Relativity and the Separation Formula

The bizarre effects of Special Relativity, introduced by Albert Einstein in 1905, are manifest as time dilation, length contraction, and varying mass. Thus, as an object moves faster, time (t) passes more slowly for it, its length in the direction of motion (l) shrinks, and its mass (m) increases. At the velocity of light, time would stand still, length in the direction of motion would shrink to zero, and mass would become infinite. These distortions occur so that the velocity of light will always appear to be a constant (c), regardless of relative motions and one's own inertial frame of reference (i.e. coordinates at a constant velocity). In Einstein's view, this simply preserves the absolute universality of the laws of nature, since the velocity of light turns out to be an artifact of Maxwell's Equations for electromagnetic interactions. Maxwell's Equations, and so the velocity of light, are equally valid for every inertial frame of reference (the original "Galilean" form of Relativity), which means however it is that one is moving, as long as one is moving at a constant velocity (which means the same speed and direction), the velocity of light (in a vacuum) will be a constant.

The change in mass itself explains why ordinary objects cannot attain the velocity of light:  They would have an infinite mass there and so would need an infinite force to accelerate themselves to that velocity. This circumstance is not always appreciated, even by great science fiction writers like Robert Heinlein, who has one character in a story ask why we can't go faster than the velocity of light and the answer is given that "we don't know" but "we'll see when we get there". Another science fiction story speculates that a ship hitting the velocity of light would be bounced back into the past.

The original formulae for the transformation of coordinates, from one frame of reference moving past another in the x axis [as given in The Universe and Dr. Einstein, by Lincoln Barnett, Mentor, 1948, 1950, 1957, footnote pp. 54-55, and Michael Berry, Principles of cosmology and gravitation, Cambridge U Press, 1976, pp. 35] are at left. These are the "Lorentz Transformations," proposed prior to Einstein by the Dutch physicst H.A. Lorentz to account for the anomaly of the Michelson-Morley experiment in 1881, where the velocity of light had not varied regardless of the direction in which it was measured. The form of the equations is for the x coordinate, and time (t). The y and z coordinates are unaffected. The equation is also given for the addition of velocities (v). Lorentz did not know, however, why this effect had occurred, so these were just ad hoc mathematical descriptions. Einstein provided the reason. Simpler equations for the length (l) contraction for the object, the dilation for a unit of time (t), and for the increase in the mass (m) of the moving object are all given at right [versions given in Physics, The Foundation of Modern Science, Jerry B. Marion, John Wiley & Sons, 1973, pp. 197-205].

The Lorentz Transformations are not mathematically very difficult, but they do not transparently relate space and time to each other, and they do not relate to any intuitive sense of why this all would happen. Another equation that does provide a better sense of things, called the "Separation Formula," is given at right, where s is the "separation" or "proper time," which is the elapsed time for a moving object, while t, x, y, and z are the changes in the coordinates in time and space as an object moves [cf. Michael Berry, Principles of cosmology and gravitation, Cambridge U Press, 1976, pp. 48-49, and Roger Penrose, The Emperor's New Mind, Oxford U Press, 1989, pp. 195-196]. At first this may not seem like an improvement over the previous equations. But, as with many equations, it can be simplified. First of all, the Greek delta, which indicates the change in the coordinates, can be left out, as in the first equation at left, giving us variables for the movement in each of four dimensions. Then we should take into account that the Separation Formula is really an extension of the Pythagorean Theorem. The basic Pythagorean Theorem is in two dimensions (x2+y2), but it can be generalized into three dimensions (x2+y2+z2). In four dimensions we get an anomaly, since all the dimensions of space are added together, while they are all subtracted from the dimension of time (t2-((x2+y2+z2)/c2)). This all by itself is revealing, since it answers the question whether time is treated exactly like a dimension of space in Relativity. It isn't. Since what concerns us is the relation between time and space, the separation formula can be simplified by replacing the x2+y2+z2 term with its simple equivalent, r2 (r = "radius"). This can then be further simplified by picking the right units. The velocity of light can be set equal to one with the choice of light years (LY) and years (y). The velocity of light is, indeed, one light year per year (that is the definition of a light year). With all those simplifications, the Separation Formula ends up as a very simple equation indeed:  s2=t2-r2. Although the velocity of light term has been eliminated, it should be remembered that its units of velocity are there and that the "separation" comes out in units of time.

Curiously enough, a full Pythagorean formula, s2=t2-(x2+y2+z2/1), has integer solutions using Pythagorean Triples, as we can see in the diagram at right.

With the simplified equation, we can inspect some Relativistic effects. The graph at left, in vertical units of years and horizontal units of light years, shows two trips in space-time. The red path is an object (a spaceship) moving at 60% of the velocity of light -- it travels 3 light years in 5 years. The blue path is the movement of light itself -- 5 light years in five years. The green line is a stationary object in this inertial frame of reference -- we could think of it as the Earth, from which the spaceship travels 3 light years away. We discover from the Separation Formula, that while 5 years have elapsed on Earth, the ship has arrived at its destination in only 4 years, according to its clocks. Meanwhile, the ray of light experiences no elapsed time -- it's separation is zero (25-25=0). That is called "light-like" separation, in comparison to the "time-like" separation of the other object. It can be seen from this that the longest path in space-time is the shortest separation. On the other hand, what if an object had gone faster than light? If so, the r term would be larger than the t term, resulting in a negative number, and the separation would be the square root of that number. The separation would therefore be an imaginary number (-1 = i). That is called "space-like" separation, and it signals us that such motion is impossible. You can't get there from here in space-time, though this does not tell us why -- we need to know about the change in mass for that. If the light ray is rotated around the vertical axis, this would generate a cone, a "light cone," which defines the area in space-time accessible from the point at the origin.

The previous graph gives rise to a paradox. If the spaceship is moving at a constant velocity away from the Earth, then it has its own inertial frame of reference, in relation to which it is the Earth that is moving away from the ship. Thus, time should be passing more slowly on Earth than in the ship. The graph at right illustrates this situation. The red line represents the ship, at "rest" in its own inertial reference frame, experiencing the passage of 4 years. At a distance of 3 light years, the Earth (green line) would only have experienced the passage of 2.65 years (s=(16-9)=7=2.65). This is peculiar, since back on Earth, everyone has experienced the passage of 5 years (purple line). What this paradox illustrates is that in Special Relativity simultaneity is relative. In the simultaneous space for the spaceship, 2.65 years have elapsed on Earth and 5 years there would be in the future, while in the simultaneous space for Earth, defined by the thin diagonal purple line to the spaceship, 5 years have passed. A graph of simultaneous space for Earth, placed on the inertial frame of reference for the spaceship, would be systematically distorted, a "Poincaré motion," so that simultaneous events in one frame of reference will be in the future or the past for another [cf. Roger Penrose, The Emperor's New Mind, pp. 199-200].

The paradox of simultaneity can be solved by reuniting the objects in a single frame of reference. For this to be done, of course, at least one object must experience an acceleration. This provides the simple rule for Special Relativity:  The "absolute" time we end up with is determined by the unaccelerated frame of reference. Thus, in the graph at right, we are back to the Earth's frame of reference, since the spaceship has turned around (which is a change in velocity and so an acceleration) and returned to the Earth. It returns at 60% of the velocity of light again and so experiences a separation of 4 years. Overall, 10 years have elapsed on Earth while only 8 years have elapsed for the spaceship. Simultaneity is reconciled. Longer trips at higher velocities produce more dramatic differences. But if the spaceship did not return to Earth, there would be nothing special about the Earth's inertial frame of reference and simultaneity between the ship and Earth would not be reconciled.

The only truly absolute frame of reference, velocity, and time in all of this is that of light. The only absolute time is none -- "light-like" separation. This sounds like nothing less than Platonism, where that which truly exists is eternal and unchanging. At the velocity of light, no time will pass at all for the entire history of the universe, which sounds like Socrates imagining that, if death is like dreamless sleep, then "all of eternity will be no more than a single night."

There is actually another paradox in the first diagram above (reproduced at left):  If, on the spaceship, it seems that 4 years have passed, then the spaceship, to travel 3 light years in 4 years, had to have been going 75% of the velocity of light (3LY/4y), not 60% (3LY/5y). The 60%c velocity is only the way the velocity would appear in the Earth's inertial frame of reference. However, this answer will not do; for if the spaceship went 4 light years in 5 years, or 80% of the velocity of light (4LY/5y), as in the diagram at right, only 3 years would have passed for the ship ((52 - 42) = 3), which would mean that the ship went 4 light years in 3 years, at a velocity then of 167% of the velocity of light (4LY/3y). But the whole point of Special Relativity is that it can't go faster than light, however we measure it. What should handle this, however, is that as the spaceship is moving, the distance (4 LY), as measured in the Earth's frame of reference, contracts (using the equation l = lo(1-v2/c2)):  4 (1-(0.8)2) = 4(1-0.64) = 40.36 = 4(0.6) = 2.4. The ship sees the distance it traverses as only 2.4 light years -- at high velocities the universe literally becomes smaller, at least in the direction of motion. 2.4 light years in 3 years is, indeed, 80% of the velocity of light (2.4LY/3y = 0.8LY/y). Similarly, the ship that goes 3 light years in 5 years (60%c), experiences that distance as 2.4 light years also, which also confirms the velocity of 60%c (2.4LY/4y = 0.6LY/y).

The equation at right is a version of the Separation Formula combined with terms for mass and the Gravitational Constant, in polar coordinates, stated as derivatives (d) of separation (s), times (t), distance (r), and angle ( and ). I have not seen the formula in print but copied it off the television screen of the cable show The Universe. I hope I haven't made any mistakes. The image could have been clearer. Since separation is stated in units of time, I'm not sure this comes out quite right.

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Copyright (c) 1998, 2011, 2012 Kelley L. Ross, Ph.D. All Rights Reserved

Historic Equations
in Physics and Astronomy

Being more of a spectator on mathematics than a participant, my favorite historic equations are those that the ignorant layman (like me) can play with just by plugging in the right values and seeing what happens. As it does happen, this doesn't get one very far with the more recent and more important equations, which often feature complex operators whose use requires a mathematical education in itself and whose solutions are often still matters of contemporary exploration and controversy. Here I give both kinds, and the mathematically literate are welcome to sneer at my rudimentary understanding of the more sophisticated material. What I can now do, however, is refer the reader to a recent examination of historic equations by the specialists themselves:  It Must be Beautiful, Great Equations of Modern Science, edited by Graham Farmelo [Granta Books, London, New York, 2002, 2003]. Here we have the likes of Roger Penrose and Steven Weinberg doing what I would really like to be doing here. Farmelo, however, has not chosen an essay about Kepler's Laws, which I think are more my speed and that I think I explain rather well.

A more recent book that is similar in purpose to my treatment is Archimedes to Hawking, Laws of Science and the Great Minds Behind Them, by Clifford A. Pickover [Oxford University Press, 2008]. Pickover gives equations, but his more general focus is on "laws of nature." Many of these laws involve very simple equations, and they are often swamped with supplemental material about the discoverers and discoveries. Towards the end of the book, when we get to Maxwell's or Schödinger's Equations, we tend to get less discussion and not much at all about the way in which the equations work. Nevertheless, Pickover's treatment can be very illuminating, as with his treatment of Coulomb's Law.

Constants for these equations are given in the table at "Physical Constants."

Equations for Kepler's Laws:  Kepler's First Law is that the orbits of the planets are ellipses, with the Sun at one focus. At right is a general equation for a conic section (circle, ellipse, parabola, or hyperbola) in polar coordinates, together with commonly used physical dimensions for the orbits of planets, asteroids, comets, etc. The "major axis" (2a) is the longest line that can be drawn in an ellipse; the "minor axis" (2b) is the shortest line that can be drawn through the center. The shortest distance from the focus to the curve (q) is "perihelion" for a planet, "perigee" for an object in orbit around the earth, or, in general, "periapsis," "closest approach," for any kind of orbit. The longest distance from a focus to the curve (Q) is "aphelion" for a planet, "apogee" for an object in orbit around the earth, or, in general, "aphapsis," "furthest approach," for any kind of orbit. The distance across the curve, through the focus, at right angles to the major axis (2d), is the "latus rectum". The periapsis (q) and semi-latus rectum (d) are physical dimensions of any conic section, not just ellipses -- though a hyperbola might be thought of as an ellipse with a negative major axis and an imaginary minor axis. The "eccentricity" (e) defines the shape of the curve:  e=0 is a circle; 0<e<1 is an ellipse; e=1 is a parabola; and 1<e is a hyperbola. Finally, the angle made by the radius (from focus to orbiting object) with the point of periapsis is the "true anomaly" ().

Kepler's Second Law is that the radius from the focus to a planet sweeps out equal areas in equal times. The mathematics of this is still not easy to deal with. Given the mean angular motion (n), which would be 360o divided by the period of the orbit (in radians: 2/p), and the time elapsed since the "epoch of perihelion" (T, a benchmark time when the planet was at perihelion), the "Mean Anomaly" (M) can be calculated, which would give the angle with perihelion if the planet had been moving with a uniform angular speed. With the mean anomaly in hand, it is not the true anomaly that is calculated first, at least for elliptical and hyperbolic orbits, but the "eccentric anomaly" (E), which is the point on a superscribed circle that corresponds to the point where the planet is on its curve. This is illustrated in the diagram at left. Calculating the eccentric anomaly from the mean anomaly is difficult because the equation M = E - e*sin E (for ellipses) cannot just be solved for E. Instead, we can write the equation as M + e*sin Em = E and, starting with Em = M, solve the equation over and over, substituting each new approximation of E for Em. This permits calculating E as accurately as desired. With the eccentric anomaly determined, the true anomaly () can be calculated directly. Equations are given at left for circular, elliptical, parabolic, and hyperbolic orbits. For circular and parabolic orbits, the true anomaly can be calculated directly either from the mean anomaly or just from the time since perihelion (t - T). The hyperbolic equations work much like the elliptical ones, usually just with some opposite signs and hyperbolic functions.

The diagram at right gives the conventions for the physical characteristics of a planetary orbit. The celestial equator is the Earth's equator projected onto the sky. The apparent path of the Sun in the sky, however, the ecliptic, is at an inclination () to the equator. The point where the Sun crosses the celestial equator and enters the northern hemisphere is the Vernal Equinox, providing the benchmark to celestial longitude, which is then measured along the ecliptic to the East, the direction in which the Sun and planets move against the background of the stars. The first physical feature of an orbit is the Ascending Node, the point where the object (planet, asteroid, comet, etc.) crosses to the north of the ecliptic. The "longitude of the ascending node" () is thus the celestial longitude of that point, measured East from the Vernal Equinox. At the ascending node there is also the angle of inclination of the oribt (i) to the ecliptic. Angles of inclination are small for the planets but can be very large, up to 90o, for asteroids and comets. If the motion of the object is retrograde, i.e. from East to West, the angle of inclination will be larger than 90o. Once on the plane of the orbit, the angle of longitude is measured to the East until the perihelion (perigee, etc.) point is reached. That angle is the "argument of perihelion" (). The longitude of the ascending node and the argument of perihelion can be added together for the "longitude of perihelion" (), but the two angles are not measured in the same plane, so the angle may differ from the true angle of separation between the Vernal Equinox and the perihelion. A similar caution holds for the "true longitude" of an object, which is the sum of the longitude of perihelion and the true anomaly (+).

Kepler's Third Law is that the square of the period of a planet's orbit is directly proportional to the cube of the semi-major axis. By the same token, the square of the mean angular motion is inversely proportional to the cube of the semi-major axis. However, the equation for the period can be considerably simplified by choosing the right units. If (Earth) years and Astronomical Units (the semi-major axis of the Earth's orbit) are chosen, then the rest of the equation can be put equal to 1, which means that the Third Law can simply be written:  p2 = a3. This was useful in Kepler's day when the true physical distances, let alone the Gravitational Constant, were unknown. It is still useful today.

It is noteworthy that Kepler's confidence in the mathematical nature of the universe was Platonic in inspiration, derived from the revival of Plato by Renaissance scholars and ultimately from the Platonism of Mistra in Romania. Kepler's hope to match planetary orbits up with the Platonic Solids, however, was not successful.

Newton's Equation for Gravity:  The force exerted between any two bodies with mass -- "G" is the gravitational constant; "m1" and "m2" are the masses of the two bodies; and "r" is the distance between them. Below the equation for force is the equation for the acceleration of gravity produced by a single body. The acceleration of gravity on the surface of the Earth is 9.8 m/s2 (aE = g).

Coulomb's Law:  The force exerted between two electrical charges -- "k" is the electrostatic force constant; "q1" and "q2" are electrical charges; and "r" is the distance between the charges. Electrical charges can be positive (+) or negative (-). Opposite charges ("+" & "-") result in a positive (attactive) force; like charges ("+" & "+" or "-" & "-" ) result in a negative (repulsive) force. The second version is the way I see this equation written now, with the "permittivity of empty space" () used instead of the electrostatic force constant. For long I had not seen it explained what the "permittivity of empty space" is supposed to mean. Factoring out 4 must have some relevance to space. Indeed, this is where Pickover's Archimedes to Hawking is of great value. The "permittivity" is "an electrical property of the medium that surrounds the two charges." The value of "k, sometimes known as Coulomb's constant, is approximately equal to 9 x 109 N.m2/C2" for empty space [p.154, boldface added]. The permitttivity is different for different materials, and Pickover helpfully gives a list of several values [p.155].

Bode's Law:  Stated by J. Bode in 1778 but discovered by J. Titius in 1766 -- and so now frequently called the "Titius-Bode Law." A simple numerical sequence that produces the mean distance from the Sun in Astronomical Units for most of the major planets. The construction of the series begins with 0 and 3; 3 is then doubled as many times as desired for subsequent terms in the series; 4 is added to each number; and then each number is divided by 10. The results are shown in the following table at left, up to the thirteenth term. The period of the orbit can be calculated using Kepler's Third Law (p2 = a3), where the mean distance (a) is in Astronomical Units and the period (p) is in Earth years. The results were very close to the planets known at the end of the 18th century, up to Uranus (discovered in 1781), with one conspicuous gap:  There was no planet corresponding to B5. However, on New Year's Eve of the year 1801 the first asteroid, Ceres, was discovered, orbiting at exactly 2.8 AU. Neptune and Pluto, discovered later (1846 & 1930) did not fit Bode's Law, however, as can be seen from the actual physical data in the table at right.


The possible physical reason for Bode's law may be seen in the relationship of the periods. Beginning with Mercury, which orbits the Sun in about three months, we approximate all the periods up to Ceres simply by doubling. What this suggests is that the orbits are in gravitational step with each other -- a harmonic resonance. That is, they reinforce rather than disturb each other's orbits. When we get out to where there are greater distances between the planets, around Neptune and Pluto, the relationship breaks down. Nevertheless, the philosopher
Hegel believed that Platonic numerology, rather than gravity, was the issue; and he tried proving in his doctoral dissertation that there could be no planet at B5 -- an argument published on the very eve of the discovery of Ceres on 1 January 1801. Karl Popper enjoys this example of Hegel's cluelessness in science.

The discovery of Ceres was, of course, only the beginning. By the late 1980's, 5000 "Minor Planets" (a term officially dropped in 2006, but still in the name of the Minor Planet Center) had been catalogued and named. That seemed like a lot; but now there are well over 250,000 of them that have been assigned a Minor Planet Number -- with over 535,000 objects that have been observed and registered, pending calculation and confirmation of their orbits. Only 16,154 so far have been named, with names that have become increasingly whimsical and peculiar. At the high end, four asteroids have been promoted to "Dwarf Planets"; and one former Planet, Pluto, has been infamously demoted down to that novel status. The Dwarf Planets are numbered in the Minor Planet sequence. The debate about the status of Pluto has gone on long enough that the number 10,000 was once suggested for it; but the whole business was so long delayed that Pluto, discovered in 1930, now bears the ignominy of being only Minor Planet 134340, which in no way distinguishes it from fellow Dwarf Planets 136108 Haumea, 136472 Makemake, and 136199 Eris, none of which was discovered before 2003. The opportunity missed here is that Pluto could retroactively be made Minor Planet Zero.

At the low end, asteroids trail off into meteoroids, which more or less by definition are simply objects that are too small to be permanently catalogued and assigned numbers. The informal cut-off used to be a size of 10 meters; but now some objects smaller than 10 meters have been numbered, so the informal cut-off is down to 1 meter. Another informal definition of meteoroids could be that they are objects too small to be noticed until they hit the atmosphere and are observed burning in the air. Then they become proper meteors. The life of a meteor, however, is brief. After ceasing to be a meteoroid, it quickly ceases to be a meteor. If it is not entirely burned up in the atmosphere, the parts that fall to earth then become meteorites and enter the realm of geology and mineralogy as much as that of astronomy or astrophysics. If the meteoroids are large enough, blast effects and craters on the ground can result. A full asteroid, of kilometer sizes, hitting the Earth can, of course, have major geological consequences, with the ends of both the Mesozoic and Paleozoic Eras dated, the former with some certainty, the latter tentatively, to asteroid impacts. The peculiar circumstance that modern meteorology is not the study of meteoroids, meteors, or meteorites, but of the weather, is due to the Greek word, metéôros, simply meaning "up in the air." Both meteors and the weather are, indeed, "up in the air."

Bode is the astronomer who originally suggested the name of the planet Uranus. Little did he know the trouble this would later cause. When I was a child, I remember pronouncing "Uranus" as you-rânus and thinking nothing about it. However, in 1974 a movie came out called The Grove Tube which had a sequence that derived considerable scatological humor from pronouncing "Uranus" as "your anus." For some reason, this seems to have cut astronomers to the quick; and it wasn't long before I began to see people like Carl Sagan pronouncing the planet as yúrin-us. I can't see this as a particular improvement, since it sounds like "urine us," which has an equally unpleasant ring to it. So the whole business has become a fiasco. My own modest proposal would be to return to something like the Greek pronunciation of the original god:  Oo-ranos. This avoids both urine and the anus.

Maxwell's Equations:  In The Emperor's New Mind (p. 186) Roger Penrose shows us Maxwell's equations, though I can't say that I understand much of the mathematics. Penrose says that the "curl" and "div" functions are "certain combinations of partial derivative operators, taken with respect to the space coordinate." The two equations on the left relate the rate of change (the partial derivative with respect to time) of the electric field (above) and the magnetic field (below) to changes in the magnetic field and electric current (above) and to changes in the electric field (below). At least the symmetry between time on the left and space on the right is evident, though I can't say that the meaning and beauty of the equations is otherwise obvious to the non-mathematician (like me). The equations on the right Penrose says are versions of the inverse square law (above) for the electric field and, for the magnetic field, the fact that there are no isolated magnetic poles (below). Paul Dirac predicted magnetic monopoles, but none have yet turned up. Penrose doesn't explain what I think is an important aspect of equations like this:  the units.  Return to "Relativity and Separation Formula".

Planck's Law:  The energy of Black Body radiation for a given temperature and wavelength. A "black body" is called that because it does not reflect any radiation; it only emits radiation because of its temperature. Stars are natural black bodies, though the effect can be duplicated by heating a box with only a small hole in it. The light that comes out of the box from the hole is black body radiation. -- "h" is Planck's Constant; "k" is Boltzmann's Constant; "c" is the velocity of light; "" is the wavelength in meters; "T" is the temperature in Kelvins; and "e" is the base of natural logarithms, Napier's Constant.

Wien's Law:  The wavelength at which Black Body radiation for a given temperature peaks. -- "c2" is the "second radiation constant"; and "T" is the temperature in Kelvins.

Stefan-Boltzmann Law:  Power emitted by a Black Body per unit area for a given temperature. Given the temperature and size of a star (and so its surface area), its total power could be calculated. -- "" is the Stefan-Boltzmann Constant; and "T" is the temperature in Kelvins. Boltzmann is also associated with "Boltzmann's Constant" in an equation for entropy, . This may be called "Boltzmann's Law," although Boltzmann did not actually write the equation, despite it being inscribed on his grave.

Einstein's Equation:  The upper equation at right is Einstein's field equation for gravity. Roger Penrose explains and discusses it in detail in It Must be Beautiful, Great Equations of Modern Science (pp.180-212), along with the Separation Formula and other things. As I understand it, whereas a "vector" is a quantity with a direction (one dimension), like velocity, a "tensor" is quantity expressed in two dimensions (a "scalar" quantity is dimensionless). It is nice to read that when Einstein got interested in tensor calculus, he "had to enlist the help of his colleague Marcel Grossmann to teach him" (p.199). This is the basis of General Relativity, where space-time curvature, Black Holes, the Big Bang, and the whole business comes from. The second equation at right is Einstein's Equation with the addition of the "cosmological constant." The negative signs in both equations indicate the attractive nature of gravity. The positive sign on the cosmological constant makes it repulsive. Einstein famously said that this was the "biggest mistake" of his life, adding a repulsive force in order to get a static universe; but it now turns out that there may very well be a cosmological constant, one big enough to make the expansion of the universe accelerate.

Relativity and the Separation Formula,

Hubble's Law:  The Red Shift from the Doppler Effect is how we actually know about the radial velocity of recession, and distance, of galaxies. The Red Shift (z) is the change in a wavelength of radiation () from the object divided by what the original wavelength was (/). This can be related to velocity (u) as:  (z+1)2=(c+u)/(c-u) or u/c=(z2+2z)/(z2+2z+2), where the velocity of light, c=1. Distance (s) then depends on Hubble's Law:  u=sH, where the Hubble Constant (H) is now taken to be 75 km/s/MPC (it is somewhere between 50 and 100 km/s/MPC). 1/H is the Hubble Time, which for the given H is 13.04 Gy. c/H, the Hubble Radius, is thus 13.04 GLY. At low values, z is virtually identical to u/c. 0.023c and z=0.023 is about the point where the two values begin to diverge. u/c cannot be larger than 1, but z can be any number up to infinity (at the velocity of light).

The June 2010 edition of Sky & Telescope reports the Hubble Constant to be 70.4 +/- 1.4 km/s/MPC [p.14]. They also report the age of the universe to be 13.75 +/- 0.11 years. It used to be that the Hubble Time would be larger than the age of the unvierse; but if the expansion of the universe is accelerating, as is now believed, then the Hubble Time is smaller than the age of the universe.

Edwin Hubble proved that spiral nebulae are external galaxies, vindicating the speculation of Immanuel Kant. At the same time, he discovered the expansion of the universe -- all this while working at the new Mt. Wilson Observatory up above Pasadena, California. In popular presentations of the history of science, I have now noticed it often said that Hubble was the first to imagine that there were external galaxies. This not only ignores Kant but it betrays complete ignorance of the intense debate about the matter all through the 19th century, culminating in a debate between Harlow Shapley and H.D. Curtis at the National Academy of Sciences in 1920. Today Mt. Wilson holds the radio and television transmission towers for much of the Los Angeles Basin and is visible, on a clear day, from the San Fernando Valley.

Schrödinger's Equation:  Roger Penrose discusses Schrödinger's Equation in The Emperor's New Mind (p. 288). This is the deterministic equation for the undisturbed wave function, , in quantum mechanics. The application of the equation ends when the wave function is "disturbed" by observations, or even inferences, that can determine the location of particles. Then the square of the wave function is interpreted, as by Heisenberg, as the probability distribution of where the particle can be found. Schrödinger, like Einstein, didn't like that indeterministic aspect of quantum mechanics:  Indeed, he said "I don't like it, and I wish I'd never had anything to do with it." Here the rate of change (the partial derivative with respect to time) applies to the "state vector," , of the wave function. The "state vector" notation is discussed by Penrose on page 257. This is multiplied by the imaginary (i) "reduced" Planck's Constant (). (My non-mathematician's impression is that imaginary numbers often occur in equations of periodic functions [note], but Penrose does not discuss the significance of the imaginary number here.) That whole side of the equation is all equivalent to the "Hamiltonian" of the state vector of the wave function. Penrose explains (p. 288) that the "classical Hamiltonian" represents the total energy of the system but that the "quantum Hamiltonian" substitutes partial differential operators with respect to the momentum for the simple occurrence of momentum in the original Hamiltonian. Thus, as in Maxwell's equations above, a great deal of the mathematics of this equation is presupposed by the symbolism. The simplicity of the equation thus conceals a level of mathematical sophistication that is rarely explained, in any way, to the lay public. Penrose's own effort to intelligibly present the details of much of this stuff, although limited and not always successful, is therefore exemplary, and in stark contrast to Stephen Hawking's A Brief History of Time, which only contained, on the publisher's advice, one equation (E = mc2).

Dirac's Equation:  The equation at right is Paul Dirac's equation for the electron. Frank Wilczek explains and discusses it in detail in It Must be Beautiful, Great Equations of Modern Science (pp.132-160, with explanation of the terms in the equation in an appendix, pp.268-270). Dirac's equation reconciles Schrödinger's Equation with Relativity in the description of the electron. Wilczek mentions that the spin of electrons (up or down, right or left) was derived by Dirac in a natural way from the equation, which also resulted in the prediction of anti-particles, which shortly thereafter were actually observed. The way the equation is written, however, the variable x is simply said to take four values, with the electron and positron each in two spin states. This makes it look like the four values are written into the equation, rather than derived from it. So I must have missed something. I also understand that Dirac predicted the existence of particles that are magnetic monopoles, i.e. have a magnetic charge that is only North or South. These have not been observed, and Wilczek doesn't seem to mention them.

Bell's Theorem, Bell's Inequality:  The equations at right are versions of John Bell's equations to test the Einstein-Podolsky-Rosen (EPR) Paradox. The form at top is Bell's own, published in 1964. Below it is a slightly rewritten form. The equations are given and discussed, with the EPR Paradox and the attendant issues, in Einstein's Moon, Bell's Theorem and the Curious Quest for Quantum Reality, by F. David Peat [Contemporary Books, Chicago, 1990, pp.111-112]. Peat thanks Bell himself (before his untimely death) for reading the manuscript, so this may be more than the typical scientific popularization. The actual equations here predict results consistent with "local reality," i.e. what Einstein wanted, with the possiblity of quantum states predetermined by "hidden variables." The two detectors (A and B) catch the "correlated" particles that are the issue in the EPR Paradox, i.e. they may have opposite spins, but which is which is undetermined, in reality as well as in knowledge, until one of them is observed. Then the spin of the other is instantaneously fixed, violating the velocity of light limitation of Special Relativity. In the second form of the equation, the probabilities should range between negative 2 and postive 2. (Since probability ranges up to 1, which is certainty, the three positive and three negative terms of the first equation cannot add up to more than 0.) However, the predictions of "non-local" quantum mechanics are going to be different. Peat does not give equations because he says that they are different for every angle. When there were experimental tests of the equations in 1982, they were at angles where a quantum correlation of 2.70 was predicted [pp.117-118]. The experimental result was 2.697, much larger than the "local reality" prediction and very close to the quantum prediction (or >2.682 and <2.712). Thus, Quantum Mechanics violates Special Relativity, and indeterminacy is established for particles when they cannot be observed or their states inferred from observations, i.e. the wave function completely specifies the reality. The Realists, like Einstein or Bell himself, were not going to like this, but, on the other hand, it still allows a Realism for those who take the wave function to be real, as did de Broglie or as is possible in a Kantian Quantum Mechanics.

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Historic Equations in Physics and Astronomy, Note


For instance, the equations at right for sine and cosine functions, which are periodic, contrast with the equations for hyperbolic sine and hyperbolic cosine functions, which are not periodic. The results of all the equations are real numbers; but for the sine and cosine we have imaginary powers of Napier's Constant (e), and, for the sine function, these imaginary powers are divided by the imaginary number itself. Also, the sine equation can alternatively be written: (because:  i-1 = -i).

These relationships go back to Euler's Theorem:  

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Imaginary Powers of Napier's Constant