The notions of space and time that dominated physics until the early twentieth century were rooted in Isaac Newton’s thought. In the seventeenth century, however, two philosophical currents clashed over their nature.
Newton defended absolute space and time as independent realities existing apart from physical phenomena.
Relationists, led by Gottfried W. Leibniz, insisted that space and time are not things in themselves but arise solely from relations among material objects.
Leibniz argued that without creatures, space and time would exist only in God’s ideas, and that motion is meaningful only as a change in relative positions. Newton countered that absolute space remains immovable and similar regardless of bodies, while absolute time flows equably without external influence.
The controversy subsided for centuries amid the success of Newtonian science, yet it lingered latently and resurfaced near the end of the nineteenth century.
A 'Shaper of Worlds' monster: the Swatterfly
Converted to
An oversized wooden fly that buzzes about on large, membranous wings of stretched bladder. Whether built by a mad Lumberjack as retaliation against rogue timber farmers or as the embodiment of his recurring nightmares, the result is the same: a mindless construct that darts into faces at the worst possible moments.
STATS
STR 45
CON 60
SIZ 25
DEX 90
INT -- (mindless)
POW 45
HP: 8
Damage Bonus: –1
Build: –1
Move: 4 / 8 flying
Armor: 2-point hardened wood. Immune to poison, disease, and all mind-affecting effects (as a construct). Takes double damage from acid. Fire deals normal damage and may ignite it (Keeper’s discretion).
Attacks per round: 2 (wing slaps) or 1 Faceswat charge
Combat
- Fighting (Wing Slap) 50% (25/10), damage 1D4 + DB (minimum 1)
- Dodge 45% (22/9)
Faceswat (special): The Swatterfly may charge and make two wing-slap attacks in the same round. If at least one attack hits, the target must succeed on a Dodge or DEX roll or become *Stunned* for 1D3+1 rounds (cannot act; all rolls suffer a penalty die if the Keeper allows limited activity). Creatures without discernible faces, or that are mindless and lack self-preservation instinct, are immune.
Rapid Movement: Creatures that are Stunned, dazzled, or otherwise impaired cannot make Fight Back maneuvers against the Swatterfly. Melee attacks against it while so impaired suffer a penalty die.
Skills: Fly 80%, Spot Hidden 30%, Stealth 50%.
Sanity Loss: 0/1D3 to see a Swatterfly.
Habitat: Urban areas, ruins, attics, abandoned workshops.
Organization: Solitary, pair, or small swarm (3–12).
Setting: A dusty, half-collapsed room in an abandoned tenement on the edge of the city. Late afternoon light filters through broken shutters. The investigators have just forced the door after hearing a persistent, insectile buzzing from within.
INVESTIGATORS (SIMPLIFIED):
- Prof. Harold Quill (academic): HP 10, Fighting (Brawl) 30%, Firearms (Handgun) 40%, Dodge 25%, DEX 50
- Det. Lena Morse (police detective): HP 11, Fighting 50%, Firearms (Handgun) 55%, Dodge 40%, DEX 70- Dr. Marcus Hale (physician): HP 11, Fighting 35%, Firearms 30%, Dodge 30%, First Aid 70%, DEX 55
All three have loaded handguns and are on alert. The Swatterfly is alone (8 HP). Initiative is roughly simultaneous; the construct is already airborne and aggressive.
ROUND 1
The wooden fly explodes out of the shadows in a buzzing charge, diving straight at Professor Quill’s face (Faceswat).
- Wing-slap attack rolls: 86 (miss), 15 (hit). Damage: 2 HP. Quill is reduced to 8 HP.
- Quill’s Dodge/DEX roll: 59 (fail). He is Stunned for 4 rounds.
Lena and Marcus react instantly, drawing pistols.
- Lena fires: 33 (success). Damage: 5 HP. Swatterfly now at 3 HP.
- Marcus fires: 66 (miss).
The construct’s rapid movement and the sudden stun leave Quill unable to act. Lena’s shot splinters one wooden wing; the fly staggers in the air but remains airborne.
ROUND 2
Still stunned, Quill can only clutch his face and stagger. The Swatterfly, damaged but relentless, wheels toward the nearest active threat--Lena--and attempts another Faceswat charge.
- Wing-slap attacks: 79 (miss), 76 (miss). Both go wide.
- Lena’s Dodge: 46 (success). She ducks cleanly.
Marcus fires again: 12 (success, Extreme). Damage: 4 HP. The Swatterfly is reduced to –1 HP and crashes to the floorboards in a heap of splintered wood and ruptured bladder-wing.
Lena, still wary, covers the wreckage while Marcus checks Quill. The buzzing has stopped.
ROUND 3
The wooden carcass twitches once (a final spasm of its damaged mechanism) then lies still. No further attacks occur.
Marcus spends the round applying First Aid to Quill (successful), restoring 1 HP and stabilizing the facial bruise. Quill’s Stun wears off at the end of the round; he is shaken, bleeding lightly from a split lip and brow, and has lost 1 Sanity point from the sudden, unnatural assault.
Lena examines the remains: clearly artificial, carved wood, stretched membrane, crude clockwork joints, but no identifying marks. The room otherwise yields only dust, old newspapers, and a faint chemical smell (perhaps the acid vulnerability residual from its original construction).
Aftermath:
The investigators have a shattered wooden fly, one mildly injured and temporarily rattled academic, and more questions than answers.
The chips fell in the investigators’ favor after the opening Faceswat, but a single successful charge nearly took Quill out of the fight for the entire encounter. In a swarm, the outcome would have been far bloodier.
The only space geometry known in Newton's time was Euclid's. Little is known of Euclid's life, and even his existence is uncertain, but he is believed to have taught in Alexandria around 300 BC.
Euclidean geometry, the system still taught in schools, is a logical structure built from definitions, postulates, and axioms. It was long regarded as an accurate description of the metric properties of space.
Its axioms and postulates appear natural and intuitive, except for the fifth postulate. That postulate states that through any point outside a given line there passes one and only one parallel. Many mathematicians found it less natural and asked whether a different geometry could be obtained by replacing it.
Karl Friedrich Gauss (1777-1855) tested the question experimentally. One consequence of the fifth postulate is that the interior angles of any triangle sum to two right angles. Gauss measured the angles of the large triangle formed by the peaks of Brocken, Hohenhagen, and Inselberg, tens of kilometers apart. The measured sum agreed with the Euclidean prediction within experimental error, showing that space is approximately Euclidean at least on ordinary scales.
Accepting Euclidean geometry, Cartesian coordinates can be introduced by means of three mutually orthogonal planes. The coordinates of a point P are the distances from P to these planes. The distance between two points is then given by Pythagoras's theorem:
d_PQ^2 = (x_P - x_Q)^2 + (y_P - y_Q)^2 + (z_P - z_Q)^2
Before Einstein, this distance was regarded as an absolute invariant, independent of the reference system.
During the long period of Aristotelian ideas before Galileo Galilei (1564-1642), people believed that a moving body needed a force acting on it. Aristotle said that a moving body stops when the pushing force stops.
Imagine a stone dropped from the top of a mast on a sailing ship. In the Aristotelian view, the stone would not fall at the foot of the mast. Once it left the mast, the force that kept it moving with the ship would disappear. So the stone would fall behind the mast.
Galileo disagreed. He said the stone keeps the motion it had before it was dropped. Therefore it continues with the ship and falls at the foot of the mast.
From 1581 to 1585 he took lessons in medicine, philosophy, and mathematics at the University of Pisa, close to the place where he was born. There he had a chair of mathematics in 1592, when he would have performed the experiment in the inclined tower to prove that the free fall motion does not depend on the properties of the bodies. In 1609 he constructed a telescope at the University of Padua, which allowed him to discover the craters of the Moon, the satellites of Jupiter, and the structure of the Milky Way. In 1613, working in the court of Toscana, he announced that the succession of the phases of Venus, and its relation with the apparent size of the planet, favored the Copernicus’ system. Because of this, Galileo was accused of heresy by the Inquisition in 1633, forced to renounce the idea that the Earth moves, and confined to house arrest at his residence of Arcetri (Florence). His last works were clandestinely taken abroad and published in Leiden (the Netherlands), exerting great influence. Galileo countered the old naive empiricism with the “arranged” experiment of modern science, which employs ad hoc equipment to interrogate Nature about the validity of a mathematically formulated physical theory. His research on inertia and the laws of motion, by means of inclined planes and pendulums, prepared the way to the work of Newton. Galileo was not the first to attain this knowledge: around 575 I. Philoponus had correctly criticized Aristotle’s thought about the fall of the bodies, the law of uniformly accelerated movement was well known from 14th century, and J. of Groot and S. Stevin had already published their own experimental verification of the fall law in 1586. However, these facts in no way mar the figure of Galileo as the father of the modern science.
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Galileo discussed this as a thought experiment (the real test was done later by Pierre Gassendi in 1641). He also studied the idea in the lab. He combined careful experiments with the method of imagining ideal conditions without extra effects like friction. Galileo rolled small balls on tables to study how motion continues (inertia). The balls always stopped because of friction with the table and air. But Galileo saw that in ideal conditions, with no friction, a ball would keep moving forever.
He wrote: "any velocity once imparted to a moving body will be
rigidly maintained as long as the external causes of acceleration
or retardation are removed, a condition which is found only on
horizontal planes; for in the case of planes which slopes downwards
there is already present a cause of acceleration, while on planes
sloping upward there is retardation; from this follows that motion
along a horizontal plane is perpetual; for, if the velocity be
uniform, it cannot be diminished or slackened, much less destroyed."
Galileo, Dialogues Concerning Two New Sciences (Leiden, 1638)
Everyday observation shows motion always stops because of friction. Yet Galileo understood that the key point is the natural tendency of motion to continue.
In the stone-and-ship thought experiment, Galileo showed that the result is the same whether the ship is tied up or sailing: the stone falls at the foot of the mast in both cases.
If a real test showed the stone falling toward the back of a moving ship, that would not prove motion fails to continue. It would only mean that air friction slowed the stone. The same thing would happen on a tied-up ship if wind blew from the front. In both situations the effect could be removed by doing the experiment inside a closed cabin fixed to the ship.
In this way Galileo ended the idea that motion always needs a force. He also showed that you cannot detect the steady motion of a ship by any experiment done inside it. The laws of physics are the same on a sailing ship and on a ship at rest.
We see this every day in cars or trains. You cannot feel the steady speed of a train while sitting inside. Looking out the window only shows motion relative to the ground. What you do notice are changes in speed or direction: when the train curves, jolts, brakes, or speeds up.
The laws of physics can be used in different reference frames, such as the earth or a moving boat. This raises a question: how do the Cartesian coordinates of a moving object change when we switch from one frame to another?
Consider two frames, S and S'.
1. Frame S' moves at a steady speed V relative to S. The frames start with their axes lined up at time t = 0.
2. The x and x' axes point in the direction of the relative motion. At any later time t, the distance between the origins O and O' is simply V times t.
From a simple diagram of the positions, the distance from O to a point P equals the distance from O to O' plus the distance from O' to P. In arcane magic symbols this is:
d_OP = d_OO' + d_O'P
Since d_OP is the coordinate x measured in S and d_O'P is the coordinate x' measured in S', we get:
x = V t + x'
or, solving for the other coordinate:
x' = x - V t
This simple rule is called a Galileo transformation. It rests on the classical idea that distances are the same no matter which frame measures them. In other words, a length measured in S equals the same length measured in S'.
The same idea must also apply to time. If time were different in the two frames, the equations would not match. Therefore classical physics also assumes that time is the same in every frame:
t' = t
The y and z coordinates do not change because the frames stay lined up the same way and distances are invariant.
So the full set of Galileo transformations is:
x' = x - V t
y' = y
z' = z
t' = t
These can be written in vector form as:
r' = r - V t
If we differentiate both sides with respect to time, we obtain the rule for how velocities transform (the Galileo addition theorem):
u' = u - V
Because the transformations were built on the idea that distances are invariant, the distance between any two points stays the same after the coordinates are changed.
Galileo said the key fact about motion is that it tends to keep going. Newton made this idea into the First Law of Motion, also called the Principle of Inertia:
Every body stays at rest or keeps moving in a straight line
at constant speed unless a force acts on it.
(I. Newton, Principia, London, 1687)
This law is not a pure experimental fact. It is hard to be sure a body has no forces acting on it. Even if we imagine only one body in the whole universe, its motion cannot be straight and steady in every possible reference frame.
However, if a body moves in a straight line at constant speed in one frame (its velocity u stays the same), then it will also move that way in any other frame that travels at a steady velocity V relative to the first frame.
This follows from the velocity-addition rule:
u' = u - V
So if the Principle of Inertia works in one frame, it works in a whole family of frames that move at constant velocity relative to each other. These special frames are called inertial frames.
Newton explained the special status of these frames by inventing absolute space.
According to Newton, the Principle of Inertia holds in a frame that is at rest in absolute space, and also in any frame that moves at constant velocity relative to that one. Absolute space itself never changes, yet it decides which paths are the natural inertial paths for bodies.
Newton wrote that we cannot see absolute space directly, so in everyday life we use relative measurements. But for deep understanding we must think about absolute space itself.
In real life we do not find inertial frames by looking at absolute space. We simply choose frames in which the Principle of Inertia and the other basic laws of physics work well enough.
Galileo’s idea of inertia means you cannot tell whether the boat in the stone-dropping experiment is moving or standing still. The stone falls at the base of the mast in both cases.
This result is stated as the Principle of Relativity: "The same basic laws of physics hold in every inertial frame."
Absolute space picks out a special family of frames (the inertial ones). The Principle of Relativity then says that all those frames are equal.
A. An experimenter inside one inertial lab cannot discover its absolute motion by any experiment done inside the lab. The laws look exactly the same as they do in any other inertial lab.
B. For the laws to look the same, they must transform correctly when we change coordinates.
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The Principle of Inertia itself works with any linear transformation of the coordinates. But the rest of Newton’s physics is tied more tightly to the Galileo transformations.
Newton’s Second Law says that a force F acting on a mass m produces an acceleration a according to:
F = m a
There are also separate laws that tell us what the force actually is (for example, Newton’s law of gravity). To check that these laws obey the Principle of Relativity under Galileo transformations, notice that acceleration is the same in every inertial frame. Differentiating the velocity-addition rule gives:
a' = a
Because acceleration does not change, the force must also stay the same if the Second Law is to hold in every inertial frame (mass is already assumed to be the same).
Forces will stay the same when they depend only on the distances between particles, because distances themselves are invariant. Gravity is an example of such a force. **When forces depend only on distances, the Galilean Principle of Relativity is satisfied.**
The earlier conclusions do not rule out forces that depend on velocity.
When a body moves through a sticky fluid, the friction force grows with the body’s speed relative to the fluid. If the body is at rest in the fluid, the friction force is zero.
Although ordinary velocity is not the same in every inertial frame, the velocity that matters here is the velocity relative to the fluid. The fluid itself becomes a special reference frame. This does not break the Principle of Relativity, because the situation naturally picks out the fluid frame.
What actually enters the physics is the relative speed between two bodies. That relative speed is the same in every inertial frame:
u_P' - u_Q' = u_P - u_Q
Isaac Newton was born an orphan in Woolsthorpe, Lincolnshire. His mother remarried when he was three and left him with his grandmother. Seven years later she returned as a widow. With help from an uncle, Newton prepared for college and entered Trinity College, Cambridge, in 1661. He studied philosophy and mathematics. When the plague struck in 1665 he went back to the family farm. There he began the work in mathematics, mechanics, astronomy, and optics that changed science forever.
He graduated in 1668 and became Lucasian Professor of Mathematics at Cambridge in 1669. Newton invented the calculus (which he called the method of fluxions). In his great book Principia (1687) he stated the laws of motion and gravity. He used them to calculate the orbits of planets and comets and to explain the tides. He also wrote Opticks (1704), where he showed that white light is made of the colors of the rainbow and built a reflecting telescope to avoid color problems with lenses.
Newton was a private, intense person who studied alchemy and religion as well as science. After a second nervous breakdown he left research in 1693. He later became Master of the Mint, a Member of Parliament, and President of the Royal Society until his death. He never married. Despite disagreements with the Church, he was buried in Westminster Abbey.
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When no outside forces act on a perfect fluid, the only force on a small piece of fluid comes from differences in pressure. Newton’s Second Law then reads:
-grad p = rho * (du/dt)
Here rho is the density, p is the pressure, and u is the velocity at each point. We also need the continuity equation that says mass is conserved:
d(rho)/dt + div (rho u) = 0
and an equation of state that links pressure to density:
p = p(rho)
To get a wave equation we make two approximations.
1. Density and pressure stay close to their rest values rho0 and p0. We write the small change in density as a perturbation s:
rho = rho0 (1 + s) (with s much less than 1)
The pressure change is then approximately:
p = p0 + cs^2 * rho0 * s
where cs^2 is a constant that depends on the fluid and has units of speed squared.
2. The fluid velocities and their changes in space are small. In the frame where the fluid is at rest on average, the acceleration simplifies to:
du/dt ≈ du/dt
(the extra term that involves velocity gradients can be ignored).
Putting these approximations into the two basic equations gives a pair of simpler relations. Taking the divergence of one and the time derivative of the other makes their right-hand sides equal. The result is the wave equation for the perturbation s:
(1/cs^2) * d^{2}s/dt^{2} - Laplacian(s) = 0
Solutions of this equation show that small disturbances travel through the fluid at the constant speed cs.
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Mechanical waves give another example of a privileged frame. e.g. Sound waves travel through air, water, or solid material, but they cannot travel through empty space. A mechanical wave is simply a disturbance of the material it travels through. Therefore the frame fixed to that material is naturally special.
The usual wave equation that describes the disturbance is valid only in the frame where the material is at rest. In particular, the wave speed that appears in the equation is set by the properties of the material. Because speed is not a Galilean invariant, the wave equation cannot hold in every inertial frame.
A. Again, this does not violate the Principle of Relativity. The wave equation can be used in any inertial frame in which the material medium happens to be at rest.
B. The wave equation itself is only an approximation, derived from the fundamental laws of mechanics, and those fundamental laws do obey Galilean relativity.
MAXWELL’S ELECTROMAGNETISM
James Clerk Maxwell (1831-1879) developed the laws of the electromagnetic field after Michael Faraday discovered electromagnetic induction in 1831.
In Maxwell’s theory the magnetic forces between charges depend on the velocities of those charges. When there are no charges or currents present, Maxwell’s equations can be rewritten as wave equations.
*** The speed that appears in those wave equations matches the measured speed of light. Maxwell therefore concluded that light itself is an electromagnetic wave.
In the nineteenth century (1800's) people still believed that distances and times are the same in every frame. Under that belief, a theory that contains both velocity-dependent forces and wave equations can only make sense if the phenomenon is mechanical and takes place inside a material medium.
Maxwell’s laws would then be valid only in the inertial frame fixed to that medium. The medium was identified with the “ether” that was already supposed to carry light waves.
We will see soon how hard people tried to detect the ether or at least measure the motion of a laboratory through it. (note: there is no ether.)
Here we show that simply using Maxwell’s laws in two different frames, together with the classical ideas of space and time, already leads to a contradiction.
Imagine two long parallel lines of charge. The upper line contains equal numbers of positive and negative charges, so it is electrically neutral overall. The lower line has a net charge. Both lines also carry electric currents in the same direction because one type of charge is moving.
1. In this situation Maxwell’s laws say there is no electric force between the lines (one of them is neutral). There is, however, a magnetic force: parallel currents attract each other.
Now switch to a new frame that moves along with the positive charges. In the new frame the upper line is still neutral (distances are assumed invariant, so the numbers of positive and negative charges remain equal).
2. But now neither line carries a current, because the charges that were moving are at rest. Maxwell’s laws applied in this frame therefore predict no electric force and no magnetic force. There is no interaction at all.
The two results 1. and 2. cannot both be true.
Either an interaction exists or it does not; that fact cannot depend on which frame we choose. The contradiction arises only because we applied Maxwell’s laws in both frames while also using Galileo transformations (which keep distances and times unchanged).
That is, as long as the classical notions of space and time are kept, Maxwell’s laws cannot be used in every inertial frame. We must first find the special frame in which those laws are valid --> the ether frame. The failure to find that frame will force us to change our ideas about space and time.
James Clerk Maxwell was born in Edinburgh in 1831. He grew up in the Scottish countryside, where his parents taught him at home. At age 10 he returned to Edinburgh to attend the Academy. By age 14 he had already presented a paper on oval curves to the Royal Society of Edinburgh. He studied mathematics at Trinity College, Cambridge, and graduated in 1854 after a four-year course.
He stayed in Cambridge until 1856. That year he published “On Faraday’s Lines of Force,” which explained how electric and magnetic fields depend on each other. He also won the Adams Prize for showing that Saturn’s rings can be stable only if they are made of many small solid particles. In 1860 he became Professor of Natural Philosophy at King’s College, London. There he calculated the speed of electromagnetic waves and found that it matched the speed of light. In 1862 he proposed that light is a wave traveling in the same medium that produces electric and magnetic effects.
Maxwell left London and returned to Scotland. In 1873 he published the final version of his electromagnetic theory. In his later years he edited the papers of Henry Cavendish and supervised the building of the Cavendish Laboratory.
Maxwell’s greatest contribution was to treat the field itself as a real physical object. He saw that the electric current Faraday observed in 1831 is caused by an electric field that exists even when no wire is present. A changing magnetic field creates an electric field, and a changing electric field creates a magnetic field. Maxwell also made important advances in the kinetic theory of gases. Working independently of Boltzmann, he showed how the speeds of molecules in a gas are distributed according to the temperature of the gas.
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Maxwell’s equations are a set of four first-order differential equations that describe how the electric field E(r, t) and the magnetic field B(r, t) behave at every point in space and time. The sources of these fields are the electric charge density rho(r, t) and the electric current density j(r, t). Once the sources and suitable boundary conditions are given, the fields are completely determined:
div B = 0
div E = rho / epsilon0
curl E + dB/dt = 0
curl B - mu0 * epsilon0 * dE/dt = mu0 * j
If we take the divergence of the last equation and use the second equation, we immediately obtain the continuity equation. This equation expresses the local conservation of electric charge:
d(rho)/dt + div j = 0
The force that the electromagnetic field exerts on a charge q that is moving with velocity u is called the Lorentz force:
F = q (E + u x B)
In regions where there are no charges or currents, each Cartesian component of the fields satisfies a wave equation. These wave equations are obtained by differentiating Maxwell’s equations with respect to time and using a standard vector identity. The results are:
(1/c^{2}) * d^{2}E/dt^{2} - Laplacian(E) = 0
(1/c^{2}) * d^{2}B/dt^{2} - Laplacian(B) = 0
Here the constant c is defined by
c = 1 / sqrt(mu0 * epsilon0)
and has the dimensions of a speed. It is the speed at which electromagnetic waves travel.
In the International System of units the value of mu0 is chosen by definition to be 4 pi x 10^{-7} N A^{-2}. An electrostatic experiment then fixes epsilon0 = 8.854187817 x 10^{-12} N^{-1} A^{2} m^{-2} s^{2}. From these two numbers one obtains
c = 2.997924580 x 10^8 m/s
Since 1983 this value of c has been adopted as an exact definition, which means that the units of length and time are no longer independent.
(In the older Gaussian system the magnetic field is redefined by the replacement B → B/c. This choice gives the electric and magnetic fields the same units, and the unit of charge is fixed by setting epsilon0 = 1/(4 pi).)
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"In Search of the Ether"
Two Models for Light: In 1704 Isaac Newton published Opticks. In that book he argued that light travels as tiny particles shot out from a source. He based this idea on the fact that light rays appear to travel in straight lines.
Using the particle picture, Newton tried to explain reflection and refraction. He imagined short-range forces that push or pull the light particles near the surface between two transparent materials. When light enters a denser material such as glass or water, the ray bends toward the perpendicular.
Newton said this happens because an attractive force speeds up the particles in the direction normal to the surface. Different colors bend by different amounts, so Newton concluded that there must be different kinds of light particles, each feeling a different force.
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The particle model could not explain every observation. Francesco Grimaldi had already shown in 1665 that shadows are not perfectly sharp; the edges show alternating bright and dark fringes (diffraction). Erasmus Bartholin discovered birefringence in calcite in 1669. Newton himself saw colored rings when he placed a curved lens on a flat glass plate (Newton’s rings). He often mixed wave ideas with particle ideas when he tried to interpret these effects.
In 1678 the Dutch scientist Christiaan Huygens presented a different view. He argued that light cannot be a stream of particles because rays from opposite directions pass through each other without colliding. Instead, he said, light is a wave that travels through a medium, just as sound travels through air.
Each point on a wave front acts as a source of secondary wavelets. The new wave front is the surface that touches all the secondary wavelets (Huygens’ Principle). Rays are the lines drawn from each secondary source to the point of tangency.
When the medium is uniform and the same in every direction, both the main wave and the secondary wavelets are spheres. If the medium changes from place to place but is still the same in every direction, the secondary wavelets remain spheres but have different radii; the rays stay perpendicular to the wave front. If the medium is the same everywhere but different in different directions, the secondary wavelets are no longer spheres and the rays are no longer perpendicular to the wave front.
Huygens’ construction also explains reflection and refraction. A plane wave hits a flat boundary. Points on the boundary become secondary sources one after another. The reflected wave leaves at the same angle as the incident wave. The transmitted wave bends toward the perpendicular if the wave speed is smaller in the second medium. This leads directly to Snell’s law:
sin i / sin r = c1 / c2 = n21
where n21 is the relative refractive index. In the wave picture, therefore, light must travel more slowly in the denser medium. The particle picture had predicted the opposite.
For more than a century the particle model remained more popular, largely because of Newton’s authority. The wave model seemed unable to explain the sharp shadows cast by ordinary objects. In addition, Huygens’ geometric construction did not yet include the ideas of amplitude and phase that are needed for interference.
The situation changed in the early nineteenth century. In 1802 Thomas Young showed that interference patterns can be understood if light is a wave.
In 1808 Étienne Malus discovered polarization by reflection; Young suggested that light must be a transverse wave. Augustin Fresnel developed a full mathematical treatment of the wave theory. He calculated diffraction patterns for many obstacles and showed that the same theory that produces soft fringes also produces sharp shadows when the obstacle is large.
By the middle of the century the wave model could account for every optical effect then known. In 1850 Léon Foucault measured the speed of light in water and found it slower than in air, exactly as the wave theory required. The particle model gradually faded away.
The First Measurement of the Speed of Light
Galileo was the first person to try to find out whether light has a finite speed. He suggested a simple experiment: two people stand far apart, each with a lantern. One uncovers his lantern; the moment the other sees the light, he uncovers his own. By timing the round trip they could calculate the speed. Galileo tried it across a distance of less than a mile but could not tell whether the return flash was instantaneous or delayed. :(
The first real measurement came in 1676 from the Danish astronomer Ole Rømer. He was working for the French Academy of Sciences on a practical problem: finding a reliable way to determine longitude at sea.
Note: Galileo had earlier suggested using the eclipses of Jupiter’s moon Io as a celestial clock. Io disappears behind Jupiter every 42.5 hours on average. If sailors had a table of the exact Paris times of those eclipses, they could compare the local time of an eclipse with the table and work out their longitude.
When Rømer examined years of careful observations, he noticed something strange. The eclipses came later than expected when Earth was moving away from Jupiter and earlier when Earth was approaching Jupiter.
The total accumulated delay from the closest approach to the farthest point was about 22 minutes. Most scientists of the day, following Descartes, still believed light travels instantly. Rømer realized the delay was caused by the extra distance light had to travel across the Earth’s orbit.
1. He reasoned that those 22 minutes were simply the time light needed to cross the diameter of Earth’s orbit.
2. Using the best estimate then available of about 293 million kilometers for that diameter, Rømer calculated a speed of roughly 222,000 kilometers per second.
3. The true delay is closer to 16 minutes 36 seconds, and modern values of the orbital size give a more accurate speed, but Rømer had the right order of magnitude.
For the first time in history, human beings knew that light does not travel infinitely fast. It has a definite, enormous, but finite speed.
Although Huygens and Newton quickly accepted Rømer’s conclusion that light has a finite speed, many scientists remained unconvinced. Full acceptance came only in 1729, when the English astronomer James Bradley announced a new effect he called the aberration of light.
Bradley had set out to measure the tiny annual shift in a star’s apparent position caused by Earth’s motion around the Sun (stellar parallax). He chose the bright star Gamma Draconis, which passes nearly overhead near London and can even be seen in daytime.
He expected the star to trace a small ellipse whose size would reveal the star’s distance. Instead he found an ellipse whose major semi-axis was about 20 arcseconds -- far larger than expected -- and whose orientation did not match the predicted parallax pattern.
Even more surprising, every star he checked produced an ellipse of the same size, independent of distance.
Bradley realized he was seeing something caused by Earth’s velocity, not its position. The star always appears slightly shifted in the direction of Earth’s motion.
A. When Earth is moving toward a star, the telescope must be tilted forward so that the incoming light travels straight down the tube.
B. Six months later the tilt is reversed, and the star appears to have moved the other way.
The result is an ellipse whose size depends only on the ratio of Earth’s orbital speed to the speed of light.
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In the simple case of a star lying perpendicular to Earth’s velocity V, the tilt angle alpha (the angle of aberration) satisfies
tan alpha = V / c
When the angle between the velocity and the light ray is a general angle theta, the relation becomes
tan alpha = (V/c * sin theta) / (1 - (V/c) * cos theta)
For the small speeds of the Earth (V much less than c) this simplifies to
alpha ≈ (V/c) * sin theta
Earth’s orbital speed is about 30 km/s. Over six months the direction of V reverses, producing a change of roughly 60 km/s. The resulting ellipse has a major semi-axis of 20.25 arcseconds and exactly what Bradley measured.
If light traveled infinitely fast there would be no aberration at all. Bradley’s discovery therefore confirmed, with an accuracy of about one percent, that light has a finite speed. The modern value of the aberration constant is 20.49552 arcseconds.
Almost two centuries after Rømer measured the speed of light using Jupiter’s moon, the first Earth-based measurement was made.
In 1849 Hippolyte Fizeau sent a beam of light through the gaps of a spinning toothed wheel. The light traveled 8.5 kilometers to a mirror, bounced back, and returned to the wheel.
A. If a tooth blocked the returning light, the observer saw darkness.
B. When the wheel spun faster, the light could pass through the next open gap and the view grew bright.
Fizeau found the brightest return signal when the wheel turned at 25 revolutions per second. With 720 teeth on the wheel, the time from one gap to the next was 1 divided by (25 times 720) seconds, or 1/18,000 of a second.
In that short time the light had covered the round-trip distance of 17 kilometers. Dividing the distance by the time gave a speed of about 306,000 kilometers per second.
The next year Léon Foucault improved the experiment by replacing the toothed wheel with a rotating mirror. Using this better method he measured the speed of light in both air and water.
*He confirmed that light travels more slowly in water than in air, just as the wave theory of light had predicted.
In the seventeenth century the wave theory and the particle theory of light made opposite predictions about what happens when light enters a material such as water.
Christiaan Huygens treated light as a wave traveling through a medium, much like sound travels through air. He reasoned that the speed of the wave should depend on the properties of the medium. When light enters a denser transparent material, the wave interacts more strongly with the matter and therefore slows down. A simple modern way to write the prediction is
v = c / n
where c is the speed of light in vacuum (about 3.00 x 10^8 m/s), v is the speed inside the material, and n is the refractive index of the material. For water n is about 1.33, so the wave theory predicted
v ≈ 3.00 x 10^8 / 1.33 ≈ 2.26 x 10^8 m/s
In other words, light should travel roughly 25 percent more slowly in water than in vacuum.
Newton’s particle theory said the opposite. Newton pictured light as tiny particles that are attracted by matter. When the particles enter a denser medium they should be pulled forward and therefore speed up.
The experimental question was clear: does light slow down or speed up when it enters water? Foucault’s 1850 measurement with a rotating mirror showed that light travels more slowly in water than in air. The result matched the wave theory and contradicted the particle theory.
Fizeau and Foucault did not invent the prediction; the wave theory had already made it. Their experiments supplied the decisive evidence. Decades later Maxwell’s electromagnetic theory supplied a deeper reason why light must travel at speed c/n inside a transparent material.
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Because light was treated as a mechanical wave, it needed a material medium to carry the disturbance. That medium was called the luminiferous ether.
Once Maxwell showed that light is an electromagnetic wave, the same ether was assumed to carry electromagnetic waves as well. The speed c of light was regarded as a property of this ether.
The discovery of polarization early in the nineteenth century proved that light waves are transverse: the disturbance moves at right angles to the direction of travel. Sound waves, by contrast, are longitudinal.
An elastic solid can carry transverse waves, so the ether was imagined to be an elastic substance. This led to a paradox. The ether had to be extremely rigid in order to produce such a high wave speed, yet it also had to be so thin that the planets could move through it without any noticeable drag.
George Stokes pointed out that rigidity can depend on the time scale of the observation. Asphalt, for example, shatters like a solid when struck sharply but flows like a liquid over long periods.
Inside transparent materials the ether was assumed to change its properties so that light travels at speed c/n, where n is the refractive index of the material.
At first people thought the ether stayed completely undisturbed by ordinary matter. It filled all space and remained at absolute rest, providing a universal reference frame in the Newtonian sense.
The aberration of light seemed to confirm this idea. In the ether frame, rays are always perpendicular to the wave fronts because the speed is the same in every direction. On the moving Earth the wave fronts keep the same orientation, but the rays (the directions in which energy actually travels) are tilted. The tilt is produced by the “ether wind” that appears when we view the situation from the Earth’s frame.
The velocity of the energy, called the ray velocity, transforms according to the ordinary Galilean rule for adding velocities. This change in ray direction is precisely the aberration observed by Bradley.
Because no one could detect the ether directly, experimenters tried instead to reveal Earth’s motion through it. Any such motion would change the speed of light relative to the laboratory, and that change should show up in optical experiments.
In 1810 François Arago performed the first of these tests.
A. He covered half the objective of a telescope with a prism. Starlight could therefore reach the eyepiece either straight through the lens or after being bent by the prism.
B. The angle between the two telescope settings equaled the deviation produced by the prism. Arago expected this angle to depend on whether the starlight was traveling with or against Earth’s orbital motion of about 30 km/s.
C. The difference should have been several tens of arcseconds.
He found only random shifts of a few arcseconds that could be blamed on experimental error. A repetition six months later gave the same null result. Earth’s motion relative to the ether refused to appear.
George Stokes offered one explanation. He argued that the Earth does not slip through the ether unnoticed. Instead it drags along the ether inside it and a thin layer stuck to its surface. Outside that layer the ether gradually returns to absolute rest.
Arago’s prism was therefore sitting in a local ether that moved with the Earth, so ordinary Snell’s law applied and no extra deviation was expected.
The difficulty with Stokes’s picture was that it made the aberration of light hard to understand. Aberration requires the rays to be tilted relative to the wave fronts; that tilt is natural only if the laboratory is moving through a stationary ether.
Augustin Fresnel proposed a milder form of dragging. He suggested that a moving transparent body carries only a fraction f of the ether inside it, where f is less than 1. The local ether therefore moves at velocity fV relative to the universal ether. Fresnel showed that if the dragging coefficient has the special value
f = 1 - 1/n²
(where n is the refractive index), two effects cancel each other. The partial drag changes the speed of light inside the prism, while aberration tilts the rays.
To first order in V/c the outgoing ray in the laboratory frame points exactly where Snell’s law says it should for a prism at rest. The absolute motion of the Earth is hidden.
Fresnel’s calculation relied throughout on the classical assumptions that distances and times are the same in every frame and that velocities add according to the Galilean rule. The partial-drag hypothesis was therefore a way of protecting those assumptions against an awkward experimental result.
What remained beyond dispute was simply that Arago’s experiment had failed to reveal any motion of the Earth through the ether.
In 1851 Hippolyte Fizeau tried to measure how much ether is dragged by moving water. He split a light beam into two rays and sent them around a closed path in opposite directions through flowing water. One ray traveled with the current; the other traveled against it. Any dragging of the ether by the water would change the travel times of the two rays and produce a measurable shift in the interference fringes.
Because the Earth itself might be moving through the universal ether at some unknown speed V, the calculation looks complicated at first. However, the closed paths of the two rays cancel all first-order effects of V. Only the known speed v of the water relative to the laboratory matters at the order the apparatus could detect. The expected time difference then reduces to a simple expression proportional to the dragging coefficient f. Stokes’s total-drag hypothesis required f = 1; Fresnel’s partial-drag hypothesis required f = 1 – 1/n² ≈ 0.43 for water. Fizeau measured a fringe shift that matched Fresnel’s value within about 15 percent.
Seventeen years later Martin Hoek performed a related experiment. Again two rays traveled opposite ways around a closed path, but this time they passed through a tube of still water in only one segment of the circuit. Because the paths were no longer symmetric, a first-order effect of Earth’s velocity V should have appeared. Turning the whole apparatus by 180 degrees would reverse the sign of the time difference and shift the fringes. No shift was seen. Once more Fresnel’s dragging coefficient made the first-order term vanish, protecting the classical notions of space and time.
In 1871 George Airy tried a different approach. He filled a telescope with water and looked for an extra tilt needed to keep a star centered in the field of view. Without dragging, the extra refraction caused by the water should have revealed Earth’s motion through the ether. With Fresnel’s partial drag, Snell’s law continues to hold inside the moving telescope (to first order in V/c), so no extra tilt is required. Airy observed none.
Arago’s, Fizeau’s, Hoek’s, and Airy’s experiments all failed to detect any motion of the Earth relative to the ether. In every case Fresnel’s partial-dragging hypothesis supplied a consistent explanation within the framework of nineteenth-century space and time.
In 1879 James Clerk Maxwell pointed out that an astronomical observation could reveal Earth’s motion through the ether at first order in V/c without any need for Fresnel’s dragging coefficient.
The delay in the eclipses of Jupiter’s moon Io is simply the time light takes to cross the diameter of Earth’s orbit. If the whole Solar System is moving through the ether at speed V, that delay should be shorter when Jupiter lies in the direction of motion and longer when Jupiter lies in the opposite direction.
The difference between the two extreme delays is proportional to V/c and could in principle be measured. The data available to Maxwell were not precise enough, but the suggestion showed that a first-order effect was still possible in astronomy.
Maxwell also noted that any laboratory experiment in which light travels a closed path must be sensitive to second-order effects of order (V/c)² ≈ 10⁻⁸. Detecting such a tiny correction requires extraordinary precision.
The American physicist Albert Michelson accepted the challenge.
Michelson designed an interferometer in which a light beam is split by a half-silvered mirror into two perpendicular paths. Each beam travels out and back, then recombines to form interference fringes. If the laboratory is moving through the ether, the travel times on the two arms differ by an amount proportional to (V/c)². Rotating the entire apparatus by 90 degrees interchanges the arms and should shift the fringes by a calculable fraction of a fringe.
In 1881 at Potsdam, Michelson performed the experiment and saw no shift. He concluded that the Earth must carry a layer of ether with it, so that the laboratory is effectively at rest in a local ether. Lorentz later pointed out a small error in Michelson’s calculation that reduced the expected shift by half and brought it close to the experimental uncertainty. A more sensitive instrument was needed.
Michelson joined Edward Morley in Cleveland. They first repeated Fizeau’s water-drag experiment with higher precision and again confirmed Fresnel’s partial-drag coefficient. Then they built a greatly improved interferometer. Multiple mirrors lengthened each optical path to about 11 meters. The whole apparatus rested on a massive granite slab floating on a mercury bearing so that it could be rotated smoothly. A compensating plate equalized the dispersive effects on the two beams, allowing the use of white light. With these improvements a fringe shift of at least 0.4 fringes was expected.
In July 1887 the experiment was run for four days. The observed shifts were at most one-twentieth of the smallest expected value. The result was again null.
Michelson and Morley believed the Earth still carried an adherent layer of ether. They suggested repeating the measurement at high altitude, where the layer might thin out. That idea, however, conflicted with both stellar aberration and the confirmed partial drag inside moving water. In 1892 Oliver Lodge looked for an ether layer dragged by a rapidly spinning wheel and found none.
By the end of the 1880s every careful attempt to detect Earth’s motion through the ether had failed. The classical picture of a stationary luminiferous ether was in serious trouble.
In 1892 the Dutch physicist Hendrik Lorentz offered a new explanation for the null result of the Michelson-Morley experiment. He proposed that the ether exerts a real physical force on any object moving through it, causing the object to contract in the direction of motion. Transverse dimensions remain unchanged. If a length that measures lo when the body is at rest becomes
l = lo * sqrt(1 - V²/c²)
when the body moves at speed V, then the two arms of the interferometer acquire identical travel times. Rotating the apparatus therefore produces no fringe shift.
The same idea had been suggested qualitatively by the Irish physicist George FitzGerald in 1889, but Lorentz gave it a precise mathematical form.
Both men regarded the contraction as an absolute, dynamical effect caused by interaction with the ether. The contracted length was supposed to be the same in every reference frame, in keeping with the classical belief that distances are invariant.
Lorentz embedded the contraction hypothesis in a broader theory of electrons. He pictured matter as made of tiny charged particles moving through a stationary ether. To first order in V/c the theory reproduced Fresnel’s partial-drag formula and avoided some of its difficulties with dispersive media. The length contraction itself, however, remained an extra assumption introduced solely to explain the Michelson-Morley result.
By the early twentieth century the ether hypothesis was in serious trouble. Repeated Michelson-Morley experiments continued to give null results. Other tests, such as searches for ether-induced birefringence or torque on a charged capacitor, were equally negative. Emission theories that tried to keep classical space and time by letting the speed of light depend on the motion of the source also failed to account for all the data.
In 1905 Albert Einstein, then a patent clerk in Bern, published a radically different solution. He took Maxwell’s equations to be fundamental laws that must hold in every inertial frame. The electromagnetic field needs no material ether to support it.
Absolute motion through a privileged frame is therefore a meaningless concept; only relative motion has physical significance. Because Maxwell’s equations are the same in every inertial laboratory, the speed of light in vacuum must be the same constant c in every direction and in every inertial frame.
That single requirement is incompatible with the Galilean addition of velocities. The classical assumptions that distances and times are absolute must therefore be abandoned.
Once those assumptions are dropped, the null results of Michelson-Morley, Trouton-Noble and similar experiments become immediate consequences: light travels at speed c on every path in every laboratory, so rotating an interferometer cannot change travel times.
Einstein’s approach also re-interprets Fresnel’s dragging coefficient. The formula is no longer the result of a dynamical interaction with an ether; it is simply the relativistic composition of the velocity of light inside a moving medium with the velocity of the medium itself. In this way the entire body of optical evidence is explained by a single change in the concepts of space and time.
The Lorentz transformations that leave Maxwell’s equations invariant had already been written down by Lorentz and Larmor. What Einstein supplied was the physical interpretation: those transformations describe the true relations between space and time measurements made in different inertial frames.
***The classical notions that had seemed self-evident for centuries had to be replaced.
By proposing a new way to regard space and time, Einstein produced in addition a deep change in the laws of Dynamics, and the allowed form of the laws for interactions (a form that Maxwell electromagnetic theory certainly has). The set of fundamental laws of classical mechanics was built on the observance of the Principle of relativity under Galileo transformations. When these laws were adjusted to satisfy the Principle of relativity under Lorentz transformations, an unsuspected consequence appeared: the possibility of turning mass into other types of energy.
POSTULATES OF SPECIAL RELATIVITY
In 1905 Einstein published the paper “On the Electrodynamics of Moving Bodies.”
In it he rebuilt the concepts of space and time from two simple postulates.
1. First, the laws of electrodynamics and optics are exactly the same in every inertial frame (the Principle of Relativity).
2. Second, light always travels through empty space at the same definite speed c, no matter how the source is moving.
These postulates place Maxwell’s equations among the fundamental laws that must hold in every inertial laboratory. As a direct result, no electromagnetic experiment can ever reveal an absolute state of motion:
“The phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.”
A finite speed that is the same in every inertial frame cannot obey the old Galilean rule for adding velocities. The Galilean transformations, and the classical assumptions that distances and times are absolute, must therefore be abandoned. The two assumptions are linked: if lengths are relative, times must be relative as well, and vice versa.
Albert Einstein (1879–1955). Born in Ulm, Germany, Albert Einstein spent his childhood in Munich, where his family owned an electromechanical business. When the family company failed, his parents and sister moved to Milan in 1894 and later to Pavia. Albert remained at a school in Munich but soon left to join his family. At that time he renounced his German citizenship and rejected all religious dogma. He finished high school in Switzerland and enrolled at the Zurich Polytechnic, graduating in 1900 as a teacher of mathematics and physics. He obtained Swiss citizenship in 1901. In 1903 he married Mileva Marić, a fellow student at the Polytechnic. They had three children: a daughter born before the marriage (whose fate is unknown) and two sons.
Unable to secure a permanent teaching post, Einstein took a job at the patent office in Bern on the recommendation of the father of his friend Marcel Grossmann. The position left him enough free time for research, even though he worked far from academic circles. While employed at the patent office in 1905, he wrote his papers on the photoelectric effect, special relativity, and Brownian motion. The last of these earned him a Ph.D. from the University of Zurich. In 1908 he obtained a position at the University of Bern. He later held professorships in Zurich, Prague, Berlin, and Princeton.
Between 1907 and 1916 he developed the relativistic theory of gravitation (general relativity), which later found applications in cosmology. He also made important contributions to statistical physics and quantum theory. In 1919 he divorced Mileva and married his cousin Elsa. In 1921 he received the Nobel Prize in Physics for his discovery of the law of the photoelectric effect.
Politically, Einstein regarded himself as an internationalist liberal Jew. In matters of religion he felt a deep “cosmic religiosity,” which he considered the strongest stimulus for scientific research: “Only those who realize the immense efforts and, above all, the devotion which pioneer work in theoretical science demands, can grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue” (1934).
In 1933, while Einstein was abroad, the Nazi regime confiscated his property in Berlin; he never returned to Germany. Rising anti-Semitism led him to support the creation of a Jewish state in Palestine. After settling in Princeton he wrote to President Roosevelt a month before the outbreak of World War II, urging him to pursue the development of a nuclear bomb in case Hitler obtained one first. Einstein remained a lifelong pacifist and deeply regretted that the bomb was ultimately used. In 1952 he declined the offer to become president of Israel. He died in Princeton and was cremated.
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To see the link clearly, call Lo the proper length of a rod measured in the frame where the rod is at rest. Suppose the same rod has a different length L when measured in a frame that is moving relative to it. Now let a particle travel from one end of the rod to the other at relative speed V. In the rod’s rest frame the trip takes time
Delta t = Lo / V
In the particle’s rest frame the rod has length L and the same trip takes time
Delta tau = L / V
(The relative speed has the same magnitude in both frames.) Dividing the two results gives
Delta t / Delta tau = Lo / L
Thus any change in length forces a corresponding change in time, and absolute lengths would force absolute times.
We call the time interval measured in the frame where two events occur at the same place the proper time of those events.
In the example above, the particle’s frame is the proper frame for the pair of events “particle passes first end” and “particle passes second end.”
In Newtonian physics both sides of the equation above equal 1. In special relativity the ratios are allowed to differ from 1. Because space is homogeneous and isotropic and time is homogeneous, each ratio can depend only on the relative speed V between the two frames. Moreover, the same function of V must govern both the length ratio and the time ratio:
Lo / L = gamma(V)
Delta t / Delta tau = gamma(V)
The explicit form of the function gamma(V) will be fixed by the requirement that the speed of light is the same in every inertial frame.
Wheel of Lunch
1. Burger King
2. Taco Bell
3. Lemon Kitchen
4, Cigarettes
Smoke break.
I can see my breath in the wind.
Today is writing day. Maybe with a pencil.
I’ve been making an ode curriculum. It’s been so long I need to build a taxonomy… again. I remember doing this in 6th grade but let it slip.
Wheel of lunch:
1. Beer
2. Taco Bell
3. Lemon kitchen
4. Cigarettes + Dr. Pepper
MISC |
IVP |
wronskian |
reduction of order |
characteristic equations |
resonance |
generalized eigenvector |
laplace shifting properties |
Transform differentiation (e.g. t*f(f))
TECHNIQUES |
seperable |
1st order linear |
exact |
non-exact made exact |
bernoulli |
homogeneous, distinct real roots |
homogeneous, repeated roots |
homogeneous, complex roots |
nonhomogeneous, polynomial forcing |
nonhomogeneous, exponential forcing |
nonhomogeneous, trigonometric forcing |
resonance |
nonhomogeneous, general forcing |
laplace, basic |
laplace, trig |
laplace, inverse |
laplace, 1st derivative |
laplace, 2nd derivative |
laplace, IVP |
laplace, partial fractions |
systems of linear, two coupled 1st order ODE |
systems of linear, matrix representation |
systems of linear, distinct real eigenvalues |
systems of linear, repeated eigenvalues |
systems of linear, defective matrice (generalized eigenvectors) |
systems of linear, complex eigenvalues
This all I can remember and thus my starting point.
Here is an easy $1 million up for grabs.
Need to keep an eye on this!
Seamus. He's coming.
This tactical research focuses on computational methods for computer-assisted simulation that can be used in tactical-level wargames.
The primary outcomes of this research are computational models for tactical-level combat simulators that enable probability-based analysis. These simulators can be used for comparative analysis of combat scenarios ranging from platoon to brigade level.
The computational models concentrate on combat effects. The models address the probability of a damaging hit, from which the resulting effects on military units are modeled using finite-state machines and Markov chains.
Furthermore, the results of these models are incorporated into event tree analysis to determine the probability of operational success.
What Makes This Interesting
The passage describes a mathematical approach to military combat simulation involving four major concepts:
1. Probability of a damaging hit: Calculating the likelihood that a weapon successfully strikes and damages a target.
2. Finite-state machines: Representing combat units through discrete states, such as operational, damaged, disabled, or destroyed.
3. Markov chains: Modeling transitions between combat states using probabilities.
4. Event tree analysis: Calculating the probability that a military operation succeeds based on sequences of possible events.
>This< is my favorite album for the next 36 minutes and 20 seconds.
Wheel of Lunch!
1. Cigarettes and Dr. Pepper
Hey, hey! I got my first reading-course curriculum: "Differential Game Theory Mastery Curriculum"
Ready... set... go!
Let us play pong. …