Exotic Matter: a New Perspective

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 interacting 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 ∂2ψ diffraction experiments [1]. In some circumstances, L. de ψ , ∆ = 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 frequency 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 {e , e }, 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 {−2}. = 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) ψ = a(n) exp [2πi(νt kr)] (1) vg dν w

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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 0 0 −1 0 0 0 −1 a     Ψ = a(x ) exp(2πi/h) φ , (7)bis  0 0 −1 0   0 0 1 0  γ0 =   , γ1 =   ,  0 1 0 0   0 1 0 0  −     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  1 0 0 0   0 0 −1 0      phase by:  0 1 0 0   0 0 0 −1  γ =   , γ =   . E = ∂t φ , p = −grad φ . (7)ter 2  0 0 −1 0  3  −1 0 0 0      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 −i  v = ∂E/∂k which corresponds to the particle’s velocity v =   g  0 0 −i 0  ∂E/∂p. Recalling (2) and (2)bis to as 1/w = n(ν)/c, λ = β =   ,  0 i 0 0  w/nν, we easily infer the Rayleigh’s formulae [11]:   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 = −γã (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)

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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. With the latter equation and taking into Simultaneously, the corresponding wave phase variation is account the classical Planck relation E = hν, we find p dφ = ∂tφdt + ∂NφdN = ∂tφ + v grad φ dt (12)bis E 1 − v2/c2 E = 0 . (17) 1 − v/w and by analogy to the classical formula (7)ter, one may write

2 However, inspection shows that the usual equation m0v m0c p = −grad φ = p , E = ∂tφ = p 1 − v2/c2 1 − v2/c2 E0 E = p (18) 1 − v2/c2 so we find   holds only if  m c2 m v2  2  0 − 0  v v dφ =  p p  dt. (13) 1 − = 1 − (19) 1 − v2/c2 1 − v2/c2 w c2 which implies Constraint B: We set the following phase synchronization: w v = c2. (20) dφ = dφ , (14) e The latter relation is satisfied provided we set which leads to: ] 2 ] m0c  2 2  E = p , (21)  m0c m0v  1 − v2/c2  p − p  dt =  1 − v2/c2 1 − v2/c2  (15) ] h p i ] m0v 2 − 2 2 p = p . (22) = m0c 1 v /c dt . 1 − v2/c2 Dividing through by dt , we retrieve the famous Planck-Laue Constraint C: ]E depends on a specific dispersive and re- equation fracting material through which the electron is tunnelling. 2 p 2 m0c 2 2 2 m0v = m0c 1 − v /c + , (15)bis Let us define this influence by a function Q(n) where n p 2 2 p 2 2 1 − v /c 1 − v /c is the refractive index of the material. Note: The variation Q n which holds provided the proper mass is slightly variable. of the proper mass is independent on ( ). Equation (21) is (see proof in Appendix A). In the frameworks of our pos- modified to as tulate, the ultra high frequency vibration imparted to the elec- ] 2 ] m0c tron can be viewed as apparently reflecting its stationary mass E = + Q(n) (23) p 2 2 variation which is likened to a fluctuation. 1 − v /c ] From now on, m0 will denote the variable rest mass of from which Eq. (22) can be expressed as: the electron so that the Planck-Laue relation becomes: h i ] v ]E − Q(n) ]m c2 ] m0v ]E 0 p = p = . (24) = p = 2 2 c2 1 − v2/c2 1 − v /c (15)ter p ] 2 ] 2 2 2 m0v Now taking into account the Doppler formulae (16), and = m0c 1 − v /c + p . 1 − v2/c2 the Planck-Laue relation (15)ter, we find

2 h] i This formulae will be required to determine the explicit form v E − Q(n) v ]E − = ]E 1 − (25) of the dispersive material which is the key point of our theory. c2 w

176 Marquet P. Exotic Matter: A New Perspective Issue 3 (July) PROGRESS IN PHYSICS Volume 13 (2017)

wherefrom is inferred subject to Constraints A, B and C, will exhibit a negative mass ! ! given by: c2 c2 Q(n) = ]E 1 − = hν 1 − (26) w v w v q ]E (]m ) = 1 − v2 /c2 < 0 , (30)bis 0 pos pos w v and with the Rayleigh formulae (4), we eventually obtain the ] ] 2 explicit form of Q(n): where E − Q(n) = Ec /vpos w) < 0 in accordance with " # Eq. (29). n∂(nν) Q(n) = ]E 1 − . (27) Let us write the mass (30)bis as: ∂ν Z ] ] √ 2.2 Specific dispersive material ( m0)pos = ( ρ0)pos −g dV, (31) Depending on the nature of the dispersive material, thus its ] index (n), it is well known that the tunelling electron’s 3- where ( ρ0)pos is the variable proper density of the positronic velocity v can be directed either in the direction of the associ- massive flow. The integral is performed over the 3-volume ] ated wave phase velocity w or in the opposite direction. The V delimiting the variable proper mass ( m0)pos boundary. We electron then moves backward through the specific material. then readily infer the familiar form of the energy density ten- Let N be the 3-unit vector directed to the wave phase di- sor in the static case rection (chosen positive) so that the wave number is given by: ] 0 ] 2 ( T0 )pos = ( ρ0)pos c , (32) Nh k = . (28) λ which is de facto negative. So, within the scheme of the wave-particle picture, we By applying the Rayleigh formulae (4) to this particular case have been able to give a consistent picture of what could be where v is opposite to the wave phase propagation, we have the united conditions to reach our goal : v < 0. Hence, from Q(n) = ]E (1 − c2/w v), we find The so-called “exotic matter” required to assemble a space-time distortion can be provided by the negative energy ]Ec2 ]E − Q(n) = (29) extracted from a stream of vibrating antifermions interacting w v with a specific dispersive refracting material adequately en- gineered. which is negative. ] p 2 2 Then, with p = m0v/ 1 − v /c , we infer from (24): 3 Concluding remarks

] ] m0 E − Q(n) Without going into details of a sound engineering, we have p = . (29)bis here only scratched the surface of a basic theory describing 1 − v2/c2 c2 the ability of a system composed of antiparticles to interact with a specific refracting and dispersive material in order to In order to maintain the variable proper mass ]m0 positive i.e. exhibit a dynamical negative mass. q ] Thus, our approach mainly relies on de Broglie’s theory ] 2 2 E m0 = 1 − v /c > 0 (30) which has been verified for the electron. w v Upon Constraints A, B, and C, we might as well consider we must have necessarily: p = −k. other heavier particles such as the antiproton to produce negative energy. 2.3 Matching the exotic matter definition Once these conditions are fulfilled, the concept of hyper- Now consider a stream of electrons and positrons placed in fast interstellar tracel is viable if one can “handle” routinely the specific material whose respective associated wave (pos- antimatter, and envision a sufficient amount of negative en- itive) direction is given by the same unit vector N (i.e. w > ergy density. These orders of magnitude are beyond the scope 0). From the Dirac theory, we kwnow that the electron mo- of this text. mentum 3-vector pelec and that of the positron momentum 3- Without any doubt, some advanced civilizations have al- vector p pos are opposed. (See proof in Appendix B). There- ready long mastered the negative energy obtained by this pro- fore we have here ppos = k, however the dispersive material cess, to achieve superluminal travels as described by space- yet imposes vpos < 0, hence, we are led to the fundamental time warp drive theories [12–14]. conclusion: For us, a huge research work is still ahead, but if we have A positron moving at the backward velocity vpos through contributed to open a small door, then the challenge is widely the specific dispersive refracting material defined above and available for physicists.

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Appendix A: The Planck-Laue relation which is satisfied when the particle is at rest, that is: v = 0 ⇒ ] ] 2 The Planck-Laue relation is a relativistic equation which has E0 = m0c . Therefore, we must always have: been derived when the proper mass is assumed to sligthly ] 2 p ] 2 ] ] m0c ] 2 2 2 m0v fluctuate. This proper mass is here denoted by m0. Under E = p = m0c 1− v /c + p . (A.6) ] 1− v2/c2 1− v2/c2 this circumstance, the relativistic dynamics of m0 can now be extended as follows. It is important to note that this variable (proper) mass, We first write the Lagrange function for an observer who ] m0, is purely intrinsic, i.e. its motion is unaffected. see the particle moving at he velocity v Equation (A.6) is known as the Planck-Laue formula. p ] 2 2 2 L = − m0c 1 − v /c Appendix B: Dirac currents so that the least action principle applied to this function is still Let us consider the real Dirac current as expressed by Ja = i ( ◦Ψ γ aΨ ) = (Ja) 1 − (Ja) 2 , Z t1 Z t1 p ] 2 2 2 where δ Ldt = δ − m0c 1 − v /c = 0 . t0 t0 a ◦ aA B a B aA ◦ (J ) 1 = i ΨA γ B Ψ , (J ) 2 = i Ψ γ B ΨA . From this principle the equations of motion a ! The charge conjugate of J is first calculated d ∂L ∂L dxa = , x˙a = , a (C) ∗ aA ∗B T A aB ∗C dt ∂x˙a ∂xa dt [(J ) 1] = i ΨA γ B Ψ = i t ΨA βB γ C Ψ are inferred, which lead to i.e. a (C) A T B T aC T ∗C [(J ) 1] = i t Ψ βA γ B Ψ . ] p d p 2 2 2 ] a = − c 1 − v /c grad m0 (A.1) From the antisymmetry of β, and remembering that the γ are dt here real, we have ] (since m0 is now variable). Hence, by differentiating the rel- T a T a a ] 2 2 ] 2 ] 2 2 γ β = − γ˜ β = β γ ativistic relation E /c = p + m0c , we obtain from which we infer ] p ] d E 2 2 2 ∂ m0 = c 1 − v /c . (A.2) a (C) A aB C ◦ ˜ A aB ◦ dt ∂t [(J ) 1] = i t Ψ γ A βB ΨC = i Ψ γ A ΨB Combining (A.1) and (A.2) readily gives hence, we see that

d ]E d ] p p d ]m [(Ja) 1](C) = (Ja) 2 − v = c2 1 − v2/c2 0 , (A.3) dt dt dt and similarly ] ] ] a (C) a where d m0/dt = ∂ m0/∂t + grad m0 is the variation of the [(J ) 2] = (J ) 1 mass in the course of its motion. On the other hand, we have therefore, we obtain the most important relation: ] · · ] d ( p v) v d p ] 2 (v/c) d(v/c) dt −(Ja)(C) = Ja (B.1) = + m0c p = dt dt 2 2 1 − v /c (A.4) ] The Dirac current orientation is opposed to that of its d p ] 2 d 2 2 ◦ = v − m0c 1 − v /c Dirac conjugate [15]. The Dirac conjugate Ψ of the plane dt dt wave spinor (6)bis is here: i.e. ◦ ◦ a Ψ = a exp −2πi (pa x ) . (B.2) d h p i ]m c2 1 − v2/c2 = dt 0 With the Dirac conjugate spinor amplitude ◦a = a∗ γ0, that is p ] p equivalent to (5)ter, we ﬁrst set the normalization condition: 2 2 2 d m0 ] 2 d 2 2 = c 1 − v /c + m0c 1 − v /c dt dt ◦ a a = m0c . (B.3) hence (A.3) can be re-written as Besides, the Dirac equation reads: d h p i ]E − v · ] p − ]m c2 1 − v2/c2 = 0 (A.5) a ◦ ◦ dt 0 (γ pa) a = m0c a . (B.4)

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Due to the property of (γa)2, Equations (7) and (B.4) are both 13. Natario J. Warp Drive with zero expansion. Class. Quantum Gravity, satisﬁed for: 2002, v. 19 , no. 6, 1157–1165. 2 2 14. Krasnikov S. The quantum inequalities do not forbid spacetime short- (pa) = (m0c) . (B.5) cuts. arXiv: gr-qc/020705. ◦ Multiplying now Equation (7) on the left with a, we obtain 15. Lichnerowicz´ A. Champ de Dirac, champ du neutrino et transforma- with (B.2) and (B.5) tions C, P, T sur un espace courbe. Ann. Institut Henri Poincar´e, sec. A: 1964, v. 3, p.233–290. ◦ a 2 2 ( a γ a) pa = (m0c) = (pa) (B.6) 16. Beretevski V.L., Lifshitz E., Pitayevski L. Electrodynamique Quan- tique. Edition Mir, Moscow, 1973. from which we infer:

◦a γa a = pa. (B.7)

The Dirac current density vector Ja = ◦Ψ γa Ψ will here yield

Ja = ◦a γa a = pa (B.8)

with a 2 µ p = m0c + p (B.9) ( [16]: compare with formulae (23.6) there). From the charge conjugate Ψ(C) corresponding to the positron plane spinor, we deﬁne the Dirac current for the positron (Ja)(C). However, it was shown that (Ja)(C) = −Ja. Therefore, assuming that (m0)elec = (m0)posit in vacuum, we must then have µ (C) µ µ (J ) = (p )posit = −(p )elect . (B.10)

This clearly means that in vacuum, vposit = −velect.

Submitted on May 15, 2017

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