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Perceptions of Physical Reality - 4 |
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by Dr. Raj Vatsya |
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Classical Theories – Special Relativity
Present article is the 4th one in a series, which continues from the 3rd: Perceptions of Physical Reality/ Classical Theories - Forces cited here as [III]; which together with the articles cited, [I] and [II], should be checked for explanations of some of the concepts appearing here. In [III], I had discussed the fundamental forces that determine the particle trajectories together with the related physical observables in the framework of the non-relativistic classical mechanics, i.e., the Newton-Galilean mechanics. The concept of forces developed into the concept of force fields. The gravitational field stood all alone but the electric and magnetic fields were unified in the form of electromagnetic fields with the associated phenomenon of electromagnetic waves. Light was recognized to be the electromagnetic waves. Consequent developments forced upon us some radical adjustments to the earlier concepts of space and time, which were thought of as the absolute independent entities. This necessitated in turn some adjustments to the earlier formulations. The pertaining issues are discussed in the present article. History of Special Relativity
It is normally assumed that the special theory of relativity was developed in its entirety by Einstein. In fact there were many contributors to the development of this theory over a period of time, most notable ones being Lorentz and Poincare. It has been said that in view of the earlier developments, the final form of the special theory of relativity was to be an immediate natural outcome. Here we mention briefly some developments preceding and about the time of Einstein’s formulation.
During the same period, Thomson, FitzGerald, Heaviside and Searle recognized that an electromagnetic field has energy and mass, and determined that the energy is directly proportional to the mass multiplied by the squared velocity of light, a precursor to the famous relativistic mass energy equivalence relation E = mc
Yet again, some physicists, e.g., Ernst Mach had suggested in about 1883 that absolute space and time are meaningless and only the relative motion is a useful concept. Poincare argued that a violation of the relativity principle can never be detected. Michelson-Morley experiments designed to detect the motion of Earth in aether produced negative result thereby disproving an existence of aether. Thus, Galileo’s principle of relativity remains intact. Lorentz Transformations
Einstein’s formulation is founded upon two basic postulates, i.e., assumptions, axioms. The first assumption is that all inertial frames of reference are equivalent for the formulation of the laws of physics, i.e., the Galilean principle of relativity is valid. The second assumption is that the velocity of light in vacuum is the same in all inertial frames of reference. Einstein called the second assumption “only seemingly at variance with the first.” Relativistic Spacetime
The Lorentz transformations introduce a new measure of invariant “length” in the four dimensional spacetime. For convenience, we use the natural units in which the velocity of light assumes value equal to one. In natural units, the square of this “length,” called the “Proper Time,” between two events is equal to the squared time difference between the events minus the squared special distance between the locations of the events, which can be positive, negative or zero. Light Cone and Causality
The Minkowskian structure defines two regions in the spacetime separated by the surface of the light cone, which is actually two cones joined at their common vertex, as shown in the figure below. The surface of the composite cone defines the path of light. Points in the interior of the cone describe the motion at speeds less than that of light and the points in the exterior describe the motion at speeds more than that of light. Time
In the Newton-Galilean formulation, time was assumed to be an absolute independent parameter. In the relativistic, Minkowskian formulation, time is just another independent dimension as is each one of the space dimensions. While the detailed description of a recent development that may alter our view of the time, and thus of spacetime radically, is beyond the scope of the present article, its brief discussion would be valuable for its significance in the present context. This development might be revisited in a later article. First we discuss some background issues.
In the same vein, let us amuse ourselves by constructing a structure out of the three-dimensional Euclidean space. A straight line segment and thus, its arclength is determined solely by the two end points of the segment. The line can be parameterized by the arclength measured from an arbitrarily selected reference point, by associating the three spatial coordinates of a point with the arclength covered up to that point. There is no need to introduce any external parameter for this purpose. This is done routinely in mathematics inducing the same in physics also. Now, we can construct a graph by plotting the coordinates with respect to the arclength, the parameter; again a standard practice to place a formulation in a more discernible, visual form. What results is the surface of a cone; in fact two cones, one being the reflected image of the other. The surface of the cone defines its interior also, which corresponds to the non-straight lines. The exterior is also naturally determined. This cone is exactly like the light cone, prompting one to identify the arclength with time. With this identification, the construction yields the light cone in the natural units. Thus, we have constructed the Minkowskian manifold from purely mathematical considerations providing the basic geometrical structure to formulate our pertaining experiences leading to the theory of special relativity in its completeness. With some further developments, this concept has been shown to provide also a formulation of the electromagnetic fields founded upon radically different axioms than the previous ones. This formulation admits further natural extensions. Mass-Energy Equivalence
In the pre-relativistic mechanics, time served as an independent invariant parameter to describe the evolution of physical systems, e.g., the motion of a particle. In the relativistic mechanics, time assumes the role of the fourth dimension of the four-dimensional spacetime complex, playing a role similar to each of the three space dimensions. Therefore, we need some other parameter to describe the evolution of physical systems in the relativistic setting. The proper time, which is an invariant of the relativistic formulation, tends to the time as the particle velocity tends to zero. Thus, proper time is a suitable, in fact a natural, extension of time to serve as the evolution parameter.
In addition to the velocity and momentum, other physical quantities, e.g., the force, are extended to the Minkowski manifold. With these extensions, Newton’s laws and the action principle admit natural Lorentz covariant extensions and the relativistic mechanics is developed out of the non-relativistic one in a natural way. Maxwell’s equations were known to be Lorentz covariant, which had stimulated early developments of the relativistic formulation. However, the formulation of electromagnetism was also streamlined by introducing the concept of the electromagnetic potentials constituting the components of a Lorentz covariant four-vector. The electromagnetic fields then assume a secondary role of the deduced quantities. The potentials also arise naturally in formulating the mechanics of a charged particle in an electromagnetic field in the framework of the action principle.
A fruit on a tree has potential energy due to the gravitation in the sense that when the fruit breaks away from the branch it falls to the ground and acquires the kinetic energy; the potential energy is converted into the kinetic energy. The total energy along its trajectory, which is the sum of the potential and the kinetic energies, remains constant at every point of its trajectory; total energy is conserved even though each of the two forms does not conserve by itself. Realized or potential kinetic energy came to be the basis of all forms of energy. Concluding Remarks
Newton’s laws introduced the concept of forces, which developed into the concept of fields, particularly the electromagnetic. Then the electromagnetic waves arose mainly out of the mathematical considerations, which were experimentally observed later. Seemingly unrelated entity, light was understood to be the electromagnetic waves; visible part covering a narrow range of frequencies. Need for a medium for waves to propagate forced upon us an untenable concept of aether, which forced us to choose between the Galilean relativity principle and the Newtonian concept of absolute space, preferred frame of reference. Experimental observation proved to be the arbiter in favor of the Galilean relativity principle. However, it left us with no medium for the electromagnetic waves to travel. For now, it was assumed that the electromagnetic waves could travel in vacuum as the fields can exist in the empty space and the electromagnetic waves are just the oscillating electromagnetic fields. This view would be superseded by yet another radical theory later.
The relativistic formulation altered our perceptions of the space-time and the workings of nature on the background of spacetime in a fundamentally radical way. The history of its development also illustrated a beautiful interplay between metaphysics together with the pertaining philosophy; theoretical physics with its basic requirement to describe nature with a set of self-consistent assumptions; experimental physics together with its power to arbitrate and mathematics together with its ability to lead to natural logical consequences that become part of our perceptions of physical reality. Such occurrences would take place in an even more profound manner in some later theories. |
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21-Oct-2013 |
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Comments on this Article
P. Rao05/20/2016 00:09 AM
Raj Vatsya05/19/2016 07:59 AM
Raj Vatsya05/19/2016 06:43 AM
P. Rao05/18/2016 06:25 AM
Dr. Varanasi Ramabrahmam10/25/2013 04:54 AM
rdashby10/21/2013 21:53 PM |
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