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Dedicated to those with the spirit of research in science that all men are free to choose, think and contribute to the best of their ability with righteous intention.
УДК 531.1
Согласование плотности массы и энергии для консервативных и
диссипативных систем
G.C. Sih
Лихайский университет, Бетлехем, PA 18015, США
Скорость свободно падающего тела в случае взаимозависимости массы и энергии равна yfgh. В случае взаимной независимости значение скорости увеличивается и достигает ^2gh. Показано, что значение времени свободного падения больше на -J2. Возможность последовательного учета законов движения от Галилея до Ньютона реализована с применением формулы E = mv1 для консервативных и диссипативных систем. Кроме того, эквивалентность энергии и массы может использоваться для математической оценки гравитационных волн не только при скорости света, но при любых скоростях. Во избежание неоднозначности персонализированные единицы следует заменить на «единицу плотности энергии». Ключевым моментом является использование гуковской «силы» в сочетании с совершаемой работой в качестве энергии. Эквивалентность массы и энергии при любой скорости получена в рамках идеомеханики — математической версии «Книги перемен». Гравитационные волны тесно связаны со сложнейшей структурой многоатомного пространства-времени.
Ключевые слова: масса и энергия, свободно падающее тело, скорость, реверсный режим, совершаемая работа, пространство-время, гравитационные волны, идеомеханика, многомасштабность, консервативный, диссипативный, плотность энергии, плотность массы, единица плотности энергии, персонализированные единицы
Concurrence of mass and energy density for conservative and dissipative systems
G.C. Sih
International Center for Sustainability, Accountability and Eco-Affordability of the Large and Small, Lehigh University, Bethlehem, PA 18015, USA
The falling body velocity is ,Jgh when mass and energy are interdependent. The overestimated yjlgh prevails when mass and energy are separately independent. The falling time is found to be greater by a factor of -Jl. A consistent account of the laws of motion from Galileo to Newton is made possible by using E = mv2 for conservative and dissipative systems. By the same token, the equivalence of energy and mass can be used for the mathematical assessment of gravitational waves at any velocities and not just at the speed of light. The personalized units should be replaced by the "energy density unit" to avoid ambiguities. The key step is the application of the Hookean "force" in conjunction with the work done as energy. The equivalence of mass and energy at any velocity were derived by using Ideomechanics, which is a mathematized verion of I-Ching. Gravitational waves are intimately related to the entanglement of multi-atom space-time.
Keywords: mass and energy, falling body, velocity, push and pull, work done, space-time, gravitational waves, ideomechanics, multiscaling, conservative, dissipative, energy density, mass density, energy density unit, personalized units
1. Introduction
A choice to distinguish matter from mass is to predicate the concurrence of mass and energy. The work done as the energy serves to scale space-time from the quantum to cosmo [1, 2], and to resolve the conundrums in physics and mechanics [3]. Matter is what nature offers. It becomes increasingly more complex as the scale range of space-time is stretched indefinitely. Interdependence of mass and energy
[4] accounts for the gravitational waves for any velocities, not limited to the speed of light. Gravitational waves are not directly explainable by force because they are generated by the interdependence of mass and energy at different velocities for different space-time scales. Moving masses and energetic collisions are the mechanisms for creating the ripples in space-time. The transmission of "energy" is made by the Hookean force of work done as energy that
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is compatible to the push and pull action. The Mutuability's law follows logically for quantifying the frequency of the ripples.
LIGO from USA dedicated "Feb. 11, 2106" to the detection of the gravitational waves were predicted by the general relativity theory of Einstein more than 100 years ago. LIGO's findings and those from eLISA from Europe shall compliment the future findings of TIANQIN [7], TAIJI and ALI from China. What is known is that gravitational waves can travel at any velocities, not just at the speed of light. The exchange of mass and energy density at any velocities was derived [4] and led to the mutuability's law that revealed the push and pull or squeeze and stretch of space-time such that the full range of frequencies can be related to the energetic collisions of masses. In this connection, the LIGO data of ground-based observation can compliment those of TIANQIN, TAIJI and ALI of space-based observation. Analytical approach involving the equality of gravitational and inertial masses and the energy-momentum tensor in the Landau theory can be found in [8]. An extensive investigation of gravitational waves has already been made at the XII International Conference on Gravitation, Astrophysics and Cosmology, PFUR, Moscow, Russia, July 2015. The foregoing findings can yield information vital to the influence of gravitational waves on teleportation [9] that can shift the scientific and political power of the world.
2. Interdependence of mass and energy via velocity squared
If the mass can turn into energy at the velocity of light, then the same process can take place at any velocity when the restriction on scale range of space-time is removed.
2.1. Exchangeable mass and energy
Equation (11) of [4] states that
W = M v2. (1)
The form of Eq. (1) is reminiscent of E = mc2 in general relativity except that v is not the speed of light c. When both sides of Eq. (1) are multiplied by the volume, the energy density W and mass density M are converted to energy E and mass m, respectively, rendering
E = mv2. (2)
Conservation and dissipation of mass and/or energy will be discussed. Rigid (point mass) and deformable bodies are automatically included since Eq. (2) applies to the full scale range of space-time.
2.2. Work done by force as energy
"No Man is an Island" implicates the need of force to accommodate the energy E such that they can be further related to matter and/or mass. The prerequisite is the work done by force f and distance d:
fd = E. (3)
More work done corresponds to more energy and vice versa regardless of conservation nor dissipation. Making use of Eq. (2), there results
fd = mv2. (4)
In passing, Eq. (4) can be shown to determine the Galileo's law of motion:
d = gt1 (5)
in which the acceleration of gravity g is found to be the proportionality constant. The derivation is not trial and was made possible using Eq. (4). Galileo's concern for the effect of mass on the time of the free falling body was not unwarranted. It was a fortunate accident that different mass were found to affect the velocity of falling bodies. The difference of the factor V2 were due to the interdependence of mass and energy. In other words, the classical falling body velocities have been overestimated by the factor -J1. The error has been swept into the scatter of data. The proof will be given subsequently in connection with the derivation of Eq. (1) and/or Eq. (2).
2.3. Exchange of mass and energy for dissipative systems A tradeoff between the mass density and energy density can be established in general for dissipative systems. Referring to the third and fourth of Eq. (6) with j = 1 and j +1 = 2, the third of Eq. (7) in [4], it can be shown that
P3MJWJ = M2W2. (6)
The factor P3 accounts for inhomogeneity and non-equilibrium. For a homogeneous system where P3 = 1, Eq. (6) reduces to Eq. (15) in [4]:
MJW = M2W2. (7)
Mass and energy density are seen to be conserved simultaneously. Replacing the distance d in Eq. (4) by h, it is found that
fh = mv2. (8)
For free falling bodies, f is mg. Hence, Eq. (8) reduces to mgh = mv2. (9)
When both mass and energy are considered concurrently the velocity can be solved from Eq. (9) as
4gh. (10) This result does not contain the factor which is differs from the classical results that mass and energy are conserved separately. In this case, Eq. (7) separates into
Mj = M2 and (11)
Wj = W2. (12)
Equations (11) and (12) indicate that mass density and energy density are conserved individually. More specifically, Eq. (12) states that the kinetic energy density is equal to the
potential energy density: j
- mv = mgh
(13)
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such that the mass m cancels out, rendering
Jlgh. (14)
To reiterate, the factor V2 does not appear in Eq. (10) when both mass and energy are conserved simultaneously. The difference implies that the free falling body time has been overestimated, while the test conditions of the falling bodies are not aware of the subtle difference in analysis.
3. Hookean force of work done as energy
The driving force of matter and/or mass should accommodate multiscaling of space-time. The association of the Hookean force with energy is accomplished through the work done fd in Eq. (3).
3.1. Force proportion to extension
Hooke proposed that force is proportional to extension moving through a distance d:
f - d. (15)
The notion in Eq. (15) led to the relation between work done as energy. The combination of Eqs. (3) and (4) was made possible by using Eq. (1), where the energy density W and mass density M were converted to energy E and mass m, respectively. Energy and mass are found to be exchangeable when mass and energy react simultaneously. The distance d in Eq. (4) can be eliminated to yield
f = t). (16) Equation (16) can be written in the conventional form without the a apriori notion of the inertia force ofNewton. Equation (17) follows from notion of Hooke and is derived from Eq. (15):
f = ma (17)
with a being the acceleration.
3.2. Force of inertia revisited
The force of inertia refers to that postulated in the laws of motion by Newton. The work done by the force of inertia can be written as
fd = m(d/t)2. (18)
Equation (18) is formally the same as Eq. (4), but they differ in meaning. The term mv2 in Eq. (18) is not the energy E of work done by the Hookean force since the inertia force ofNewton cannot accommodate for the concurrence of mass density and energy density at any velocities that is an inherent characteristic of gravitational waves.
4. Push and pull action of work done as energy
Moving masses entail energy of gravitational waves that stretch and squeeze space-time. The push and pull in spacetime p is assumed to be proportional to 1/r, with the magnitude E.
4.1. Engulfment of gravitational waves
Gravitational waves are said to excite mass interactions. A two-body system may be referred to as "local" and the
other as "global". Equation (1) in terms of energy and mass may be written separately for the local with mass m and global with mass M as
El = mvL and EG = mvG. (19)
The form of Eq. (2) is reminiscent of E = mc2 in General Relativity except that v is not the speed of light c. Both sides of Eq. (2) can be multiplied by the volume to relate the energy density to the mass density as given by Eq. (1).
4.2. Mutuabilitys law
The mutuability's law may be stated as: The local gravity pull or push p is equal to the global gravity push or pull p such that their respective energetic colliding fields EL and Eg are inversely proportional to the interactive distance r in accordance with the product:
(20)
2 _ EL EG
P =-
r r
The third law of motion Newton of equal and opposite reaction for two colliding bodies is automatically included in the mutuability's law. Substituting Eq. (19) into Eq. (20), it is found that
p2 = mMg 2G2
r 2Q4
Solving for p, there renders gGr4mM.
P =±-
rQ2
(21)
(22)
The + and - stand for push and pull or squeeze and stretch of space-time giving the appearance of ripples as the fabric of space-time. The g and G correspond to characteristic parameters. The distance between m and M is r. The full range of Q are covered by Eq. (22).
5. A drop in the bucket
At the cosmological time scale, 100 years is a drop in the bucket. The number of drops that will take to fill the bucket will depend on the characteristics of the drops. There are a few bright spots of the first drop, despite the remaining uncertainties.
5.1. Energy density unit
Personalized units such as a Newton of force or a Joule of energy cannot provide the consistency for characterizing gravitational waves. The Hookean force fits into the model of work done as energy such that "energy density" becomes the common unit that all physical quantities can be related to. The positive definiteness of energy density devoids ambiguities of mathematical and physical modeling.
5.2. Teleportation
The atoms in space-time are said to be entangled. The gravitational waves are said to occupy the space-time. It follows that the gravitational waves and the entangled
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atoms are interactive. This interdependence of mass and energy rules out the "single atom" model of classical physics and mechanics. Entanglement prevails for multi-atom space-time. Experiments have shown that three entangled particles of a nitrogen atom locked in a diamond crystal and two electrons were teleported three meters [9]. This is a positive proof that matter can dematerialize and beamed to a target, where it is rematerialized into matter. Gravitational waves will affect teleportation, a topic of future research.
5.3. Ideomechanics
Ideomechanics is a mathematized version of I-Ching available to those at the time of Newton. His associates, however, were more interested in the mathematics of binary progression rather than the physics of mass and energy. Although ideomechanics is a self-independent discipline, it is overly complex for solving the problems of gravitational waves. There is no need to use an elephant gun to annihilate a flea. The ideograms of four ideons suffice to determine the interchange of mass and energy density as given by Eq. (2). In a nut shell, it can be said that "gravitational waves can now be assessed mathematically at any velocities by Eq. (22) using Eq. (2) and not just at the speed of light".
It should be further emphasized that Eq. (2) applies to dissipative systems where energy is not conserved. The conservation of energy would yield Eq. (13) which is a statement of the kinetic energy density being equal to the potential energy density. Equation (2) applies to non-homogeneity, non-equilibrium, and dissipation. This capability is found only in ideomechanics.
The mystery of the Universe remains well hidden. Surprises and new discoveries are expected. Gravitational waves may not just be the push and pull or squeeze and stretch of space-time. The answer lies in the last drop when the bucket is full.
References
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Поступила в редакцию 24.02.2016 г.
Сведения об авторе
George C. Sih, Prof., Lehigh University, USA, gcs@ecust.edu.cn
