Relativity Calculator

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Sun, 11 Feb 2007 13:38:32 -0500

You are about to enter a voyage of knowing and understanding your physical world. It is a voyage built upon the knowledge and understandings handed down to us by ancient Greek philosophers and thinkers, astronomers and mathematicians, right straight down thru to Einstein and other modern men and women of wisdom and insight into nature's ultimate secrets.
How, for instance, was the earth's circumference determined by simply pacing out steps on the earth's surface while at the same time thinking about this problem? Or, how was the distance to the sun and other planets determined with only their light coming into the eyes of those wise enough to understand and interpret its significance? Why, anyways, is it so significant to understand the nature of light and its distant travels? And how, by the way, do we literally count time and distances when simply standing on the earth's surface? Simply standing on either the moon or mars does not alter this basic question because their distances to the galaxies are not significantly different from that of the earth to these far off filaments in the universe.
As each of these questions - simple and not so simple - came into the minds of great thinkers past, an arduously built staircase of knowledge of experimental and mathematical latices came to be constructed. And what questions were heretofore never even imagined by the Ancients such as whether we exist in a multi-verse system of bubbling, interacting and competing universes, are now being asked by modern cosmologists possessing newer and more powerful tools of theoretical mathematics and applied experimentation.
Finally, of what importance is all this knowledge and understanding to human existence? This is not a minor question.

The Challenge of Plato's Cave

Sun, 11 Feb 2007 13:38:28 -0500

What is time?

This is not an easy question to answer. We all intuitively "understand time" by the clocks and calendars we invent to "measure it". In a sense, therefore, the "concept of time" is given an "operational definition" by these invented instruments which ultimately relate back to earth's axis of rotation as well as its orbital transit about the solar system's sun.

Perhaps a more elegant definition of time would involve the relative motion of objects [ or event-objects ] such as the statement "this event-object occurred before that event-object" in a volume of human-perceived 3-d space.

And the "units of time" are suggested to the human mind by the regularity of recurring event-objects such as the periodicity of the sun's rising and falling. Other units of time were also suggested to the early Greek astronomer-philosophers as they studied the celestial movements of stars and planets.

Dictionary definition of time according to Webster's College Dictionary, 2001: " 1. the system of those sequential relations that any event has to any other, as past, present, or future; indefinite and continuous duration regarded as that in which events succeed one another."

A more precise yet simpler definition of time given by this author: time is an accounting of the relative motions of bodies. Or, time is a system of accounting for the relative motion of bodies. And how we keep "score" is by means of human - invented clocks and calendars.

Early Models of the Universe

Sun, 11 Feb 2007 13:38:23 -0500

"What are the uniform and ordered movements, by the assumption of which the apparent movements of the planets can be accounted for?" - Plato 4th Century B.C.

The Anthropic Principle - From the Greek word for human being ( ἄνθρωπος - ánthrōpos ), the Anthropic Principle has come to mean that there exists in the universe finely tuned constants and forces such as space - time curvature, K, the fine - structure constant∗ or the Summerfeld fine-structure constant, which may or may not actually be constant, introduced into physics in 1916 by German theoretical physicist Arnold Summerfeld ( 1868 - 1951 ) ( consisting of the speed of light, Planck's Constant, elementary charge, and the permittivity of free space or space's ability as an intermediary to transmit forces ), and the four fundamental forces of nature ( strong nuclear force, weak nuclear force, electromagnetism, and gravity force ) together with dark energy, all of which conspire to produce the configuration of the present universe and thence the carbon necessary for intelligent human life on Planet Earth to have successfully evolved. Change one constant or one force, and intelligent human life on Planet Earth ceases to exist. [ note: whether intelligent human life continues successfully on Planet Earth is definitely up for debate ] This latter is also sometimes referred to as "carbon chauvinism" as there may actually be alternative bio-chemistries in the universe as the basis for the development and evolution of intelligent life forms different from our own. Nevertheless we are what we are in this carbon-based world of ours, and the Anthropic Principle still stands as a unifying philosophical statement as to why everything exists as it does.

The Earth's Circumference

Sun, 11 Feb 2007 13:38:17 -0500

Or, how far is Paris from Ceyenne, French Guiana? This was a critical question for Giovanni Cassini in 1672 in his determination of the Mars Parallax.

The earliest Greek philosophers and mathematicians considered the sphere to be among the most perfect of all geometric figures. With a few exceptions such as Homer and Anaximenes, practically all Greek Era thinkers beginning with Pythagoras ( 569 - 475 BC ) and Aristotle ( 384 - 322 BC ) believed that Earth was God's creation of the perfect celestial body and hence was created as the perfect spherical object in the heavens. Plato guessed Earth's circumference as ≈ 40,000 miles while Archimedes ( 287 - 212 BC ) estimated it as ≈ 30,000 miles. Only Hellenistic philosopher and mathematician Eratosthenes ( 276 - 194 BC ), born in Cyrene ( modern Libya ), but working, studying and dying in Alexandria, Ptolemaic Egypt, first did devise the mathematics for determining Earth's circumference and hence its diameter. Besides studying in Athens and Alexandria, Eratosthenes was the 2nd appointed head in 236 BC of the Great Library at Alexandria succeeding Zenodotos. Cleomedes's "On the Circular Motion of the Celestial Bodies" gives the first reasonable description of Eratosthenes's Earth circumference mathematics where in 240 BC at the summer solstice on June 21st, the sun shone directly overhead in Syene ( now Aswan, southern Egypt ), but yet cast a shadow upon the ground at noontime by the Alexandria Spire equal to ≈ 1/8th the height of the large tower.

Phenomenon of Parallax

Sun, 11 Feb 2007 13:38:12 -0500

Against a very distant seemingly "fixed star" background, any nearby star will display an apparent movement as the Earth revolves in its orbit about the Sun. This apparent movement produces an angle of perception called "angle of Parallax" which is used to measure the distance to the star from Earth. This phenomenon is similar as to when an object is held at arm's length and seems to move against a more distant background as each eye is opened and closed. However at extreme distances the angle of Parallax becomes infinitesimally small and hence unusable.

Kepler's 1st Law ( Planetary Law of Ellipses )

Sun, 11 Feb 2007 13:38:08 -0500

Kepler's 1st Law ( Planetary Law of Ellipses ) - All planetary orbits are ellipses with the Sun at one of the two foci.

Kepler's 2nd Law: Equal Areas in Equal Times.

Sun, 11 Feb 2007 13:38:04 -0500

The Law of Equal Areas states that a line joining a planet and its star sweeps out equal areas during equal intervals of time.

Kepler's 3rd Law ( Harmonic Law )

Sun, 11 Feb 2007 13:37:59 -0500

Kepler's 3rd Law ( Harmonic Law ) - The square of a planet's orbital period is directly proportional to the cube of the planet's mean distance ( semi-major axis of the planet's elliptical orbit ) from the

The Earth-Sun Distance

Sun, 11 Feb 2007 13:37:54 -0500

Or, how far are the planets from the Sun?

To solve this problem, it was necessary in the History of Science for the following to have occurred:

• Tycho Brahe's planetary data, especially that of Mars
• The mathematics of Parallax utilized by Giovanni Cassini in 1672 at the Royal Observatory, Paris
• Kepler's Equations of Planetary Motion

Olaf Roëmer and the speed of light

Sun, 11 Feb 2007 13:37:49 -0500

Almost from ancient times, light has been acknowledged as the transmitter of information to human eyes. Without light hardly anything is known about the "external world". Aristotle ( 384-322 BC ) conceived of light as ubiquitous and instantaneous whereas Empedocles of Acragas ( 492-432 BC ), Sicilian philosopher-physician-poet and Socrates contemporary, philosophized that the time of transit of light thru an intervening space and bringing information to an observer's eye is finite.

More recently, however, the first acknowledged quantitative determination of the finite speed of light was performed by Danish Olaf Roëmer ( 1644-1710 ) in December 7, 1676 and officially published in "Philosophical Transactions of the Royal Society, No. 136; June 25, 1677, by Olaf Roëmer".

Distance and the John Harrison time clocks which measure it

Sun, 11 Feb 2007 13:37:43 -0500

"The clock is a piece of machinery whose 'product' is seconds and minutes" - Louis Mumford ( 1895 - 1990 )

"The clock, not the steam engine, is central to the industrial revolution. The clock is the crowning achievement that all other machines aspire to."
Louis Mumford ( 1895 - 1990 )

The Luminiferous Aether Wind

Sun, 11 Feb 2007 13:37:38 -0500

Or, what is the nature of light and how does it comes to our eyes?

According to Isaac Newton ( 1642 - 1727 ) in his "Opticks" ( 1704 ) light is transmitted at a finite speed by the "Luminiferous Aether Wind" which is the common or universal medium of propagation. This aether wind was further hypothesized as an absolute and stationary "frame of reference" for the measurement of time, especially as James Clerk Maxwell's ( 1831 - 1879 ) equations for electromagnetic radiation - including light - required it.

The Luminiferous Aether was imagined by physicists since Isaac Newton as the invisible "vapor" or "gas aether" filling the universe and hence as the carrier of heat and light to our eyes, although we now understand by Maxwell's equations that these are both a unified electromagnetic phenomena.

A Deeper Understanding of the Mathematics of the Michelson-Morley Experiment

Sun, 11 Feb 2007 13:37:33 -0500

Ok, so how does the FitzGerald-Lorentz Contraction Equation solve the Michelson-Morley null result?

Remember, at this stage of physics understanding ( 1895 ) the idea of the hypothetical luminiferous aether is still being maintained and that the FitzGerald-Lorentz Contraction Equation is being utilized in an ad-hoc fashion in order to rescue the concept of the aether! That is, the interpretation given by Lorentz for this "contraction effect" was wrong, nevertheless the equation was correct and it would shortly enter into a pantheon of other equations developed by Poincaré and others leading ultimately to the tensor mathematics of Special Relativity.

In truth the FitzGerald-Lorentz Contraction Equation only temporarily postponed the eventual abandonment of the concept of an invisible luminiferous aether. It will be Albert Einstein's publication in 1905 of the tensor mathematics of Special Relativity and Einstein's concept of space-time with the Lorentz Transformation Equations ( as the FitzGerald-Lorentz Contraction Equation came to be known ) at its core foundation to have finally "solved" the Michelson-Morley null result.

The FitzGerald-Lorentz Solution to the Michelson-Morley Null Result

Sun, 11 Feb 2007 13:37:29 -0500

The FitzGerald-Lorentz Solution to the Michelson-Morley Null Result

Summary: The FitzGerald-Lorentz Solution to the Michelson-Morley Null Result

Sun, 11 Feb 2007 13:37:24 -0500

Summary: The FitzGerald-Lorentz Solution to the Michelson-Morley Null Result

Stationary vs. Moving Clocks

Sun, 11 Feb 2007 13:37:19 -0500

§ Assume: S' system is moving "inside" stationary S system with velocity v and is carrying the following clock consisting of a light-flash source and a receiving photocell. One "clock tick" consists of a roundtrip light-flash and photocell reception:

The Heart of Special Relativity: Lorentz Transformation Equations

Sun, 11 Feb 2007 13:37:14 -0500

The Relativity Postulates:

1). The Principle of Relativity - All the laws of physics in their simplest reduced form are transformable and hence invariant as between an infinite number of moving reference
systems ( inertial systems ), each one of which is moving uniformly and rectilinearly with respect to any other system and where no one system
is privileged or preferred over any other reference ( inertial ) system when measurements of length or time are taken.

2). The Principle of the Constancy of the Speed of Light - The speed of light in empty ( vacuo ) space is a universal constant as measured in any reference ( inertial ) system when
measured with rods and clocks of the same kind. This is always true notwithstanding any "relativistic effects" of either the
Lorentz length contraction or time dilation as earlier revealed by the Michelson-Morley Experiment (1887 ).

Solution to Equation (7a) - Lorentz Transformation Equations

Sun, 11 Feb 2007 13:37:09 -0500

Solution to Equation (7a) - Lorentz Transformation Equations

Solution to Equation (9) - Lorentz Transformation Equations

Sun, 11 Feb 2007 13:37:05 -0500

Solution to Equation (9) - Lorentz Transformation Equations

A Simpler Lorentz Transformation Derivation

Sun, 11 Feb 2007 13:37:00 -0500

§ Assume strictly and only from the Michelson - Morley Experiment ( 1887 ) - that is, we know nothing more than that which is obtained from this famous experiment and its
ad-hoc solution:

Some Results of the Lorentz Transformation Equations

Sun, 11 Feb 2007 13:36:54 -0500

I. Result 1 - clock rates:

Ia. Corollary - space-time:

II. Result 2 - "The Failure of Simultaneity at great distances":

III. Result 3 - Length Contraction:

IV. Result 4 - Time Dilation ( Time Contraction ):

Time as the 4th Dimension

Sun, 11 Feb 2007 13:36:49 -0500

Ok, so we now know about the Lorentz Transformation Equations. So why is time the 4th dimension?

Addition of Relativistic Velocities

Sun, 11 Feb 2007 13:36:44 -0500

The Problem:

At velocities approaching the speed of light, mass-particles and other object bodies contract in the direction of motion as well as the measurement of time dilates ( contracts ) as seen by an outside ( relatively ) stationary observer. In fact, the speed of light itself determines the very upper limit of velocity at which any object body or mass-particle can attain because otherwise the frame of reference of such an object body ( or mass-particle ) would "outrun" any light propagation from itself and would thus violate the Lorentz Transformation Equations upon which all has been derived up to this point in our discussion of Special Relativity by giving imaginary number results as can be viewed directly from the Lorentz equations themselves.

However for velocities simply approaching the speed of light, nevertheless no simple Galilean addition of velocities of two or more frames of references of bodies will suffice because of physical body distance contraction and time dilation effects.

How to solve this problem of Addition of Relativistic Velocities will now form the following text.

Nature's Laws of Conservation

Sun, 11 Feb 2007 13:36:38 -0500

Mass: Mass is the measure of resistance of an object body [ or mass-particle ] to change(s) in its velocity ( acceleration ).

Conservation of Mass: The total amount of mass contained in any aggregate of object bodies [ or mass-particles ] before any change(s) in motion will be equal to the total amount of aggregate mass after any change(s) in motion of this aggregation of object bodies [ or mass-particles ].

Conservation of Momentum:

According to Newton's 3rd Law ( Principle of Equality of Action and Reaction ) for every action there is an equal and opposite reaction.

The binomial expansion

Sun, 11 Feb 2007 13:36:32 -0500

The binomial expansion is integral for eventually deriving Einstein's E=mc2 and hence is of vital importance!

Mass as a Variable Quantity

Sun, 11 Feb 2007 13:36:27 -0500

Mass as a Variable Quantity

E=mc2, Law of Inertia of Energy

Sun, 11 Feb 2007 13:36:22 -0500

E=mc2, Law of Inertia of Energy

Some Consequences of E=mc2

Sun, 11 Feb 2007 13:36:16 -0500

Because of different relative velocities for different observers in different frames of reference, the values of p and E ( momentum and energy respectively ) will accordingly be
different for different observers residing in different systems.

However ...

Heisenberg's Uncertainty Principle

Sun, 11 Feb 2007 13:36:10 -0500

• The subatomic world of nature at its essentially reduced smallest is comprised of localized packets of energy best shown as waves comprised of a variety of wavelengths.
• Because these localized packets of energy are comprised of a variety of wavelengths of different waves, therefore a wave-particle also possesses a variety of momenta since ...

• We can now plainly see that in order to maintain constancy in the above equation, that as the composite variety of wave lengths for a mass-particle becomes overall shorter, the overall composite magnitude of the variety of momenta must become greater.

Compton Effect

Sun, 11 Feb 2007 13:35:54 -0500

The Compton Effect is whenever an incoming gamma- photon ( monochromatic beam of x rays or gamma rays ) hits a stationary electron, part of the photon's energy is transferred to the released electron with the consequent result of the scattered gamma-prime- photon having a lower energy level as well as concomitant lower frequency and longer ( increased ) wavelength. The collision will have enough energy to make a new wave-particle of the same type, in this case another electron. This is all summarized by Planck's relationship ...

COSMOLOGY: The Universe's Invisible Hand

Sun, 11 Feb 2007 13:35:46 -0500

Dark energy does more than hurry along the expansion of the universe. It also has a stranglehold on the shape and spacing of galaxies

Cosmos: Dark Energy's Hand in the Universe

Sun, 11 Feb 2007 13:35:36 -0500

The universe gives up its deepest secret
It is the invisible material that makes up most of the cosmos. Now, scientists have created the first image of dark matter

"Parallel Universes" by Max Tegmark

Sun, 11 Feb 2007 13:35:29 -0500

Survey of physics theories involving parallel universes, which form a natural four-level hierarchy of multiverses allowing progressively greater diversity.

"Parallel Universes" by Max Tegmark, Scientific American, 2003

Sun, 11 Feb 2007 13:35:24 -0500

Infinite earths in parallel universes really exist.

"The Universe on a String", by Brian Greene, New York Times, 2006

Sun, 11 Feb 2007 13:35:17 -0500

The combined equations of general relativity and quantum mechanics yield similar problems. While the conflict rears its head only in environments that are both extremely massive and exceptionally tiny — black holes and the Big Bang being two primary examples — it tells of a fissure in the very foundations of physics.

"Village May Have Housed Builders of Stonehenge", New York Times, 2007

Sun, 11 Feb 2007 13:35:07 -0500

New excavations near Stonehenge have uncovered hearths, timbers and other remains of what archaeologists say was probably the village of workers who erected the monoliths on Salisbury Plain in England.

The archaeologists announced yesterday that the 4,600-year-old ruins appear to form the largest Neolithic village ever found in Britain. The houses at the site known as Durrington Walls were constructed in the same period that Stonehenge, less than two miles away, was built as a religious center, presumably for worshipers of the sun and for their ancestors.

"Early Astronomical ‘Computer’ Found to Be Technically Complex", New York Times, 2006

Sun, 11 Feb 2007 13:34:59 -0500

Decoding the Ancient Greek Astronomical Calculator Known as the Antikythera Mechanism (Nature)

But a century ago, pieces of a strange mechanism with bronze gears and dials were recovered from an ancient shipwreck off the coast of Greece. Historians of science concluded that this was an instrument that calculated and illustrated astronomical information, particularly phases of the Moon and planetary motions, in the second century B.C.

The instrument, the Antikythera Mechanism, sometimes called the world’s first computer, has now been examined with the latest in high-resolution imaging systems and three-dimensional X-ray tomography. A team of British, Greek and American researchers deciphered inscriptions and reconstructed the gear functions, revealing “an unexpected degree of technical sophistication for the period,” it said.

The researchers, led by the mathematician and filmmaker Tony Freeth and the astronomer Mike G. Edmunds, both of the University of Cardiff, Wales, are reporting their results today in the journal Nature.

They said their findings showed that the inscriptions related to lunar-solar motions, and the gears were a representation of the irregularities of the Moon’s orbital course, as theorized by the astronomer Hipparchos. They established the date of the mechanism at 150-100 B.C.

Quantum Quirk: Stopped Laser Pulse Reappears a Short Distance Away

Sun, 11 Feb 2007 13:33:59 -0500

Harvard University researchers have halted a pulse of laser light in its tracks and revived it a fraction of a millimeter away. Here's the twist: they stopped it in a cloud of supercold sodium atoms, known as a Bose-Einstein condensate (BEC), and then restarted it in a second, distinct BEC as though the pulse had spookily jumped between the two locations.

Relative Motion of Solitons in the Light-Carrying Ether

Sat, 10 Feb 2007 18:57:22 -0500

Several errors in the mathematical part of the special theory of relativity are found. The transformations of space and time are obtained provided all elementary particles, of which material bodies consist, are soliton formations of the light-carrying ether. The reverse transformations, and also the formula for the composition of velocities are obtained (these formulae differ from the appropriate formulae of the non-ether theory of relativity). It is shown that, in accordance with the proposed point of view, the interference pattern in the experiments with Michelson's interferometer must not depend on the orientation of the device. The formula for Doppler's effect is obtained provided photons are soliton formations of the ether.