-tensor density = √gg and its inverse − = √g gab, we can collecting our results in Table III:

19 Table III. The premetric Maxwell equations with the Maxwell-Lorentz spacetime relation for vacuum

Physics law Math. expression Amp`ere-Maxwell law ∂ H D˙ = J × − Coulomb-Gauss law ∂ D = ρ · Faraday induction law ∂ E + B˙ =0 × conserved magnetic ﬂux ∂ B =0 · permittivity of vacuum D = ε0 g E H 1 1 permeability of vacuum = µ0− g− B

Our scheme is consistent with the Tonti diagram [ELE3] on electromagnetism in vector notation, see Tonti [165, p. 412]. However, our Maxwell equations are genuinely metric-free. Gronwald et al. [51] studied this formalism and its relation to gauge ﬁeld theories.

6 A legacy of Kottler: premetric fundamental physical constants versus the speed of light

6.1 Vacuum admittance λ0, vacuum speed of light c The Maxwell-Lorentz law invites us to look at the physical dimensions involved in this expression. A physical quantity Q is by deﬁnition Q = nume- { rical value [unit], see Wallot [171], Stille [159], and Flowers [45]. A place- holder for} the × unit is the physical dimension [Q], which leaves the choice of the system of units open, even though we will use here for convenience SI (speak: “the International System of Units”). In SI, a law in physics can be expressed as a quantity equation, that is, as an equation relating diﬀerent physical quantities.5

5In contrast, one could use purely numerical equations in physics; they are only valid in a predetermined system of units, see Jackson [78] who oscillates in his whole book between numerical equations for SI and those for the Gaussian system. However, quantity equations, like those in our Table III, are universally valid for an arbitrary choice of a unit system. If a fundamental physical equation, which is not related to spherical symmetry, a “π” appears, you can be sure that it is a numerical equation valid only in a speciﬁc unit

20 From elementary electrostatics we know that the physical dimensions of the electric excitation and the electric field strength are el.charge q coulomb C [ a]= = =SI = and (55) D area l2 meter2 m2 voltage mag.flux φ weber Wb [E ]= = = =SI = , (56) b length length time lt meter second m s × × respectively. For determining the physical dimension of ε0, we turn to orthonormal frames, since we now must have a metric gab, since a carries an a Dab upper index and Eb a lower one; then, we can compute E := g Eb and are able, since [gab] = 1, to compare it with a. We find D a [ ] qt [λ0] SI second s [ε0]= D = = = = . (57) [Eb] φl v ohm meter Ω m × SI The vacuum admittance λ0, with [λ0] := q/φ = 1/ohm, has the dimension of a reciprocal resistance. Similarly, in magnetostatics we find for magnetic excitation and field strength, respectively,

current q SI C b mag.ﬂux φ SI Wb [ a]= = = and [B ]= = = . (58) H length lt m s area l2 m2 Thus, the dimension of the magnetic constant turns out to be

b [B ] 1 SI Ω s [µ0]= = = . (59) [ a] [λ ]v m H 0 It is surprising, the innocently looking non-relativistic analysis within electro- and magnetostatics led us to the physical dimensions [ε0] and [µ0] of ε0 and µ0, see (57) and (59), respectively. They contain a clear message: the square-roots of their quotient and of their product carry a simple physical interpretation:

ε0 SI 1 1 SI 8 m λ0 := , c := 3 10 . (60) µ0 ≈ 377Ω √ε0µ0 ≈ × s r system. For a collection of numerous diﬀerent unit systems in electrodynamics, see Carron [19].

21 The electrodynamic vacuum is characterized by the vacuum resistance (impedance) and by the vacuum speed of light.6 These are the two moduli of the electrodynamic “æther,” that is, the vacuum admittance λ0 and the vacuum speed of light c are two “moduli” specifying the electromagnetic properties of the vacuum.

6.2 Absolute dimension of a physical quantity The absolute dimension of a quantity is the physical dimension of its corresponding exterior diﬀerential form. As exterior form, these quantities are scalars under diﬀeormorphisms and under frame transformations. Therefore, the notion of an absolute dimension has a covariant meaning and is, thus, of fundamental importance. Since the 3d Levi-Civita symbol ǫabc = 1, 0 represents sheer numbers, ± we have, using (55) and (58), for the absolute dimensions of the excitations and , D H 1 a b c a 1 [ ]= εabc[ dx dx ]= q and [ ]= a dx = qt− . (61) D 2 D ∧ H H Accordingly, we arrive for the four-dimensional excitation H simply at the electric charge q:

[H] = [dt ] + [ ]= charge = q =SI C = coulomb . (62) ∧ H D For the ﬁeld strength E and B we have analogously

a 1 1 a b c [E] = [Ea dx ]= φt− and [B]= ǫabc[B dx dx ]= φ (63) ∧ 2 ∧ or, for the four-dimensional ﬁeld strength F ,

[F ]= [dt E] + [B]= magnetic ﬂux = φ =SI Wb = weber . (64) − ∧ If we take the theory of the absolute dimensions, with [H]= q and [F ]= φ, to its logical conclusion, we can extract information by computing the

6The speed of light c, or rather √2 c—because of diﬀerent units and conventions— had already been measured electromagnetically by Weber & Kohlrausch in 1855, before Maxwell’s equations were derived in 1865. In fact, Maxwell used this result in his considerations in an essential way. It seems, however, that Weber & Kohlrausch had not been aware that the constant they determined is related to the speed of light.

22 quotient and the product of them, with h as the dimension of an action,7

[H] q 1 = = [λ ] =SI , [H] [F ]= qφ = action = h =SI J s = joule second . [F ] φ 0 Ω ∧ × (65) Accordingly, the absolute dimensions of the excitation H and the field strength F led us to the 4d scalars electric charge q and magnetic flux φ, their quotient to an admittance with dimension [λ0]= q/φ and their product to an action with dimension h = qφ. In SI, we have, respectively, C, Wb, 1/Ω, and Js. These notions, as well as their SI expressions, are 4d diffeomorphism invariant scalars. This is the message of classical electrodynamics, facts, which are hardly mentioned in the textbook literature. Note in particular that the magnetic flux and the action feature in a purely classical situation, no quantum aspects were involved. Particularly noteworthy is that the speed of light does not occur here. This is clear since we know that the speed of light changes at the presence of a gravitational field—in contrast to electric charge and magnetic flux, which are unaffected by gravity. In this sense, c must not enter here—and it does not! These facts haven been already noted by Fleischmann [44], for example: there are premetric diffeomorphism invariant 4-scalars in physics, let us call them 4d-scalars, namely q, φ, [λ0], h, and those related to the existence of a metric and connected to space and time differences, with the speed of light, c, as an example. We call c as a P-scalar, since c is only a scalar under the Poincarégroup of special relativity. We consider the premetric 4d-scalars as more fundamental than the metric-dependent P-scalar c. The premetric 4d-scalars, which we mentioned here, were extracted from mechanics (h) and electrodynamics (q, φ, [λ0]= q/φ) alone. From thermody- namics, the entropy S should be added to this list of 4d-scalars.

6.3 Counting procedures, the Josephson constant KJ and the von Klitzing constant RK How can we understand our 4d covariant results from a more intuitive point of view? Well, we just have to look at an axiomatics of classical electrodynamics

7 The 4d scalars, electric charge q, magnetic ﬂux φ, and impedance 1/λ0, all have rightfully own names in SI, namely C, Wb, and Ω, respectively. They really do merit own names. However, the action has none. We suggest Pℓ = planck. Then, planck = coulomb weber = joule second, or, Pℓ = CWb = AsVs = VAs2 = Ws2 = J s. × ×

23 [58]. The inhomogeneous Maxwell equation (13), dH = J, results from the postulated conservation of electric charge Q = 3 J. As is known from a proof of the Stokes theorem, no metric concepts are involved, rather—if applied R to the charge 3-form—the counting of charges, and addition or subtraction depending on their sign, is all that is required. If we prescribe an arbitrary three dimensional domain (a 3d “volume”), we have just to know how many charges are contained therein. The net change of the charge in the course of time has to flow through the 2-boundary encircling the 3d domain. We describe classical electrodynamics. But, of course, in the back of our mind is the knowledge that in nature there are quantized elementary charges SI 19 with electric charge of e 1.602 10− C. This makes us sure that the ≈ × counting process has a solid basis in experimental physics. In this way, the concept of electric charge enters electromagnetism. The homogeneous Maxwell equation (12), dF = 0, emerges from a careful 4d discussion of Faraday’s induction law [58]. It turns out that this law can be understood as a conservation law of the magnetic flux Φ = 2 F . Please note that this conservation law refers to the 2-form F , whereas usually we R relate a conservation law to a 3-form, see the case of the electric charge. A geometrically trained physicist has no difficulty to understand this fact, but those attached to 3d vector calculus may find this counterintuitive and may go astray to the outdated magnetic charge concept instead. But how is it with the counting of elementary flux (lines)? Certainly, in classical theory, there is no discrete flux. But already Faraday spoke of lines of force, a line being a discrete entity. We know from experimental physics that under certain circumstances, namely in type II (two) superconductors, magnetic flux is quantized in terms of flux units (Abrikosov vortices) with SI 15 Φ0 := h/(2e) 2.068 10− Wb, with h as the Planck constant; the factor 2 occurs because≈ a Cooper× pair in the superconductor is built up from two electrons. Certainly, in nature the magnetic flux is not quantized in general. Mag- netic fluxes of currents in our brain can be appreciably smaller that the flux quantum. Still, it is clear that the notion of counting flux lines exists in nature under special circumstances. This speaks in favor of the reasonableness of the counting procedure also for the magnetic flux in general. After all, we know that the Faraday law is very well fulfilled in nature: This is the way the magnetic flux emerges in electrodynamics. Having electric charge and magnetic flux now at our disposal and having

24 recognized that the corresponding counting procedures are based on the elementary charge e and the ﬂux quantum h/(2e), it is clear that physically the quantized nature of the charge (e is a 4d scalar) and of the action (h is a 4d scalar8) are the raisons d’ˆetre for the possibilities of counting. Accordingly, the dimensions q of charge and h of action are fundamentally distinguished from other physical dimensions, such as mass m, length l, and time t. If q and h are premetric scalars, the same must be true for the expressions

qn1 hn2 = 4d scalar . (66)

Strictly, by our arguments n1 and n2 are not required to be integers. However, examples for such dimensionful 4-scalars with integers are h h q electric charge , magnetic ﬂux , electric resistance .... → q → q2 → (67) And here we only observe n1 = 1, 2 and n2 =0, 1. We discussed these results previously,± − see [62]. If we pick for q the elementary charge e and for h the Planck constant h, we immediately arrive at the the Josephson [81, 2] and the von Klitzing [84, 80] constants of modern metrology—for reviews, see [46, 108, 109, 134, 53]—which provide highly precise measurements9 of e and h,

2e SI h SI K = 0.483 PHz/V, R = 25.813kΩ; (68) J h ≈ K e2 ≈ for new experiments to the quantum Hall effect see Weiss & von Klitzing [173] and for the theory Bieri & Fröhlich [11], and for quantum metrology generally, Göbel & Siegner [47]. The Planck units of 1899 are discussed by Kiefer [83]. Note that the reciprocal of the Josephson constant KJ has the absolute dimension φ of a magnetic flux, the von Klitzing constant RK, of course, that of an electric resistance (impedance). Incidentally, the premetric nature of RK has helped us to predict [61] that the quantum Hall effect is not influenced by gravity (that is, the Hall resistance is independent of the gravitational field). So far, our dimensional analysis in the subsections 6.2 and 6.3 did not make use of the metric. Moreover, it is generally covariant and as such valid

8Post [132] argued that the action could be, perhaps, a pseudoscalar instead. 9In SI: kilo=k=103, mega=M=106, giga=G=109, tera=T=1012, peta=P=1015, exa=E=1018.

25 in particular in general relativity as well as in special relativity. And, on top of that, our considerations do not depend on any particular choice of the system of physical units. Whatever your favorite system of units may be, our results will apply to it. In short: Our dimensional analysis so far is premetric, generally covariant, and valid for any system of units—and it led straightforwardly to the 4d scalars q, φ, [λ0]= q/φ, h.

6.4 The vacuum speed of light at last But why did the speed of light not occur in our analysis of the absolute dimensions? Simply because the measurement of a velocity requires a spatial and a temporal distance concept, that is, the additional existence of a metric. In connection with measuring a velocity, besides counting, distances are relevant. For the analysis of the premetric structure of the Maxwell equations, the counting procedure was all we needed. And it led to the premetric fundamental constants with the dimensions of q, φ, [λ0], h. More is required for the understanding of the constitutive law of Maxwell-Lorentz, see Table III. The Maxwell-Lorentz law describes the electromagnetic properties of the vacuum, as we saw in Sec. 6.1. Clearly, this vacuum is assumed to be homogeneous and isotropic. Consequently, the Euclidean group, as a semidirect product of the translations T (3) with the rotations SO(3), underlies the geometry of 3d space. In the 4d Minkowski spacetime of special relativity, the group of motion is the Poincar´egroup, in which T (3) is generalized to T (4) and SO(3) to the Lorentz group SO(1, 3). The existence of a unique isotropic speed of light attests to the isotropy of spacetime. In an anisotropic spacetime,10 the velocity of light would depend on the direction of propagation. Accordingly, we will be able to ﬁnd an unique speed of light in the homogeneous and isotropic Minkowski spacetime. Let us recall of how anisotropy of light propagation can be achieved. We recall the constitutive vacuum law in electrostatics: D = ε0 g E, or, in com- a ab ponents, = ε0 √gg Eb, see Table III. For a local and linear dielectric material,D which is additionally homogeneous and isotropic, this law generalizes to D = εε0 g E, with the (dimensionless) permittivity ε. If the dielectric a is anisotropic, as was known already to Maxwell, ε becomes a tensor ε b such

10Incidentally, attempts were made to develop a corresponding generalization of electrodynamics and of special relativity on the basis of the Finsler geometry [13, 14].

26 that we ﬁnd a a cb ab = ε c ε √gg Eb =(ε √gε )Eb . (69) D 0 0 The expression within the parentheses we call the permittivity tensor density. Note that the 3d metric and its scalar density √g is included in this expression. Analogous consideration, also already performed by Maxwell, can be a ab made in magnetostatics with the permeability tensor density, B =(µ0√gµ ) Hb. We can additionally allow for cross-terms between the magnetic and the electric eﬀects, which occur in so-called magneto-electric media, see O’Dell [127]. Then the four-dimensional generalization of our anisotropic law, gen- ⋆ eralizing the isotropic law H = λ0 F , can be written as

1 kl kl kl lk Hij = κij Fkl , with κij = κji = κij (70) 2 − − as the doubly antisymmetric local and linear response tensor. It has 36 in- kl dependent components and the dimension [κij ] of an admittance [λ0]. Also here, the metric and its scalar density as well as the electric and the magnetic constants are included within the response tensor density. In Sec. 9 we will turn to a discussion of this law. kl For a speciﬁc choice of the response tensor κij , we recover the Maxwell- Lorentz vacuum case (53), namely for [58, p. 303]

kl mk nl ε0 κij = λ ǫijmn √ gg g , λ = . (71) 0 − 0 µ r 0

Here ǫijkl = 1, 0 is the totally antisymmetric Levi-Civita tensor density and ± gkl is the metric of the Minkwoski spacetime. Then, of course, the vacuum response tensor is determined, besides by the vacuum admittance λ0, only by the components gkl of the metric of the Minkwoski spacetime. We substitute the vacuum constitutive law (70) cum (71) into the premetric Maxwell equations, which, in components, read (compare with Einstein [36]) ∂[iHjk] = Jijk , ∂[iFjk] =0 . (72) Subsequently, we can derive the wave equation and discover that the propagation of electromagnetic waves is ruled by the speed of light c := 1/√ε0µ0. This is how the speed of light enters the scene. Not before the vacuum law (71) is assumed. The vacuum admittance λ0, a premetric concept, is already

27 kl incorporated in (70) in the form of the physical dimension [κij ] = [λ0] of kl κij . The speed of light requires for its emergence the Minkowski metric in (71). The speed of light c—in contrast to the vacuum admittance λ0—is a metric-dependent quantity willy nilly. This is also one of the lessons one learns in looking at electrodynamics with the eyes of Kottler. Let us recall how Einstein constructed SR (special relativity), neglecting gravity, in 1905, for the time being. In “The Meaning of Relativity” [37], we read: ...the Maxwell-Lorentz equations have proved their validity in the treat- ment of optical problems in moving bodies. No other theory has satisfactorily explained the facts of aberration, the propagation of light in moving bodies (Fizeau), and phenomena observed in double stars (De Sitter). The consequence of the Maxwell-Lorentz equations that in a vacuum light is propagated with the velocity c (“principle of the constancy of the velocity of light”),11 at least with respect to a definite inertial system K, must therefore be regarded as proved. According to the principle of special relativity, we must also assume the truth of this principle for every other inertial system. Thus, SR, according to Einstein, is based (i) on the special relativity principle (equivalence of all inertial frames of reference) and (ii) on the principle of the constancy of the speed of light. Because of (ii), the speed of light is, in SR, a scalar under Poincarétransformations (a P -scalar) by construction. Needless to say that Einstein’s conception was highly successful and, in 1945, the Alamogordo bomb was the final and widely visible proof of the validity special relativity. In GR (general relativity), one postulates the existence of a constant c0 = 1 299 792.458 km s− . However, c0 is not an actual speed of light in the presence of the gravitational field. Recall, that in a medium with the refractive index n, the light propagates with the velocity c0/n, when the medium is at rest and we neglect gravity. In inhomogeneous media, the refractive index is a function of the coordinates, n = n(x). The gravitational field is a special type of an “inhomogeneous medium” with the permittivity, permeability, and the magneto-electric moduli constructed from the components of the spacetime metric g. The latter contains information about the true gravitational field as well as about the inertial field arising due to the motion of the reference frame. Accordingly, in GR the actual speed of light c depends on this field g; symbolically, c = c(g).

11Note that the phrase in the parentheses, which we printed in roman letters, was not translated into English, even though it appeared in the German original. Incidentally, in English, one often talks about the speed of light, referring to its modulus.

28 This fact is well established from a great number of observations. In particular, we see that the light of distant stars is deflected by the gravitational field g of the Sun, and space missions reveal numerous time delay effects for light and radio signals. When gravity is present, which is always the case due to its universality, c0 cannot be measured optically as speed of light in GR. For the measurement of a velocity, one needs two different points in spacetime; and always there are tidal gravitational forces between those 2 points, which cannot be transformed to zero even in a freely falling laboratory. The speed of light is a P -scalar, but it is not a four-dimensional spacetime scalar, it is not diffeomorphism invariant. The c0 is by assumption a four-dimensional scalar, but it is no longer the speed of light. Still, in all textbooks one reads that c0, confusingly, is called the speed of light. In other words, the principle of the constancy of the velocity of light loses its general validity. Einstein [35] stated that (our translation) The principle of the constancy of the velocity of light allows also here [when gravity is present] the definition of simultaneity, provided one restricts oneself to very small light paths. That is, the principle of the constancy of the velocity of light is only applicable in a certain limiting case. One could argue that the scalar nature under diffeomorphism of c0 is a new principle of GR, which generalizes the principle of the constancy of of the speed of light c in SR. For a discussion of c0 and the cosmological constant, see Dadhich [23] and for Einstein’s introduction of the cosmological constant Sauer [148]. In a recent paper, Braun, Schneiter, and Fischer [17] discuss how precisely one can measure the speed of light. They find quantum effects that limit the corresponding precision in principle. Here we similarly argue that we have already classical effects of gravity, which likewise limit the precision of the determination of the speed of light. As we can see, the division of universal constants in four-dimensional (diffeomorphism) scalars and P -scalars marks a decisive qualitative difference between the universal constants. In modern metrology [109, 156] the value of the constant c0 = 299792.458 1 km s− is exact, and the role of c0 is reduced to a conversion factor that is used to define an SI unit of length: the meter is introduced as the length which the light passes in vacuum during a time that equals 1/299 792 458 of a second. The possibility of a similar definition of a kilogram, by making use of the fundamental physical constants, is now widely discussed, see Haddad et al. [53].

29 D = − −m −r gr 4π r3

Figure 3: Gravitational excitation Dgr: We integrate ∂ Dgr = ρm over a ball of radius r and apply the Gauss theorem. For a point· particle− with mass m, this yields a gravitational excitation that is proportional to m/r2.

7 Kottler’s program for Newton’s gravity

When in electrodynamics the electric charge density ρel does not depend on time and neither electric currents nor magnetic ﬁelds are present, the Maxwell equations (39) reduce to the electrostatic law: ∂ D = ρ . The · el electric charge density ρel, a scalar density, creates the electric excitation D. Kottler noticed that Newton’s gravity can be treated similarly to electrostatics. Now, instead of the electric charge density, the mass density ρm, also a scalar density, induces the gravitational excitation Dgr with physical 2 dimension [Dgr]= m/l : ∂ D = ρ . (73) · gr − m The minus sign appears in gravity because of the gravitational attraction of masses as opposed to the repulsion of electrically equally charged particles.

Dgr is the active part of the gravitational ﬁeld, created by the mass density scalar ρm. Thus, the conserved mass (Lavoisier) is the source of Newtonian gravity. In electrodynamics, D is a measurable quantity, since we have positive and negative charges and D can be measured by the Maxwell double plates (because of the charge separation). However, in gravity, we have only positive masses and the gravitational excitation does not seem to have a direct operational interpretation. Thus, Dgr is of a more formal nature than D. The fact

30 that mass has only one sign, in contrast to the two signs in the electric case, accounts for this basic distinction between gravity and electromagnetism.12 The passive aspect of Newton’s gravitational field can be described as follows: In electrostatics, a force on a test charge e is given by F = eE and the corresponding force density by F = ρel E. This determines, in an operational way, the electric field strength E. Similarly, the gravitational field strength Γ is defined via the force density l F = ρ Γ , [Γ]= = a = acceleration . (74) gr m t2 According to Newton’s attraction law, Γ is conservative: ∂ Γ =0 or Γ = ∂φ. (75) × − Following Kottler, we recognize that the equations (73) and (75) are formulated independently of the metric of the Euclidean 3-space! This premetric structure is even more clearly visible in exterior calculus: for the 2-form gr of the gravitational excitation, the 3-form ̺m of the mass density, and theD gravitational field strength 1-form Γ, we have

d = ̺ ; f α =(eα Γ) ̺ ; dΓ=0 , Γ= dφ . (76) Dgr − m gr ⌋ m − Here eα is the local triad and denotes the interior product of vector with ⌋ an exterior form. Analogously, in vector calculus, we find ∂ D = ρ ; F = ρ Γ ; ∂ Γ =0 , Γ = ∂φ. (77) · gr − m gr m × − These are, according to Kottler, the premetric fundamental laws in Newton’s quasi-field theory of gravity. The local and linear constitutive law relating the excitation Dgr to the ab field strength Γ requires a 3-dimensional Euclidean metric g . Since Dgr is a a vector density with components gr and Γ a covector with components Γb, we find (a, b =1, 2, 3) D

a ab 1 f force = εg √ gg Γb , εg = , [εg]= = . (78) Dgr − 4πG v4 velocity4

12If one requires the energy densities of the electromagnetic and the gravitational field to be positive, this entails that electromagnetism must be a vector and gravity a scalar or a tensor field of 2nd rank. Because of light deflection a pure scalar gravitational field is excluded, whereas a scalar admixture to the tensor field it is allowed for gravity (Jordan- Brans-Dicke theory). For these questions one should compare Deser & Pirani [34, 33].

31 11 3 2 Here Newton’s gravitational constant enters, G =6.67 10− m /kg s . The × constitutive law (78) Dgr = εg g Γ transforms Kottler’s 3 premetric axioms in (77) into a predictive physical theory, namely the quasi-ﬁeld theoretical version of Newton’s gravitational theory. Our results in (77) and (78) are consistent with the Tonti-diagram of the classical gravitational ﬁeld in [165, p.401]. Substitution of the constitutive relation (78) into the Coulomb-Gauss law (77) yields the Poisson equation

∂ D = ∂(εg g Γ)= εg ∂ g ∂ φ = ρ or εg ∆φ = ρ . (79) · gr · − · − m m Here ∆ is the Laplace operator. In exterior calculus, with the 3-dimensional metric-dependent Hodge star operator ⋆,

⋆ = εg Γ . (80) Dgr ⋆ ⋆ ⋆ d = d(εg Γ) = εg d dφ = ̺ or εg ∆φ = ̺ , (81) Dgr − − m m with the Laplacian for scalar ﬁelds ∆ := ⋆d ⋆d . The Hodge star is the only metric dependent quantity in these equations. For its dimension, see footnote13.

8 Gravitomagnetism and critical assessment of Kottler’s approach in gravity

In electrodynamics, there are two types of sources: electric charges and electric currents (moving charges). Accordingly, there are two types of phenomena—the electric and magnetic ones. Their existence and interrelation has a deep consequence: instead of the clearly unphysical instantaneous action, electromagnetism is propagating with a finite speed. The natural idea that gravitation should propagate with a finite speed, too, and that such a propagation process is substantiated by a certain physical medium, is very old. Already Laplace in his “Méchanique Céleste” (1799) developed the model of a “gravific fluid,” see the discussion of Whittaker [174, Vol. 1, pp. 207–208]. In fact, even earlier, Newton himself [113] wrote: “That

13If φ is a p-form and [ds2] = l2, then in n dimensions [⋆ φ] = [φ]ln−2p. In particu- ⋆ −3 lar, [ ρ] = [ρ] l ; this coincides with the dimension of the left-hand-side εg∆φ, namely, (f/v4)l−2v2 = m/l3, qed.

32 gravity should be innate inherent & essential to matter so that one body may act upon another at a distance{ through} a vacuum without the medi- ation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe{ } no man who has in philosophical matters any competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting consta ntl y according to certain laws, but whether this agent be material{ } or immaterial{ } is a question I have left to the consideration of my readers.” One should also compare the fairly recent historical investigation by Roseveare [141]. Electrodynamics provided useful hints for the corresponding studies of gravity. Among the early attempts to modify Newton’s gravity accordingly, it is worthwhile to mention the investigations of Holzmüller [70] and Tis- serand [163, 164] who analyzed the possible extension of Newton’s gravity along the lines of Weber’s electrodynamics. Weber introduced velocity- and 2 q1q2 r r˙ r r¨ acceleration-dependent corrections to the Coulomb force 3 1 2 + 2 . 4πε0 r − 2c c Doing the same for Newton’s gravitational force, it was possible to solve the the problems of celestial mechanics and even to find the effect on the pre- cession of a planetary orbit (perihelion shift). However, for the Mercury perihelion the result turned out to be incorrect, see [174, Vol.1, pp. 207– 208]. The problem was also addressed by Heaviside who asked himself [55]: “Now what is there analogous to magnetic force in the gravitational case? And if it has its analogue, what is there to correspond with electric current? At first glance it might seem that the whole of the magnetic side of electromagnetism was absent in the gravitational analogy.” Taking Maxwell’s electrodynamics (and not Weber’s one) as a guide, Heaviside came up with the proposal to model the gravitational current as

Jm = ρmu , (82) with the flow velocity u of the massive matter, and suggested to relate this current (with a correction due to a “displacement current” ) to the gravitomagnetic field H via the equation: curl H = J + D˙ . gr gr − m gr The beginning of the modern development of what is generally known as the gravitoelectromagnetism goes back to the work of Thirring [162], who noticed that the geodesic equation of a massive test particle in general relativity can be recast into the form of the equation of motion under the action of the Lorentz force due to the gravitoelectric and gravitomagnetic fields. A

33 modern and easily accessible account can be found in Rindler [139, Sec. 15.5], see particularly Rindler’s table from Eqs.(15.77) to (15.81). In the weak field linear approximation Einstein’s gravitational equation can be formally written in a “premetric” form, compare Table III, ∂ H D˙ = J , ∂ D = ρ , (83) × gr − gr − m · gr − m ∂ E + B˙ = 0 , ∂ B =0 , (84) × gr gr · gr where the mass density ρm and the mass current Jm of (82) are the sources of the gravitational field. The gravitoelectric E = ∂ φ A˙ and the grav- gr − − itomagnetic B = ∂ A field strengths are linked to the gravitoelectric and gr × gravitomagnetic excitations Dgr and Hgr, respectively, by the constitutive relations 1 Dgr = εg Egr , Hgr = Bgr . (85) µg The gravitolectric and the gravitomagnetic constants, 1 16πG ε = , µ = , (86) g 4πG g c2 like the electric and magnetic constants in Maxwell’s electrodynamics (54), satisfy an analogous relation √εgµg =2/c. When A = 0, we find Bgr = 0 and the system (83)-(85) reduces to Kottler’s premetric formulation of Newton’s gravity which we described in Sec. 7, where we denoted Egr = Γ. In order to make the scheme complete, we have to add the result of Thirring [162] for the force density on test matter: F = ρ E + J B . (87) m gr m × gr The complete analogy between electromagnetism, see Table III, and gravitational theory, as represented in (83, 84, 85), is stunning. However, one has to be aware that in premetric electrodynamics the electromagnetic fields E, B and D, H are pieces of the fundamental metric-free objects: F and H, respectively. In general relativity (“Einstein gravity”), the gravitolectromagnetic fields Egr, Bgr are constructed from the components of the spacetime metric di- 2 rectly: φ = g00 c , a = g0a. The excitations Dgr, Hgr are actually components of− theA Riemann− curvature tensor, which is found by dif- ferentiation of the connection (Christoffel symbols). The Christoffels themselves are determined by differentiating the metric g, namely connection ∼ 34 ∂ ( metric ). Thus, symbolically, gravitoelectromagnetism arises in linearized Einstein gravity in the following way:

E , B ∂ ( metric ), D , H ∂ ( connection ) ∂∂ ( metric ) . (88) gr gr ∼ gr gr ∼ ∼ In other words, in Einstein gravity the Riemannian metric is deeply rooted in the gravitomagnetism, and the construction of the premetric formulation of the gravity theory does not seem to be possible. On the other hand, in the framework of gauge theories of gravity, it is well-known that there is a so-called teleparallel equivalent of general relativity, GR (spoken “GR teleparallel”), see Blagojevićet al. [12, p.2, fron- || tispiece], Itin [72], Obukhov & Pereira [125], and Maluf [104]. For ordinary matter, but excluding fermionic fields, GR is equivalent to Einstein gravity. || However, the gravitational potentials (belonging to the translations) are the coframe (“tetrad”) and the connection becomes a teleleparallel connection, both quantities that can be defined premetrically. The metric-dependent constitutive assumption would then be that the frame is chosen orthonormal. Therefore, in the end the metric comes in. However, the Kottler type framework with (83) and (84) is uphold. The details of this GR framework || need to be worked out in detail, but the general structures are well-known and the program looks feasible to us. Accordingly, even in gravity, it seems, Kottler was also on the right track. Back to Einstein gravity: the gravito-electromagnetic approach—for a general overview, compare the textbooks [153, 147]—can be extended also beyond the weak field linear approximation [79, 103, 22, 102, 15]. It plays an important role in modern gravitational theory, in particular with respect to the discussion of the inertial drag and spin effects [143, 105].

9 Linear and local electrodynamics

After this somewhat complicated case with gravity, let us come back to elec- 1 kl trodynamics. In Sec. 6 we saw, see (70), Hij = 2 κij Fkl, that, on the premetric level, we can assume a medium with local and linear response in order to complete the fundamental equations of electrodynamics to a fully predictive theory: in components, it comprises the Maxwell equations (72) and the local and linear law (70). Our conclusion contradicts, however, the verdict of Tonti [166]. Whereas he agrees that the Maxwell equations can be formulated premetrically, he

35 insists that “all constitutive equations require the metric.” No doubt, he is mostly correct with this stipulation. The collection of constitutive laws assembled in Tonti’s book [165] seems to attest to his belief. There are interesting exceptions known, though: (i) The phenomenological constitutive law that describes the quantum Hall eﬀect within (1+2)- dimensional electrodynamics, see Bieri & Fr¨ohlich [11, Eq.12] and Hehl & Obukhov [58, Eq.(B.4.60]. This should not come as a surprise when in con- densed matter circles the notion of the topological superconductors is preva- lent. (ii) The axionic part of the constitutive law of the multiferroic chromium sesquioxide Cr2O3 is premetric, see [64, Eq.(17)]. Accordingly, these laws are topological and independent of the metric. Other examples are known. In any case, Tonti’s belief that all constitutive law require a metric is not universally valid. This will be also demonstrated in Sec. 11 where a metric is derived from a premetric law by requiring certain supplementary postulates.

9.1 A look at the history...... of our subject may widen our view. Already in 1910, Bateman [10] investigated local and linear materials. In his own notation, his constitutive equations were [10, Eq.(II)]

B = κ H + κ H + κ H + κ D + κ D + κ D , − 1 11 1 12 2 13 3 14 1 15 2 16 3 ......

E3 = κ61H1 + κ62H2 + κ63H3 + κ64D1 + κ65D2 + κ66D3 , (89) with the symmetry requirement κrs = κsr, r, s = 1,... 6, which leaves only 21 independent components. Besides the permeabilities (here κ11, κ12, κ13) and the permittivities (here κ64, κ65, κ66), also magneto-electric cross terms (the rest of the terms) were assumed, the latter of which were not discovered before 1961 by Astrov [4] (here κ61, κ62, κ63) )and by Rado & Folen [136] (here κ14, κ15, κ16), see also O’Dell [127]. Nowhere did Bateman mention the metric explicitly. Also his Maxwell (vacuum) equations had a premetric form. Thus, he already anticipated Kottler’s program to some extend. However, he did not think that his as- sumptions were related to physics [10]: “These conditions [the constitutive relations (89)] may not correspond to anything occurring in nature; nevertheless their investigation was thought to be of some mathematical interest on account of the connection which is established between line-geometry and

36 the theory of partial diﬀerential equations.” Then, “The general Kummer’s surface appears to be the wave surface for a medium of a purely ideal charac- ter...” To this last remark, we will come back in the next section, see Baekler et al. [5].

In 1925, Tamm [160] used modern 4-dimensional tensor analysis, and he ij ijpq assumed the local and linear law f = s Fpq, which is equivalent to (70). In our notation, it reads ij 1 ijkl ˇ = χ Fkl , (90) H 2 ˇij 1 ijkl with the excitation tensor density := 2 ǫ Hkl and the electromagnetic ijkl 1 ijmn kl H moduli χ := 2 ǫ κmn . In his ansatz, Tamm, like Bateman, had 21 independent components. In his actual applications, he used, however, only 4 of them. This state of the art was summarized by the monographs of Post [130] (see also Post [131]) and O’Dell [127], with the only proviso that Post killed, by mistake, one of the 21 independent components of χijkl (“Post constraint”). ijkl The symmetry of Bateman’s κrs or, equivalently, the symmetry χ = χklij was dropped for the ﬁrst time by Serdyukov, Semchenko, Tretyakov, and Sihvola [155], as far as we know. Thus, they arrived at the 36 independent components postulated in (70). In this way dissipative processes can be included. A systematic study of the 36 components of χijkl was performed by two of us [58]. In particular, an irreducible decomposition was achieved for χijkl for the ﬁrst time. It was cut into a principal part (20 components), a skew part (15 independent components), and an axion part (1 independent component). Incidentally, the Maxwell-Lorentz vacuum law in contained in the principal part in the form of

χijkl =2λ √ ggi[kgl]j , (91) 0 − see (71). Subsequently, χijkl with 36 independent moduli were investigated, amongst others, by Lindell [98, 99, 100], Favaro [39], Ni [120], and Pfeifer and Siemssen[129]. If one has moving bodies around, also the 4-velocity of the body could be included into such considerations, as was early observed by Schmutzer [150], see in this context also Balakin et al. [8, 6].

37 9.2 Local and linear response After this look back in the history of the constitutive law, let us come back to (70). It reads 1 H = κ klF . (92) ij 2 ij kl It can be recast into the decomposed form

b b a = C a Eb + Bba B , (93) Ha − ba a b = A Eb + Db B . (94) D − The generalized permittivity matrix Aba and the generalized impermeability b a matrix Bba and the magneto-electric matrices C a and Db are constructed from the components of the constitutive tensor, 1 1 Aba = ǫbcdκ 0a, B = ǫ κ cd, (95) 2 cd ba 2 bcd 0a 1 Cb = κ 0b, D a = ǫacdǫ κ ef . (96) a 0a b 4 bef cd kl 2 The twisted constitutive tensor κij (x) of type [2] has 36 independent components. We can decompose this object into several irreducible parts. Deﬁne k kl κi := κil , (97) with 16 independent components. The second contraction yields the twisted scalar function k kl κ := κk = κkl (98) (also called pseudo- or axial-scalar). The traceless piece

k k 1 k κi := κi κ δ (99) 6 − 4 i has 15 independent components. Three irreducible pieces then read:

kl (1) kl (2) kl (3) kl κij = κij + κij + κij

(1) kl [k l] 1 k l = κij +2 κ δ + κ δ δ . (100) 6 [i j] 6 [i j] (1) kl Besides the principal part κij , we have the skewon and the axion ﬁelds

j 1 j 1 Si := κi , α := κ. (101) 6 − 2 6 12 38 Accordingly, the general local and linear constitutive relation (70) reads explicitly 1 (1) kl k Hij = κij Fkl +2S Fj k + α Fij . (102) 2 6 [i ] kl Thus, we split κij according to 36 = 20+15+1. Let us look into the properties of the three irreducible pieces. First, we (1) kl study the principal part. By construction, κij is totally traceless:

(1) kl κil =0. (103)

It is more nontrivial to verify the double-duality property 1 (1)κ kl = ǫ ǫklmn (1)κ pq, (104) ij 4 ijpq mn but in fact this is a direct consequence of (103). Following (95) and (96), we introduce the 3-dimensional blocks of the principal part

ba 1 bcd(1) 0a 1 1 (1) cd ε = ǫ κcd , µ− = ǫbcd κ a , (105) − 2 ba 2 0 1 γb = (1)κ 0b = ǫbcdǫ (1)κ ef . (106) a 0a 4 aef cd Taking into account (103) and (104), we can then straightforwardly demonstrate the symmetry properties

ab ba 1 1 c ε = ε , µab− = µba− , γ c =0. (107) Thus we indeed verify that the total number of independent components (εab, 1 a (1) kl µab− , γ b) of the principal part κij is 6 + 6 + 8 = 20. (2) kl Turning to κij , it is convenient to parametrize the skewon part by

c a j sc m Si = − a . (108) 6 nb sb a a The total number of parameters (m , nb, sb ) correctly yields 3 + 3 + 9 = 15 skewon degrees of freedom. We thus eventually reveal the ﬁne structure of the constitutive tensor

b 1 b b c c b γ a µab− sa + δasc ǫˆabcm δa 0 κ = ab a + − abc −a a c + α a .(109) ε γ b ǫ nc sb δ sc 0 δ − − b b principal part 20 comp. skewon part 15 comp. axion part 1 comp.

| {z } | 39{z } | {z } Accordingly, (93) and (94) read explicitly as follows:

1 c b b b b c a = µab− ǫâbcm B α Ea + γ a + sa δasc Eb , (110) Ha ab − abc − a − a a − a c b = ε ǫ nc Eb + α B +( γ b + sb δ sc ) B . (111) D − − b ab ba 1 1 c Recall that ε = ε , µab− = µba− , and γ c = 0. Incidentally, α is a c 4-dimensional (axial) scalar, whereas sc (summation over c) is only a 3- dimensional scalar. The physical meaning of the various pieces of the general linear constitu- ab 1 tive law is clear: ε stands for a generalized permittivity, µab− encodes the a impermeability properties, and the cross-term γ b accounts for the magnetoelectric moduli and is related to the Fresnel-Fizeau effects. The skewon con- c tributions m , nc are responsible for electric and magnetic Faraday effects, b respectively, whereas the skewon terms sa describe optical activity. We un- derline that nowhere in our considerations a metric was used. This analysis is completely premetric. The premetric approach of electrodynamics has also been developed over the years by Delphenich [30, 31, 32] and, more recently, by Cabral & Lobo [18]. Rumpf [144] and Zhang et al.[176], for instance, discuss realistic engineering applications for (110,111), but without using the the skewon part. Similar constitutive laws are applied by de Carvalho [21] to the relativistic electron gas.

9.3 Axion part of the magnetoelectric moduli found in the multiferroic Cr2O3 The decomposition of the constitutive law, as manifest in (110) and (111), immediately suggests to look for experimental evidence of the 4d pseudoscalar α, the axion part of the constitutive tensor, see [60, 123]. It is a speciﬁc magnetoelectic piece, that is, after the discovery of magneto-electricity in 1961, it was within the reach of experimentalists. Two of us joint forces with two experimental colleagues [64, 65, 66], who had the corresponding measurements already in their drawer. It was the multiferroic chromium 4 sesquioxide Cr2O3 that carries a small axion piece of at most α 10− λ0 at around 275K. Thus, in 2008, the Post constraint was disproved,∼ that is, an axion part, even if rare and small, can exist. Model calculations for the axion piece were made by Essin, Moore, and Vanderbilt [38]. New experiments with Cr2O3 were described, for example,

40 by Seki et al. [154] and wave propagation studied by Lindell, Sihvola, and Favaro [101]. No other material than Cr2O3 has been described in the meantime with a nonvanishing α. Rosch [140] has suggested to one of us that the copper selenide Cu2OSeO3, an insulator, due its symmetries, could possibly carry an α = 0. Linear optical properties of this chiral magnet Cu2OSeO3 were recently6 investigated by Versteeg et al. [169]. Because of the chiral properties of this material, the existence of a nonvanishing axion piece would be plausible. If vacuum electrodynamics is amended by an axion piece, one speaks of axion electrodynamics. This theory goes back to ideas that Ni developed in the 1970s [114, 115]. Corresponding applications were discussed recently by Balakin [7], for example. Axion electrodynamics can also be used to discuss the response of materials to photons at low energy. Optical eﬀects, like the Faraday and the Kerr rotation can be calculated with its help, see Wu et al. [175]. As we described in [64], the axion piece α is akin to a Tellegen material. Some new insight in this connection is provided by Prodenˆcio et al. [133].

9.4 Premetric reciprocity analysis In order to demonstrate further the power of our approach, we will show that already on this premetric level the notion of reciprocity can be applied. We recall that, in a very general sense, the reciprocity of the electromagnetic media means that after changing the source of the ﬁeld with the measuring device, the reading of the measuring device remains the same. Let us consider a premetric generalization of the “reaction” quantity [145]:

((1)E (2)J (2)E (1)J). (112) ∧ − ∧ VZ Here the integral is taken over the spatial volume V , and the 1-form of the (1) (1) a electric field E = Eadx is created by the source 2-form of the electric (1) (1) a (2) (2) current J = J ǫa, whereas the field E is created by the current J. The concept of reaction in the electromagnetic theory is used for the analysis of electromagnetic wave propagation in media and it has important applications for the formulation of the boundary value problem and compu- tation of the scattering and transmission coefficients.

41 Using the premetric Maxwell equations, see Table III, we derive

(1)E (2)J = d((1)E (2) ) (2) (1)B˙ (1)E (2) ˙ . (113) ∧ − ∧ H − H ∧ − ∧ D Making use of the constitutive relations (93) and (94), we ﬁnd

((1)E (2)J (2)E (1)J)= ((2)E (1) (1)E (2) ) V ∧ − ∧ ∂V ∧ H − ∧ H R [ba](1) (2) R (1) a(2) b + 2A Ea E˙ b 2B[ba] B B˙ V − h R b b (2) (1) a (1) (2) a +(C a + Da )( Eb B˙ Eb B˙ ) ǫ. (114) − i The reciprocity requires that the “reaction” (112) vanishes for all sources. Usually, V is the whole space, and the surface integral over the boundary ∂V disappears by assumption. As a result, we ﬁnd that the reciprocal media are characterized by the following properties of the constitutive matrices:

ab ba b b A = A , Bab = Bba, C a + Da =0. (115)

Recalling the irreducible decomposition of the constitutive tensor, we thus see that neither of the three irreducible parts is totally reciprocal. In a the principal part, the magneto-electric susceptibilities γ b are nonreciprocal, a whereas in the skewon part the Faraday-responsible parameters m , na are nonreciprocal. The axion, being a special (4-dimensional isotropic) case of magneto-electricity, is also nonreciprocal. Schematically, we have

n r r n n 0 (1)κ = , (2)κ = , (3)κ = . (116) r n n r 0 n Here “r” and “n” stands for reciprocal and non-reciprocal, respectively. In general, a reciprocal medium is characterized by the electric and mag- a netic permeabilities ε,µ and by the skewonic susceptibilities s b. When the former are purely real and the latter is purely imaginary, the matter is non- dissipative. It is sometimes called a chiral bi-anisotropic medium. Imaginary parts of permittivity and permeability and the real part of the skewon are responsible for dissipation. It was demonstrated for complex media (see [106, 82], e.g.) that reciprocity is related, in the broad sense, to the Onsager- Casimir time-reversal invariance [128, 20] of the equations of motion of a physical system. However the complete understanding of the details of this

42 relation is still absent, cf. [135, 167] and [96]. Nonreciprocity of magnetoelectric materials is featured, in particular, in [90, 91]. For a general discussion of the reciprocity in bi-anisotropic media one can refer to the books [3, 98, 100], for applications see also Monzon [111, 112].

10 Waves, rays, generalized Fresnel equation

Having set up the premetric framework with the Maxwell equations and the 36 independent components of the electromagnetic response tensor, we can now draw the consequences of this layout. We take here the version (90) for the response law. The immediate reaction of most physicists would be to look for the wave phenomena related to the theory under investigation and—turning to the geometrical optics limit—to study the emerging light rays. And this is exactly what we will do; for a classical text, see Born & Wolf [16].

10.1 Hadamard’s method, Tamm-Rubilar tensor density We implemented this idea by using a standard method due to Hadamard [52] by considering a wave surface with a continuous electromagnetic ﬁeld strength F , but the derivative of F has a jump. The direction of the jump is given by the wave covector qi. By integration, we can create thereby the wave vectors as tangents to the rays [58]. This was done roughly since the year 2000 and the results have been reviewed in the book of two of us [58], see also Obukhov et al. [121], Hehl et al. [67], Rubilar [142] (and for nonlinear electrodynamics Obukhov et al. [126]). The outcome is a generalized Fresnel equation, see [58, Eq.(D.2.23)], which is (algebraically) quartic in the wave covector qi: ijkl [χ] qiqjqkql =0 . (117) G This equation is governed by the Tamm-Rubilar tensor density, which is cubic in χ, see [58, Eq.(D.2.22)]:14

ijkl 1 c(i j ad k l)b [χ] := ⋄χab χ | | χ⋄ cd . (118) G 3! 14 ⋄ kl 1 abkl We use here the left diamond dual χij := 2 ǫijabχ and the right diamond dual ⋄ ij 1 ijcd χ kl := 2 χ ǫcdkl as convenient abbreviations.

43 -4 x y 6 -2 4 0 2 2 4 0

z 0

-5

Figure 4: This is Fresnel surface for the anisotropic medium without skewon part. It is a Kummer surface, see Obukhov et al. [122].

Figure 5: Here the skewon part is present: (2)χijkl = 0. The two branches merge into one, and the points where the old branches6 in Fig. 2 touched are replaced by the holes—the wave vectors do not go though the holes. This illustrates the typical eﬀect of a skewon: damping of wave propagation in certain directions (or everywhere), see Obukhov et al. [122].

44 Note that ijkl is totally symmetric ijkl = (ijkl); thus, it has 35 independent components.G Moreover, the axionG part (3)Gχijkl = χ[ijkl] drops out of (118) and, accordingly, also of (117), that is, the axion part cannot be seen in the geometric optics limit. Curiously, ijkl = (ijkl), which is cubic in χijkl, as well as (χijkl (3)χijkl), have the sameG numberG of independent components, − namely 35—this is probably not by chance. The generalized Fresnel equation (117) spans the Fresnel surface. We present three typical cases for visualizing it, see also Favaro [40]. In Fig. 4 a normal Fresnel surface for an anisotropic crystal, in Fig. 5 the presence of a hypothetical skewon part is allowed, and in Fig. 6 the extreme case of a hypothetical metamaterial with moduli [42], namely

10 0 1 0 0 1 1 εab = (1 √3) 01 0 + 0 1 0 , (119) 4   2   − 0 0 2 0 0 1 −  10 0   1 0 0  1 1 1 µ− = (1 √3) 01 0 0 1 0 , (120) ab 4   2   − − 0 0 2 − 0 0 1 −  1 0 0    √3 γa = (1 + √3) 0 1 0 . (121) b 4   0− 0 0   This yields the maximal possible number of 16 real singular points of a Kummer surface. Note that in this material the skewon part vanishes by assumption. It seems that the Kummer surfaces are destroyed by the skewon part, see Fig. 4, but we have no general proof for that. Schuller et al. [152] started from the doubly antisymmetric and local and linear response tensor density χijkl, with (2)χijkl = 0, which arises so naturally in electrodynamics. They promoted it to a corresponding area metric for an arbitrary n-dimensional manifold and extended their idea also to string theory. For recovering a Fresnel type equation, they used a generalized Tamm- Rubilar tensor, see [152, Eq.(30)]. See also some developments by Dahl [25]. A fresh approach has been set up recently by Ho & Inami [69].

10.2 Eikonal method Itin [74] has developed a geometric optics analysis of premetric Maxwell’s iσ equations on the basis of the eikonal ansatz Fij = fije , where fij are slowly

45 ab 1 Figure 6: The permittivity ε , the impermeability µab− , and the magnetoelec- a tric moduli γ b of the hypothetical metamaterial are speciﬁed in Eqs.(119, 120, 121), see Favaro et al. [42].

46 changing amplitude functions of the spacetime coordinates. By making use of strictly algebraic methods, he found the following dispersion relation (this is a synonym for the generalized Fresnel equation),

i1(ij)j1 i2abj2 i1(ia)j1 i2(jb)j2 i3cdj3 ǫii1i2i3 ǫjj1j2j3 χ χ +4χ χ χ qaqbqcqd =0 . (122) Surprisingly, this expression looks diﬀerent from the generalized Fresnel equation (117) cum (118). However, Obukhov [74, App.] was able to show, by means of heavy algebraic manipulations, that (122) and (117), apart from a numerical factor of 72, are just the same expressions. This is, of course, − what one had to expect. Subsequently Itin has demonstrated that (122) can be transformed into diﬀerent equivalent forms, the most symmetric form probably being

ii1jj1 i2abj2 i3cdj3 ǫii1i2i3 ǫjj1j2j3 χ χ χ qaqbqcqd =0 . (123) Exactly this dispersion relation, in a version using multiforms and dyadics, was also found by Lindell [100, Eq.(5.244)], bringing this subject of dispersion relations to a ﬁnal close.

10.3 Kummer tensor density of electrodynamics The problem of the generalized Fresnel equation/dispersion relation for the local and linear medium was, as we saw, solved. Still, since the year 1910 it was clear, see Bateman [10], that the Fresnel surface should be a Kummer surface. Originally Kummer had started to investigate optical ray bundles of second order, that is, those ray bundles that have, like in a birefringent crystal, at one point two independent rays. The tool for investigating these ray bundles was projective geometry—a geometry without any metric—in which a ray in 4-dimensional space is projected into a real projective 3-dimensional space RP3. These investigations led Kummer around 1864 to a certain quartic wave surface with, in general, 16 singular points [93], see our Fig. 6. This wave surface was later called Kummer surface by Hudson [71], who wrote an authoritative review of such surfaces. The proposal of the Kummer surface constituted a major breakthrough in a ﬁeld that is nowadays called algebraic geometry (see [49]). Some more details can be found in Baekler et al. [5]. Favaro [39], in 2012, provided a critical and thorough evaluation of the premetric electrodynamics with a local and linear response law. Eventually,

47 Baekler, Favaro, et al. [5] introduced a new premetric tensor density of 4th rank, the Kummer tensor of local and linear electrodynamics, in order to bring the property of Fresnel surfaces to be ‘Kummer’ into focus:

ijkl aibj ckdl [χ] := ⋄χ⋄ χ χ . (124) K acbd We immediately recognize the symmetry of this premetric definition, ijkl[χ]= klij[χ] . (125) K K Note that the definition (124) corresponds to Itin’s most symmetric case (123) and to the analogous expression of Lindell [100]. If ijkl[χ] is understood as a symmetric 16 16 matrix, it can be read off thatK it has 136 independent components. An× irreducible decomposition of the Kummer tensor density under the GL(4, R) was provided in [5]. For the Tamm-Rubilar tensor density, we find 1 ijkl[χ]= (ijkl)[χ] . (126) G 6 K Accordingly, the totally symmetric piece of the Kummer tensor density with its 35 independent components (it is one of the irreducible pieces) can, up to a factor, observed in crystal optics. It has an operational interpretation, exactly like ijkl. The name “Kummer tensor” was coined by Zund in the context of generalG relativity [178], see also Zund [177]. In his definition, instead of χijkl, he used the Riemann curvature tensor Rijkl = Rjikl = − Rijlk of general relativity. However, Rijkl, in contrast to χijkl, has the −additional symmetry Rijkl = Rklij and, thus, carries only 20 independent components (see the Post constraint!). Since Kummer surfaces were first discovered in the context of optics, we find the new name Kummer tensor density perfectly appropriate in electrodynamic considerations. So far, we were able to show that, for vanishing skewon part, the Fresnel surfaces are Kummer surfaces, exactly as Bateman [10] had anticipated. This is a decisive step forward in the understanding of Fresnel surfaces. Then, our favorite form of the generalized Fresnel equation eventually turns out to be

ijkl [χ] qiqjqkql =0 . (127) K What is left to understand is the nature of the Fresnel surfaces for a nonvanishing skewon part (2)χijkl = 0 of the response tensor density and the 6 operational interpretation of the remaining 136 35 = 101 remaining components of the Kummer tensor density. −

48 11 Spacetime metric derived

Wave propagation in electrodynamics in materials with a local and linear response law exhibits the phenomenon of birefringence (double refraction), see Born & Wolf [16]. In the geometric optics approximation, this is manifest in the fourth order Fresnel equation (117) for the wave covector, see our Figs. 4 and 5. Such a picture is typical for anisotropic and magnetoelectric materials. Let us now consider the physical vacuum. It is natural to assume that the physical vacuum is a non-birefringent continuum. In other words, if we start with a local and linear medium and then, in case we want to recover the physical vacuum, forbid birefringence, then we can hope to recover the light cone of relativity theory—and thus the metric up to a conformal factor. We will follow here partly our presentation in [57].

11.1 Suppression of birefringence in vacuum: the light cone It was shown by Lämmerzahl et al. [97, 59], see also the review [63], that the postulate of vanishing birefringence reduces the general Fresnel equation to one for the light cone. Thus, up to a conformal factor, the second rank spacetime metric gij is recovered from the local and linear fourth rank re- ijkl kl sponse tensor density χ or κij , respectively. These considerations have been improved by Itin, Favaro, and Bergamin [73, 41, 24]. Originally, we found the light cone by postulating certain reciprocity constraints for the response tensor [58], see also Sec. 9.4. However, these arguments appear to be more mathematical, whereas birefringence has a very intuitive meaning. Moreover, in the cosmos birefringence is excluded with high accuracy, compare the observations of Polarbear [1] and the discussion of Ni et al. [118, 107, 119]. Looking at Fig. 4, for instance, it is clear that we have to take care that both shells in each Fresnel wave surface become identical spheres. Then light propagates like in vacuum. For this purpose, we can solve the quartic Fresnel equation (117) with respect to the frequency q0, keeping the 3–covector qa fixed. One finds four solutions, for the details please compare [97, 73]. To suppress birefringence, one has to demand two conditions. In turn, the quartic equation splits into a product of two quadratic equations proportional

49 ij to each other. Thus, we ﬁnd a light cone g (x) qiqj = 0 at each point of spacetime. Perhaps surprisingly, we derived also the Lorentz signature, see [58, 77, 76]. Because of the importance of this result, we will discuss it separately in Sec. 12.

11.2 Dilaton, metric, axion At the premetric level of our framework, we found that, besides the principal piece, the skewon and the axion fields emerged. Only subsequently the light cone was brought up. The skewon field was phased out by our insistence of the vanishing birefringence in vacuum. Accordingly, the light cone and the axion field are the lone survivors of the collapse of birefringence. By the light cone the metric gij is only defined up to an arbitrary function λ(x), the dilaton field, see also Ni et al. [117]. Thus, the vanishing birefringence leads to the following response law:

ij ik jl ijkl ˇ = [ λ(x) √ gg (x) g (x) + α(x) ǫ ] Fkl . (128) H − dilaton metric axion In exterior calculus, it| is{z even} | more{z compact} by| the{z} use of the metric dependent Hodge star ⋆ operator:

H = [λ(x) ⋆ + α(x)]F. (129)

We recognize that the three ﬁelds λ(x), gij(x), and α(x) are all descendants of electromagnetism and they emerge with a straightforward physical interpretation. All experimental searches for the axion A0 turned out to be negative. Thus, at least for the time being, α = 0. Furthermore, under conventional circumstances, the dilaton ﬁeld seems to be constant, λ(x)= λ0, where λ0 is the vacuum admittance with a value of about 1/377Ω. Accordingly, we end up with the Maxwell-Lorentz law

ij = λ √ g F ij or H = λ ⋆F, (130) H 0 − 0 which coincides with (53).

50 12 The sign of the electromagnetic energy, the Lenz rule, and the emergence of the Lorentz signature

The premetric Maxwell’s equations, dH = J and dF = 0, are universally valid in a four-dimensional manifold, on which time and space coordinates are arbitrarily introduced by means of the (1+3)-foliation [58]. Any physical object, say a form Ψ of a rank p, can be uniquely decomposed Ψ = dτ Ψ +Ψ ∧ ⊥ into the longitudinal (p 1)-form Ψ and the transversal p-form Ψ. ⊥ Although the (1 + 3)− splitting is unique, the interpretation of the longitudinal and transversal parts of physical object in Maxwell’s theory admits a certain freedom. The resulting three-dimensional forms of premetric electrodynamics correspond to the different basic physical laws such as positivity versus negativity of the energy, attraction versus repulsion between two charges, and so on. In this section we present a brief analysis of these different possibilities and give a justification of the sign conventions presented in Sec. 3. For the proofs and the details, see [58, 77, 76]. The analysis of the (1 + 3)-decomposition of the electric current J shows that all the possibilities of the sign factors are merely conventional and the splitting has the ordinary form J = j dτ + ρ , (131) − ∧ where the standard notations for the 3-dimensional current j := J and ⊥ charge density ρ := J are used. The uniqueness here is due to the fact that for a 3-form J, the spatial exterior differential of the transversal 3-form ρ vanishes identically. Turning to the excitation 2-form H, we arrive at the decomposition

H = hT dτ + . (132) H ∧ D The sign factor hT = 1 remains undetermined, reﬂecting the freedom to ± identify the magnetic excitation either with the longitudinal part H or ⊥ with H . Thus, the inhomogeneousH Maxwell equations read − ⊥ hT d + ˙ = j, d = ρ . (133) H D − D Quite similarly, the decomposition of the 2-form F of the electromagnetic ﬁeld strength is given by

F = fT E dτ + B. (134) ∧ 51 and the ﬁeld equations by

fT d E + B˙ =0 , d B =0 , (135) with another undetermined sign factor fT = 1. ± In order to ﬁx the free factors hT and fT we turn to an additional ingredi- ent of premetric electrodynamics—the energy-momentum current 3-form, see [58]. The “time” (transversal) component of this current 3-form represents the energy density of the ﬁeld. In our setting it takes the form 1 1 u = fT E hT B. (136) 2 ∧ D − 2 H ∧ Up to this point, everything was premetric. Now we have to take recourse to the vacuum constitutive relations. Substituting the Maxwell-Lorentz constitutive relation for vacuum (54) into (136), we obtain

1 ab hT 1 a b u = fTε0√g g EaEb gabB B ǫ . (137) 2 − µ0 √g The ﬁrst term in (137) represents the pure electric part of the energy density while the second part is the pure magnetic part. We requite that both parts of the energy are non-negative. With the Euclidean 3d metric gab, we arrive to the conditions fTε0 > 0 , hTµ0 < 0 . (138) We then readily recognize that for positive values of the electric and magnetic constants ε0 and µ0, the ordinary sign factors fT = +1 and hT = 1 of Maxwell’s electrodynamics are recovered. Nevertheless, one can speculate− about various possibilities arising for negative values of the vacuum constants. Let us list some basic physical consequences of the inequalities (138): Dufay’s law: In our setting, the electric force between two static charges • comes with an additional factor of fT, see [77]. In particular, fT = +1 yields attraction between opposite charges and repulsion between charges of the same sign: this is Dufay’s law. In contrast, fT = 1 yields anti-Dufay law with attraction between charges of the same sign and− repulsion between opposite charges.

Lenz’s rule: For hT = 1, we have the ordinary relative sign in Faraday’s law,• that is usually described− by Lenz’s rule. It is responsible for the pulling

52 of a ferromagnetic core into a solenoid independently on the direction of the current. For hT = +1, the situation is opposite and the corresponding anti-Lenz rule holds. Lorentz signature: By considering the electromagnetic waves in our con- • struction we arrive to the ordinary wave equation for the ﬁeld E, for instance. The sign factors in this equation are completely canceled and the square of 2 the wave velocity is given by the usual formula c = 1/(ε0µ0). The form of the wave equation manifests the properties of the underlying geometry. Thus, the ordinary case ε0µ0 > 0 corresponds to the Lorentzian signature of the 4-metric, whereas the case εµ < 0 corresponds to the Euclidean signature. Our analysis shows that in the framework of Kottler’s premetric approach we are able to derive the basis physical properties of electromagnetism in vacuum. For this, the ordinary vacuum constitutive relation must be assumed and the positivity of the electric and magnetic energy must be required. Let us extend now the constitutive relation Eq.(54) to isotropic media. As a result, the vacuum constants are replaced by the permittivity ε and permeability µ parameters of the medium. On can even speculate that the media parameters are not necessary positive, which can be the case for the artiﬁcially designed materials – metamaterials. We collect all possible sign options with their physical consequences in the following table:

Table IV. Classiﬁcation of physical media by sign factors

fT = +1 , hT = 1 Lenz rule Dufay law Lorentzian − ε> 0 , µ> 0 signature fT = 1 , hT = 1 Lenz rule anti-Dufay law Euclidean ε< 0−, µ> 0− signature fT = +1 , hT = +1 anti-Lenz rule Dufay law Euclidean ε> 0 , µ< 0 signature fT = 1 , hT = +1 anti-Lenz rule anti-Dufay law Lorentz ε< 0−, µ< 0 signature

This analysis is based on the assumption of the positivity of the electric and magnetic energy densities. In contrast, the ordinary description of metamaterials allows for the negative energies. In this case, the standard Dufay and Lenz rules are recovered in our model, too.

53 Let us examine now the most general premetric linear constitutive relation (93)–(94). We ﬁnd

a ba a b E = EaD ǫ = A EaEb + Db EaB ǫ , (139) ∧D − and a b a b a B = HaB ǫ = C a EbB + Bba B B ǫ (140) H ∧ − Consequently, the energy density (137) takes the form

1 ab a b a a b u = fT A EaEb + hT Bab B B (fT Db + hT C b) EaB ǫ . (141) −2 − In the first two terms we recognize the electric and magnetic energies, respectively. For the standard Lorentzian signature with fT = +1 and hT = 1, the ab − matrix A must be negative definite and the matrix Bab must be positive definite. This result yields positive definiteness of the matrices εab and µab.

13 Concluding remarks

We should mention that by introducing generalized concepts of stress and hyperstress, one can set up a kind of a premetric elasticity theory, as shown by Gronwald et al. [50]. Only if one relates this framework to the classical Euler stress, which is symmetric, one has to eventually to introduce the metric tensor. Moreover, Favaro (private communication) has pointed out that, in non equilibrium statistical mechanics, the Fokker-Planck equation can be formulated in a premetric way. The Kottler program was intended to remove the metric from the fundamental equations in physics as far as possible. Since the metric is the gravitational potential in general relativity theory, Kottler’s program does not apply here. However, in the teleparallel version of gravity, it seems possible to carry through the Kottler program, although the details will have to be seen. In classical electrodynamics, Kottler’s path led to a resounding suc- cess and, see Sec. 11, to a new understanding of the emergence of the metric tensor together with a dilaton, an axion, and a skewon field. All these fields, together with the metric, come into existence in an electrodynamic context. These remarkable facts are not yet completely understood. In particular, only a first attempt was started to develop a theory for a skewon field [68], for which some cosmic limits were set by Ni [116].

54 Acknowledgments We would like to extend our sincere thanks to Wei-Tou Ni (Hsin-chu) for his encouragement to write up this article. Quite a number of colleagues con- tributed appreciably to our understanding of the subject of our investigation. We are most grateful to the following people: To Alberto Favaro (London) for many detailed discussions on the foundations of electrodynamics and on Kummer surfaces; to Enzo Tonti (Trieste) for advice on the operational interpretation of electrodynamics; to Naresh Dadhich (Pune) and to Claus Kiefer (Cologne) for insight on fundamental constants; to Reinhard Meinel (Jena) on gravitomagnetism; to Hubert Goenner (Göttingen) and, particularly, to Tilman Sauer (Mainz) (Göttingen) for help in historical questions; to Daniel Braun (Tübingen) for discussions on the precision with which the speed of light can be measured; to Ari Sihvola (Espoo) for explanations to [155]; and to Achim Rosch (Cologne) for decisive hints to copper selenide.

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