Category Archives: Physics

Inertial frame of reference (Wikipedia)

January 14, 2018 – 9:23 AM

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Reference: Disturbance Theory

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Parts from Wikipedia article are quoted in black. My comments follow in bold color italics.

Inertial frame of reference – Wikipedia

An inertial frame of reference, in classical physics, is a frame of reference in which bodies, whose net force acting upon them is zero, are not accelerated, that is they are at rest or they move at a constant velocity in a straight line. In analytical terms, it is a frame of reference that describes time and space homogeneously, isotopically, and in a time-independent manner. Conceptually, in classical physics and special relativity, the physics of a system in an inertial frame have no causes external to the system. An inertial frame of reference may also be called an inertial reference frame, inertial frame, Galilean reference frame, or inertial space.

As described in the paper, The Electromagnetic Cycle, “The electromagnetic cycles collapse into a continuum of very high frequencies in our material domain, which provides the absolute and independent character to the space and time that we perceive.”

The “inertial frame of reference” of classical physics describes only the space and time occupied by matter. It does not describe the space and time that is not occupied by matter.

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces to be negligible, one can find a set of inertial frames that approximately describe that region.

As described in the paper, The Problem of Inertia, “The uniform drift velocity results naturally from the innate acceleration of disturbance balanced by its inertia. The higher is the inertia, the smaller is the velocity. Matter may be looked upon as a “disturbance” of large inertia. Therefore, black holes of very large inertial mass shall have almost negligible velocity. On the other hand, bodies with little inertial mass shall have higher velocities.”

These inertial frames are valid for the material domain only. They are described by the Newton’s Laws of motion. The velocities in this domain are extremely small compared to the velocity of light. These material velocities are described in relation to each other by simple Galilean transformations. Acceleration applied to a body changes its velocity only for the duration of that acceleration. In the absence of acceleration the original velocity restores itself if the inertia of the body has not changed.

In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces. In contrast, systems in non-inertial frames in general relativity don’t have external causes, because of the principle of geodesic motion. In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the Earth is rotating, which means the frame of reference of an observer on Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

An accelerating non-inertial frame is changing in inertia. Therefore, additionalforces appear in that frame to balance that additional inertia.

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Introduction

The motion of a body can only be described relative to something else—other bodies, observers, or a set of space-time coordinates. These are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary. For example, suppose a free body that has no external forces acting on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, a coordinate system could describe the simple flight of a free body in space as a complicated zig-zag in its coordinate system. Indeed, an intuitive summary of inertial frames can be given as: In an inertial reference frame, the laws of mechanics take their simplest form.

It is not true that the motion of a body can only be described relative to something else. A body’s absolute motion may be described in terms of its inertia. In the absence of externally applied forces, the velocities of two bodies differ because of difference in their inertia. The velocities become equal when the difference in inertia is balanced by externally applied forces.

The motion of any non-accelerating body may be chosen as a frame of reference. Another body with the same motion will appear at rest in this frame of reference. Such arbitrary frames of references provide fictitious motion on a relative basis. Absolute motion may be perceived only in a frame of zero inertia.

In an inertial frame, Newton’s first law, the law of inertia, is satisfied: Any free motion has a constant magnitude and direction. Newton’s second law for a particle takes the form…

All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In an inertial frame, a body has no acceleration. Its absolute motion is determined by its inertia. Its apparent velocity and direction is determined by the frame of reference being used. The body is imparted acceleration by a force. The acceleration is proportional to the force applied. The proportionality constant is called the “mass” of the body.

The “mass” of the body is an aspect of its inertia. It shows how “pinned” the body is in space. We know from experience that any rotating motion pins a body in space. Therefore, the “mass” of a body may be looked upon as representing some rotating frame of reference.

In Newton’s time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton’s laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force. Two interesting experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket. In both cases, application of Newton’s second law would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

The fixed stars represent a reference frame of infinite inertia. The absolute motion of such a reference frame is almost zero. An object with lesser inertia will be seen to be in motion in this reference frame. The lesser is the inertia of an object the greater shall be its motion. A rotating frame of reference shall also be rotating with respect to the fixed stars. In that rotating frame of reference there will be inertial forces that are not fictitious, but real, such as, parabolic water surface in the case of the rotating bucket.

As we now know, the fixed stars are not fixed. Those that reside in the Milky Way turn with the galaxy, exhibiting proper motions. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to expansion of the universe, and partly due to peculiar velocities. The Andromeda galaxy is on collision course with the Milky Way at a speed of 117 km/s. The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based upon the simplicity of the laws of physics in the frame. In particular, the absence of fictitious forces is their identifying property…

The identification of an inertial frame is based upon the absence of unexplained force or acceleration.

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Background

A brief comparison of inertial frames in special relativity and in Newtonian mechanics, and the role of absolute space is next.

A set of frames where the laws of physics are simple

According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K’ moving in uniform translation relatively to K.
— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

The special principle of relativity seems to consider the inertial frames of references in material domain only.

This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes. The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel and also Blagojević.

The laws of Newtonian mechanics do not always hold in their simplest form…If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a ‘force’ pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton’s laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity: The laws of mechanics have the same form in all inertial frames.
— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

Only the laws of mechanics have the same form in all inertial frames because they operate on a relative basis in the material domain (the continuum of very high frequencies).

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment (otherwise the differences would set up an absolute standard reference frame). According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.

Absolute motion shall be visible only in a frame of reference of zero inertia. Newtonian mechanics uses light as the reference point of “infinite velocity” for material domain. This is adequate except on cosmological scale where the finite speed of light generates anomalies. Special relativity accounts for the finite velocity of light and explains the cosmological anomalies. Special relativity is adequate except on atomic scale where the finite inertia of light generates anomalies.

Absolute space

Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called “relativists” by Mach), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

As explained in the paper, The Electromagnetic Cycle, space and time may be treated as absolute in the material domain only. The doubts entered only where cosmic dimensions were involved in which light’s finite velocity could not be ignored.

Indeed, the expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton’s definitions of “absolute space and time” by a more operational definition. As translated by Iro, Lange proposed the following definition:

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

A discussion of Lange’s proposal can be found in Mach.

The inadequacy of the notion of “absolute space” in Newtonian mechanics is spelled out by Blagojević:

  • The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.
  • Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
  • Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.
— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

“Absolute space” is actually the reference frame of zero inertia. Newton approximated “absolute space” as the background of fixed stars. But fixed stars provide a reference frame of infinite inertia and not of zero inertia.

The utility of operational definitions was carried much further in the special theory of relativity. Some historical background including Lange’s definition is provided by DiSalle, who says in summary:

The original question, “relative to what frame of reference do the laws of motion hold?” is revealed to be wrongly posed. For the laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.
— Robert DiSalle Space and Time: Inertial Frames

Inertial frames are frames of constant inertia. Acceleration is always related to change in inertia. The physicists have overlooked the concept of zero inertia

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Newton’s inertial frame of reference

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton’s first law of motion is valid. However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton’s first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion relative to the fixed stars; that is, neither rotating nor accelerating relative to the stars. Today the notion of “absolute space” is abandoned, and an inertial frame in the field of classical mechanics is defined as:

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

An inertial frame of reference has its true basis in zero inertia of EMPTINESS, and not in the infinite inertia of fixed stars.

Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton’s first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant speed. Newtonian inertial frames transform among each other according to the Galilean group of symmetries.

In material domain, the level of inertia is so high that compared to it, the differences in inertia of material bodies, and its effect on their velocities can be ignored.

If this rule is interpreted as saying that straight-line motion is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then we have to be able to determine when zero net force is applied. The problem was summarized by Einstein:

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.
— Albert Einstein: The Meaning of Relativity, p. 58

The weakness here is the implicit assumption that the relative uniform motion remains constant in the absence of external forces. This assumption ignores the influence of inertia on motion.

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so we have only to be sure that a body is far enough away from all sources to ensure that no force is present. A possible issue with this approach is the historically long-lived view that the distant universe might affect matters (Mach’s principle). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is that we might miss something, or account inappropriately for their influence, perhaps, again, due to Mach’s principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when we shift reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames, and have complicated rules of transformation in general cases. On the basis of universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces…

A source of “force” is body’s inertia, from which the body cannot be separated. We cannot assume all inertial frames to have the same inertia.

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Separating non-inertial from inertial reference frames

Theory

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces, as explained shortly.

The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….
— Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame:

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.
— V. I. Arnol’d: Mathematical Methods of Classical Mechanics Second Edition, p. 129

Bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames…

The “fictitious” forces are essentially due to the inertia of the reference frame. The influence of inertia can be fully accounted only in a frame of reference of zero inertia.

Applications

Inertial navigation systems used a cluster of gyroscopes and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it. Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of dead reckoning that requires no external input, and therefore cannot be jammed by any external or internal signal source…

The “inertial space” is set by the orientation of spinning gyroscopes.

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Newtonian mechanics

Classical mechanics, which includes relativity, assumes the equivalence of all inertial reference frames. Newtonian mechanics makes the additional assumptions of absolute space and absolute time. Given these two assumptions, the coordinates of the same event (a point in space and time) described in two inertial reference frames are related by a Galilean transformation…

Newtonian mechanics applies only to the material domain of very high inertia, where differences in the inertia of material bodies can be ignored.

Special relativity

Einstein’s theory of special relativity, like Newtonian mechanics, assumes the equivalence of all inertial reference frames, but makes an additional assumption, foreign to Newtonian mechanics, namely, that in free space light always is propagated with the speed of light c0, a defined value independent of its direction of propagation and its frequency, and also independent of the state of motion of the emitting body. This second assumption has been verified experimentally and leads to counter-intuitive deductions including:

  • time dilation (moving clocks tick more slowly)
  • length contraction (moving objects are shortened in the direction of motion)
  • relativity of simultaneity (simultaneous events in one reference frame are not simultaneous in almost all frames moving relative to the first).

These deductions are logical consequences of the stated assumptions, and are general properties of space-time, typically without regard to a consideration of properties pertaining to the structure of individual objects like atoms or stars, nor to the mechanisms of clocks…

From this perspective, the speed of light is only accidentally a property of light, and is rather a property of spacetime, a conversion factor between conventional time units (such as seconds) and length units (such as meters).

Incidentally, because of the limitations on speeds faster than the speed of light, notice that in a rotating frame of reference (which is a non-inertial frame, of course) stationarity is not possible at arbitrary distances because at large radius the object would move faster than the speed of light.

As described in the paper, The Electromagnetic Cycle, “The electromagnetic cycles collapse into a continuum of very high frequencies in our material domain, which provides the absolute and independent character to the space and time that we perceive.

“This is the Newtonian domain of space and time. Einsteinian length contraction and time dilation does not occur in this Newtonian domain. It occurs at much lower electromagnetic frequencies.”

Special relativity allows frames of references that are outside the material domain. The error is to consider them equivalent to those in material domain.

 

General relativity

General relativity is based upon the principle of equivalence:

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.
— Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

General relativity acknowledges inertia in the context of a field.

This idea was introduced in Einstein’s 1907 article “Principle of Relativity and Gravitation” and later developed in 1911. Support for this principle is found in the Eötvös experiment, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 1011. For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.

Inertial and gravitational mass are equivalent.

Einstein’s general theory modifies the distinction between nominally “inertial” and “noninertial” effects by replacing special relativity’s “flat” Minkowski Space with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

The inertial frames of general relativity acknowledge the differences in their inertia and start to account for it.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play. Consequently, modern special relativity is now sometimes described as only a “local theory”. The study of double-star systems provided significant insights into the shape of the space of the Milky Way galaxy. The astronomer Karl Schwarzschild observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the solar system. Schwarzschild pointed out that that was invariably seen: the direction of the angular momentum of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.

Special relativity uses light as its reference frame. This is different from a reference frame of zero inertia. This introduces an error that is carried forward into General relativity.

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By vinaire | Also posted in Inertia, Wikipedia | Comments (2)

Comments on Thermodynamic temperature

November 26, 2017 – 6:14 AM

Reference: Disturbance Theory

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Thermodynamic temperature

Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics.

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. The absolute measure of temperature is one of the principal parameters of thermodynamics.

Thermodynamic temperature is defined by the third law of thermodynamics in which the theoretically lowest temperature is the null or zero point. At this point, absolute zero, the particle constituents of matter have minimal motion and can become no colder. In the quantum-mechanical description, matter at absolute zero is in its ground state, which is its state of lowest energy. Thermodynamic temperature is often also called absolute temperature, for two reasons: one, proposed by Kelvin, that it does not depend on the properties of a particular material; two that it refers to an absolute zero according to the properties of the ideal gas.

The concept of absolute zero provides a reference point of the ground state of matter, which is the state of lowest energy. It is the same for all matter.

The International System of Units specifies a particular scale for thermodynamic temperature. It uses the kelvin scale for measurement and selects the triple point of water at 273.16 K as the fundamental fixing point. Other scales have been in use historically. The Rankine scale, using the degree Fahrenheit as its unit interval, is still in use as part of the English Engineering Units in the United States in some engineering fields. ITS-90 gives a practical means of estimating the thermodynamic temperature to a very high degree of accuracy.

The thermodynamic temperature provides an absolute scale.

Roughly, the temperature of a body at rest is a measure of the mean of the energy of the translational, vibrational and rotational motions of matter’s particle constituents, such as molecules, atoms, and subatomic particles. The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, make up the total internal energy of a substance. Internal energy is loosely called the heat energy or thermal energy in conditions when no work is done upon the substance by its surroundings, or by the substance upon the surroundings. Internal energy may be stored in a number of ways within a substance, each way constituting a “degree of freedom”. At equilibrium, each degree of freedom will have on average the same energy: kBT/2 where kB is the Boltzmann constant, unless that degree of freedom is in the quantum regime. The internal degrees of freedom (rotation, vibration, etc.) may be in the quantum regime at room temperature, but the translational degrees of freedom will be in the classical regime except at extremely low temperatures (fractions of kelvins) and it may be said that, for most situations, the thermodynamic temperature is specified by the average translational kinetic energy of the particles.

TEMPERATURE = a measure of the mean of the energy of the kinetic motions (translational, vibrational and rotational) of matter’s particles. For most situations, the thermodynamic temperature is specified by the average translational kinetic energy of the particles. The Boltzmann constant (kB) is a physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas.

INTERNAL ENERGY = the full variety of kinetic motions along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these.

DEGREES OF FREEDOM = the number of ways in which the internal energy may be stored within a substance. At equilibrium, each degree of freedom will have on average the same energy = kBT/2

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By vinaire | Also posted in Wikipedia | Comments (0)

Comments on Einstein Solid

November 22, 2017 – 8:47 AM

Escv

Heat capacity of an Einstein solid as a function of temperature. Experimental value of 3Nk is recovered at high temperatures.
Reference: Disturbance Theory

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Einstein Solid – Wikipedia

The Einstein solid is a model of a solid based on two assumptions:

  • Each atom in the lattice is an independent 3D quantum harmonic oscillator
  • All atoms oscillate with the same frequency (contrast with the Debye model)

While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves or phonons, collective modes involving many atoms. In the Einstein model, however, each atom oscillates independently. Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics.

A 3D quantum harmonic oscillator would be made up of high frequency, compacted cycles of electromagnetic field that have slowed down considerably due to their high inertia. Their motion is no longer linear, but a combination of circular, rotational and linear, which shows up as oscillatory.

Einstein treated each atom as an independent 3D harmonic oscillator, whose energy could only increase in quantum intervals of ‘hω’. Einstein assumed the same frequency for all atoms for the sake of simplicity.

The original theory proposed by Einstein in 1907 has great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit law was required by classical mechanics, the specific heat of solids should be independent of temperature. But experiments at low temperatures showed that the heat capacity changes, going to zero at absolute zero. As the temperature goes up, the specific heat goes up until it approaches the Dulong and Petit prediction at high temperature.

The classical mechanics predicts the heat capacity of solids to be independent of temperature. It did not explain the observed dependence at lower temperatures. Einstein could show this dependence with his quantum model even if not very accurately.

By employing Planck’s quantization assumption, Einstein’s theory accounted for the observed experimental trend for the first time. Together with the photoelectric effect, this became one of the most important pieces of evidence for the need of quantization. Einstein used the levels of the quantum mechanical oscillator many years before the advent of modern quantum mechanics.

The continuous change in properties, which is a feature of classical mechanics, is seen in context of normal dimensions. When we view properties at atomic dimensions, as in the case of black body radiation, the quantum effects of frequency cycles become prominent.

In Einstein’s model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a T3 power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.

Einstein demonstrated the quantum effects on specific heats of solids at low temperatures which classical mechanics could not explain. Einstein did make simplified assumptions as regard the frequency of atoms, which were modified later in Debye model.

When Walther Nernst learned of Einstein’s 1906 paper on specific heat, he was so excited that he traveled all the way from Berlin to Zürich to meet with him.

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By vinaire | Comments (0)

Comments on Rest Mass

November 19, 2017 – 7:18 PM

rest mass

Reference: Disturbance Theory

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Rest Mass – Wikipedia

The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is that portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system’s total energy and momentum that is the same in all frames of reference related by Lorentz transformations. If a center of momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that “rest frame”. In other reference frames, where the system’s momentum is nonzero, the total mass (a.k.a. relativistic mass) of the system is greater than the invariant mass, but the invariant mass remains unchanged.

Due to mass-energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.

The word “rest” means that mass is not being pushed through the surrounding field. The surrounding field is a continuation of mass. When the mass is pushed through the surrounding field there is the resistance of inertia and acceleration. When there is no manifestation of acceleration the mass is “at rest”. A mass moving at uniform velocity is “rest mass”. When a mass is accelerating, there is force and energy in addition to the mass. This may be looked upon as “equivalent additional mass”.

The Lorentz transformations look at field from the viewpoint of matter and gives it a “mass” that is equivalent to its energy.

Systems whose four-momentum is a null vector (for example a single photon or many photons moving in exactly the same direction) have zero invariant mass, and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.

A field is defined as having cycles and not mass (tight cycles at the upper end of the electromagnetic scale). Therefore, a field is massless but not “cycle-less” or “inertia-less”.  To be able to move faster than light, a particle must have less inertia than a photon. The above description in terms of “four-momentum” is part of a mathematical theory.

If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects’ rest masses. This is also equal to the total energy of the system divided by c2. See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system’s invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.

Here the definition of “invariant” or rest mass is based on a center of momentum frame. An absolute definition of “rest mass” is possible only from the reference point of zero inertia.

For an isolated massive system, the center of mass of the system moves in a straight line with a steady sub-luminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center of momentum frame, the total momentum is zero, and the system as a whole may be thought of as being “at rest” if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c2. This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.

An isolated massive system moving at uniform velocity has zero acceleration same as a system at rest.  This is the center of momentum frame. The uniform velocity is not relevant because it is based on an arbitrary reference frame.

Note that for reasons above, such a rest frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or “rest frame” if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.

The “rest mass” basically boils down to a measure of INERTIA in the reference frame of Emptiness, which provides the reference point of zero inertia.

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By vinaire | Also posted in Wikipedia | Comments (1)

Comments on Mass

November 18, 2017 – 5:51 AM

Mass

Reference: Disturbance Theory

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Mass – Wikipedia

Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. It also determines the strength of its mutual gravitational attraction to other bodies. The basic SI unit of mass is the kilogram (kg).

Mass is the inertial property of matter [see Comments on Inertia]. It manifests in very high frequency regions of the field, where cycles are squeezed very tightly together. The greater is the mass the higher is the frequency gradient with respect to the surrounding field. This frequency gradient acts as force during interactions.

In physics, mass is not the same as weight, even though mass is often determined by measuring the object’s weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.

Mass is the tightness of cycles at very high frequencies. Weight appears when the frequency gradient of mass interacts.

In Newtonian physics, mass can be generalized as the amount of matter in an object. However, at very high speeds, special relativity states that the kinetic energy of its motion becomes a significant additional source of mass. Thus, any stationary body having mass has an equivalent amount of energy, and all forms of energy resist acceleration by a force and have gravitational attraction. In modern physics, matter is not a fundamental concept because its definition has proven elusive.

Very high speeds are meaningless if the associated acceleration is zero. They have meaning only when there is acceleration or deceleration during interactions. This little fact modifies the theory of relativity.

There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured:

  • Inertial mass measures an object’s resistance to being accelerated by a force (represented by the relationship F = ma).

During acceleration, matter moves through the surrounding field. The interaction of its frequency gradient with the surrounding field appears as the “resistance” called inertia. This “resistance” is equal to the force generating the acceleration. The ratio of force to acceleration provides a measure of inertial mass.

  • Active gravitational mass measures the gravitational force exerted by an object.

The gravitational force occurs between two material objects separated by field. This force is manifestation of the frequency gradients of masses with the surrounding field. Newton’s formula for gravitation then provides a measure of gravitational mass.

  • Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field.

Gravitational force is essentially a measure of the frequency gradient between the mass region and the surrounding field.

The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe the same properties of physical bodies at the qualitative and quantitative level respectively, by other words, the mass quantitatively describes the inertia. According to Newton’s second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A body’s mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass mA is placed at a distance r (center of mass to center of mass) from a second body of mass mB, each body is subject to an attractive force Fg = GmAmB/r2, where G = 6.67×10−11 N kg−2 m2 is the “universal gravitational constant”. This is sometimes referred to as gravitational mass. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been entailed a priori in the equivalence principle of general relativity.

Mass exists in a region of the field because of the tightness of the very high frequency cycles. The frequency gradient of this mass with the surrounding low frequency field determines the force required to move it through the field. This is perceived as inertia.

There is definite relationship between two masses and the relative frequency gradient, which determines the gravitational force between them. The distance between them is part of that combined frequency gradient.

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