Adonis Diaries

Posts Tagged ‘Riemann

Einstein speaks on theoretical physics; (Nov. 18, 2009)

The creative character of theoretical physicist is that the products of his imagination are so indispensably and naturally impressed upon him that they are no longer images of the spirit but evident realities. Theoretical physics includes a set of concepts and logical propositions that can be deduced normally. Those deductive propositions are assumed to correspond exactly to our individual experiences.  That is why in theoretical book the deduction exercises represent the entire work.

Newton had no hesitation in believing that his fundamental laws were provided directly from experience.  At that period the notion of space and time presented no difficulties: the concepts of mass, inertia, force, and their direct relationship seemed to be directly delivered by experience.  Newton realized that no experience could correspond to his notion of absolute space which implicates absolute inertia and his reasoning of actions at distance; nevertheless, the success of the theory for over two centuries prevented scientists to realize that the base of this system is absolutely fictive.

Einstein said “the supreme task of a physician is to search for the most general elementary laws and then acquire an image of the world by pure deductive power. The world of perception determines rigorously the theoretical system though no logical route leads from perception to the principles of theory.” Mathematical concepts can be suggested by experience, the unique criteria of utilization of a mathematical construct, but never deducted. The fundamental creative principle resides in mathematics.

Logical deductions from experiments of the validity of the Newtonian system of mechanics were doomed to failures. Research by Faraday and Maxwell on the electro-magnetic fields initiated the rupture with classical mechanics. There was this interrogation “if light is constituted of material particles then where the matters disappear when light is absorbed?” Maxwell thus introduced partial differential equations to account for deformable bodies in the wave theory. Electrical and magnetic fields are considered as dependent variables; thus, physical reality didn’t have to be conceived as material particles but continuous partial differential fields; but Maxwell’s equations are still emulating the concepts of classical mechanics.

Max Plank had to introduce the hypothesis of quanta (for small particles moving at slow speed but with sufficient acceleration), which was later confirmed, in order to compute the results of thermal radiation that were incompatible with classical mechanics (still valid for situations at the limit).  Max Born pronounced “Mathematical functions have to determine by computation the probabilities of discovering the atomic structure in one location or in movement”.

Louis de Broglie and Schrodinger demonstrated the fields’ theory operation with continuous functions. Since in the atomic model there are no ways of locating a particle exactly (Heisenberg) then we may conserve the entire electrical charge at the limit where density of the particle is considered nil. Dirac and Lorentz showed how the field and particles of electrons interact as of same value to reveal reality. Dirac observed that it would be illusory to theoretically describe a photon since we have no means of confirming if a photon passed through a polarizator placed obliquely on its path. 

      Einstein is persuaded that nature represents what we can imagine exclusively in mathematics as the simplest system in concepts and principles to comprehend nature’s phenomena.  For example, if the metric of Riemann is applied to a continuum of four dimensions then the theory of relativity of gravity in a void space is the simplest.  If I select fields of anti-symmetrical tensors that can be derived then the equations of Maxwell are the simplest in void space.

The “spins” that describe the properties of electrons can be related to the mathematical concept of “semi-vectors” in the 4-dimensional space which can describe two kinds of elementary different particles of equal charges but of different signs. Those semi-vectors describe the magnetic field of elements in the simplest way as well as the properties electrical particles.  There is no need to localize rigorously any particle; we can just propose that in a portion of 3-dimensional space where at the limit the electrical density disappears but retains the total electrical charge represented by a whole number. The enigma of quanta can thus be entirely resolved if such a proposition is revealed to be exact.

Critique

            Till the first quarter of the 20th century sciences were driven by shear mathematical constructs.  This was a natural development since most experiments in natural sciences were done by varying one factor at a time; experimenters never used more than one independent variable and more than one dependent variable (objective measuring variable or the data).  Although the theory of probability was very advanced the field of practical statistical analysis of data was not yet developed; it was real pain and very time consuming doing all the computations by hand for slightly complex experimental designs. Sophisticated and specialized statistical packages constructs for different fields of research evolved after the mass number crunchers of computers were invented. 

            Thus, early theoretical scientists refrained from complicating their constructs simply because the experimental scientists could not practically deal with complex mathematical constructs. Thus, the theoretical scientists promoted the concept or philosophy that theories should be the simplest with the least numbers of axioms (fundamental principles) and did their best to imagining one general causative factor that affected the behavior of natural phenomena or would be applicable to most natural phenomena.

            This is no longer the case. The good news is that experiments are more complex and showing interactions among the factors. Nature is complex; no matter how you control an experiment to reducing the numbers of manipulated variables to a minimum there are always more than one causative factor that are interrelated and interacting to producing effects.

            Consequently, the sophisticated experiments with their corresponding data are making the mathematician job more straightforward when pondering on a particular phenomenon.  It is possible to synthesize two phenomena at a time before generalizing to a third one; mathematicians have no need to jump to general concepts in one step; they can consistently move forward on firm data basis. Mathematics will remain the best synthesis tool for comprehending nature and man behaviors.

            It is time to account for all the possible causatives factors, especially those that are rare in probability of occurrence (at the very end tail of the probability graphs) or for their imagined little contributing effects: it is those rare events that have surprised man with catastrophic consequences.

            Theoretical scientists of nature’s variability should acknowledge that nature is complex. Simple and beautiful general equations are out the window. Studying nature is worth a set of equations! (You may read my post “Nature is worth a set of equations”)

Einstein speaks on General Relativity; (Nov. 20, 2009)

I have already posted two articles in the series “Einstein speaks on…” This article describes Einstein’s theory of restricted relativity and then his concept for General Relativity. It is a theory meant to extend physics of fields (for example electrical and magnetic fields among others) to all natural phenomena, including gravity. Einstein declares that there was nothing speculative in his theory but it was adapted to observed facts.

The fundamentals are that the speed of light is constant in the void and that all systems of inertia are equally valid (each system of inertia has its own metric time). The experience of Michelson has demonstrated these fundamentals. The theory of restrained relativity adopts the continuum of space coordinates and time as absolute since they are measured by clocks and rigid bodies with a twist: the coordinates become relative because they depend on the movement of the selected system of inertia.

The theory of General Relativity is based on the verified numerical correspondence of inertia mass and weight. This discovery is obtained when coordinates posses relative accelerations with one another; thus each system of inertia has its own field of gravitation. Consequently, the movement of solid bodies does not correspond to the Euclid geometry as well as the movement of clocks. The coordinates of space-time are no longer independent. This new kind of metrics existed mathematically thanks to the works of Gauss and Riemann.

Ernst Mach realized that classical mechanics movement is described without reference to the causes; thus, there are no movements but those in relation to other movements.  In this case, acceleration in classical mechanics can no longer be conceived with relative movement; Newton had to imagine a physical space where acceleration would exist and he logically announced an absolute space that did not satisfy Newton but that worked for two centuries. Mach tried to modify the equations so that they could be used in reference to a space represented by the other bodies under study.  Mach’s attempts failed in regard of the scientific knowledge of his time.

We know that space is influenced by the surrounding bodies and so far, I cannot think the general Relativity may surmount satisfactorily this difficulty except by considering space as a closed universe, assuming that the average density of matters in the universe has a finite value, however small it might be.

Einstein speaks his mind processes on the origin of General Relativity; (Nov. 21, 2009)

This article is on  how Einstein described his mind processes that lead to the theory of restricted relativity and then his concept for General Relativity. In 1905, restricted relativity discovered the equivalence of all systems of inertia for formulating physics equations.

From a cinematic perspective, there was no way to doubting relative movements. Still, there was the tendency among physicists to physically extend privileged significance to system of inertia.  The question was “if speed is relative then, do we have to consider acceleration as absolute?”

Ernest Mach considered that inertia did not resist acceleration except when related to the acceleration toward other masses. This idea impressed Einstein greatly.  Einstein said: ” First, I had to establish a law of gravitation field and suppress the concept of absolute simultaneity. Simplicity urged me to maintain Laplace’s “scalar gravity potential” and fine tune Poisson’s equation.

Given the theorem of inertia of energy then, inertia mass must be depended on gravitation potential; but my research left me skeptical. In classical mechanics, vertical acceleration in a vertical field of gravity is independent of the horizontal component of velocity; it follows that vertical acceleration is exercised independently of the internal kinetic energy of the body in movement.

I discovered that this independence did not exist in my draft theory; this evidence did not coincide with the affirmation that all bodies submit to the same acceleration in a gravitational field. Thus, the principle that there is equality between inertia mass and weight grew with striking significance. I was convinced of its validity, though I had no knowledge of the results of experiments done by Eotvos.”

Consequently, the principle of equality between inertia mass and weight would be explained as follows: in a homogeneous gravitational field, all movements are executed in relation to a system of coordinates accelerating uniformly as if in absence of gravity field. I conjectured that if this principle is applicable to any other events then it can be applied to system of coordinates not accelerating uniformly.

These reflections occupied me from 1908 to 1911 and I figured that the principle of relativity needed to be extended (equations should retain their forms in non uniform accelerations of coordinates) in order to account for a rational theory of gravitation; the physical explanation of coordinates (measured by rules and clocks) has to go.

I reasoned that if in reality “a field of gravitation used in system of inertia” did not exist it could still be served in the Galilean expression that “a material point in a 4-dimensional space is represented by the shortest straight line”. Minkowski has demonstrated that this metric of the square of the distance of the line is a function of the squares of the differential coordinates.  If I introduced other coordinates by non linear transformation then the distance of the line stay homogeneous if coefficients dependent on coordinates are added to the metric (this is the Riemann metric in 4-dimension space not submitted to any gravity field). Thus, the coefficients describe the field of gravity in the selected system of coordinates; the physical significance is just related to the Riemannian metric. This dilemma was resolved in 1912.

Two other problems had to be resolved from 1912 to 1914 with the collaboration of Marcel Grossmann.

The first problem is stated as follows: “How can we transfer to a Riemannian metric a field law expressed in the language of restrained relativity?”  I discovered that Ricci and Levi-Civia had answered it using infinitesimal differential calculus.

The second problem is: “what are the differential laws that determine the coefficients of Riemann?”  I needed to resolve invariant differential forms of the second order of Riemann’s coefficients. It turned out that Riemann had also answered the problem using curb tensors.

“Two years before the publication of my theory on General Relativity” said Einstein “I thought that my equations could not be confirmed by experiments. I was convinced that an invariant law of gravitation relative to any transformations of coordinates was not compatible with the causality principle. Astronomic experiments proved me right in 1915.”

Note:  I recall that during my last year in high school my physics teacher, an old Jesuit Brother, filled the blackboard with partial derivatives of Newton’s equation on the force applied to a mass; then he integrated and he got Einstein’s equation of energy which is  mass multiplied by C square. At university, whenever I had problems to solve in classical mechanics on energy or momentum conservation I just applied the relativity equation for easy and quick results; pretty straightforward; not like the huge pain of describing or analyzing movements of an object in coordinate space.


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