Classical Kalb-Ramond field theory in curved spacetimes

For that purpose, our starting model is Maxwell-Lorentz electrodynamics, whose structure we summarize to then transpose it to the KR theory. Interactions between electric sources are mediated by the electromagnetic field, which, in terms of exterior algebra, can be encapsulated within four postulates. The first two posit the existence of two closed forms in four-dimensional spacetime, a 2-form ${\sf F}$ for the field and a 3-form ${\mathscr{J}}$ for the sources, that is

	$\displaystyle d{\sf F}$	$\displaystyle=$	$\displaystyle 0,$		(2)
	$\displaystyle d{\mathscr{J}}$	$\displaystyle=$	$\displaystyle 0,$		(3)

where $d$ is the exterior derivative (for an account of exterior calculus in connection with electrodynamics, see fumeron2020improving ). These two forms are thus locally exact, meaning that there exists a 1-form ${\sf A}$ and a 2-form ${\sf G}$ such that

	$\displaystyle{\sf F}$	$\displaystyle=$	$\displaystyle d{\sf A},$		(4)
	$\displaystyle{\mathscr{J}}$	$\displaystyle=$	$\displaystyle d{\sf G}.$		(5)

Equations (2) and (5) are Maxwell equations, the first one being the Bianchi identity (or structure equation), and the second the “equation of motion”. (4) is the definition of the field strength and (3) is the charge conservation equation. Translated into ordinary vector language, this leads (in flat spacetime) to four equations ${\boldsymbol{\nabla}}\times{\bf E}+\partial_{t}{\bf B}=0$, ${\boldsymbol{\nabla}}\cdot{\bf B}=0$, ${\boldsymbol{\nabla}}\cdot{\bf D}=\rho$ and ${\boldsymbol{\nabla}}\times{\bf H}-\partial_{t}{\bf D}={\bf j}$, involving four independent vector fields, ${\bf E}$, ${\bf B}$, ${\bf D}$ and ${\bf H}$ which enter the definitions of ${\sf F}$ and ${\sf G}$, and sources $\rho$ and ${\bf j}$ which define the current 3-form ${\mathscr{J}}$.

The system of equations is not closed, and one needs further hypotheses, the constitutive equations, fixed by postulating an action,

$$S_{\rm EM}[A]=-\int{\textstyle\frac{1}{2}}\mu_{0}^{-1}{\sf F}\wedge\star{\sf F}+{\sf A}\wedge\star{\sf J}.$$

(6)

This expression is justified by the requirement that one needs, under the integral, a Lorentz invariant scalar quantity (for a Lorentz invariant theory), or more generally a scalar under arbitrary diffeomorphisms. Indeed, the exterior product of a $p$-form ${\sf u}$ with its dual (or with the dual of another $p$-form) ${\sf u}\wedge\star{\sf u}$ is such a quantity, proportional to a full index contraction $u_{\mu}u^{\mu}$ and to the volume-form $d\hbox{vol}$. Therefore, a term in ${\sf F}\wedge\star{\sf F}$ in the field Lagrangian density, being quadratic in the gauge field derivatives, has the good properties. Also, rather than the 3-form $\mathscr{J}$, let us mention that it is customary to introduce in the action a current 1-form ${\sf J}$: for the same reason as above, one is then confident in the fact that ${\sf A}\wedge\star{\sf J}$ is a scalar invariant multiplying the volume-form.

Furthermore, the free field term displays gauge invariance as it is obviously unchanged in the transformation

$${\sf A}\to{\sf A}+d\upchi,$$

(7)

with $\upchi$ a zero-form. Owing to the charge conservation, this transformation also leaves the interaction term unchanged, up to boundary terms. Indeed, one can write the modification of this term under the gauge transformation (7) as $d\upchi\wedge\star{\sf J}=d(\upchi\wedge\star{\sf J})+\upchi\wedge\star d^{\dagger}{\sf J}$. The first term contributes to the action functional by a surface term which we can neglect by assuming that there are not sources at infinity and, using the properties of the coderivative $d^{\dagger}$, the second term is modified into $\upchi\wedge d{\mathscr{J}}=0$, where use was made of equation (3) and the relation ${\mathscr{J}}=-\star{\sf J}$, still to be shown. Note that the sign of the coupling between the charges and the gauge field is essentially arbitrary and we follow here reference bertlmann_1996 .

Being written in differential forms formalism, the theory is linear, coordinate-free, and displays general covariance. All these properties make it a “pleasant” theory.

Minimization of the action (6) leads to the equation of motion in one of the following forms, in terms of the exterior derivative, $d$, or of the coderivative, $d^{\dagger}$,

$$\mu_{0}^{-1}d^{\dagger}{\sf F}=-{\sf J}\quad\hbox{or}\quad\mu_{0}^{-1}d\star{\sf F}=-\star{\sf J}.$$

(8)

The relation among the current 1-form ${\sf J}$ and the current 3-form ${\mathscr{J}}$ is just ${\mathscr{J}}=-\star{\sf J}$, as one can read from comparing (5) and (8), and this also provides the identification of

$${\sf G}=\mu_{0}^{-1}\star{\sf F}$$

(9)

which is precisely the constitutive relation announced and which closes the system of equations.

A complete theory for electrodynamics also requires the dynamics of the sources given by the Lorentz force, but we don’t need it for the present purpose. It must be remarked that despite Maxwell theory being 150-year-old, the question of its foundations is still debated and our choice of axioms is not unique (for other proposals, see for instance hughes1992feynman ; ObukhovHehl ; sobouti2015lorentz ; heras2016axiomatic ).

One might appreciate having the link with the tensor form of Maxwell equations. The action in this formalism,

	$\displaystyle S_{\rm EM}[A]$	$\displaystyle=$	$\displaystyle\int d^{4}x\ \!{\mathscr{L}}_{\rm EM}$		(10)
		$\displaystyle=$	$\displaystyle-\int d^{4}x\ \!\sqrt{-g}\ \!\Bigl{(}{\textstyle\frac{1}{4}}\mu_{0}^{-1}F_{\mu\nu}F^{\mu\nu}+A_{\mu}j^{\mu}\Bigr{)}$

is probably more usual. It obeys the requirements of being covariant under diffeomorphisms, gauge covariant, and invariant under parity reversal (this latter condition avoids a contribution in $F_{\mu\nu}{\mathscr{F}}^{\mu\nu}$ with the dual field strength tensor ${\mathscr{F}}^{\mu\nu}$ introduced below). It follows that

$$-\frac{\partial{\mathscr{L}}_{\rm EM}}{\partial A_{\rho}}=\sqrt{-g}j^{\rho}$$

(11)

which suggests, together with the Euler-Lagrange equation, to define the constitutive relation in the form of

$$G^{\sigma\rho}=-\frac{\partial{\mathscr{L}}_{\rm EM}}{\partial(\partial_{\sigma}A_{\rho})}=\mu_{0}^{-1}\sqrt{-g}F^{\sigma\rho}$$

(12)

for the equation of motion to be

$$\frac{1}{\sqrt{-g}}\partial_{\sigma}G^{\sigma\rho}=\frac{1}{\mu_{0}}\frac{1}{\sqrt{-g}}\partial_{\sigma}(\sqrt{-g}F^{\sigma\rho})=j^{\rho}$$

(13)

with the conservation of the electric charge

$$\partial_{\rho}(\sqrt{-g}j^{\rho})=0$$

(14)

imposed by the antisymmetry of $G^{\sigma\rho}$.

In tensor form, it is common to introduce the dual field strength tensor

$${\mathscr{F}}^{\mu\nu}={\textstyle\frac{1}{2}}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$

(15)

(with $\epsilon^{\mu\nu\rho\sigma}$ the Levi-Civita tensor) in terms of which the Bianchi identity reads as (in Minkowski spacetime $\mathbb{M}^{4}$)

$$\partial_{\mu}{\mathscr{F}}^{\mu\nu}=0.$$

(16)

From the similarity of equations (2) and (16) on one hand, and of (5) and (13) on the other hand, it is tempting to associate ${\sf F}$ to ${\mathscr{F}}^{\mu\nu}$ and ${\sf G}$ to $G^{\mu\nu}$. However, this is misleading, because (2) and (5) are in fact three-indices equations, e.g. $d{\sf F}=\frac{1}{3!}(d{\sf F})_{\lambda\mu\nu}dx^{\lambda}\wedge dx^{\mu}\wedge dx^{\nu}$. Hence, the proper form of the Bianchi equation is better written as a third rank tensor equation

$$\partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}=0$$

(17)

which is true, even in curved spacetimes, as soon as one has defined the “electromagnetic curvature” as

$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$

(18)

This being said, it is now clear that ${\sf F}$ corresponds to $F_{\mu\nu}$. Accordingly, we can also rewrite the equation of motion as a third rank tensor form, introducing the dual of $G_{\mu\nu}$,

$${\mathscr{G}}^{\mu\nu}={\textstyle\frac{1}{2}}\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma},$$

(19)

as

$$\partial_{\lambda}{\mathscr{G}}_{\mu\nu}+\partial_{\mu}{\mathscr{G}}_{\nu\lambda}+\partial_{\nu}{\mathscr{G}}_{\lambda\mu}=-\sqrt{-g}{\mathscr{J}}_{\lambda\mu\nu}.$$

(20)

This equation is not as usual as (13), but this again shows the correct correspondence, which is between ${\sf G}$ and ${\mathscr{G}}_{\mu\nu}$.

Up to now, we have implicitly assumed a certain number of conventions. It is probably time to become more explicit with the choices used in this paper. We consider objects (tensors) defined on a 4-dimensional metric manifold ${\cal M}$ equipped with a metric tensor $g_{\mu\nu}$. We use the signature convention with $g_{00}\geq 0$, the sign of the antisymmetric Levi-Civita tensor is chosen such as $\epsilon_{0123}=+1$ in $\mathbb{M}^{4}$ (where it coincides with the Levi-Civita symbol) and the orientation of the volume-form is $d\hbox{vol}=\sqrt{-g}\ \!dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}$ with $g=\hbox{det}\ \!(g_{\mu\nu})$. Differential forms are written in sanserif Latin or upright Greek, and their Hodge dual in calligraphic. In this paper, we also mostly follow the notations and conventions of the excellent books Differential geometry, gauge theories, and gravity by Göckeler and Schücker gockeler_schucker_1987 and Bertlmann’s Anomalies in quantum field theory bertlmann_1996 . In particular, it means that we use the Hodge star operation such that it discriminates the forms according to their degree as $\star\star{\sf u}=-{\sf u}$ for even degree while $\star\star{\sf u}={\sf u}$ for odd degree, and the coderivative is simply $d^{\dagger}=\star\ \!d\ \!\star$. Among the other implicit choices, we should specify that the vector fields ${\bf E}$, ${\bf B}$, ${\bf D}$ and ${\bf H}$ have their usual meaning if we define ${\sf F}=-{\sf E}\wedge dt-{\sf B}$ and ${\sf G}=-c{\sf D}+{\sf H}\wedge cdt$ with ${\sf E}$, ${\sf B}$, ${\sf D}$ and ${\sf H}$ differential forms in 3-dimensional space. The signs are imposed by the signature choice. Similarly, the current 3-form is ${\mathscr{J}}=-\rho c+{\sf j}\wedge cdt$. We have decided, contrary to common use, to keep the constants $c$ and $\mu_{0}$, to make explicit the difference between $F^{\mu\nu}$ and $G^{\mu\nu}$, even in flat spacetime.

A very instructive generalization of Maxwell electrodynamics is provided by Kalb-Ramond extension to higher-order tensors (a succinct treatment of the generalization to higher order tensors and arbitrary spacetime dimensions can be found in the book of Ortín, Ortin ). This is an example which illustrates the introduction (besides, or instead of, the Faraday tensor $F_{\mu\nu}$), of new fields, $M_{\lambda\mu\nu}$, which couple to new types of matter sources.

Let us thus assume the existence of a field strength 3-form ${\sf M}$ and a generalized conserved charge 2-form acting as a source term ${\mathscr{S}}$, both forms being closed,

	$\displaystyle d{\sf M}$	$\displaystyle=$	$\displaystyle 0,$		(21)
	$\displaystyle d{\mathscr{S}}$	$\displaystyle=$	$\displaystyle 0.$		(22)

Therefore, there exists a gauge field 2-form ${\sf B}$ and a 1-form ${\sf N}$ such that

	$\displaystyle{\sf M}$	$\displaystyle=$	$\displaystyle d{\sf B},$		(23)
	$\displaystyle{\mathscr{S}}$	$\displaystyle=$	$\displaystyle d{\sf N}.$		(24)

Equations (21) and (24) are equivalent to Maxwell equations (2) and (5). We are now facing the same problem as with Maxwell theory, and we need some constitutive relations to close the system of equations. For that purpose, let us assume an action functional given by an obvious generalization of (6),

$$S_{\rm KR}[B]=-\int{\textstyle\frac{1}{2}}{g_{0}}^{-1}{\sf M}\wedge\star{\sf M}+{\sf B}\wedge\star{\sf S}$$

(25)

with ${g_{0}}$ an unknown dimensional parameter and ${\sf S}$ introduced in the action to describe the sources in analogy with the Maxwell case. The variation of $S_{\rm KR}[B]$ leads to

$\displaystyle\delta S_{\rm KR}[B]$

$\displaystyle=$

$\displaystyle-\int\delta{\sf B}\wedge({g_{0}}^{-1}\star d^{\dagger}{\sf M}+\star{\sf S})+\hbox{BT}$

(26)

such that, up to the boundary terms (BT), the equation of motion follows in either of the two forms

$$d^{\dagger}{\sf M}=-{g_{0}}{\sf S}\quad\hbox{or}\quad d\star{\sf M}=-{g_{0}}\star{\sf S}.$$

(27)

The analogy with (8) is transparent. Using the duality relation for the sources, ${\mathscr{S}}=-\star{\sf S}$, the constitutive relation follows,

$${\sf N}={g_{0}}^{-1}\star{\sf M}.$$

(28)

Although the approach using differential forms has the advantage of a high degree of generality, valid in particular in the case of curved spacetimes, it is convenient to write also these expressions in their tensorial forms, or even in a very traditional vector shape (this was briefly done in flat spacetime in the Appendix of the original paper Kalb1974classical ). Thus, we have an antisymmetric third rank field strength tensor $M_{\lambda\mu\nu}$ which derives from an antisymmetric second-rank gauge tensor $B_{\mu\nu}$

$$M_{\lambda\mu\nu}=\partial_{\lambda}B_{\mu\nu}+\partial_{\mu}B_{\nu\lambda}+\partial_{\nu}B_{\lambda\mu}.$$

(29)

$M_{\lambda\mu\nu}$ has four independent components while $B_{\mu\nu}$ has six. This definition leads to the Bianchi equation

$$\partial_{\kappa}M_{\lambda\mu\nu}-\partial_{\lambda}M_{\mu\nu\kappa}+\partial_{\mu}M_{\nu\kappa\lambda}-\partial_{\nu}M_{\kappa\lambda\mu}=0.$$

(30)

Gauge invariance of the Kalb-Ramond theory is guaranteed by the following transformation of the gauge field

$$B^{\mu\nu}\to{B^{\prime}}^{\mu\nu}=B^{\nu\mu}+\partial^{\mu}\xi^{\nu}-\partial^{\nu}\xi^{\mu}$$

(31)

as it can easily be shown.

The free field Lagrangian density is built in analogy to the Maxwell case, $-\frac{1}{12}{g_{0}}^{-1}M_{\lambda\mu\nu}M^{\lambda\mu\nu}$. The coupling of the gauge field $B_{\mu\nu}$ to external sources requires an antisymmetric second rank tensor describing the matter currents to saturate the Lorentz indices, $-{\textstyle\frac{1}{2}}B_{\mu\nu}S^{\mu\nu}$. Such sources describe extended objects and Kalb-Ramond theory has a natural application in the context of string theories as we said in the introduction. The Kalb-Ramond action reads as

	$\displaystyle S_{\rm KR}[B]$	$\displaystyle=$	$\displaystyle\int d^{4}x\ \!{\mathscr{L}}_{\rm KR}$
		$\displaystyle=$	$\displaystyle-\int d^{4}x\sqrt{-g}\ \!\left({\textstyle\frac{1}{12}}{g_{0}}^{-1}M_{\lambda\mu\nu}M^{\lambda\mu\nu}+{\textstyle\frac{1}{2}}B_{\mu\nu}S^{\mu\nu}\right).$

The Euler-Lagrange equation is easily applied to (LABEL:eq_KRAction), leading to the equation of motion

$$\frac{1}{g_{0}}\frac{1}{\sqrt{-g}}\partial_{\sigma}(\sqrt{-g}M^{\sigma\mu\nu})=S^{\mu\nu},$$

(33)

where the antisymmetry (29) of the Kalb-Ramond field strength tensor has been used.

This equation of motion, together with the antisymmetry of $B^{\mu\nu}$, leads to the conservation of the second-rank tensor current,

$$\partial_{\mu}(\sqrt{-g}\ \!S^{\mu\nu})=0.$$

(34)

Rephrased now in the language of ordinary functions and vectors, we can write equations that look very similar to the usual Maxwell equations. Let us introduce an effective $3$-metric $\gamma_{ij}$ (with $i,j=1,2,3$) via a space-time decomposition

$$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{00}(dx^{0}+(g_{0i}/g_{00})dx^{i})^{2}-\gamma_{ij}dx^{i}dx^{j}$$

(35)

with $\gamma_{ij}=-g_{ij}+g_{0i}g_{0j}/g_{00}$ and $-g=\gamma g_{00}$ with $g$ the $4$-spacetime metric determinant and $\gamma$ that of the $3$-metrics. We define a density $\kappa$ (for “Kalb”) and a pseudo-vector ${\bf R}$ (for “Ramond”) corresponding to two purely spatial differential forms, respectively a 3-form ${\sf K}$ and a 2-form ${\sf R}$ such that ${\sf M}={\sf R}\wedge cdt+{\sf K}$ (here we do not impose negative signs like in the Maxwell case since there are no standardized definitions of the fields ${\bf R}$ and $\kappa$). This enables us to write the second Bianchi equation as two equations, $d_{3}{\sf R}+\frac{1}{c}\partial_{t}{\sf K}=0$ and $d_{3}{\sf K}=0$ (here $d_{3}$ is the purely spatial exterior derivative). The components of $\kappa$ and ${\bf R}$ are obtained from the identification $\kappa=\frac{1}{6}\frac{\epsilon^{ijk}}{\sqrt{\gamma}}K_{ijk}$ with $K_{ijk}=M_{ijk}$ and $R^{i}=\frac{1}{2}\frac{\epsilon^{ijk}}{\sqrt{\gamma}}M_{0jk}$. The first Bianchi identity is then written as

$$\frac{1}{c}\frac{1}{6\sqrt{\gamma}}\epsilon^{ijk}\partial_{t}K_{ijk}-\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}R^{i})=0.$$

(36)

In vector form, we obtain an expression which looks very similar to that of flat spacetime Kalb1974classical ,

$$\frac{1}{c}{\mathscr{D}}_{t}{\kappa}-\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}{\bf R}=0,$$

(37)

with here the divergence operator defined in curved space LandauLifshitz as $\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}{\bf R}=(1/\sqrt{\gamma})\partial_{i}(\sqrt{\gamma}R^{i})$ and the time derivative as ${\mathscr{D}}_{t}{\kappa}=(1/\sqrt{\gamma})\partial_{t}(\sqrt{\gamma}\kappa)$. The second equation following from Bianchi identity written in $3$-dimensional forms does not have any counterpart there, except that $\kappa$ is a density.

The inhomogeneous equations also have their vector counterparts, calling for the introduction of two additional fields, a vector field ${\bf T}$ and a scalar field ${\lambda}$, associated respectively to a 1-form ${\sf T}$ and a zero-form $\uplambda$ which constitute ${\sf N}=g_{0}^{-1}\star{\sf M}={\sf T}+\uplambda\wedge cdt$. The external sources ${\mathscr{S}}=\upsigma+{\sf s}\wedge cdt$ are represented by a pseudo-vector $\boldsymbol{\sigma}$ (corresponding to the 2-form $\upsigma$) and a vector, ${\bf s}$ (the 1-form ${\sf s}$). We get then the equations of motion $d_{3}{\sf T}=\upsigma$ and $d_{3}\uplambda+\frac{1}{c}\partial_{t}{\sf T}={\sf s}$. In terms of ordinary fields ${\bf T}$ and ${\lambda}$, one has

	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}{\bf T}=\boldsymbol{\sigma}},$		(38)
	$\displaystyle{\color[rgb]{0,0,0}\frac{1}{c}\mathscr{D}_{t}{\bf T}-\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}{\lambda}={{\bf s}}},$		(39)

themselves analogs of

	$\displaystyle{\boldsymbol{\nabla}}\cdot{\bf D}=\rho,$		(40)
	$\displaystyle{\boldsymbol{\nabla}}\times{\bf H}-\partial_{t}{\bf D}={\bf j}$		(41)

in Maxwell theory. The components of the vector ${\bf T}$ are those of the one-form ${\sf T}$, $T_{i}=\frac{1}{2g_{0}}\sqrt{\gamma}\sqrt{g_{00}}\epsilon_{ijk}M^{0jk}$ and the scalar field $\lambda$ is defined as $\lambda=-\frac{1}{6g_{0}}\sqrt{\gamma}\sqrt{g_{00}}\epsilon_{ijk}M^{ijk}$. The components of the sources $\boldsymbol{\sigma}$ are $\boldsymbol{\sigma}|^{i}=\sqrt{g_{00}}s^{0i}$ and those of ${\bf s}$ are built from the space components of the two-form ${\sf s}$, ${\bf s}|_{i}=\sqrt{g_{00}}s_{i}=\frac{1}{2}\sqrt{g_{00}}\sqrt{\gamma}\epsilon_{ijk}{\mathscr{S}}^{jk}$. The curl of a vector field in curved space is defined as LandauLifshitz

$$\left.\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}{\bf T}\right|^{i}=(1/\sqrt{\gamma}){\epsilon^{ijk}}\partial_{j}T_{k},$$

(42)

and the gradient operator is defined ordinarily as $\left.\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}{\lambda}\right|_{i}=\partial_{i}\lambda$.

There is an interesting observation made by Landau and Lifshitz LandauLifshitz in the case of Maxwell electrodynamics:

The reader should note the analogy (purely formal, of course) with Maxwell equations for the electromagnetic field in material media. In particular, in a static gravitational field the quantity $\sqrt{\gamma}$ drop out of the terms containing time derivatives (…) We may say that with respect of its effect on the electromagnetic field a static gravitational field plays the role of a medium with electric and magnetic permeabilities $\epsilon=\mu=1/\sqrt{g_{00}}$.

It also simplifies the problem here, because then most of the flat spacetime formulas of vector analysis apply and the differential operators ${\mathscr{D}}_{t}$ and $\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}$ or $\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}$ commute with each other.

The conservation of the Kalb-Ramond charges follows from the equations of motion and reads as $d_{3}\upsigma=0$ and $d_{3}{\sf s}+\frac{1}{c}\partial_{t}\upsigma=0$. In vector formalism, equation (38) demands that $\boldsymbol{\sigma}$ is divergence free and, taking the curl of (39), we get a continuity equation of the form (44)

	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}}\boldsymbol{\sigma}=0,$		(43)
	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}}{{\bf s}}{\color[rgb]{0,0,0}-}\frac{1}{c}\partial_{t}{\boldsymbol{\sigma}}=0.$		(44)

The definition of gauge potentials is suggested by the form of the Bianchi equations above. They suggest the introduction of a potential 2-form $\upalpha$ and a potential 1-form $\upphi$ such that ${\sf K}=d_{3}\upalpha$ and ${\sf R}=-d_{3}\upphi-\frac{1}{c}\partial_{t}\upalpha$. In vector notation, these translate into two gauge vector potentials $\boldsymbol{\alpha}$ and $\boldsymbol{\phi}$ such that the Kalb-Ramond fields are defined in terms of these as

	$\displaystyle{\color[rgb]{0,0,0}\kappa=\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}\boldsymbol{\alpha},}$		(45)
	$\displaystyle{\color[rgb]{0,0,0}{\bf R}=-\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}\boldsymbol{\phi}-\frac{1}{c}\partial_{t}\boldsymbol{\alpha},}$		(46)

with $\alpha^{i}=\frac{1}{2}\frac{\epsilon^{ijk}}{\sqrt{\gamma}}\alpha_{jk}$ the components of the vector $\boldsymbol{\alpha}$ in terms of those of the $2$-form $\upalpha$.

We know that in vector notation, Maxwell theory has in the vacuum, and in the absence of a gravitational field, the nice property that there is a simple proportionality between ${\bf E}$ and ${\bf D}$ on one hand and between ${\bf B}$ and ${\bf H}$ on the other hand, these are the constitutive relations in 3$D$ formalism. Therefore, the four coupled first-order partial differential Maxwell equations decouple into two second-order equations known as wave equations. This property in fact hides a duality relation in terms of forms. The same strategy can be used in Kalb-Ramond theory, wherefrom ${g_{0}}^{-1}\star({\sf R}\wedge cdt)={\sf T}$ and ${g_{0}}^{-1}\star{\sf K}=\uplambda cdt$, and one deduces the constitutive relations in the form

	$\displaystyle{g_{0}}^{-1}{\bf R}={\bf T},$		(47)
	$\displaystyle{g_{0}}^{-1}\kappa=\lambda.$		(48)

This leads to the Kalb-Ramond equations, now written only in terms of the fields ${\bf R}$ and $\kappa$,

	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}}{\bf R}{\color[rgb]{0,0,0}-}\frac{1}{c}\partial_{t}{\kappa}=0,$		(49)
	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}}{\bf R}={g_{0}}\boldsymbol{\sigma},$		(50)
	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}\kappa}{\color[rgb]{0,0,0}-}\frac{1}{c}\partial_{t}{\bf R}={g_{0}}{\bf s}.$		(51)

One may wonder why only three equations are required in Kalb-Ramond theory whereas four are needed in Maxwell’s (once constitutive relations have been used). This is a natural consequence of Helmholtz decomposition theorem: Maxwell theory involves two dynamic vector fields (each known from its curl and divergence), whereas Kalb-Ramond’s involves one dynamic vector field and one dynamic scalar field.

Taking the time partial derivative of (49) and using (51) we get, for the scalar field $\kappa$, a non-homogeneous wave equation

$${\color[rgb]{0,0,0}\Delta_{\rm LB}}{\kappa}-\frac{1}{c^{2}}\partial_{t}^{2}{\kappa}={\color[rgb]{0,0,0}g_{0}}{\boldsymbol{\nabla}}\cdot{\bf s},$$

(52)

with the Laplace-Beltrami operator

$$\Delta_{\rm LB}\kappa=\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}(\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}\kappa)=\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}\gamma^{ij}\partial_{j}\kappa).$$

(53)

The wave equation takes the standard form $\Box\kappa=0$ in flat empty space, meaning that the scalar field propagates in the form of waves at the light velocity $c$. Now, taking the gradient of (49), the curl of (50), and combining with (51) also leads to a non-homogeneous wave equation for the vector field ${\bf R}$,

	$\displaystyle{\color[rgb]{0,0,0}\operatorname{{\mathop{\mathbf{missing}}{Grad}\ \!}}(\operatorname{{\mathop{\mathrm{missing}}{Div}\ \!}}{\bf R})-\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}(\operatorname{{\mathop{\mathbf{missing}}{Curl}\ \!}}{\bf R})}$	$\displaystyle-$	$\displaystyle\frac{1}{c^{2}}\partial_{t}^{2}{\bf R}$
			$\displaystyle={\color[rgb]{0,0,0}-}g_{0}({\boldsymbol{\nabla}}\times{\boldsymbol{\sigma}}+\frac{1}{c}\partial_{t}{\bf s}).$

Again, one retrieves $\Box{\bf R}=0$ in source-free flat space.

Let us first contemplate the case of a cosmic string crossing the origin of a Schwarzschild blackhole spacetime. This is an example of a spacetime characterized by a singular curvature.

We use the Schwarzschild coordinates $ct,r,\theta,\varphi$ and the line element

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\Bigl{(}1-\frac{2M}{r}\Bigr{)}c^{2}dt^{2}$		(58)
			$\displaystyle-\Bigl{(}1-\frac{2M}{r}\Bigr{)}^{-1}dr^{2}-r^{2}d\theta^{2}-\alpha^{2}r^{2}\sin^{2}\theta d\varphi^{2}\quad$

where $\alpha$ is the cosmic string defect-angle parameter ($\alpha^{2}=1-4G\mu/c^{2}$ with $\mu$ the string stress tensor amplitude). Instead of the local basis $(cdt,dr,d\theta,d\varphi)$, the tetrad formalism allows to write the line element in an orthonormalized basis $({\sf e}^{0},{\sf e}^{1},{\sf e}^{2},{\sf e}^{3})$,

$$ds^{2}=({\sf e}^{0})^{2}-({\sf e}^{1})^{2}-({\sf e}^{2})^{2}-({\sf e}^{3})^{2}$$

(59)

with Minkowski metric. Here, the coframe basis vectors are obtained by inspection, ${\sf e}^{0}=\Bigl{(}1-\frac{2M}{r}\Bigr{)}^{1/2}cdt$, ${\sf e}^{1}=\Bigl{(}1-\frac{2M}{r}\Bigr{)}^{-1/2}dr$, ${\sf e}^{2}=rd\theta$, and ${\sf e}^{3}=\alpha r\sin\theta d\varphi$. Using the equivalent of Gauss law, (38), which, due to the form degree is more like an Ampère law for Kalb-Ramond fields, we can write

$$\int_{\Sigma}d{\sf T}=\int_{\partial\Sigma}{\sf T}=\int_{\Sigma}\upsigma$$

(60)

and call $\Phi_{\sigma}$ this quantity, by analogy with a flux. Let us mention again that this quantity is produced by some charge density $\upsigma$ which is not specified more, and in particular, which is not the source of the metric (58). Equation (60) is a counting procedure of the Kalb-Ramond charges crossing the compact domain $\Sigma$: as underlined by ObukhovHehl , (60) defines a purely topological quantity and must not depend explicitly on the spacetime geometry (physical parameters $M$ and $\alpha$). Assuming cylindrical symmetry in the Minkowskian coframe, we choose for the surface $\Sigma$ a disk of radius $\rho=r\sin\theta$ with contour $\partial\Sigma$. The total KR charge crossing the disk is characterized by the flux denoted as $\Phi_{\sigma}$. The 1-form ${\sf T}$ can be expressed either on the local basis, or using the dual tetrad basis $({\sf e}^{0},{\sf e}^{1},{\sf e}^{2},{\sf e}^{3})$ where cylindrical symmetry imposes an expression of the form ${\sf T}=T_{3}{\sf e}^{3}$. Then, using the relation to the local basis, ${\sf e}^{3}=\alpha\sin\theta d\varphi$, one gets

$${\sf T}=T_{3}{\sf e}^{3}=T_{3}\alpha r\sin\theta d\varphi=T_{\varphi}d\varphi$$

(61)

with $T_{3}=T_{3}(\rho)=T_{3}(r\sin\theta)$. Therefore one deduces from (60)

$$T_{3}(r\sin\theta)=\frac{\Phi_{\sigma}}{2\pi\alpha r\sin\theta}=\frac{\Phi_{\sigma}}{2\pi\alpha\rho},\quad\hbox{and}\quad T_{\varphi}=\frac{\Phi_{\sigma}}{2\pi}.$$

(62)

Now, in absence of the field ${\sf K}$, using the relations

$${\sf M}={\sf R}^{\prime}\wedge{\sf e}^{0}={\sf R}\wedge cdt=g_{0}\star{\sf N}={g_{0}}\star{\sf T}$$

(63)

and the expression of the Hodge dual in the local co-tetrad frame (with metric tensor there denoted as $\eta_{ab}$),

$$\star{\sf T}=\sqrt{|\eta|}{\eta}^{33}T_{3}\epsilon_{3012}{\sf e}^{0}\wedge{\sf e}^{1}\wedge{\sf e}^{2},$$

(64)

we can read that

	$\displaystyle g_{0}\star{\sf T}$	$\displaystyle=$	$\displaystyle g_{0}T_{3}{\sf e}^{1}\wedge{\sf e}^{2}\wedge{\sf e}^{0}$		(65)
		$\displaystyle=$	$\displaystyle g_{0}\frac{T_{\varphi}}{\alpha\sin\theta}dr\wedge d\theta\wedge cdt.$

The first line leads to the “Minkwoskian” identification $R^{\prime}_{12}=g_{0}T_{3}$, while in the local basis, we can interpret our result as a non-trivial vacuum polarization, since we can write

$$R_{r\theta}=g(r\sin\theta)T_{\varphi}=\frac{g_{0}}{\alpha\sin\theta}\frac{\Phi_{\sigma}}{2\pi}$$

(66)

with $g(r\sin\theta)=\frac{g_{0}}{\alpha\sin\theta}$ the analogue of a relative permittivity.

One may wonder about the robustness of this result with respect to a change of local coordinates. For that purpose, we will now consider a cosmic string in a Schwarzschild spacetime with Israel coordinates mueller2010catalogue . The main interest is that the metric tensor is not diagonal in this case. The Schwarzschild plus string metric in Israel coordinates $(y,x,\theta,\varphi)$ reads

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle-r_{S}^{2}\Bigl{[}4dx\Bigl{(}\frac{y^{2}dx}{1+xy}+dy\Bigr{)}$		(67)
			$\displaystyle\qquad+(1+xy)^{2}(d\theta^{2}+\alpha^{2}\sin^{2}\theta d\varphi^{2})\Bigr{]}$

where $x$ and $y$ are linked to Schwarzschild coordinates via $t=r_{S}(1+xy+\ln(y/x))$ and $r=r_{S}(1+xy)$. We have also added the cosmic string with defect-angle $\alpha$. The (opposite of the) metric tensor takes the form

$$(-g_{\mu\nu})=\begin{pmatrix}0&2r_{S}^{2}&0&0\\ 2r_{S}^{2}&\frac{4r_{S}^{2}y^{2}}{1+xy}&0&0\\ 0&0&r_{S}^{2}(1+xy)^{2}&0\\ 0&0&0&r_{S}^{2}(1+xy)^{2}\alpha^{2}\sin^{2}\theta\end{pmatrix}.$$

(68)

and its determinant is

$$-g=4r_{S}^{8}(1+xy)^{4}\alpha^{2}\sin^{2}\theta.$$

(69)

The cotetrad vectors are given by

	$\displaystyle{\sf e}^{0}=r_{S}\frac{\sqrt{1+xy}}{y}dy,$		(70)
	$\displaystyle{\sf e}^{1}=r_{S}\left(\frac{2y}{\sqrt{1+xy}}dx+\frac{\sqrt{1+xy}}{y}dy\right),$		(71)
	$\displaystyle{\sf e}^{2}=r_{S}(1+xy)d\theta,$		(72)
	$\displaystyle{\sf e}^{3}=r_{S}(1+xy)\alpha\sin\theta d\varphi,$		(73)

with again Eq. (59) satisfied by this choice. Note that we specify (by choice) that the $0$ coordinate is along $y$. Again, in the orthonormalized basis, one has

$$\int{\sf T}=\Phi_{\sigma}=\int T_{3}(\rho)\ \!{\sf e}^{3}$$

(74)

which requires that

$$T_{3}(\rho)=\frac{\Phi_{\sigma}}{2\pi r_{S}(1+xy)\alpha\sin\theta}=\frac{\Phi_{\sigma}}{2\pi\alpha\rho}$$

(75)

and, like in Schwarzschild coordinates, $T_{\varphi}=\Phi_{\sigma}/(2\pi)$ in the coordinate basis. The Hodge dual $\star{\sf T}$ is the same as the first line of equation (65) in the orthonormalized basis, but it delivers, in the present coordinate basis, the following expression

$$R_{x\theta}=\frac{2g_{0}r_{S}^{2}}{\alpha\sin\theta}\frac{\Phi_{\sigma}}{2\pi},$$

(76)

hence a different “permittivity”. The consequences of the choice of coordinate basis will be commented in the discussion.

Cosmic strings may have a structure, and the case of the “wiggly” cosmic string is particularly interesting for our purpose, since the line element acquires a further space dependence. The presence of wiggles indeed generates a far gravitational field contribution and averaging the effect of these perturbations along the string increases the linear mass density ${\tilde{\mu}}$ and decreases the string tension $\tilde{T}$, (with an equation of state ${\tilde{\mu}}\>\tilde{T}=\mu^{2}$), leading the wiggly string to exert a gravitational field at large distances. We can thus address here the question of the form of the vacuum polarization in such a case.

In the weak gravitational field approximation, the linearized line element in the presence of a wiggly string oriented along the $z$-axis is given by Vilenkin1990 ; Vachaspati1991 ; Vilenkin1994 ; PhysRevD.96.084047 :

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\left(1+8\varepsilon\ln\left({\rho}/{\rho_{0}}\right)\right)c^{2}dt^{2}-d\rho^{2}$		(77)
			$\displaystyle\qquad-\alpha^{2}\rho^{2}d\varphi^{2}-\left(1-8\varepsilon\ln\left({\rho}/{\rho_{0}}\right)\right)dz^{2},$

where the conical deficit-angle associated to the string is now $\alpha^{2}=1-4G({\tilde{\mu}}+\tilde{T})/c^{2}$ (with $4G({\tilde{\mu}}+\tilde{T})/c^{2}\ll 1$). The parameter $\varepsilon$ defines the excess of mass-energy density, $2\varepsilon=G({\tilde{\mu}}-\tilde{T})/c^{2}$. The value of $G(\tilde{\mu}+\tilde{T})$ accounts for the discrepancy between flat and conical geometries, whereas $G({\tilde{\mu}}-\tilde{T})$ accounts for the discrepancy between straight and wiggly strings. The constant $\rho_{0}$ denotes the effective string radius patrick1994 . We also consider the limit where $\varepsilon\ln\left({\rho}/{\rho_{0}}\right)\ll 1$.

The calculation of ${\sf T}$ follows the same lines of reasoning as in the previous section, and one just has a slight modification due to the use of a different ordering of the local (cylindrical) coordinates, $(ct,\rho,\varphi,z)$, which now demands that ${\sf T}=T_{2}{\sf e}^{2}$, with ${\sf e}^{2}=\alpha\rho d\varphi$, thus

$${\sf T}=\frac{\Phi_{\sigma}}{2\pi\alpha\rho}{\sf e}^{2}=\frac{\Phi_{\sigma}}{2\pi}d\varphi.$$

(78)

The dual follows,

$$\star{\sf T}=\frac{\Phi_{\sigma}}{2\pi\alpha\rho}dz\wedge d\rho\wedge cdt,$$

(79)

hence $R_{z\rho}=g_{0}T_{\varphi}/(\alpha\rho)$.

Probably more interesting is the case of a non-diagonal metric associated with non-zero torsion, axially localized. As an example, we consider a chiral cosmic string for which the line element is given by

$$ds^{2}=c^{2}dt^{2}-d\rho^{2}-\rho^{2}d\varphi^{2}-(\beta d\varphi+dz)^{2},$$

(80)

where $\beta$ is the Burgers parameter of the cosmic string. Like in the previous example, the string at the origin of the form of the metric is not the source of the KR field that we calculate. We can also write down the metric tensor

$$(g_{\mu\nu})=\begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-(\rho^{2}+\beta^{2})&-\beta\\ 0&0&-\beta&-1\end{pmatrix},$$

(81)

which allows building the dual tetrad basis ${\sf e}^{0}=cdt$, ${\sf e}^{1}=d\rho$, ${\sf e}^{2}=\rho d\varphi$ and ${\sf e}^{3}=\beta d\varphi+dz$.

Due to the use of the same system of local coordinates, the 1-form ${\sf T}$ takes the same simple form as for the wiggly string, ${\sf T}=T_{2}(\rho){\sf e}^{2}$, and its line integral along the contour of radius $\rho$ is again transparent

$$\int{\sf T}=\int T_{2}\ \!{\sf e}^{2}=\int T_{2}(\rho)\rho d\varphi=2\pi\rho T_{2}(\rho)=\Phi_{\sigma}.$$

(82)

It follows the same expression (78) on the local coordinate basis. Then we obtain

	$\displaystyle g_{0}\star{\sf T}$	$\displaystyle=$	$\displaystyle g_{0}T_{2}{\sf e}^{3}\wedge{\sf e}^{1}\wedge{\sf e}^{0}$		(83)
		$\displaystyle=$	$\displaystyle g_{0}\frac{\Phi_{\sigma}}{2\pi\rho}(dz\wedge d\rho-\beta d\rho\wedge d\varphi)\wedge cdt$

and, as a result of the non-diagonal character of the metric, the spacetime of the chiral cosmic string induces a non-zero vacuum polarization, with anisotropic, space-dependent polarizability and two non-vanishing components for the 2-form ${\sf R}$

	$\displaystyle R_{z\rho}=\frac{g_{0}}{\rho}{T_{\varphi}},$		(84)
	$\displaystyle R_{\rho\varphi}=-\frac{g_{0}\beta}{\rho}{T_{\varphi}}.$		(85)