Neutrino currents in the absolute spacetime: Relating the refractive index of the aether to the OPERA excess velocity

doi:10.1209/0295-5075/97/39003

Neutrino currents in the absolute spacetime: Relating the refractive index of the aether to the OPERA excess velocity

R. Tomaschitz

R. Tomaschitz 2012 EPL 97 39003
doi:10.1209/0295-5075/97/39003
Received 12 October 2011, accepted for publication 2 January 2012
Published 6 February 2012

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Abstract

The superluminal neutrino velocity measured by the OPERA experiment is explained in a non-relativistic spacetime conception. Spacetime is viewed as a permeable medium of wave propagation. The neutrino wave equation is coupled to a permeability tensor, like electromagnetic fields in dielectric media. The inertial frame in which this tensor is isotropic defines a distinguished frame of reference, the rest frame of the aether. The dispersion relation of the spinorial wave modes gives rise to a superluminal group velocity of the energy flux. The Gordon decomposition of spinor currents in a refractive and dispersive spacetime is performed with finite as well as zero rest mass. The convective and spin components of the superluminal neutrino current are related to the permeability tensor. The refractive index of the aether depends on the neutrino energy, and is inferred in the 10 to 50GeV range from the measured excess velocity. Implications of the superluminal speed of signal transfer regarding relativity principles and causality are discussed.

Introduction

The OPERA experiment [1] measured a neutrino excess speed of υ/c-1 ≈2.48±0.6×10⁻⁵, at an average neutrino energy of 17GeV. This was inferred from a time-of-flight measurement, with a muon neutrino beam over a baseline of 730km between the source at CERN and the OPERA detector at the underground Gran Sasso Laboratory. We explain this excess speed in an absolute spacetime conception, by coupling the neutrino wave equation to the permeability tensor of the aether [2,3].

We discuss the coupling of Dirac spinors to a frequency-dependent permeability tensor, the relation to the Klein-Gordon equation in a dispersive spacetime, as well as the eikonal limit. The permeable spacetime gives rise to a superluminal group velocity. We analyze spinor currents in the aether, and perform the Gordon decomposition into a convection and spin component. The spin current is described in terms of a magnetization and polarization field. We relate the energy flux to the superluminal convection current and show that it propagates with group velocity in the aether.

Finally we identify the superluminal group velocity of a neutrino current with the OPERA excess velocities at three different frequencies, and find, by interpolation, the frequency variation of the refractive index of the aether for neutrinos in the 10–50GeV interval. In the conclusion, we further elaborate on the absolute spacetime, in particular on causality, inertial frames, and the rest frame of the aether as manifested by the isotropic cosmic microwave background radiation.

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Dirac equation coupled to the permeability tensor of the aether

We start with the Dirac Lagrangian

$\begin{equation*} \begin{align*} \begin{equation} L = \frac{1}{{\rm{i}}}\left(\frac{1}{2}\bar \psi \gamma_\mu \eta^{\mu \nu } \psi_{,\nu}-\frac{1}{2}\bar \psi_{,\mu } \eta^{\mu \nu } \gamma_\nu \psi + m\bar \psi \psi\right), \end{equation} \end{align*} \tag{1}\end{equation*}$

where we substitute a real symmetric permeability tensor for the Minkowski metric, $\eta^{\mu \nu } \to g^{\mu \nu }$ . First, we assume that g^μν is constant. Standard Euler variation gives the Dirac equation and its adjoint,

$\begin{equation*} \begin{align*} \begin{equation} \gamma_\mu g^{\mu \nu} \psi_{,\nu}+ m\psi= 0,\quad \bar\psi_{,\mu}g^{\mu \nu} \gamma_\nu-m\bar \psi = 0. \end{equation} \end{align*} \tag{2}\end{equation*}$

ψ^† indicates transposition and complex conjugation, and $\bar \psi = \psi^{\rm{\dagger }} \gamma^0$ is the adjoint 4-spinor. A subscript comma followed by an index denotes a partial derivative, $\bar \psi_{,\mu } = \partial_\mu \bar \psi$ . Indices are raised and lowered with the Minkowski metric, η_μν=diag(-1, 1, 1, 1), and we have put $\hbar=c=1$ . Greek indices run from 0 to 3, Latin ones from 1 to 3. The inverse of the permeability tensor is denoted by g_μν⁻¹, so that g_μα⁻¹g^αν=δ_μ ^ν, whereas g_μν=η_μα g^αβ η_βν. The Dirac matrices satisfy γ_μ γ_ν+γ_ν γ_μ=2η_μν, where γ₀ is antihermitian and the γ_i are hermitian [4].

We will consider a homogeneous and isotropic permeability tensor, so that g^μν is diagonal,

$\begin{equation*} \begin{align*} \begin{equation} g^{00} = -\varepsilon,\quad g^{ik} = \delta^{ik} /\mu ,\quad g^{0k} = 0, \end{equation} \end{align*} \tag{3}\end{equation*}$

where ε and μ are real positive permeabilities, assumed to be constant at this point. The rest frame of the aether defines a distinguished frame of reference, a frame of absolute rest. This frame is physically manifested as the rest frame of the cosmic microwave background radiation, which is isotropic in this frame, with vanishing dipole anisotropy [5,6]. The Solar system barycenter is moving in this frame with a speed of 369km/s toward (l,b) ≈ (264.0°,48.26°) in Galactic coordinates [7–10]. The angle between the baseline of the neutrino beam and the velocity vector of the barycenter is periodically changing. This affects the superluminal neutrino speed measured in the rest frame of the barycenter, which can substantially differ from the neutrino speed in the aether frame. At superluminal velocities much higher than the speed of light, this angular dependence can even affect the causality interpretation. However, since the neutrino excess speed is very small and the velocity of the barycenter much smaller than the speed of light, we can assume that the measurement of the neutrino velocity is carried out in the rest frame of the microwave background, cf. the conclusion.

We perform a time separation of the wave field, using the factorization $\psi = \hat{\psi} (\omega ){\rm e}^{-{\rm i}\omega t}$ , $\bar{\psi}= \skew5\bar{\skew5\hat{\psi}} (\omega ){\rm e}^{{\rm i}\omega t}$ , where $\hat{\psi}$ is time independent, and use the permeability tensor (3) to write the Lagrangian (1) as

$\begin{equation*} \begin{align*} \begin{equation} \hat L = \omega \varepsilon \skew5\bar{\skew5\hat{\psi}} \gamma_0 \hat{\psi} + \frac{1}{{2{\rm{i}}\mu }}(\skew3\bar{\skew5\hat{\psi}} \gamma_i \hat{\psi}_{,i} -\skew5\bar{\skew5\hat{\psi}}_{,i} \gamma_i \hat{\psi} ) + \frac{m}{{\rm{i}}}\skew5\bar{\skew5\hat{\psi}} \hat{\psi}. \end{equation}\noindent \end{align*} \tag{4}\end{equation*}$

Time-separated quantities will be denoted by a hat.

Applying the Dirac operator γ_μ g^μν ∂_ν+m onto the first equation in (2), we recover the Klein-Gordon equation coupled to the permeability tensor,

$\begin{equation*} \begin{align*} \begin{equation} (h^{\mu \nu } \partial_\mu \partial_\nu -m^2 )\psi = 0, \end{equation}\noindent \end{align*} \tag{5}\end{equation*}$

where h^μν=g^μα η_αβg^βν. The diagonal g^μν in (3) gives h⁰⁰=−ε², h^ik=δ^ik /μ², h^0k=0. When time-separated, this reads

$\begin{equation*} \begin{align*} \begin{equation} (\Delta + k^2 )\hat{\psi} = 0,\quad k^2 = \mu^2 (\varepsilon^2 \omega^2 -m^2), \end{equation} \end{align*} \tag{6}\end{equation*}$

where Δ is the Euclidean Laplace operator, and the time-separated Dirac equations (2) are

$\begin{equation*} \begin{align*} \begin{eqnarray} &&\gamma_i \hat{\psi}_{,i}\!+\!({\rm{i}}\gamma_0 \mu \varepsilon \omega\!+\!\mu m)\hat{\psi}\!=\! 0,\quad \skew5\bar{\skew5\hat{\psi}}_{,i} \gamma_{i}\!-\!\skew5\bar{\skew5\hat{\psi}} ({\rm{i}}\gamma_0 \mu \varepsilon \omega\!+\! \mu m)\!=\! 0.\nonumber\\ \end{eqnarray} \end{align*} \tag{7}\end{equation*}$

These equations can directly be derived by Euler variation of the Lagrangian (4). From now on, we take $\hat L$ in (4) and the Dirac equations (7) as starting point (instead of (1) and (2)), and substitute frequency-dependent permeabilities ε (ω) and μ (ω). Moreover, applying the operator γ_i ∂_i+(iγ₀ ωμ ε+μm) onto the first equation in (7), one can derive the time-separated Klein-Gordon equation (6) with frequency-dependent permeabilities. If the spinor is a plane-wave solution $\hat{\psi} = \tilde \psi {\rm{e}}^{{\rm{i}}{\bf{kx}}}$ of (7), where $\tilde \psi$ is a constant 4-spinor, it also satisfies (6), which requires the dispersion relation

$\begin{equation*} \begin{align*} \begin{equation} k^2 = \mu^2 (\omega )(\varepsilon^2 (\omega )\omega^2 -m^2 ), \end{equation} \end{align*} \tag{8}\end{equation*}$

connecting the wave number k(ω )=|k| with the permeabilities. A positive squared wave number implies the frequency condition ε (ω )>m/|ω |. Dispersion is modeled by frequency-dependent permeabilities in time-separated quantities [11,12]. The dispersion relation (8) also follows from the Hamilton-Jacobi equation, h^μν S_,μ S_,ν+m²=0, since ψ∝ e^iS, with the action S=k(ω )x-ωt, so that the semiclassical approximation of (5) is exact: the phase of the quantum-mechanical elementary waves is the classical action or eikonal.

The permeability tensor g^μν in the Dirac equation (2) is unrelated to electromagnetic permeabilities, even though the formalism resembles that of electromagnetic fields in dielectrics. Electromagnetic radiation is not affected by the permeability tensor of neutrinos. In fact, this would even contradict the Michelson-Morley experiment. Different particles couple with different permeability tensors to the aether.

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Convection and spin currents in the dispersive spacetime

We start with a constant permeability tensor g^μν, and consider two solutions ψ and phi of the Dirac equation (2), so that the current $j^\mu= -q\bar \varphi \gamma_\nu g^{\nu \mu } \psi$ is conserved, ∂_μ j^μ=0. q is some charge carried by the current. The charge density is defined by ρ=j⁰. Using the diagonal permeability tensor (3) (frequency independent), we find $\rho = q\varepsilon \bar \varphi \gamma_0 \psi$ , ${\bf{j}} = -(q/\mu )\bar \varphi \gamma_n \psi$ , and the continuity equation ∂ρ /∂t+div j=0.

To derive the continuity equation in the case of frequency-dependent permeabilities, we employ the time-separated Dirac equations,

$\begin{equation*} \begin{align*} \begin{equation}\quad\quad\gamma_i \hat{\psi}_{,i} (\omega ) + [{\rm{i}}\gamma_0 \mu (\omega )\varepsilon (\omega )\omega + \mu (\omega )m]\hat{\psi} (\omega ) = 0,\quad \end{equation} \end{align*} \tag{9}\end{equation*}$

$\begin{equation*} \begin{align*} \begin{equation} \skew3\bar{\skew5\hat{\varphi}}_{,i} (\omega')\gamma_i -\skew3\bar{\skew5\hat{\varphi}} (\omega')[{\rm{i}}\gamma_0 \mu (\omega')\varepsilon (\omega')\omega' + \mu (\omega')m] = 0, \end{equation} \end{align*} \tag{10}\end{equation*}$

so that the divergence $(\skew3\bar{\skew5\hat{\varphi}} \gamma_i \hat{\psi} )_{,i}$ vanishes at ω'=ω. The spinors $\hat{\psi} (\omega )$ and $\skew3\bar{\skew5\hat{\varphi}} (\omega')$ also satisfy the Klein-Gordon equation (6) with dispersion relation (8). The time-dependent Fourier modes are $\psi = \hat{\psi} (\omega ){\rm{e}}^{-{\rm{i}}\omega t}$ and $\bar \varphi = \skew3\bar{\skew5\hat{\varphi}} (\omega'){\rm{e}}^{{\rm{i}}\omega't}$ . If not indicated otherwise, $\hat{\psi}$ means $\hat{\psi} (\omega )$ and $\skew3\bar{\skew5\hat{\varphi}}$ stands for $\skew3\bar{\skew5\hat{\varphi}} (\omega')$ .

We use the time separation $\rho = \hat \rho {\rm{e}}^{{\rm{i}}(\omega'-\omega )t}$ and ${\bf{j}} = {\bf{\hat j}}{\rm{e}}^{{\rm{i}}(\omega'-\omega )t}$ , and define the current as

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}} = -\frac{q}{2}\left({\frac{{\skew3\bar{\skew5\hat{\varphi}} (\omega')\gamma_k \hat{\psi} (\omega )}}{{\mu (\omega')}} + \frac{{\skew3\bar{\skew5\hat{\varphi}} (\omega')\gamma_k \hat{\psi} (\omega )}}{{\mu (\omega )}}} \right), \end{equation} \end{align*} \tag{11}\end{equation*}$

which coincides with ${\bf{\hat j}} = -(q/\mu )\skew3\bar{\skew5\hat{\varphi}} \gamma_n \hat{\psi}$ stated above for constant permeabilities. In the first term, we substitute the adjoint Dirac equation (10), and use (9) in the second term, to decompose the current into a convection and a spin current, ${\bf{\hat j}} = {\bf{\hat j}}_{\rm{C}} + {\bf{\hat j}}_{\rm{S}}$ , cf. [13,14],

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{C}} = -\frac{q}{{2m}}\left({\frac{{\skew3\bar{\skew5\hat{\varphi}}_{,k} \hat{\psi} }}{{\mu^2 (\omega')}}-\frac{{\skew3\bar{\skew5\hat{\varphi}} \hat{\psi}_{,k} }}{{\mu^2 (\omega )}}} \right),\hphantom{00000,}\end{equation}\noindent \end{align*} \tag{12}\end{equation*}$

$\begin{equation*} \begin{align*} \begin{eqnarray} {\bf{\hat j}}_{\rm{S}}&=&-\frac{q}{{2m}}\left[\frac{{\skew3\bar{\skew5\hat{\varphi}}_{,i} \sigma_{ik} \hat{\psi}}}{{\mu^2 (\omega')}}-\frac{{\skew3\bar{\skew5\hat{\varphi}} \sigma_{ki} \hat{\psi}_{,i} }}{{\mu^2 (\omega)}}\right.\nonumber\\ &&\left.-{\rm{i}}\left(\frac{{\varepsilon (\omega')\omega'}}{{\mu (\omega')}}-\frac{{\varepsilon (\omega)\omega }}{{\mu (\omega)}}\right)\skew3\bar{\skew5\hat{\varphi}} \sigma_{0k} \hat{\psi} \right],\nonumber\\ \end{eqnarray}\noindent \end{align*} \tag{13}\end{equation*}$

where σ_μν=(γ_μ γ_ν−γ_ν γ_μ )/2, so that γ_μ γ_ν=η_μν+σ_μν. We also split the charge density into a convection and a spin density, $\hat \rho = \hat \rho_{\rm{C}} + \hat \rho_{\rm{S}}$ , and use the time separation $\rho_{{\rm{C}},{\rm{S}}} = \hat \rho_{{\rm{C}},{\rm{S}}} {\rm{e}}^{{\rm{i}}(\omega'-\omega )t}$ , ${\bf{j}}_{{\rm{C}},{\rm{S}}} = {\bf{\hat j}}_{{\rm{C}},{\rm{S}}} {\rm{e}}^{{\rm{i}}(\omega'-\omega )t}$ . The time derivatives of $\hat \rho_{\rm{C}}$ and $\hat \rho_{\rm{S}}$ are defined by separate continuity equations, ∂ρ _C /∂t=−div j_C, ∂ρ _S /∂t=−div j_S, so that a time integration gives

$\begin{equation*} \begin{align*} \begin{equation} \hat \rho_{{\rm{C}},{\rm{S}}} = {\rm{i}}\frac{{{\rm{div}}\,{\bf{\hat j}}_{{\rm{C}},{\rm{S}}} }}{{\omega'-\omega }}. \end{equation} \end{align*} \tag{14}\end{equation*}$

Here, we substitute the current component (12), and perform the limit $\omega' \to \omega$ by expanding in powers of ω'-ω, to find the convection density

$\begin{equation*} \begin{align*} \begin{equation} \hat \rho_{\rm{C}} = \frac{{{\rm{i}}q}}{{2m}}\left( {\skew3\bar{\skew5\hat{\varphi}} \hat{\psi} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{{k^2 }}{{\mu^2 }}-\skew3\bar{\skew5\hat{\varphi}}_{,k} \hat{\psi}_{,k} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\mu^2 }}} \right), \end{equation} \end{align*} \tag{15}\end{equation*}$

where k² is defined in (8), and both fields $\hat{\psi}$ and $\skew3\bar{\skew5\hat{\varphi}}$ are taken at ω. The second term in (15) can be rewritten via the Klein-Gordon equation (6),

$\begin{equation*} \begin{align*} \begin{equation}\hat \rho_{\rm{C}} = \frac{{{\rm{i}}q}}{m}\frac{k}{{\mu^2 \upsilon_{{\rm{gr}}} }}\skew3\bar{\skew5\hat{\varphi}} \hat{\psi} -\frac{{{\rm{i}}q}}{{2m}}(\skew3\bar{\skew5\hat{\varphi}}_{,k} \hat{\psi} )_{,k} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\mu^2 }},\hphantom{0,} \end{equation} \end{align*} \tag{16}\end{equation*}$

$\begin{equation*} \begin{align*} \begin{equation} \qquad\frac{1}{{\upsilon_{{\rm{gr}}} }} = \frac{{\mu^2 }}{k}\left[ {\omega \varepsilon^2 \left( {1 + \frac{{\varepsilon'}}{\varepsilon }\omega + \frac{{\mu'}}{\mu }\omega } \right)-\frac{{\mu'}}{\mu }m^2 } \right], \end{equation} \end{align*} \tag{17}\end{equation*}$

where υ_gr is the group velocity

$\begin{equation*} \begin{align*} \begin{equation} \upsilon_{{\rm{gr}}} = \frac{{{\rm{d}}\omega }}{{{\rm{d}}k}} = \frac{{2k}}{{{\rm{d}}k^2 /{\rm{d}}\omega }}, \end{equation} \end{align*} \tag{18}\end{equation*}$

with k(ω) in (8). The second term in (16) is a divergence, which drops out in a spatial integration or for plane waves. The phase velocity is υ_ph=ω/k, and the refractive index is defined as the inverse phase velocity, n_r=k/ω.

We consider plane-wave solutions of the Dirac equations (9) and (10) at ω'=ω: $\hat{\psi} (\omega ) = \skew3\tilde \psi {\rm{e}}^{{\rm{i}}{\bf{kx}}}$ and $\skew3\bar{\skew5\hat{\varphi}} (\omega ) = \skew4\bar{\tilde{\varphi}} {\rm{e}}^{-{\rm{i}}{\bf{kx}}}$ , where k=k(ω )k₀, cf. (8). We find, cf. (12) and (16),

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{C}} = \frac{{{\rm{i}}qk}}{{m\mu^2 }}\skew4\bar{\tilde{\varphi}} \tilde \psi {\bf{k}}_0 ,\quad \hat \rho_{\rm{C}} = \frac{{{\rm{i}}q}}{{m\mu^2 }}\frac{k}{{\upsilon_{{\rm{gr}}} }}\skew4\bar{\tilde{\varphi}} \tilde \psi, \end{equation} \end{align*} \tag{19}\end{equation*}$

and ${\bf{\hat j}}_{\rm{C}} = \upsilon_{{\rm{gr}}} \hat \rho_{\rm{C}} {\bf{k}}_0$ , so that the charge transfer happens at group velocity.

The charge density of the spin current ${\bf{\hat j}}_{\rm{S}}$ in (13) is obtained in like manner, via the time integration (14),

$\begin{equation*} \begin{align*} \begin{equation} \hat \rho_{\rm{S}} = -\frac{{{\rm{i}}q}}{{2m}}\left[ {\skew3\bar{\skew5\hat{\varphi}}_{,k} \sigma_{ki} \hat{\psi}_{,i} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\mu^2 }}-{\rm{i}}(\skew3\bar{\skew5\hat{\varphi}} \sigma_{0k} \hat{\psi} )_{,k} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{{\omega \varepsilon }}{\mu }} \right], \end{equation} \end{align*} \tag{20}\end{equation*}$

where the adjoint field $\skew3\bar{\skew5\hat{\varphi}} (\omega )$ is taken at ω'=ω. We write $\skew3\bar{\skew5\hat{\varphi}}_{,k} \sigma_{ki} \hat{\psi}_{,i}$ as a divergence, symmetrize, and use the Dirac equations (9) and (10) to obtain

$\begin{equation*} \begin{align*} \begin{eqnarray} \hat \rho_{\rm{S}}&=&\frac{{\rm{i}}q}{2m}\left[\frac{1}{2}(\skew3\bar{\skew5\hat{\varphi}} \hat{\psi}_{,k} + \skew3\bar{\skew5\hat{\varphi}}_{,k} \hat{\psi} )_{,k} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\mu^2 }} \nonumber\right.\\ &&\left. -{\rm{i}}(\skew3\bar{\skew5\hat{\varphi}} \sigma_{0k} \hat{\psi} )_{,k} \left(\omega \mu \varepsilon \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\mu^2 }}-\frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{{\omega \varepsilon }}{\mu }\right)\right]. \end{eqnarray} \end{align*} \tag{21}\end{equation*}$

The spin density $\hat \rho_{\rm{S}}$ is the divergence of a polarization field, $\hat \rho_{\rm{S}} = -{\rm{div}}\,{\bf{P}}$ ,

$\begin{equation*} \begin{align*} \begin{eqnarray} {\bf{P}}&=&\frac{q}{{2m}}\frac{\varepsilon }{\mu }\left({1 + \omega \frac{{\varepsilon'}}{\varepsilon } + \omega \frac{{\mu'}}{\mu }} \right)\skew3\bar{\skew5\hat{\varphi}} \sigma_{0k} \hat{\psi} \nonumber\\ &&+\frac{{{\rm{i}}q}}{{2m}}\frac{{\mu'}}{{\mu^3 }}(\skew3\bar{\skew5\hat{\varphi}} \hat{\psi}_{,k} + \skew3\bar{\skew5\hat{\varphi}}_{,k} \hat{\psi}). \end{eqnarray} \end{align*} \tag{22}\end{equation*}$

The spin current ${\bf{\hat j}}_{\rm{S}}$ in (13) at ω'=ω is a magnetization current,

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{S}} = \frac{q}{{2m\mu^2 }}(\skew3\bar{\skew5\hat{\varphi}} \sigma_{ki} \hat{\psi} )_{,i} = {\rm{rot}}\,{\bf{M}}, \end{equation} \end{align*} \tag{23}\end{equation*}$

where the magnetization field reads

$\begin{equation*} \begin{align*} \begin{equation} {\bf{M}} = \frac{q}{{2m\mu^2 }}\skew3\bar{\skew5\hat{\varphi}} {\boldsymbol{\sigma }}\hat{\psi} ,\quad \sigma_k = \frac{1}{2}\varepsilon_{kij} \sigma_{ij}. \end{equation} \end{align*} \tag{24}\end{equation*}$

Here, we defined a spin vector $\boldsymbol{\sigma}=\sigma_{k}=\varepsilon_{kij}\gamma_{i}\gamma_{j}/2$ , where ε_kij is the Levi-Civita symbol. ${\bf{\hat j}}_{\rm{S}}$ vanishes for plane waves, and so does $\hat \rho_{\rm{S}}$ . If we refrain from performing the limit $\omega' \to \omega$ in (14), a polarization current ∂P/∂t has to be added to rotM in (23); at ω'=ω the polarization field (22) is time independent.

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Conservation of energy in the aether

In the case of a constant permeability tensor g^μν, the energy-momentum tensor of the Dirac field is obtained by standard Lagrange formalism [15,16],

$\begin{equation*} \begin{align*} \begin{equation} T_\mu^\nu= \frac{{\rm{1}}}{{{\rm{2i}}}}(\bar \varphi \gamma_\alpha g^{\alpha \nu } \psi_{,\mu } -\bar \varphi_{,\mu } \gamma_\alpha g^{\alpha \nu } \psi). \end{equation} \end{align*} \tag{25}\end{equation*}$

This tensor satisfies the conservation law T_μ,
ν^ν=0, provided that the fields ψ and $\bar \varphi$ solve the Dirac equation and the permeability tensor g^αν is constant. The components T₀^ν define the energy density T₀⁰ and energy flux T₀ⁱ. We specify the permeability tensor as in (3) (frequency independent) and split off time, writing $\psi = \hat{\psi} (\omega ){\rm{e}}^{-{\rm{i}}\omega t}$ , $\bar \varphi = \skew3\bar{\skew5\hat{\varphi}} {\rm{(}}\omega'{\rm{)e}}^{{\rm{i}}\omega't}$ and $T_\mu^\nu= \hat T_\mu^\nu {\rm{e}}^{{\rm{i}}(\omega'-\omega )t}$ . The fields $\hat{\psi} (\omega )$ and $\skew3\bar{\skew5\hat{\varphi}} {\rm{(}}\omega'{\rm{)}}$ satisfy the time-separated Dirac equations (9) and (10). We thus find

$\begin{equation*} \begin{align*} \begin{equation} \begin{array}{l} \hat T_0^0 = \varepsilon \displaystyle\frac{{\omega + \omega'}}{2}\skew3\bar{\skew5\hat{\varphi}} (\omega')\gamma_0 \hat{\psi} (\omega ), \\ \hat T_0^n = -\displaystyle\frac{{\omega + \omega'}}{{2\mu }}\skew3\bar{\skew5\hat{\varphi}} (\omega')\gamma_n \hat{\psi} (\omega ). \end{array}\end{equation} \end{align*} \tag{26}\end{equation*}$

Next, we consider a dispersive spacetime with frequency-dependent permeabilities ε (ω) and μ (ω), and define the flux vector as

$\begin{equation*} \begin{align*} \begin{equation} \hat T_0^n = -\frac{1}{2}\left( {\frac{{\omega'}}{{\mu (\omega')}} + \frac{\omega }{{\mu (\omega )}}} \right)\skew3\bar{\skew5\hat{\varphi}} (\omega')\gamma_n \hat{\psi} (\omega ).\end{equation} \end{align*} \tag{27}\end{equation*}$

At ω'=ω, we may write $\hat T_0^n = (\omega /q){\bf{\hat j}}$ , with current ${\bf{\hat j}}$ in (11). Here, we substitute the Gordon decomposition ${\bf{\hat j}} = {\bf{\hat j}}_{\rm{C}} + {\bf{\hat j}}_{\rm{S}}$ , cf. (12) and (13). We obtain the energy density $\hat T_0^0$ by applying the time integration (14) and performing the limit $\omega' \to \omega$ ,

$\begin{equation*} \begin{align*} \begin{equation} \hat T_0^0 = \frac{\omega }{q}(\hat \rho_{\rm{C}} + \hat \rho_{\rm{S}}),\quad\hat T_0^n = \frac{\omega }{q}({\bf{\hat j}}_{\rm{C}} + {\bf{\hat j}}_{\rm{S}}), \end{equation} \end{align*} \tag{28}\end{equation*}$

with the convection and spin densities $\hat \rho_{{\rm{C,S}}}$ in (16) and (21). On substituting plane-wave solutions as in (19), we find $\hat \rho_{\rm{S}} = 0$ , ${\bf{\hat j}}_{\rm{S}} = 0$ , as well as $\hat T_0^n = \upsilon_{{\rm{gr}}} \hat T_0^0 {\bf{k}}_0$ , where k₀ is the unit wave vector. The energy flux propagates in the dispersive aether with group velocity υ_gr, cf. (17).

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Spinorial wave propagation in the aether with zero rest mass

In the massless limit, the convection and spin densities in (16) and (21) become singular, and we have to proceed differently. We put m=0 in the Dirac equations (9) and (10), and multiply by γ₀. The spinors $\hat{\psi} (\omega )$ and $\skew3\bar{\skew5\hat{\varphi}} (\omega')$ satisfy the Klein-Gordon equation (6) with dispersion relation k²=ε² (ω )μ² (ω )ω², cf. (8).

We start with current ${\bf{\hat j}}$ in (11). In the first term, we replace $\skew3\bar{\skew5\hat{\varphi}} (\omega')$ by way of the adjoint Dirac equation (10), and in the second term we substitute (9) for $\hat{\psi} (\omega )$ , to split the current into a convection and spin component, ${\bf{\hat j}} = {\bf{\hat j}}_{\rm{C}} + {\bf{\hat j}}_{\rm{S}}$ , $\hat \rho = \hat \rho_{\rm{C}} + \hat \rho_{\rm{S}}$ , where, cf. (12) and (13),

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{C}} = {\rm{i}}\frac{q}{2}\left( {\frac{{\skew3\bar{\skew5\hat{\varphi}}_{,k} \gamma_0 \hat{\psi} }}{{\varepsilon (\omega')\mu^2 (\omega')\omega'}}-\frac{{\skew3\bar{\skew5\hat{\varphi}} \gamma_0 \hat{\psi}_{,k} }}{{\varepsilon (\omega )\mu^2 (\omega )\omega }}} \right), \end{equation} \end{align*} \tag{29}\end{equation*}$

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{S}} = {\rm{i}}\frac{q}{2}\left( {\frac{{\skew3\bar{\skew5\hat{\varphi}}_{,i} \gamma_0 \sigma_{ik} \hat{\psi} }}{{\varepsilon (\omega')\mu^2 (\omega')\omega'}}-\frac{{\skew3\bar{\skew5\hat{\varphi}} \gamma_0 \sigma_{ki} \hat{\psi}_{,i} }}{{\varepsilon (\omega )\mu^2 (\omega )\omega }}} \right). \end{equation} \end{align*} \tag{30}\end{equation*}$

The charge densities $\hat \rho_{{\rm{C}},{\rm{S}}}$ are calculated via the time integration (14) and by performing the limit $\omega' \to \omega$ and making use of the Klein-Gordon equation (6). The convection density reads

$\begin{equation*} \begin{align*} \begin{equation} \hat \rho_{\rm{C}} = \frac{q}{\mu }\frac{1}{{\upsilon_{{\rm{gr}}} }}\skew3\bar{\skew5\hat{\varphi}} \gamma_0 \hat{\psi} -\frac{q}{2}(\skew3\bar{\skew5\hat{\varphi}}_{,k} \gamma_0 \hat{\psi} )_{,k} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\omega \varepsilon \mu^2}}, \end{equation} \end{align*} \tag{31}\end{equation*}$

where υ_gr is the group velocity in (17) taken at m=0, so that 1/υ_gr=εμ+(εμ )'ω. Using plane waves $\skew3\bar{\skew5\hat{\varphi}} = \skew4\bar{\tilde{\varphi}} {\rm{e}}^{-{\rm{i}}{\bf{kx}}}$ , $\hat{\psi} = \tilde \psi {\rm{e}}^{{\rm{i}}{\bf{kx}}}$ , we find

$\begin{equation*} \begin{align*} \begin{equation}{\bf{\hat j}}_{\rm{C}} = \frac{q}{\mu }\skew4\bar{\tilde{\varphi}} \gamma_0 \tilde \psi {\bf{k}}_0,\qquad\hat \rho_{\rm{C}} = \frac{q}{\mu }\frac{1}{{\upsilon_{{\rm{gr}}} }}\skew4\bar{\tilde{\varphi}} \gamma_0 \tilde \psi, \end{equation} \end{align*} \tag{32}\end{equation*}$

and thus ${\bf{\hat j}}_{\rm{C}} = \upsilon_{{\rm{gr}}} \hat \rho_{\rm{C}} {\bf{k}}_0$ . We may replace $\skew3\bar{\skew5\hat{\varphi}} \gamma_0 = \hat \varphi^{\rm{\dagger}}$ , where phi ^† denotes the transposed and conjugated spinor. The charge transfer happens with group velocity, and the refractive index for zero rest mass is n_r=k/ω=εμ, cf. [3,17].

The charge density of the spin current ${\bf{\hat j}}_{\rm{S}}$ in (30) reads, cf. (14),

$\begin{equation*} \begin{align*} \begin{equation} \hat \rho_{\rm{S}} = -\frac{q}{2}\skew3\bar{\skew5\hat{\varphi}}_{,k} \gamma_0 \sigma_{ki} \hat{\psi}_{,i} \frac{{\rm{d}}}{{{\rm{d}}\omega }}\frac{1}{{\omega \varepsilon \mu^2 }}. \end{equation} \end{align*} \tag{33}\end{equation*}$

We symmetrize the divergence $(\skew3\bar{\skew5\hat{\varphi}}_{,k} \gamma_0 \sigma_{ki} \hat{\psi} )_{,i}$ , and use the masslessDiracequations (9)and (10)aswellas $(\!\skew3\bar{\skew5\hat{\varphi}} \gamma_i \hat{\psi}\!)_{,i}=0$ , to write this as a polarization density $\hat \rho_{\rm{S}} = -{\rm{div}}\,{\bf{P}}$ ,

$\begin{equation*} \begin{align*} \begin{equation} {\bf{P}} = \frac{q}{4}\frac{1}{{\varepsilon \mu^2 \omega^2 }}\left(1 + \frac{{\varepsilon'}}{\varepsilon }\omega + 2\frac{{\mu'}}{\mu }\omega\right)(\skew3\bar{\skew5\hat{\varphi}}_{,i} \gamma_0 \hat{\psi} + \skew3\bar{\skew5\hat{\varphi}} \gamma_0 \hat{\psi}_{,i}). \end{equation} \end{align*} \tag{34}\end{equation*}$

The spin current ${\bf{\hat j}}_{\rm{S}}$ in (30) is a magnetization current,

$\begin{equation*} \begin{align*} \begin{equation} {\bf{\hat j}}_{\rm{S}} = -{\rm{i}}\frac{q}{2}\frac{{(\hat \varphi^{\rm{\dagger}} \sigma_{ki} \hat{\psi} )_{,i} }}{{\omega \varepsilon \mu^2 }} = {\rm{rot}}\,{\bf{M}},\quad{\bf{M}} = -{\rm{i}}\frac{q}{2}\frac{{\hat \varphi^{\rm{\dagger}} {\boldsymbol{\sigma }}\hat{\psi} }}{{\omega \varepsilon \mu^2 }}, \end{equation} \end{align*} \tag{35}\end{equation*}$

with spin vector $\boldsymbol{\sigma}$ defined in (24).

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Refractive index and neutrino excess velocity

The goal is to relate the OPERA excess velocity to the refractive index n_r of the aether, cf. after (18) and (32). In the massless limit, the group velocity (17) reads

$\begin{equation*} \begin{align*} \begin{equation} \frac{1}{{\upsilon_{{\rm{gr}}} (\omega )}} = n_{\rm{r}} (\omega ) + \omega n'_{\rm{r}} (\omega), \end{equation} \end{align*} \tag{36}\end{equation*}$

where n_r=ε (ω )μ (ω), cf. (3) and (8). A possible neutrino rest mass of a few eV is negligible given the GeV energy range of the OPERA experiment. I also mention that a negative mass-square is not an option in the Dirac equation, as mass enters linearly and an imaginary mass destroys the hermiticity of the energy operator [4].

At 28.2GeV, an excess speed of υ_gr−1 ≈ (2.51 ± 0.6)×10⁻⁵ was measured for muon neutrinos. At 13.8GeV and 40.7GeV, the excess speed was υ_gr−1 ≈ 2.25×10⁻⁵ and υ_gr−1 ≈ 2.80×10⁻⁵, respectively, inferred from the time-of-flight differences δt_28.2 ≈ 61.1 ns, δt_13.8 ≈ 54.7 ns, and δt_40.7 ≈ 68.1 ns, where δt=τ_c−τ_ν is the difference of the photon and neutrino time of flight [1]. A baseline estimate of 730.085km was used, resulting in a photon time of flight of τ_c ≈ 2.43530×10⁶ ns. The excess velocities are found via υ_gr−1=δt/(τ_c−δt). The independent MINOS experiment [18] measured a neutrino excess speed of υ_gr−1 ≈ (5.1 ± 2.9)×10⁻⁵ at a peak energy of 3GeV. The neutrino flux from supernova SN1987A at 10MeV did not exhibit a deviation from the vacuum speed of light [19–21].

We use the OPERA excess velocities to determine the frequency variation of the refractive index n_r (ω) in the 10–50GeV range. To this end, we expand n_r (ω) at the average energy ω₀ ≈ 28.2 GeV into a truncated Taylor series (or Laurent series if needed),

$\begin{equation*} \begin{align*} \begin{equation} n_{\rm{r}} \approx n_{\rm{r}}^{{\rm{(0)}}} + n_{\rm{r}}^{{\rm{(1)}}} (\omega -\omega_0 ) + \frac{{n_{\rm{r}}^{{\rm{(2)}}} }}{2}(\omega -\omega_0 )^2 , \end{equation} \end{align*} \tag{37}\end{equation*}$

and substitute this into (36). We obtain three linear equations for the expansion coefficients n_r^(0,1,2) (ω₀), if we use for ω the neutrino energies of 28.2GeV, 13.8GeV and 40.7GeV, and for υ_gr−1 the respective excess velocities. Solving this system, we find 1-n_r⁽⁰⁾ (ω₀ ) ≈ 2.27×10⁻⁵, as well as the derivatives n_r⁽¹⁾ (ω₀ ) ≈−8.61×10⁻⁸ GeV⁻¹ and n_r⁽²⁾ (ω₀ ) ≈−1.27×10⁻⁹ GeV⁻². These small derivatives justify expansion (37) in the 10–50GeV range. On substituting the coefficients n_r^(0,1,2) (ω₀) into (37), we obtain the frequency variation of the refractive index in this interval. For instance, 1-n_r (28.2 GeV)=2.27×10⁻⁵, 1-n_r (13.8 GeV )=2.16×10⁻⁵, and 1-n_r (40.7 GeV)=2.38×10⁻⁵, in accordance with the mild variation of the excess velocities at these energies. If we extrapolate further down to 3GeV, the energy of the MINOS experiment, we find, via (36) and (37), υ_gr−1 ≈ 2.11×10⁻⁵ at 3GeV. This is still marginally consistent with the MINOS excess speed (5.1 ± 2.9)×10⁻⁵.

The mentioned SN result at 10MeV is a null result consistent with the speed of light [21]. At this energy, the permeability tensor (3) coincides with the Minkowski metric. Accordingly, there is a substantial frequency variation of the permeabilities between 3GeV and 10MeV. As there are no data points available in this energy range, I do not specify the permeabilities in this range, which have to be inferred from an empirical fit. If υ_gr (ω) is known over an extended energy range, one can obtain the refractive index by a numerical quadrature of (36), $\omega n_{\rm{r}} (\omega )-\omega_0 n_{\rm{r}} (\omega_0 ) = \int_{\omega_0 }^\omega {{\rm{d}}\omega /} \upsilon_{{\rm{gr}}} (\omega )$ , where the integration constant n_r (ω₀) is calculated locally as above, by means of a truncated Taylor expansion.

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Conclusion

The OPERA excess velocity amounts to superluminal signal transfer, which is causality violating in a relativistic spacetime view based on the relativistic interpretation of the Lorentz symmetry of Maxwell's equations [22,23]. Causality implies that every effect has a cause, that the cause precedes the effect, and that the distinction of cause and effect is unambiguous. The third condition means that every observer can arrive at the same conclusion as to what is cause and effect. If two events (cause and effect) are related by the emission and absorption of a superluminal signal, this establishes a spacelike connection. One can always find a Lorentz boost which overturns the time order of this connection. Two inertial observers related by this Lorentz boost will see a different time order in their proper time, and arrive at the opposite conclusion as to what is cause and effect. This does not depend on the specific physical mechanism of the signal transfer. Superluminal signal transfer is tantamount to a spacelike connection, and the time order in spacelike connections is ambiguous, as it can be altered by a Lorentz transformation. Thus, any kind of superluminal signal transfer is causality violating in a relativistic context, since, according to the relativity principle, the contradictory conclusions of the two inertial observers are equally valid. Therefore, a distinguished frame of reference is required, to which all uniformly moving inertial observers can relate and arrive at an unambiguous causality interpretation of their observations, even though the cosmic time order of events may be inverted in their proper time. We have identified this distinguished frame as the rest frame of the aether, in which the cosmic microwave background is isotropic, and which determines the absolute cosmic time order [5,6,24]. This discussion can be made quantitative quite easily, as the causality of superluminal signal transfer does not depend on the specific realization of the signal.

The coordinates of the rest frame of the aether are denoted by (t,x). The world lines of source and detector read ${\bf{x}}_{\rm{S}} (t) = {\bf{\tilde x}}_0 + {\boldsymbol{\upsilon }}t$ and ${\bf{x}}_{\rm{D}} (t) = {\bf{\tilde x}}_1 + {\boldsymbol{\upsilon }}t$ , respectively. The rest frame of source and detector is denoted by primed coordinates (t',x'), and connected to the aether frame by a proper orthochronous Lorentz boost,

$\begin{equation*} \begin{align*} \begin{equation} \begin{array}{l} t' = \gamma t-\displaystyle\sqrt {\gamma^2 -1} {\boldsymbol{\upsilon }}_0 {\bf{x}},\\ {\bf{x'}} = -\displaystyle\sqrt {\gamma^2 -1} {\boldsymbol{\upsilon }}_0 t + {\bf{x}} + (\gamma -1){\boldsymbol{\upsilon }}_0 ({\boldsymbol{\upsilon }}_0 {\bf{x}}), \end{array} \end{equation} \end{align*} \tag{38}\end{equation*}$

where $\boldsymbol{\upsilon}_{0}$ is the velocity unit vector of source and detector, $\boldsymbol{\upsilon}=\upsilon\boldsymbol{\upsilon}_{0}$ , and γ=(1-υ²)^−1/2, so that $\gamma \upsilon = \sqrt {\gamma^2 -1}$ . The source and detector world lines (t'_S ,x'_S) and (t'_D ,x'_D) in the primed frame can be parametrized by

$\begin{equation*} \begin{align*} \begin{equation} \begin{array}{l} t'_{{\rm{S,D}}} = \gamma (t-{\boldsymbol{\upsilon}}{\bf x}_{{\rm{S,D}}} (t)),\\ {\bf x'}_{\!{\rm S,D}} = {\bf{\tilde x}}_{0,1} + (\gamma -1){\boldsymbol{\upsilon }}_0 ({\boldsymbol{\upsilon }}_0 {\bf{\tilde x}}_{0,1}). \end{array} \end{equation} \end{align*} \tag{39}\end{equation*}$

In the aether frame, a superluminal signal is emitted at (t₀ ,x_S (t₀ )), and absorbed at a later instant t₁>t₀ at (t₁ ,x_D (t₁ )). The speed of this signal is assumed to be constant, $\boldsymbol{\upsilon}_{{\rm T}}=({\bf x}_{{\rm D}}(t_{1})-{\bf x}_{{\rm S}}(t_{0}))/(t_{1}-t_{0})$ , and tachyonic, υ_T> 1, so that a spacelike connection is established between emission and absorption. In the primed frame, the emission and absorption events take place at (t'₀ ,x'_S) and (t'₁ ,x'_D), respectively, where t'₀=t'_S (t₀) and t'₁=t'_D (t₁), cf. (39). The time interval between emission t'₀ and absorption t'₁ in the rest frame of source and detector is

$\begin{equation*} \begin{align*} \begin{equation} t'_1 -t'_0 = (t_1 -t_0 )(1-{\boldsymbol{\upsilon}\boldsymbol{\upsilon}}_{\rm{T}} )\gamma. \end{equation} \end{align*} \tag{40}\end{equation*}$

We may write ${\boldsymbol{\upsilon}\boldsymbol{\upsilon}}_{\rm T}=\upsilon \upsilon_{\rm T}{\rm cos}\,\theta$ , where $\boldsymbol{\upsilon}$ and $\boldsymbol{\upsilon}_{\rm T}$ denote velocities in the aether frame, $\boldsymbol{\upsilon}$ being the velocity of the source and the detector and $\boldsymbol{\upsilon}_{\rm T}$ the tachyonic (superluminal) signal speed. υ_T corresponds to υ_gr in (36). A subluminal source moving with velocity $\boldsymbol{\upsilon}$ in the aether frame can only emit superluminal signals of velocity $\boldsymbol{\upsilon}_{\rm T}$ if the condition ${\boldsymbol{\upsilon}\boldsymbol{\upsilon}}_{\rm T}< 1$ is satisfied. This condition is required by causality, in addition to kinematic constraints due to energy-momentum conservation, and is also necessary for absorption. If an observer moves with velocity $\boldsymbol{\upsilon}_{\rm obs}$ in the aether frame, he will see the time order of emission and absorption inverted in his proper time, ${\rm d}t_{\rm obs}={\rm d}t(1-\boldsymbol{\upsilon}_{\rm obs}\boldsymbol{\upsilon}_{\rm T})\gamma$ , if the angle between $\boldsymbol{\upsilon}_{\rm obs}$ and $\boldsymbol{\upsilon}_{\rm T}$ is acute and the superluminal signal speed sufficiently high.

The causality of superluminal signal transfer is solely determined by the time order in the aether frame, in particular it does not depend on the energy concept. In the rest frame of the aether, sub- as well as superluminal particles or radiation quanta have positive energy, and energy-momentum conservation applies. The positivity of energy of a subluminal particle (with positive mass square) is preserved under a Lorentz transformation, which is not the case for tachyonic quanta, so that superluminal emission can result in energy gain in the rest frame of the emitting subluminal particle, if causality is violated [25].

The speed of the superluminal signal in the primed rest frame of source and detector is $\boldsymbol{\upsilon}'_{\rm T}=({\bf x}'_{\rm D}-{\bf x}'_{\rm S})/(t'_{1}-t'_{0})$ , where, cf. (39),

$\begin{equation*} \begin{align*} \begin{eqnarray} &&{\bf x}'_{\rm D} -{\bf x}'_{\rm S} = [{\boldsymbol{\upsilon }}_{\rm{T}} + ((\gamma -1){\boldsymbol{\upsilon }}_0 {\boldsymbol{\upsilon }}_{\rm{T}} -\sqrt {\gamma^2 -1} ){\boldsymbol{\upsilon }}_0 ](t_1 -t_0).\nonumber\\ \end{eqnarray}\noindent \end{align*} \tag{41}\end{equation*}$

Here, we split the superluminal speed $\boldsymbol{\upsilon}_{\rm T}=\upsilon_{{\rm T},\parallel} \boldsymbol{\upsilon}_{0}+\boldsymbol{\upsilon}_{{\rm T},\bot}$ into components parallel and orthogonal to the unit vector $\boldsymbol{\upsilon}_{0}$ , and substitute (40) for t₁−t₀ to obtain

$\begin{equation*} \begin{align*} \begin{equation}{\boldsymbol{\upsilon}}'_{\rm T} = \frac{{{\boldsymbol{\upsilon }}_{{\rm{T}},{\bot}} + (\upsilon_{{\rm{T}},\parallel} -\upsilon )\gamma {\boldsymbol{\upsilon }}_0 }}{{(1-\upsilon \upsilon_{{\rm{T}},\parallel} )\gamma }}.\end{equation} \end{align*} \tag{42}\end{equation*}$

We multiply with $\boldsymbol{\upsilon}_{0}$ to find $\upsilon_{{\rm T},\parallel}=(\upsilon+\boldsymbol{\upsilon}_{0}\boldsymbol{\upsilon}'_{\rm T})/(1+\boldsymbol{\upsilon\upsilon}'_{\rm T})$ , substitute this into (42) and solve for $\boldsymbol{\upsilon}_{{\rm T},\bot}$ . Thus, $\boldsymbol{\upsilon}_{\rm T}=\boldsymbol{\upsilon}_{{\rm T},\parallel}+\boldsymbol{\upsilon}_{{\rm T},\bot}$ , where

$\begin{equation*} \begin{align*} \begin{equation} {\boldsymbol{\upsilon }}_{{\rm{T}},\parallel} = \frac{{{\boldsymbol{\upsilon }} + {\boldsymbol{\upsilon }}_0 ({\boldsymbol{\upsilon }}_0 {\boldsymbol{\upsilon}}'_{\rm{T}} )}}{{1 + {\boldsymbol{\upsilon \upsilon}}'_{\rm{T}} }},\quad{\boldsymbol{\upsilon }}_{{\rm{T}},{\bot}} = \frac{{{\boldsymbol{\upsilon}}'_{\rm{T}} -{\boldsymbol{\upsilon }}_0 ({\boldsymbol{\upsilon }}_0 {\boldsymbol{\upsilon}}'_{\rm{T}})}}{{(1 + {\boldsymbol{\upsilon \upsilon}}'_{\rm{T}} )\gamma }}. \end{equation} \end{align*} \tag{43}\end{equation*}$

By means of this addition law, we can infer the superluminal speed $\boldsymbol{\upsilon}_{\rm T}$ in the aether frame from the measured speed $\boldsymbol{\upsilon}'_{\rm T}$ in the rest frame of the source (and detector) and the measured speed $\boldsymbol{\upsilon}$ of this rest frame in the aether. Expanding (43) in powers of υ, we find

$\begin{equation*} \begin{align*} \begin{eqnarray}\boldsymbol{\upsilon}_{\rm T} &\sim&\boldsymbol{\upsilon}'_{\rm T} + \upsilon [\boldsymbol{\upsilon}_0 -\boldsymbol{\upsilon}'_{\rm T} (\boldsymbol{\upsilon}_0 \boldsymbol{\upsilon}'_{\rm T})] \nonumber\\ &&+ \upsilon^2 [\boldsymbol{\upsilon}'_{\rm T} (\boldsymbol{\upsilon}_0 \boldsymbol{\upsilon}'_{\rm T})^2 -\frac{1}{2}\boldsymbol{\upsilon}_0 (\boldsymbol{\upsilon}_0 \boldsymbol{\upsilon}'_{\rm T})-\frac{1}{2}\boldsymbol{\upsilon}'_{\rm T}], \end{eqnarray} \end{align*} \tag{44}\end{equation*}$

up to terms of O(υ³). We square this and substitute $\boldsymbol{\upsilon}_{0}\boldsymbol{\upsilon}'_{{\rm T},0}={\rm cos}\,\theta'$ to obtain

$\begin{equation*} \begin{align*} \begin{eqnarray} \upsilon_{{\rm T}} &\sim&\upsilon'_{{\rm T}} \left[1-\frac{\upsilon}{\upsilon'_{{\rm T}}}(\upsilon_{{\rm T}}'^{2}-1){\rm cos}\,\theta'\right.\nonumber\\ &&\left.-\frac{1}{2}\frac{\upsilon^2}{\upsilon_{{\rm T}}'^{2}}(\upsilon'^{2}_{{\rm T}}-1)(1-(2\upsilon'^{2}_{{\rm T}} + 1){\rm cos}^2\,\theta')\right], \end{eqnarray} \end{align*} \tag{45}\end{equation*}$

where $\boldsymbol{\upsilon}'_{\rm T}=\upsilon'_{\rm T}\boldsymbol{\upsilon}'_{{\rm T},0}$ . In the example discussed in this article, υ'_T is the measured neutrino speed, in units of the speed of light, and υ is the speed of the Solar system barycenter or neutrino baseline (neglecting the Earth's relative motions, which are at least one order of magnitude smaller than υ) in the aether. Since υ'_T−1 ≈ 2.5×10⁻⁵ and υ≈ 369 kms⁻¹/c≈ 1.2×10⁻³, we find υ_T ≈υ '_T (1-6×10⁻⁸cosθ'). Thus, in view of the current error bound on υ'_T−1 ≈ (2.5 ± 0.6)×10⁻⁵, the superluminal speed υ_T in the aether frame can be identified with the actually measured speed υ'_T, irrespectively of the periodically varying angle cosθ' between the neutrino baseline and the velocity of the barycenter. However, this angle becomes significant if the error bound is reduced by two orders. In any case, if υ'_T>> 1 or υ~1, the velocities $\boldsymbol{\upsilon}_{\rm T}$ and $\boldsymbol{\upsilon}'_{\rm T}$ can substantially differ depending on this angle, cf. (45).

Wave fields propagating in the aether are coupled by permeability tensors, which are a manifestation of the microscopic space structure [3,22]. Here, we have discussed the permeability tensor for neutrinos, and explained the coupling of the Dirac equation to this tensor. We have studied freely propagating spinorial wave modes in the aether, and shown that the frequency-dependent refractive index gives rise to superluminal energy transfer. The OPERA excess velocities measured at GeV energies allow to derive a quantitative estimate of the refractive index of the aether for superluminal neutrinos in this energy range.

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References

[1]: Adam T. et al 2011 arXiv:1109.4897
[2]: Tomaschitz R. 1998 Chaos, Solitons Fractals 9 1199
[3]: Tomaschitz R. 2001 Class. Quantum Grav. 18 4395
[4]: Tomaschitz R. 2005 Eur. Phys. J. D 32 241
[5]: Tomaschitz R. 2003 Physica A 320 329
[6]: Tomaschitz R. 2007 Ann. Phys. (N. Y.) 322 677
[7]: Kogut A. et al 1993 Astrophys. J. 419 1
[8]: Fixsen D. J. et al 1996 Astrophys. J. 473 576
[9]: Hinshaw G. et al 2009 Astrophys. J., Suppl. Ser. 180 225
[10]: Jarosik N. et al 2011 Astrophys. J., Suppl. Ser. 192 14
[11]: Landau L. D. and Lifshitz E. M. 1984 Electrodynamics of Continuous Media (Oxford: Pergamon)
[12]: Tomaschitz R. 2010 Eur. Phys. J. C 69 241
[13]: Bjorken J. D. and Drell S. D. 1964 Relativistic Quantum Mechanics (New York: McGraw-Hill)
[14]: Halzen F. and Martin A. D. 1984 Quarks and Leptons (New York: Wiley)
[15]: Wentzel G. 1949 Quantum Theory of Fields (New York: Interscience Publishers)
[16]: Bjorken J. D. and Drell S. D. 1965 Relativistic Quantum Fields (New York: McGraw-Hill)
[17]: Tomaschitz R. 2009 Physica B 404 1383
[18]: Adamson P. et al 2007 Phys. Rev. D 76 072005
[19]: Hirata K. et al 1987 Phys. Rev. Lett. 58 1490
[20]: Bionta R. M. et al 1987 Phys. Rev. Lett. 58 1494
[21]: Longo M. J. 1987 Phys. Rev. D 36 3276
[22]: Tomaschitz R. 1996 Chaos, Solitons Fractals 7 753
[23]: Tomaschitz R. 2000 Celest. Mech. Dyn. Astron. 77 107
[24]: Tomaschitz R. 2004 Chaos, Solitons Fractals 20 713
[25]: Tomaschitz R. 2000 Int. J. Mod. Phys. A 15 3019

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References

[1]: Adam T. et al 2011 arXiv:1109.4897 Preprint
[2]: Tomaschitz R. 1998 Chaos, Solitons Fractals 9 1199 CrossRef
[3]: Tomaschitz R. 2001 Class. Quantum Grav. 18 4395 IOPscience
[4]: Tomaschitz R. 2005 Eur. Phys. J. D 32 241 CrossRef
[5]: Tomaschitz R. 2003 Physica A 320 329 CrossRef
[6]: Tomaschitz R. 2007 Ann. Phys. (N. Y.) 322 677 CrossRef
[7]: Kogut A. et al 1993 Astrophys. J. 419 1 CrossRef
[8]: Fixsen D. J. et al 1996 Astrophys. J. 473 576 IOPscience
[9]: Hinshaw G. et al 2009 Astrophys. J., Suppl. Ser. 180 225 IOPscience
[10]: Jarosik N. et al 2011 Astrophys. J., Suppl. Ser. 192 14 IOPscience
[11]: Landau L. D. and Lifshitz E. M. 1984 Electrodynamics of Continuous Media (Oxford: Pergamon)
[12]: Tomaschitz R. 2010 Eur. Phys. J. C 69 241 CrossRef
[13]: Bjorken J. D. and Drell S. D. 1964 Relativistic Quantum Mechanics (New York: McGraw-Hill)
[14]: Halzen F. and Martin A. D. 1984 Quarks and Leptons (New York: Wiley)
[15]: Wentzel G. 1949 Quantum Theory of Fields (New York: Interscience Publishers)
[16]: Bjorken J. D. and Drell S. D. 1965 Relativistic Quantum Fields (New York: McGraw-Hill)
[17]: Tomaschitz R. 2009 Physica B 404 1383 CrossRef
[18]: Adamson P. et al 2007 Phys. Rev. D 76 072005 CrossRef
[19]: Hirata K. et al 1987 Phys. Rev. Lett. 58 1490 CrossRef PubMed
[20]: Bionta R. M. et al 1987 Phys. Rev. Lett. 58 1494 CrossRef PubMed
[21]: Longo M. J. 1987 Phys. Rev. D 36 3276 CrossRef
[22]: Tomaschitz R. 1996 Chaos, Solitons Fractals 7 753 CrossRef
[23]: Tomaschitz R. 2000 Celest. Mech. Dyn. Astron. 77 107 CrossRef
[24]: Tomaschitz R. 2004 Chaos, Solitons Fractals 20 713 CrossRef
[25]: Tomaschitz R. 2000 Int. J. Mod. Phys. A 15 3019

View all references

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Neutrino currents in the absolute spacetime: Relating the refractive index of the aether to the OPERA excess velocity

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