_{1}

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Based on an assumption of that the cosmological constant term in Einstein field equation can be modeled as a material medium, we propose the concept of cosmological acoustic wave (CAW). In the process, we will see: 1) By the dual framework of electromagnetic theory, it is possible to consider some implications of CAW, such as, the acoustic charge quantization, the unit spin of phonon, the mechanical meaning of acoustic current and the acoustic radiation. It shows that, besides the transverse and longitudinal waves, there should be a sort of adjoint waves without contribution to energy flows. 2) The electromagnetic-acoustic equations are applied in homogeneous and isotropic material media, which can help us to describe uniformly the electromagnetic-acoustic phenomena. 3) It will be verified that, the acoustic interaction is identical to the gravitational one in classical physics. This decision suggests the so-called gravitational wave could be indeed understood as an acoustic radiation in cosmological space.

Following a general line of reasoning inspired by the electromagnetic theory, an interesting procedure was proposed to describe the acoustic wave propagating in homogeneous and isotropic material, which can predict successfully the macroscopic behavior of long-wavelength sound propagation in such a medium [

The paper is organized as follows. Through reevaluating cosmological vacuum [

In massive electrodynamics, a set of generalized Maxwell equations(GMEs) for massive electromagnetic fields

Meanwhile, due to being usually written in the form of

According to the relativistic cosmological model [

The energy meaning of

The quantum theory can provide an explanation for the mechanical properties of CBM. According to the theory, vacuum is not empty, but filled with a large number of virtual particle-antiparticle pairs [

and emphasize,

To illustrate CAW, we draw an electromagnetic analogy to introduce acoustic fields

On the other hand, the electromagnetic interaction essence of material elasticity and tension also suggests, the acoustic equations should be the same with GMEs in structure. Therefore, to unify electromagnetism and acoustics, we reedit (2.1) in the dualized form of [

Then, replacing the dual terms by acoustic quantities

Classification | Cosmological medium | Conventional medium |
---|---|---|

Basic unit | Virtual particle-antiparticle pair | Atom or molecular |

Investigation system | Large number virtual pairs | Large number molecules |

Leading interaction | Virtual Van der Waals force | Van der Waals force |

Inertia indexes | Vacuum mass density | Mass density |

Elasticity indexes | Vacuum elastic modulus | Elastic modulus |

Propagated waves | Cosmological acoustic wave | Conventional acoustic wave |

gives

with the related fields defined by

where,

To present the dual invariance of Equation (2.10), we define the following complex arrays [

and write the dual transformation relation

The dualized tensor

followed by the dualized continuity equation

Furthermore, by the stress-energy tensor of compound fields

we get

This equation has the obvious dual invariance, that is, no matter whether a particle carries charge

or charge

all the physical results would be unanimous. At the same time, the dual symmetry also requires the co-quantize- tion relationship of

In the below, we will verify the acoustic charge carried by a moving particle is equal to the product of its mass m and Hubble constant H, i.e.

Following the above practice, we write Equation (2.14) in 5-dimensional d’Alembert’s form

which has the retarded solution

for current

followed by a general solution for free wave

It implies that, as the quantum of acoustic wave, phonon (just like photon) would possess a unit spin (

These two tend to c together, only as

For simplicity, we neglect the weak

charge to be conserved strictly), and adopt electrodynamic method [

Then, the acoustic equations in (2.10) become a 2-component form

Among which, there are eight equations to be independent

where,

with potentials

By which, an arbitrary potential

Now, it is easy to calculate two acoustic current stresses (the

with

The right-hand side of the first (second) equation in (3.8) is a sink (source) term that transfer energy from (to) the acoustic fields to (from) the particles of interacting with the corresponding fields. Specifically, the rate

This desired emphasizes that: the positive (negative) mechanical work done by field

Moreover, the fact of acoustic currents (

It suggests, an acoustic current is always equivalent to a drag force on the surrounding medium,

Making use of (3.11) can help us to get the following inhomogeneous wave equations

These equations are more challenging to solve than their homogeneous counterparts, but the introduction of acoustic potentials can help us to get a set of simplified equations

being inhomogeneous equations also, but with much simpler source terms. It is very advantageous for problem-solving that the alternate stress appears on the right-hand sides, which can be used to calculate the acoustic fields produced by dynamically the forced vibration matter system.

If a point time-harmonic acoustic charge

This current is nothing but a drag force density with transverse and longitudinal components:

To use the longitudinal component of (3.16) and integrate the Lorenz gauge condition gives

Which combining with (3.4) can help us to get

as well as the energy flows

Here,

A striking feature of the obtained above is the coexistence of various terms with different algebraic dependence on the radial distance r from the source. If

It consists of an oscillating dipole field

This is nothing but a special standing wave with a vibrating amplitude of

By definition, when a compact source radiates its Poynting flow into a differential element of solid angle

Correspondingly, the total powers radiated transversely and longitudinally read

which represent the transverse and longitudinal Poynting fluxes through a spherical surface at infinity.

As presented by ref [

with the electromagnetic fields redefined by

followed by the gauge conditions [

Correspondingly, the stress on current

together with that on

Clearly, the equations in first row describe the usual electromagnetic radiation, the remainders the longitudinal and shadow waves. Specifically, although the shadow wave characterized by

To study the electromagnetic properties of matter, we can apply directly the 2-component GMEs (analogous to (3.2)) in a material medium, but need to use two effective electromagnetic interaction ranges

of

termine two modified frequent dispersions

we get

Right now, drawing on an analogy with Equation (4.8), we are allowed to apply (3.11) in a macroscopically homogeneous and isotropic unbounded medium with mass density

followed by

Then, Equation (4.10) becomes

which, in the case of no driving source, can bring us a set of equations

What the equations describe are very the usual transverse and longitudinal mechanical waves we are familiar with.

Furthermore, we can also express (4.9) and (4.11) in the dualized d’Alembert’s form

with

This unified form would provide us a physical framework to study the electro-acoustic coupling.

Domain | Electromagnetism | Acoustics | ||
---|---|---|---|---|

Alignment | Transverse | Longitudinal | Transverse | Longitudinal |

Permittivities | ||||

Permeabilities | ||||

Currents | ||||

Continuity equations | ||||

Fields | ||||

Adjoint fields | ||||

Quanta of fields | Transverse photon | Longitudinal photon | Transverse phonon | Longitudinal phonon |

Motion equations (in d’Alembert form) | ||||

Gauges | ||||

Free wave equations | ||||

Shadow wave equations | ||||

Energy flows | ||||

Energy densities | ||||

Quantized charges |

In electromagnetism, the electric current flowing in conductor is a complex phenomenon, whose microscopic description requires arguments from statistical physics and quantum mechanics. Nevertheless, it is well established phenomenologically that the current density in many systems obeys Ohm’s law:

The equality makes it possible for the particle to achieve its acoustic charge

Clearly we see that, the greater mass m and the shorter collision time

Hereby, we borrow Ohm law from electromagnetism and bring it to acoustics. As such, the acoustic Ohm law formally reads

Importantly, the mechanical explanation of acoustic current allows us to examine the acoustic effect from the perspective of cosmology. To this end, we now recall a full velocity concept [

Following the velocity is the definition of full momentum

The conservation theorem requires

with

with an average collision time reading the cosmological time

we can find, the acoustic charge of a moving particle is just equal to the product of its mass m and Hubble constant H, namely

identical to the result of (5.3). The result suggests, a mass m at the same time is also an acoustic charge

What it reproduces is nothing but the gravitation, and this identity encourages us to treat the gravitation as a sort of acoustic interaction (see

With regard to the CAW, there is a simply designed experiment to detect its presence, by observing the sympathy phenomenon between two simple pendulums (under high vacuum condition). It is shown as

Interaction | Acoustic interaction | Gravitation |
---|---|---|

Interaction constants | ||

Charges | m | |

Interaction laws | ||

Fields |

In which, the second pendulum (with mass

According to mechanics, this driving force can compel the second pendulum to oscillate, whose oscillating amplitude finally reads

Choosing

The result shows, as long as the difference between two cord lengths

The radiation is so weak that it is extremely difficult to detect.

This work is dedicated to establish a general theoretical framework for the unified description of electromagnetic-acoustic phenomena. In summary we can write:

1) On the stress-energy meaning of the

2) It is suggested that, the electromagnetic-acoustic analogy is a degenerate version of a much deeper one. This deeper version not only provides a consistent approach to examine synthetically the general properties of electromagnetic and acoustic motions, but also can help us to study the intrinsic mechanism of electro-acoustic coupling in physics.

3) The acoustic charge carried by a moving particle has been verified to be proportional to its mass with a scale coefficient of Hubble constant, by which the determined acoustic interaction is just equal to the gravitational force. Therefore, the decision allows us to understand the gravitational interaction as the acoustic interaction, and the gravitational wave as CAW. Finally, the feasibility of measuring this acoustic wave (referring to its shadow component) is discussed.

Qiankai Yao, (2016) A Deeper Analogy between Electromagnetism and Acoustics. Open Access Library Journal,03,1-16. doi: 10.4236/oalib.1102280