Theoretical, Numerical, and Experimental Evidence of Superluminal Electromagnetic and Gravitational Fields Generated in the Nearfield of Dipole Sources (Presented at Days on Diffraction Conference, St. Petersburg, Russia, June 2005)
William D. Walker Örebro University, Department of Technology, Sweden Research papers [1]
[email protected]
Abstract Theoretical and numerical wave propagation analysis of an oscillating electric dipole is presented. The results show that upon creation at the source, both the longitudinal electric and transverse magnetic fields propagate superluminally and reduce to the speed of light as they propagate about one wavelength from the source. In contrast, the transverse electric field is shown to be created about 1/4 wavelength outside the source and launches superluminal fields both towards and away from the source which reduce to the speed of light as the field propagates about one wavelength from the source. An experiment using simple dipole antennas is shown to verify the predicted superluminal transverse electric field behavior. In addition, it is shown that the fields generated by a gravitational source propagate superluminally and can be modeled using quadrapole electrodynamic theory. The phase speed, group speed, and information speed of these systems are compared and shown to differ. Provided the noise of a signal is small and the modulation method is known, it is shown that the information speed can be approximately the same as the superluminal group speed. According to relativity theory, it is known that between moving reference frames, superluminal signals can propagate backwards in time enabling violations of causality. Several explanations are presented which may resolve this dilemma.
Introduction The electromagnetic fields generated by an oscillating electric dipole have been theoretically studied by many researchers using Maxwell’s equations and are known to yield the following well-known results (MKS units): Variable definitions System differential equation Fig. 1
z p
x
Er θ r
System PDE Bφ
Eθ
1 ∂ 2V − ρ ∇V− 2 2 = c ∂t εo
y
2
φ
(1)
Field analysis
Er = Radial electric field Eθ = Transverse electric field Bφ = Transverse magnetic field V = Scalar potential ρ = Charge density εo = Free-space permittivity ∇2 = Laplacian c = Speed of light t = Time p = Dipole (q d) ω = Angular frequency k = Wave number
Solving the homogeneous equation (Eq. 1) for a dipole source yields [2, 3, 4, 5, 6]: 1 pk 2 i i (kr −ωt ) (2) where: N = V = N Cos (θ ) + e 2 4πε o kr (kr ) The fields can then be calculated using the following relations [6]:
B=
ω
(r × ∇V )
c2 yielding:
Eθ =
[{
E=
pCos (θ ) [1 − i(kr )]e i (kr −ωt ) Er = 3 2πε o r
}
]
pSin(θ ) 2 1 − (kr ) − i (kr ) e i (kr −ωt ) 3 4πε o r
Bφ = 1
ic 2
ω
(∇ × B)
ωp Sin(θ ) [− kr − i ]e i (kr −ωt ) 4πε o c 2 r 2
(3)
(4)
Phase analysis The general form of the electromagnetic fields generated by a dipole is: Field ∝ (x + iy ) ⋅ ei [kr −ωt ] If the source is modeled as Cos (ω t ) , the resultant generated field is: Field ∝ Mag ⋅ Cos[{kr + ph} − ω t ] = Mag ⋅ Cos (θ − ω t ) where: Mag = x 2 + y 2 It should be noted that the formula describing the phase is dependent on the quadrant y of the complex vector. y x
θ 1 = kr + Tan −1
x2 + y2
θ 2 = kr − Cos −1
x
θ1
x
(5)
θ2
Phase speed analysis
Phase speed can be defined as the speed at which a wave composed of one frequency propagates. The phase speed (cph) of an oscillating field of the form Sin(ωt − kr ) , in which k = k (ω , r ) , can be determined by setting the phase part of the field to zero, differentiating the resultant equation, and solving for ∂ r ∂ t . ∂r ω ∂ ∂r ∂k ∂r = ∴ c ph = (6) (ω t − kr ) = 0 ∴ω − k −r =0 ∂k ∂t ∂t
∂r ∂t
∂t
Differentiating θ ≡ − kr with respect to r yields:
k+r
∂k ∂θ = −k − r ∂r ∂r
∂r
(7)
Combining these results and inserting the far-field wave number (k = ω/co) yields: c ph = − ω
Group speed analysis
∂θ ∂θ = − co k ∂r ∂r
(8)
The group speed of an oscillating field of the form: Sin(ωt − kr ) , in which k = k (ω , r ) , can be calculated by considering two Fourier components of a wave group [7]: Sin(ω 1t − k1 r ) + Sin(ω 2 t − k 2 r ) = Sin(∆ωt − ∆kr ) Sin(ωt − kr ) (9) ω + ω2 k − k2 k + k2 ω − ω2 in which: ∆ω = 1 , ∆k = 1 , ω= 1 , k= 1 2 2 2 2 The group speed (cg) can then be determined by setting the phase part of the modulation component of the field to zero, differentiating the resultant equation, and solving for ∂ r ∂ t : ∆ω ∂r ∂ ( ∆ω t − ∆kr ) = 0 ∂t
∴ ∆ω − ∆k
∂r ∂ ∆k ∂ r −r =0 ∂t ∂r ∂t
Differentiating ∆θ ≡ − ∆kr with respect to r yields:
∴ cg =
=
∂ ∆k ∂r ∂ ∆θ ∂ ∆k = − ∆k − r ∂r ∂r ∂t
∆k + r
(10) (11)
Combining these results and using the far-field wave number (k = ω/co) yields: c g = − ∆ω
∂ ∆θ ∂ ∆θ =− ∂r ∂ r ∆ω
−1
∂ 2θ ∴cg = − lim ∆θ ∂ r∂ ω small ∆ω
−1
∂ 2θ = −c o ∂ r∂ k
−1
(12)
It should be noted that other derivations of the above phase and group speed relations are available in previous publications by the author [8, 9, 10, 11, 12, 13] and in the following well-known reference [14].
2
In addition, in order for the group speed to be valid, a signal should not distort as it propagates. It is known from electronic signal theory that in order to minimize signal distortion, the phase vs. frequency curve must be approximately linear over the bandwidth of the signal and the amplitude vs. frequency curve must be approximately constant over the bandwidth of the signal [15]. It is shown below that the amplitude vs. frequency curve can even be approximately linear over the bandwidth of the signal, provided the ratio of the slope of the curve to the signal amplitude is small. Assuming that the amplitude vs. frequency curve is increasing and approximately linear over the bandwidth of a modulated carrier signal, each signal magnitude (A) Fourier component (wm) will be increased by (u) and the Fourier component symmetric about the carrier (wc) will be reduced by (u):
1 [( A − u ) Sin( wc t − wm t ) + ( A + u ) Sin( wc t + wm t )] 2 = A ⋅ Cos ( wm t ) Sin( wc t ) + u ⋅ Sin( wm t )Cos ( wc t )
A
u
u
wc
(13)
Wm Wm
The two Fourier components form an amplitude modulated signal where the magnitude of the carrier is:
A 2 ⋅ Cos 2 ( wm t ) + u 2 ⋅ Sin 2 ( wm t ) ≈ A ⋅ Cos ( wm t )
(14)
It should be noted that distortions to the magnitude are minimal provided: u2/A2 0) the signal propagation speed must be less than the speed of light (w
