ton Law of Masses and informational origin of gravity since changes in ..... Consequently, we find the Newton law of attraction (scalar equation). F = G. Mm. R2.
Waves and gravity - mathematical formulations that allow reinterpretations and openings in technology Constantin Udri¸ste, Florin Munteanu, Dorel Zugr˘avescu UNESCO Chair in Geodynamics, ”Sabba S. S¸tef˘anescu” Institute of Geodynamics, Romanian Academy Abstract The purpose of this paper is twofold: (i) a description of standing waves, (ii) revealing the informational origin of gravity. Waves are achieved by imposing a special form of solutions of the system of partial differential equations (PDE) describing electromagnetic waves. One obtains a frequency dependent EDP system. This system can be solved using two methods: (i) eigenvalues eigenvectors, (ii) three-dimensional Fourier transform. Information theory is a branch of applied mathematics, electrical engineering, bioinformatics, and computer science involving the quantification of information. A key measure of information is entropy, which is usually expressed by the average number of bits needed to store or communicate one symbol in a message. Entropy quantifies the uncertainty involved in predicting the value of a random variable. Entropy is a measure of unpredictability or information content. From entropy, energy and temperature we obtain the Newton Law, the Newton Law of Masses and informational origin of gravity since changes in the amount of information, measured by entropy, can lead to a force.
Mathematics Subject Classification 2010: 74J99, 94A15. Keywords: equilibrium wave distribution, information, gravity, Newton Laws.
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Systems optimization
The systems optimization is based on finding an alternative with the most or highest achievable performance under the given constraints, by maximizing desired factors and minimizing undesired ones. Particularly, the antenna optimization aims at creating advanced and complex electromagnetic devices that must be competitive in terms of performance, serviceability, and cost effectiveness. Our paper highlights the most controversial ideas (standing waves, and passing from entropy to Newton Law) that can change the traditional technology into nano-technology. It is aimed at defining certain technologies (ortho-technologies) based on control of ”quantum vacuum” dynamics, using geometric control functions (patterns) resonant to the structure induced in Dirac space (vacuum) [2], [4].
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Time-independent wave PDE
The propagation of the electromagnetic waves in a medium free of charges and currents is described by the wave equation ∂2 2 ∇ −µ ˆ ˆ 2 F (t, ~r) = 0. ∂t Here, µ ˆ is the permeability and ˆ is the permittivity of the medium and F (t, ~r) stands for the electromagnetic fields, E(t, ~r) or B(t, ~r). The relative permeability and permittivity µ and are defined by µ=
ˆ µ ˆ , = , µ0 0
where µ0 0 = 1/c2 and c is the speed of light in free space. These two parameters are generally complex, space-, and frequency-dependent corresponding to absorbing or active, non-uniform and dispersive medium. The time independent wave equation, for oscillatory electromagnetic fields of the form F (t, ~r) = F0 (~r)e−iωt , becomes a frequency dependent PDE, ω2 2 ∇ + µ(ω, ~r) (ω, ~r) 2 F0 (~r) = 0. c 2
Assuming that the permittivity and the permeability are piecewise constant, and adding boundary conditions, we have a problem of proper eigenvectors and eigenvalues of Laplacian. It is known, and our calculations confirm this, that there is a discrepancy between the time-dependent and the frequencydependent solutions of the wave equation in an active medium. Since ω2 lim (∇ + µ(ω, ~r) (ω, ~r) 2 ) = ∇2 , ω→0 c 2
we obtain the PDE for steady-state solutions ∇2 F0 (~r) = 0. The solutions represent the equilibrium wave distribution (they describes standing waves that do not move at all). Postulate The time waves are born on a bed of standing waves.
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Eigenvalues and eigenvectors of the Laplacian in a parallelepiped
Let us explain what we understand by eigenvalues and eigenvectors of the Laplacian in a parallelepiped ([6]). For each natural number n, the function un (x) = sin
nπx a
satisfies the second order ODE u00n (x)
n2 π 2 + 2 un (x) = 0 a
with the boundary condition un (0) = un (a) = 0. It follows that the functions un1 n2 n3 (x, y, z) = c sin
n1 πx n2 πy n3 πz sin sin , c = const a1 a2 a3
satisfy the PDE ∆u(x, y, z) + λu(x, y, z) = 0, λ = π 3
2
n21 n22 n23 + 2+ 2 a21 a2 a3
and vanish on the boundary of the parallelepiped Ω : 0 < x < a1 , 0 < y < a2 , 0 < z < a3 . In this way, the functions un1 n2 n3 (x, y, z) are eigenvectors, and the numbers 2 n1 n22 n23 2 λn1 n2 n3 = π + 2+ 2 a21 a2 a3 are eigenvalues for the Dirichlet problem attached to the Laplace operator on a parallelepiped. The system un1 n2 n3 (x, y, z) is orthogonal in L2 (Ω); it is also orthonormal if we select 3
c = 22 √
1 . a1 a2 a3
Moreover, our system is complete in L2 (Ω). Theorem: Let n1 , n2 , n3 be fixed integers. The minimum value of 1 n21 n22 n23 λ + 2 + 2, = n n n π2 1 2 3 a21 a2 a3 at constant volume a1 a2 a3 = the edge lengths ai =
23 , c2
2
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is 43 (n1 n2 n3 ) 3 c 3 . This value is attained for 2ni 1
2
(n1 n2 n3 ) 3 c 3
, i = 1, 2, 3.
Hint. AM-GM inequality.
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Fourier Transform
The wave PDE can be solved using the Fourier transform. First and foremost, a Fourier transform of a signal tells you what frequencies are present in your signal and in what proportions [1]. From spatial variable to momentum Let x = (x1 , ..., xn ) be a spatial variable and ξ = (ξ 1 , ..., ξ n ) as momentum (a ”frequency variable”). The Fourier transform of f (x), x ∈ Rn is the function F f (ξ), or fˆ(ξ), defined by Z Ff (ξ) = e−2πi f (x)dx. Rn
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The inverse Fourier transform of a function g(ξ), ξ ∈ Rn is Z −1 F g(x) = e2πi g(ξ)dξ. Rn
The Fourier transform, or the inverse transform, of a real-valued function is (in general) complex valued. From (spatial variable, time) to (momentum, angular frequency) Let (x, t) be the pair (spatial variable,time) and (ξ, ω) the pair (momentum, angular frequency). The Fourier transform of f (x, t), x ∈ Rn , t ∈ R is the function F f (ξ, ω), or fˆ(ξ, ω), defined by Z Z e−2πi(+ωt) f (x, t)dx dt. Ff (ξ, ω) = R
Rn
The inverse Fourier transform of a function g(ξ, ω), ξ ∈ Rn , ω ∈ R is Z Z −1 F g(x, t) = e2πi(+ωt) g(ξ, ω)dξ dω. R
Rn
The Fourier transform, or the inverse transform, of a real-valued function is (in general) complex valued.
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Using the Fourier Transform to solve a wave PDE
Let u(x; t) be the displacement from equilibrium of a string at position x and time t. If the string is undergoing small amplitude transverse vibrations, then 2 ∂ 2u 2∂ u (x, t) = c (x, t), ∂t2 ∂x2 for a constant c. We are now going to solve this PDE by the well-known method of Fourier transform. By multiplying both sides with eikx and integrating with respect to x, that is, taking the Fourier transform Z ∞ uˆ(k; t) = u(x, t)e−ikx dx, −∞
with respect to the spatial variable x, we find a considerable simplification ∂ 2 uˆ (x; t) = −c2 k 2 uˆ(x; t). ∂t2 5
We now have, for each fixed k, a constant coefficient, homogeneous, second order ordinary differential equation with the unknown uˆ(k; t). The general solution ikct ˆ uˆ(k; t) = Fˆ (k)e−ikct + G(k)e To recover u(x; t) we just need to take the inverse Fourier transform, finding u(x; t) = F (x − ct) + G(x + ct). This is called the D’Alembert form of the solution of the wave equation.
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From entropy to Newton Law
Information theory is a branch of applied mathematics, electrical engineering, bio-informatics, and computer science involving the quantification of information. A key measure of information is entropy, which is usually expressed by the average number of bits needed to store or communicate one symbol in a message. Entropy quantifies the uncertainty involved in predicting the value of a random variable. Entropy is a measure of unpredictability or information content. E. Verlinde ([3] put forward one such idea which has taken the world of physics by storm. It suggested that gravity is merely a manifestation of entropy in the Universe. His idea is based on the second law of thermodynamics, that entropy always increases over time. It suggests that differences in entropy between parts of the Universe generates a force that redistributes matter in a way that maximises entropy. This is the force we call gravity (see also, [5]).
Figure 1: Gravity Emerges from Quantum Information
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Let us explain how we can pass from entropy to Newton Law. Postulate 1: The change of the entropy S, associated to the information on the boundary, equals |∆S| = 2πkB , when ||∆x|| ≤
~ . mc
Postulate 2: The change in entropy near the screen is linear in the displacement ∆x, i.e., ∆S = 2πkB
mc < e, ∆x >, ~
where e is a versor of acceleration type. It follows that maximum of the entropy is attained on boundary, |∆S| ≤ 2πkB . Postulate 3: When a particle has an entropic reason to be on one side of the membrane and the membrane carries a temperature T , it will experience an effective force (entropic force) F such that < F, ∆x >= T ∆S. Postulate 4: The acceleration a and the temperature T are related by kB T e =
1 ~ a. 2π c
It follows the Newton Law (vectorial equation) F = ma.
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Informational Approach of Newton Law of Gravity
From entropy, energy and temperature we obtain the Newton Law of masses since changes in the amount of information, measured by entropy, can lead to 7
a force. The informational origin of gravity is justified adding the following Postulates: Postulate 5: The number N of bits is proportional to the area A, i.e., N=
Ac3 , G~
where G is a new constant (Newton constant). Postulate 6: The total energy E, the number N of bits and the temperature T are related by 1 E = N kB T. 2 Approximation The mass m from the Einstein formula E = mc2 is connected to the Newton mass m0 by m0 m= q 1−
. v2 c2
Since v c, we approximate m ' m0 . To find the Newton law of attraction, we insert A = 4πR2 . It follows
1 Ac3 1 N kB T, mc2 = kB T 2 2 G~ 1 Ac 2πR2 c m= kB T, m = kB T 2 G~ G~ Mm 2πckB T G 2 =M = M ||a|| = F. R ~ Consequently, we find the Newton law of attraction (scalar equation) mc2 =
F =G
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Mm . R2
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Inertia and Newton Potential
Since each bit carries an energy 12 kB T , the number of bits n is obtained from mc2 =
1 nkB T. 2
It follows the variation of the entropy ∆S < a, ∆x > = kB . n 2c2 Principle: inertia is a consequence of the fact that a particle in rest will stay in rest because there are no entropy gradients. Now, introducing a Newton potential Φ, we write the acceleration vector as a gradient a = −∇Φ. Since ∆Φ =< ∇Φ, ∆x >, we find ∆Φ ∆S = −kB 2 . n 2c Acknowledgements We thank Prof. Dr. Ionel Tevy for stimulating conversations as well as the manuscripts anonymous reviewers for constructive comments. Partially supported by University Politehnica of Bucharest, by UNESCO Chair in Geodynamics, ”Sabba S. S¸tef˘anescu” Institute of Geodynamics, Romanian Academy and by Academy of Romanian Scientists.
References [1] Using the Fourier Transform to Solve PDEs, INTERNET, 2013. [2] H. Bahlouli, A. D. Alhaidari, A. Al Zahrani, E. N. Economou, Study of electromagnetic wave propagation in active medium and the equivalent Schr¨odinger equation with energy-dependent complex potential, INTERNET, 2013. [3] E. Verlinde, On the origin of gravity and laws of Newton, JHEP04(2011) 029.
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[4] T. E. Bearden, Extracting and using electromagnetic energy from the active vacuum, CEO, CTEC, Inc. [5] J.-W. Lee, H.-C. Kim, J. Lee, Gravity from Quantum Information, arXiv:1001.5445v3 [hep-th] 13 Apr 2013. [6] C. Udriste, Multitime optimal control with second order PDEs constraints, AAPP - Atti della Accademia Peloritana dei Pericolanti Classe di Scienze Fisiche, Matematiche e Naturali / Physical, Mathematical, and Natural Sciences (ISSN 1825-1242); http://dx.doi.org/10.1478/AAPP.911A2.
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