Volume 13 (2017) PROGRESS IN PHYSICS Issue 3 (July) Exotic Matter: A New Perspective Patrick Marquet 18 avenue du President´ Wilson, 62100 Calais, France E-mail: [email protected] In this paper we suggest a possible theoretical way to produce negative energy that is required to allow hyperfast interstellar travels. The term “Exotic Matter” was first coined by K. Thorne and M. Morris to identify a material endowed with such energy in their famous traversable space-time wormhole theory. This possibility relies on the wave-particle dualism theory that was originally predicted by L. de Broglie and later confirmed by electrons scattering experiments. In some circumstances, an electron in- teracting with a specific dispersive and refracting medium, has its velocity direction opposite to that of the phase velocity of its associated wave. However, it is here shown that a positron placed in the same material exhibits a negative mass. Generalizing the obtained equations leads to an energy density tensor which is de facto negative. This tensor can be used to adequately fit in various “shortcut theories” without violating the energy conditions. Introduction propagates along the direction given by the unit vector N. Introduction In this paper we show that it is possible to ob- Here k is the 3-wave vector, kr = φ is the wave spatial phase, tain a negative energy provided the associated proper parti- and n is the refractive index of the medium. Equation (1) is a cle’s mass is variable. The basis for this study starts with solution of the wave propagation equation the associated wave that was originally detected on electrons 1 @2 diffraction experiments [1]. In some circumstances, L. de ; ∆ = 2 2 2 (1)bis Broglie showed that a particular homogeneous refractive and w c @t dispersive material may cause the tunnelling particle to re- where w is the wave phase velocity of the wave moving in verse its velocity with respect to its wave phase propagat- a dispersive medium whose refractive index is n(ν) generally ing velocity [2]. In this case, and under the assumption that depending of the coordinates, and which is defined by: the proper mass of the particle is subject to a ultra high fre- quency vibration synchronized with the wave frequency, it is 1 n(ν) = : (2) formally shown that an anti-particle exhibits a negative mass w c (energy). This energy could be extracted to sustain for ex- In our study, the medium is assumed to be homogeneous ample the space-time wormhole, set forth by K. Thorne and but it can be anisotropic and ir will depend on the fequency ν. M. Morris [3, 4]. To be physically viable, it is well known In this material, the phase φ of the wave is progressing along that it requires a so-called exotic matter endowed with a neg- the given direction with a separation given by a distance ative energy density which violates all energy conditions [5]. However, if the exotic matter threading the inner throat of w c λ = = (2)bis the wormhole is likened to the specific dispersive material ν nν wherein circulates a stream of antiparticles, our model does- not conflict with classical physics restrictions and can be fully called the wavelength. Consider now the superposition of two applied. stationary waves along the x-axis having each close frequen- cies ν0 = ν + δν and close velocities w0 = w + (dw=dν)δν, so Notations that their superposition can be expressed by: ! In this paper we will use a set of orthonormal vector basis νx ν0 x denoted by fe ; e g, where the space-time indices are a; b = sin 2π νt − + sin 2π ν0t − = 0 a w w0 0; 1; 2; 3, while the spatial indices are µ, ν = 1; 2; 3. The " # νx ν d ν ν space-time signature is {−2g. = 2 sin 2π νt − cos 2π δ t − x δ : w 2 dν w 2 1 Proper mass variation The resulting wave displays a wave packet (or beat) that 1.1 Phase velocity and group velocity varies along with the so-called group velocity (v = vµ): It is well known that the classical wave with a frequency n 1 d ν − = : (3) = a(n) exp [2πi(νt kr)] (1) vg dν w 174 174 Issue 3 (July) PROGRESS IN PHYSICS Volume 13 (2017) The wave mechanics shows that the momentum 3-vector of electron can be approximated to a plane wave spinor without an electron of a rest mass m0 (in vacuum) is given by the de loss of generality [10]: Broglie relation a a h ΨA = a (x ) exp 2πi (pa x ) ; (6)bis p = m v = : (4) 0 λ where which completes the Einstein relation E = hν. a µ pa x = Et − pµ x : (6)ter 1.2 The plane wave spinor The 4-vector pa is the 4-momentum of the electron . The a Since we deal here with a spin 1/2-fermion, we must intro- spinor “amplitude” a(x ) satisfies the Dirac equation duce the four components wave function ΨA expressed with a γ (pa)elec a = [(m0)elec c] a (7) the non local 4 × 4 Dirac trace free matrices γa (capital latin spinor indices are A = B = 1; 2; 3; 0). They display here the a where the operator [γ (pa)elec] is here substituted to the Dirac following real components [8]: a operator γ @a. We now re-write (6)bis as 0 1 0 1 0 0 0 −1 0 0 0 −1 a B C B C Ψ = a(x ) exp(2πi=h) φ ; (7)bis B 0 0 −1 0 C B 0 0 1 0 C γ0 = B C ; γ1 = B C ; B 0 1 0 0 C B 0 1 0 0 C − @B AC @B AC where the global phase is φ = h[ν (αx + βy + γz)/λ] t (here 1 0 0 0 −1 0 0 0 α, β, γ are the direction cosines). The energy and momentum of the electron located at xa are then related with the wave 0 1 0 0 0 1 0 0 0 −1 0 1 B C B C phase by: B 0 1 0 0 C B 0 0 0 −1 C γ = B C ; γ = B C : E = @t φ ; p = −grad φ : (7)ter 2 B 0 0 −1 0 C 3 B −1 0 0 0 C @B AC @B AC 0 0 0 −1 0 −1 0 0 Now, if the electron moves at a velocity v = β c within a slight variation β, β + δβ, corresponding to the frequency These matrices are said standard representation as opposed interval ν; ν + δν, w and ν are functions of β. The wave phase for example to the Majorana representation. Moreover, they velocity (in vacuum) can be expressed as w = c2=v = c/β and verify 2 p 2 since ν = (1=h) m0c = 1 − β , it is easy to infer that: γaγb + γbγa = −2ηabI (5) dν 1 where ηab is the Minkowski tensor and I is the unit matrix. vg = = β c = v : (8) ∗ dβ d ν In what follows, Λ is the complex conjugate of an arbitrary dβ w matrix Λ, TΛ is the transpose of Λ, and Λ˜ is the classical adjoint of Λ. The group velocity vg of the wave packet associated with the electron of rest mass m , coincides with its velocity v. The Introducing now the Hermitean matrix β = iγ0 0 group velocity is thus also expressed by the Hamiltonian form 0 0 0 0 −i 1 v = @E=@k which corresponds to the particle’s velocity v = B C g B 0 0 −i 0 C @E=@p. Recalling (2) and (2)bis to as 1=w = n(ν)=c, λ = β = B C ; B 0 i 0 0 C w=nν, we easily infer the Rayleigh’s formulae [11]: @B AC i 0 0 0 1 1 1 @nν @ λ which verifies β2 = I, we derive the important relation = = : (9) vg c ∂ν ∂ν −1 β γa β = −γ˜a (5)bis 1.3 Making the electron vibrate with β and the spinor Ψ, we form the Dirac conjugate [9] In the framework of the special theory of relativity, the proper frequency ν0 of a plane monochromatic wave is transformed ◦Ψ = t Ψ˜ β ; (5)ter as ν0 ν = p : (10) where t is the time orientation. or the electron, the Dirac equa- 1 − v2=c2 tion is written as Constraint A: We assume that the electron is subject to an ultra high stationary vibration having a proper frequen- [W − (m0)elec c] Ψ = 0 ; (6) cy ν0. a A where W = γ B @a is the Dirac operator and it is customary When moving at the velocity v, this frequency is known a to omit the spinor indices A; B by simply writing γa = γa B to transform according to: a so that this operator becomes γ @a, or in the slash notation p 2 2 (Feynman), −@a. The monochromatic wave associated with the νe = ν0 1 − v =c : (11) Marquet P. Exotic Matter: A New Perspective 175 Volume 13 (2017) PROGRESS IN PHYSICS Issue 3 (July) We clearly see that its frequency νe differs from that of its 2 Exotic matter associated wave denoted here by ν. 2.1 Dynamics in a refracting material If N is the unit vector normal to the associated wave 2 Let us first recall the relativistic form of the Doppler formu- phase, the electron subject to the frequency ν0 = m0c =h has traveled a distance dN during a time interval dt, so that we lae: ν (1 − v=w) may define an electronic phase φe which has changed by: ν0 = p ; (16) 1 − v2=c2 p p − 2 2 2 − 2 2 dφe = hν0 (1 v =c ) dt = m0c (1 v =c ) dt: (12) where as before, ν0 is the wave’s frequency in the frame at- tached to the electron.