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“A short history of nearly everything” by Bill Bryson

Physics, the quantification of Earth, and the Universe

The physicist Michio Kaku said: “In some sense, gravity does not exist; what moves the planets and stars is the distortion of space and time.”

Gravity is not a force but a byproduct of the warping of space-time, the “ultimate sagging mattress”.

This new understanding of the universe that time is an intrinsic dimension as space was offered by Albert Einstein through his Special Theory of Relativity.

Among other principles, Einstein realized that matter is energy that can be released under specific conditions so that energy is defined as the product of mass and the square of the speed of light c = 300,000 km/s.

In his attempt to unify classical and relativity laws, Einstein offered his General Theory of Relativity and introduced a constant in the formula to account for a stable Universe.  Einstein declared that this constant was “the blunder of his life”, but scientists are now trying to calculate this constant because the universe is not only expanding but the galaxies are accelerating their flight away from the Milky Way.

In 1684, Edmond Halley, a superb scientist in his own right and in many disciplines, and the inventor of the deep-sea diving bell, visited Isaac Newton at Cambridge and asked him what is the shape of the planetary paths and the cause of these specific courses.  Newton replied that it would be an ellipse and that he did the calculation, but could not retrieve his papers.  The world had to wait another two years before Newton produced his masterwork: “Mathematical Principles of natural Philosophy” or better known as the “Principia”.

Newton set the three laws of motion and that for every action there is an opposite and equal reaction.  His formula stated that force is proportional to the product of the masses and inversely proportional to the square of their corresponding distances.  The constant of gravity was introduced, but would wait for Henri Cavendish to calculate it.

It is to be noted that most of his life, Newton was more serious in alchemy and religion than in anything else.

Henry Cavendish was born from a dukes families and was the most gifted English scientist of his age; he was shy to a degree bordering on disease since he would not meet with anyone and, when he visited the weekly scientific soirees of the naturalist Sir Joseph Banks, guests were advised not to look him straight in the face or address him directly.

Cavendish turned his palace into a large laboratory and experimented with electricity, heat, gravity, gases, and anything related to matter.  He was the first to isolate hydrogen, combine it with oxygen to form water.  Since he barely published his works many of his discoveries had to wait a century for someone else to re-discover the wheel.

For example, Cavendish anticipated the law of the conservation of energy, Ohm’s law, Dalton’s law of partial pressures, Richter’s law of reciprocal proportions, Charles’ law of gases, and the principles of electric conductivity. He also foreshadowed the work of Kelvin on the effect of tidal friction on slowing the rotation of the earth, and the effect of local atmospheric cooling, and on and on.  He used to experiment on himself as many scientists of his century did, such as Benjamin Franklin, Pilate de Rozier, and Lavoisier.

In 1797, at the age of 67, Cavendish assembled John Michell’s apparatus that contained two 350-pound lead balls, which were suspended beside two smaller spheres. The idea was to measure the gravitational deflection of the smaller spheres by the larger ones to calculate the gravitational constant of Newton.

Cavendish took up position in an adjoining room and made his observations with a telescope aimed through a peephole.  He evaluated Earth weight to around 13 billion pounds, a difference of 1% of today’s estimate and an estimate that Newton made 110 years ago without experimentation.

John Michell was a country parson who also perceived the wavelike nature of earthquakes, envisioned the possibility of black holes, and conducted experiments in magnetism and making telescopes. Michell died before he could use his apparatus which was delivered to Cavendish.

The 18th century was feverish in measuring Eart: its shape, dimensions, volumes, mass, latitude and longitude, distance from the sun and planets and they came close to the present measurement except its longivity, and had to wait till 1953 for Clair Patterson (a male geologist) to estimate it to 4,550 million years using lead isotopes in rocks that were created through heating.

Neutrinos are elementary particles hypothesized a couple of days to having no mass.  Neutrinos easily cross matters and the atomic core at the speed of light.  Yes, neutrinos are allocated three “flavors” or types:  neutrino-electron, neutrino-muon, and neutrino-tau (guess relating to the kinds of nuclear reactions emitting the neutrinos).  The short story is that a team of scientists announced on Mai 31, 2010 that they observed neutrino-muon oscillating (mutating) into neutrino-tau type.

Why this big fuss? Now that physicists believe that, after 80 years of experimentation, they observed one neutrino flavor mutates into another flavor then, neutrino should have a mass.  Again, why this fuss?  Physicists feel now more confident to explaining how we ended up living in a world of matter instead of anti-matter. The theory of the Big Bang would like us to believe that a universal fight of annihilation “for existence” between these two types of matters ended up with matter as victor (in our world).  So, why this fuss?  Apparently, if neutrinos have masses (infinitesimally small) then they contributed in this universal struggle for matter winning the battle!

Hold on a second.  This is not the end of the wonderful story.  There is a laboratory close to Rome called Gran Sasso, 1,400 meters below ground to shelter its rooms from cosmic rays.  In Geneva, and exactly 732 kilometers from Gran Sasso, there is this monster of nuclear accelerator of particles, 22 kilometers long, used by a European conglomerate CERN.  In CERN, protons were accelerated on a graphite target; the collision produced all kinds of particles at huge quantities.  Among these elemental particles we have neutrinos-muon by the trillions that reached Gran Sasso within 2.3 millisecond.  Of all the neutrinos-muon, a SINGLE neutrino-tau was detected (caught hand in the bag).

So much investment to proving a theory of the Big Bang that may also greatly interest superpower military complexes anxious to mass kill mankind while preserving the materials.   In the meanwhile, millions are dying of hunger, thirst, and common diseases every day for lack of a wretched single dollar to survive.  Just another point of view.

Note:  An Austrian physicist hypothesized the existence of neutrinos in 1930.  Neutrinos were detected 30 years later.  Every second, each square centimeter is bombarded by 65 billion neutrinos from stars and a variety of nuclear reactions.

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”)

568.  theoretical physicsspeaks on theoretical physics; (Nov. 18, 2009)

 

569.  I am mostly the other I; (Nov. 19, 2009)

 

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

 

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

 

572.  Everyone has his rhetoric style; (Nov. 22, 2009)

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.

 

Article #52, (September 12, 2006)

“Mathematics: a unifying abstraction for Engineering and Physics Phenomena”

A few examples of mechanical and electrical problems will demonstrate that mathematical equations play a unifying abstraction to various physical phenomena of entirely different physical nature.

Many linear homogeneous differential equations with constant coefficients can be solved by algebraic methods and their solutions are elementary functions known from calculus such as the examples in article 51.  For the differential equations with variable coefficients, the functions are non elementary and they fall within two classes and play an important role in engineering mathematics.

The first class consists of linear differential equations such as Bessel, Legendre, and the hyper geometric equations; these equations can be solved by the power series method.

The second class consists of functions defined by integrals which cannot be evaluated in terms of finite many elementary functions such as the Gamma, Beta, and error functions (used in statistics for the normal distribution) and the sine, cosine, and Fresnel integrals (used in optics and antenna theory); these functions have asymptotic expansions in the sense that their series may not converge but numerical values could be computed for large values of the independent variable.

Entirely different physical systems may correspond to the same differential equations, not only qualitatively, but even quantitatively in the sense that, to a given mechanical system, we can construct an electric circuit whose current will give the exact values of the displacement in the mechanical system when suitable scale factors are introduced.

The practical importance of such an analogy between mechanical and electrical systems may be used for constructing an electrical model of a given mechanical system. In many cases the electrical model provides essential simplification because it is much easier to assemble and the values easily measured with accuracy while the construction of a mechanical model may be complicated, expensive, and time-consuming.

An RLC-circuit offers the following correspondence with a mechanical system such as: Inductance (L) to mass (m), resistance (R) to damping constant (c), reciprocal of capacitance (1/C) to spring modulus (k), derivative of electromotive force to the driving force or input force, and the current I(t) to the displacement y(t) or output.

Here are a few elementary examples:

5)      Ohm’s law: Experiments show that the voltage drop (E) in a close circuit when an electric current flows across a resistor (R) is proportional to the instantaneous current (I), or E = R* I.

Also, that the voltage drop across an inductor (L) is proportional to the instantaneous time rate of change of the current, or E = L*dI/dt.

Also, the voltage drop across a capacitor (C) is proportional to the instantaneous electric charge (Q) on the capacitor, or E = Q*1/C.  Note that I(t) = dQ/dt.

6)    Kirchhoff’s second law:  The algebraic sum of all the instantaneous voltage drops around any closed loop is zero, or the voltage impressed on a closed loop is equal to the sum of the voltage drops in the rest of the loop. Thus,

E(t) = R*I + L*dI/dt + Q*1/C.

For example, a capacitor (C = 0.1 farad) in series with a resistor (R = 200 ohms) is charged from a source (E = 12 volts).  Find the voltage V(t) on the capacitor, assuming that at t = 0 the capacitor is completely uncharged.

7)    Hooke’s Law: Experiments show that when a string is stretched then the force generated from the string is proportional to the displacement of the stretch,

or F = k*s.  If a mass (M) is attached to a string, then when the string is stretched further more (y) after the system is in a static equilibrium, then: F = -k*s(0) – k*y.

Newton’s second law for the resultant of all forces acting on a body says that:

Mass * Acceleration = Force, or My” = -k*y.

Furthermore, if we connect the mass to a dashpot, then an additional force come into play, which is proportional to the rate of change of the displacement due to the viscous substance with constant (c).  The equation is then a homogeneous second order differential equation: M*y” + c*y’ + k*y = 0.  Depending on the magnitude of (c) we have 3 different solutions: either 2 distinct real rots, 2 complex conjugate roots, or a real double root [c(2) = 4*M*k)} corresponding respectively to the conditions of  over damping, under dumping, or critical damping.

For example, determine the motions of the mechanical system described in the last equation, starting from y = 1, initial velocity equal zero, M = 1 kg, k =1 for the various damping constant: c = 0, c = 0.5, c = 1, c =1.5, and c = 2.

8)    Laplace’s equation is one of the most important partial differential equations because it occurs in connection with gravitational fields, electrostatic fields, steady-state heat conduction, and incompressible fluid flow.  The solutions of the Laplace equation fall within the potential theory.

For example, find the potential of the field between two parallel conducting plates extending to infinity which are kept at constant potentials; or the potential between two coaxial conducting cylinders; or the complex potential of a pair of opposite charged sources lines of the same strength at two points.

 


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