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A Fluid Model of Matter, Forces, and Spacetime

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This paper describes theoretical work done in an attempt to find a common physical mechanism for the gravitational, electric, and magnetic forces, thus allowing further investigation into a theory of quantum gravity. Starting from known physical
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   A Fluid Model of Matter, Forces, and Spacetime Dahl Winters 8/15/2018   ABSTRACT This paper describes theoretical work done in an attempt to find a common physical mechanism for the gravitational, electric, and magnetic forces, thus allowing further investigation into a theory of quantum gravity. Starting from known physical constants, we re-interpret these constants in the context of acoustic radiation forces (Bjerknes forces). These forces operate within a vacuum that can be interpreted as having properties of a high-pressure ideal gas with fluid properties. Vacuum parameters are derived for use in both the primary and secondary acoustic radiation force equations to demonstrate the equivalency of these equations to Coulomb’s law and Newton’s gravitational force law, respectively. In particular, this paper will demonstrate the following: 1) spacetime can be modeled as an ideal gas under high pressure, giving it fluid qualities 2) the electric and gravitational forces are acoustic radiation forces operating within this fluid 3) the magnetic force is linked to the presence of a small nonzero viscosity within the fluid. Key   words: Acoustic radiation forces, Bjerknes forces, quantum gravity I. INTRODUCTION Acoustic radiation (AR) forces are radiation forces exerted by an acoustic field on gas bubbles in a fluid. They are a highly important set of forces in the fields of sonoluminescence, cavitation, acoustic degassing, and medical ultrasonics [1]. There are two AR forces, called primary and secondary. The primary force is due to an external sound field and results in the attraction or repulsion of single bubbles at a pressure node or antinode. The secondary force is due to the sound fields emitted by other bubbles and also results in mutual attraction or repulsion [2], though repulsion here is a special case. The forces are also named Bjerknes forces after C. A. Bjerknes and his son V. F. K. Bjerknes, the first to describe their effects [1]. This paper makes a direct link between the electric force and the primary AR force, and with the gravitational force and the secondary AR force. The link is created by recasting existing physical constants in a new light, using them to define values for vacuum pressure and density. We then show that the same values can be used in the primary and secondary AR equations to yield Coulomb’s law and Newton’s gravitational force law, respectively. This method introduces no novel values, just re-expressions of existing ones to allow us to interpret the electric and gravitational forces as two sides of the same mechanism: acoustic radiation forces. In doing so, we open up a new line of research into quantum gravity since the mechanism of acoustic radiation forces provides a common basis for the electric and gravitational forces.    2 Table 1. Physical Constants Used in this Paper Symbol Description Value h  Planck’s constant 6.62607E-34 J*s   Planck’s constant divided by 2    (h-bar) 1.05457E-34 J*s c  Speed of light 2.99792458E+08 m*s-1 o    Permittivity of free space 8.85419E-12 A2*s4*kg-1*m-3 o    Permeability of free space 1.25664E-06 m*kg*s-2*A-2 o  Z   Impedance of free space 3.76730E+02 kg*m2*s-3*A-2  B k   Boltzmann’s constant 1.38065E-23 J/K G  Gravitational constant 6.67408E-11 m^3*kg-1*s-2 e m  Electron mass 9.10938E-31 kg  pl  m  Planck mass 2.17647E-08 kg e q  Electron charge 1.60218E-19 C  pl  q  Planck charge 1.87555E-18 C c    Compton angular frequency 7.76344E+20 Hz    Planck frequency 1.85489E+43 Hz    Fine structure constant 7.29735E-03  g     Gravitational coupling constant 1.75175E-45 c r   Compton radius, e hm c  2.42631E-12 m  pl  r   Planck length, 3 Gc   1.61623E-35 m  A  Vacuum pressure amplitude , 24 4  pl o c qr     , defined in our first paper [1] 9.12245E+20 Pa c  P   Compton pressure, 2  A   which is 24 2  pl o c qr     5.73180E+21 Pa  pl   P   Planck pressure, 72 cG   4.63324E+113 Pa , bh c T   Compton black hole temperature, 3 8 e B cm k G    , temperature of a black hole of electron mass 1.34690E+53 K , bh pl  T   Planck black hole temperature, 3 8  pl B cm k G    , temperature of a black hole with Planck mass 5.63730E+30 K c    Vacuum kinematic viscosity, c cc  Pr c     7.27390E-04 m^2/s    3 II. THEORY If spacetime can be modeled as a compressible fluid under high pressure and thus having a nonzero viscosity, then two things must be true: 1) transverse pressure waves must arise to transmit non-instantaneous forces [9][10] and 2) it must be able to support the formation of longitudinal (acoustic) waves within itself. In a compressible fluid of nonzero viscosity, the transmission of forces is not instantaneous but occurs through the motion of transverse pressure waves. Since the pressure waves take a finite amount of time to travel from one location to another, interesting effects are said to occur when flow speed is close to wave propagation speed [9]. Transverse waves are a characteristic of electromagnetic waves, and these “interesting effects” seem very much like relativistic effects. Indeed, the expression for Mach number is equivalent to that of gamma in the formulas of length contraction and time dilation. Acoustic waves differ from transverse pressure waves in that they are longitudinal. These acoustic waves would in turn permit the action of acoustic radiation forces on objects within the fluid that are of a different density than the fluid itself. An example of such objects are bubbles. Below, the two types of AR forces are described in greater detail. 1. Primary AR Force The primary AR force is central to studies of sonoluminescence, whereby an air bubble inside a flask insonified by a strong external sound field is made to suddenly collapse, emitting light. The  bubble is trapped at the center of the flask by the primary AR force, and this trapping is a combined effect of the sound field and nonlinear bubble oscillations [4]. A body of time-averaged volume V in a liquid under a time-averaged acoustic pressure gradient  P    experiences a force 1  B  F V P      (1.1) [4][5]. For a spherical bubble, the volume would be that of a sphere of a particular equilibrium radius, though the above equation is not limited to spherical bodies. The physical mechanism of the AR force is illustrated in greater detail in [5], but a brief description is as follows. This description is for the case of small driving pressures and drive frequencies below the bubble’s natural resonance frequency since at larger pressures and higher frequencies, nonlinear effects occur. The primary AR force arises from a pressure gradient across the bubble – a slight difference in  pressure exerted on either side of the bubble’s surface. During the tensile phase of the sound field, the pressure force directs the bubble toward the pressure antinode. During the compressive  phase, the pressure force directs the bubble away from the pressure antinode. However, because the bubble is smaller during the compressive phase than the tensile phase, the pressure force is smaller during the compressive phase, so the time-averaged force is in the same direction as that of the tensile phase–toward the pressure antinode. In the case of bubbles driven above their    4 natural resonance frequency, as is the case here, a different phase response occurs that drives  bubbles away from the pressure antinode and toward a node [5]. 2. Secondary AR Force The secondary AR force results in the mutual interaction between oscillating gas bubbles in a sound field: 22102022222212 2()()  B  A R R F  L             (1.2) where A = acoustic pressure amplitude, ω = the acoustic driving frequency, R 10 and R 20 are the equilibrium radii of the two bubbles in question, ρ is the fluid density, L is the distance between  bubbles, and ω 1 and ω 2 are the resonance frequencies of the two bubbles, respectively [3]. Weakly driven bubbles of a fixed equilibrium radius R 10 or R 20 show a maximum response at their linear resonance frequency ω 1 or ω 2 . If the driving frequency lies between ω 1 and ω 2 the bubbles will repel each other; otherwise the force is attractive [2]. III. DESCRIPTIONS OF WORK We will now show how spacetime can be modeled as an ideal gas under pressure (at least at speeds well below the speed of light) and how the primary and secondary AR forces lend themselves to explanations of the electric and gravitational forces, respectively. Lastly we show that spacetime can be modeled as having a small but nonzero viscosity and that this viscosity is closely connected to the magnetic force. Part A. Spacetime as an Ideal Gas Under High Pressure 1. The Combined Gas Law of Spacetime In this section, the relationship between the Compton and Planck scales is supplied in the form of a combined gas law, which is as follows: 33,, c c bh c pl pl bh pl   Pr T P r T    (1.3) Equation 1.3 demonstrates a relationship between the Compton and Planck scales. The electron (Compton scale) is seen to be the lowest energy state of a dilatating wave center in the ideal gas, while the Planck mass (Planck scale) is the highest energy state. To see two ways in which the Compton pressure can be derived, please see Appendix A and B. 2. The Ideal Gas Law of Spacetime The ideal gas law of spacetime is as follows:    5 3, 8  pl pl B bh pl   P r k T      (1.4) A connection is seen with the combined gas law: the left side of this ideal gas equation is part of the right-hand side of the combined gas law equation. We see here that the ideal gas law of spacetime takes the same form as the ideal gas law already known from chemistry and physics,  PV nRT    or, identically,  B  PV Nk T    where N = amount of substance. In this case, the amount of substance is 8   , a number we see many times in  physics – particularly in Einstein’s field equation relating gravity to energy and momentum: 4 182 G R Rg g T c            (1.5) Equation 1.4 demonstrates that spacetime can be defined by a three-dimensional ideal gas under high pressure. The electron (Compton scale) can be seen to be the lowest energy state of a dilatating wave center in the ideal gas, while the Planck mass (Planck scale) is the highest energy state. Part B. Electric and Gravitational Forces as Acoustic Radiation Forces in the Ideal Gas 3. The Electric Force We have derived the following equivalence between Coulomb’s law and the primary AR force for two elementary charges 12 e q q q    separated a distance of c r   apart: 121122 42 e Bo c q q F F n nr         (1.6) Where 321 c c B c cc c  Pr  PV  F V P Pr r r           , 11 e qnq  , and 22 e qnq  . We see that the electric force is equal to a term involving 1  B  F  , the primary AR force. Since we have just described how spacetime can be modeled as an ideal gas, then acoustic waves must be  possible. This then allows the production of acoustic radiation forces. 4. The Gravitational Force We have also derived the following equivalence between Newton’s gravitational force law and the secondary AR force between two electron masses 12 e m m m   separated a distance of c r   apart:   122122  g Bc Gm m F F n nr     (1.7)
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