Special Relativity
                    created 9/06
                         revised 11/4/13
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Relativity introduction
        Special relativity postulates that all laws of physics, including the value measured for the speed of light, are exactly the same for all observers who are moving at constant speeds relative to each other. The implication of this is that the time is not the same for all observers. Special relativity provides a fundamental understanding of what otherwise appears to be difficult to understand (if not bizarre) experimental results:

          * You can't get particles to go faster than 3x10^8 m/sec regardless of how much
                      energy you give them.
          * You can increase the energy (apparently without limit) carried by particles moving
                      near 3x10^8 m/sec even though they barely go any faster.
         *  When light is emitted from a moving source, the speed of the source is not added
                     to (nor does it have any effect on) the speed of the emitted light without need
                     for a stationary medium (ether) to carry the light.
          * High speed, unstable particles (muons) with only 2 usec lifetimes, created in the
                    upper atmosphere by cosmic rays, are (somehow) able to hit the ground in large
                    numbers even though their with short lifetimes they should almost all decay
                    high in the atmosphere.

As a bonus, special relativity explains that mass and energy are (in some sense) equivalent and that a magnetic field is just a relativistic artifact of the electric field from moving electrons.

Four relativistic concepts from just two experiments (2/10/11)

1) Acceleration of electrons with high voltage

        Classical physics predicts that electrons should be able to be accelerated to any speed, limited only by how much voltage is available, but tests show (see below) this is not true. What happens is that at a fairly modest voltage, as electron speeds approach [c = 3 x 10^8 m/sec], the speed curve begins to flatten and no amount of energy can accelerate the electrons above this speed limit.

        Thermal measurements show that the energy carried by the moving electrons does not flatten and can (apparently) be increased without limit. Higher acceleration voltages drive the (kinetic) energy carried by the electrons higher and higher even though speeds near 'c' barely change. Classical physics has no explanation for this.

2) Short lifetime muons travel all the way through the atmosphere
        Muons (heavy, unstable electron-like subatomic particles) are continually created in the upper atmosphere by fast moving cosmic rays (mostly protons). Muons are unstable with a measured lifetime of 2 usec (round numbers). Since the fastest any particle can move is 'c' (1 nsec/ft or 1 usec/thousand feet), travel time from 50,000 feet to the ground is (at least) 50 usec. This is 25 muon lifetimes, so classical physics would predict that only a tiny fraction of muons would survive long enough to reach the ground, but this is not what tests show. What is found is that a substantial fraction of the muons are (somehow) able to survive long enough to travel all the way through the atmosphere and reach the ground.
        'Hyperphysics' site (using slightly different numbers, which it says agree with historical muon experiments) calculates that about 5% of muons created 6 miles up (10 km) and traveling only at .98c will reach the ground, vs less than 1 in a million for no time dilation, so muon time dilation is NOT a subtle effect.
        These two simple experiments demonstrate something very strange happens as objects approach [3 x 10^8 m/sec], which is known to be the speed of light in a vacuum. In fact they demonstrate all four key relativistic concepts: hard speed limit, mass increase, time dilation and length contraction. The 'relativistic correction factor' is the same for all three parameters: [1/sqrt{1 - (v/c)^2}], which is 1.41 at 70.7% speed of light.

Hard speed limit
        Electron velocity is seen to flatten and limit at [3 x 10^8 m/sec], the speed of light ('c'), even though the acceleration voltage though which they travel is raised higher and higher. This should pump more energy into the electrons, and thermal measurements show it does, but still the speed barely increases. Very weird, but it's an experimental fact, this is how mother nature is!

Increase in mass
       Measurement of the heat generated by crashing high speed electrons into a barrier confirms that even though the speed limits at the speed of light, the energy carried by moving electrons does not limit and can (apparently) be increased without limit. So if velocity near 'c' can barely increase, how can the kinetic energy, which is classically [KE = (1/2) m v^2], increase? The special relativity explanation is that the as 'v' begins to flatten out 'm' begins to increase, multiplied by the relativity factor. (For v = 99.92%c mass increase is x25).

Muon penetration through atmosphere
        (In round numbers and simplified physics) measured lifetime of a muons is 2 usec. Muons are continually created in the upper atmosphere traveling at speed near 'c'.  In classical physics muon's traveling near 'c' would be expected to penetrate only about 2,000 feet (@ c = 1 nsec/foot or 1,000 ft/usec) into the atmosphere, but instead a significant fraction of muons penetrate all the way to the ground (approx 50,000 ft or 10 miles).

Time dilation
        From the point of view of earthmen the logical explanation for this is that moving muons appear to have a stretched out decay time, and in fact an adjustment to muon time is the relativistic explanation. The period of the clock on a moving muon as measured from earth is seen to be increased by the relativity factor (x25 idealized). Time on a moving object is seen to be dilated.

Length contraction
        Muon men agree that the muons are hitting the ground, but they don't agree with the earthmen as to why. They do not agree that the muon clock is running slow. When they measure the muon decay time, it is 2 usec like always. So what relativistically has changed for them? It's the thickness of the earth's atmosphere. From their perspective the moving earth with its atmosphere is rushing at them with a speed near 'c', and the muon men see it squashed, here by a factor of x25 (from 50,000 ft to 2,000 ft), so of course they can make it through a 2,000 ft thick atmosphere and hit the ground before decaying. The length of a fast moving object (along its direction of motion) is seen to be contracted.

Visualizing time dilation
        Below I sketch how a muon light clock would look to muon men (left) and earthmen (center). A light clock is a photon bouncing back and forth between two mirrors. In special relativity everybody measures the speed of a photon as 'c'. Muon men just see the photon bouncing back and forth. To keep the numbers simple let's say the (muon) mirrors are 1,000 ft apart, then a muon man counts two tick (2 usec) before the muon decays. The earthmen can clearly see the clock make two ticks too, yet from the shape of the photon path it is clear to them that the clock is running slow compared to what muon men will see.

Muon men (left) see two (1 usec) ticks before decay.
Earthmen (center) same as muon men also see two ticks of the light clock,
but due to mostly downward travel of photon (@ c) from their perspective the two ticks take 50 usec
allowing the muon to travel 50,000 feet down through the atmosphere (@ 99.92% c).

        In other words the earth men's explanation for the muon experiment results is that they 'see' (experience) time on the moving muons as slowed (time dilation), while the muon men's explanation is that they see the thickness of the atmosphere rushing at them as compressed (length contraction).

Speed of light --- nature's speed limit
        Acceleration tests on electrons done over 50 years ago showed that as the accelerating voltage increased the speed initially increased as the square root of energy, which is consistent with Newtonian mechanics where [kinetic energy = 1/2  m vel^2].  However, a strange thing happened as the speed neared 3 x 10^8 m/sec. The speed increases got smaller and smaller, until finally no amount of energy could get them to go any faster. The speed at which the electrons maxed out in the lab (3 x 10^8 m/sec) is equal to the speed of light in a vacuum. Bizare indeed.

Does mass increase with speed?
        As the voltage of the machine increased, the heat measurement of the electrons hitting the target showed that the electrons were indeed carrying all the energy applied. How is this possible if the speed is barely increasing? Well, there is another term in the kinetic energy equation (E = 1/2 m vel^2) and that's mass. The almost inescapable conclusion is that at speeds near the speed of light the electron mass must be increasing, and it must be increasing in proportion to the increases in electron energy. As energy goes into the electrons (Energy ==> mass), and when the electrons hit the barrier (mass ==> energy) in the form of heat.

        What's the formula for how mass increases with speed?  Well, the relativistic 'fudge factor' for both time dilation and length contraction is 1/sqrt{1 - (v/c)^2}. Let's try it with mass and see how it works out. 1/sqrt{1 - (v/c)^2} goes from one at zero speed to larger and larger values as speed approaches the speed of light. We want mass to increase with speed, so we multiply the (rest) mass by the relativity factor. In the equation below the rest mass (normal, low speed mass) is represented by the symbol 'mo'.

    guess                                       m = mo  x 1/sqrt{1 - (v/c)^2} = mo x {1 - (v/c)^2}^-1/2

Since (v/c)^2 is always less than 1, we can expand this formula into a power series and as long as speeds are moderately below the speed of light only the first two terms are significant.

    power series             (1 - a)^-1/2 = 1 + 1/2 a + higher order terms
     set                                               a = (v/c)^2
     then                                           m  = mo  + 1/2 mo (v/c)^2
    multiply by c^2                     m c^2  = mo c^2  + 1/2 mo v^2

          This looks encouraging. If m c^2 is interpreted as relativistic energy, then at low speeds we find it is equal to 1/2 mo v^2, which is just Newtonian kinetic energy, plus a constant term (mo c^2).  In classical mechanics when we lift or drop or apply force to an object we only are concerned with changes in energy, so if mass did have constant energy associated with it it would never be noticed!  The constant term (moc^2) is very large since it is mass multiplied by the speed of light squared. How can we interpret this term? Well, it's called the rest mass energy and it's famous.

                       1/2 mo  v^2  =>   classical kinetic energy
                        mo c^2        =>   rest mass energy

          I think we made a good guess for the formula of mass vs speed. At low speeds it is compatible with classical mechanics, and at high speeds it agrees with experiment. Here's data showing how mass of electrons increases as their speed nears the speed of light. The smooth line is supposed to be m/mo = 1/sqrt{1 - (v/c)^2}.  Let's check it at v/c=0.9 to be sure. With v/c = 0.9  m/mo = 1/sqrt{1 - 0.81} = 1/0.436 = 2.29, which agrees with the plotted smooth curve.

Inertial mass of electrons vs speed. Special Relativity by A.P. French, MIT Introductory Physics Series, 1968

Mass as c is approached (update)
        So how does mass increase up near the speed of light? We take the formula above and plug in v = (c - delta), where delta is a small fraction of c, for example for v = .9c delta = .1c and for v = .99c delta = .01c.

                            (v/c)^2 = ((c - delta)/c)^2 = (1 - delta/c)^2 = 1 - 2 delta/c + (delta^2 term than we can neglect)
                   [1 -  (v/c)^2] = 1 - (1 - 2 delta/c) = 2 delta/c
                                    m = mo  x 1/sqrt{1 - (v/c)^2}
                                    m = mo  x 1/sqrt{2 delta/c} = mo  x  sqrt{c/2 delta}
                                         For delta = .1c    m = sqrt{5}
                                         For delta = .01c  m = sqrt{50}

So the mass goes up as the 1/2 square root of the delta of the speed below c.

E=m c^2
        Notice the constant rest mass energy term (mo c^2) that has popped out of our (assumed) relativistic energy equation is the famous Einstein equation E=m c^2. It is a prediction of relativity that  mass <=> energy, more specifically that:

         Mass (scaled by c^2) is (somehow) equivalent to energy

        It took a while for all the implications of this equivalence between mass and energy to be understood, but it turned out to be a key to understanding atomic fission, fusion, and radioactive decay. Here we have one of the most famous equations in physics (E=m c^2), and by just extrapolating from experiments with accelerated electrons we found it. Pretty damn good.

Time dilation and length contraction
        When the equations are worked out for how you 'see' a moving body (moving at a fixed speed relative to you) you see it affected by two strange 'distortions' that are generally called time dilation and length contraction. These are not illusions, but real effects demonstrable by experiment. These effects result because the only information you get from a moving body (in the form of light) travels at a fixed speed, speed of light. A spaceman sitting on a moving body also 'sees'  us affected by time dilation and length contraction. Einstein explained all this by showing how the speed of light limit means that we and a moving spaceman will not agree on whether events we both see are simultaneous. In other words we don't have the same time!

Time dilation
        Probably the easiest way to calculate, and visualize, time dilation is with a light clock.. A light clock is an idealize, simple clock that is just a single light photon bouncing back and forth continuously between  two parallel mirrors.

        Suppose the light clock is on a spacecraft that goes zipping by the earth at speed v. (Technically v is the relative speed between the frame of the earth and the frame of the spacecraft). The light clock is oriented so the photon is bouncing up & down perpendicular to the direction of motion of the spacecraft. Earthmen measure the same distance between mirrors as the spacemen because the (Lorenz) contraction of the spacecraft is only along the axis of motion. Whereas the spacement see the light photon just bouncing up and down, the earthmen see the photon moving diagonally, the combination of the up/down motion and the forward motion of the spacecraft.

        The notes below calculate the photon time to travel mirror to mirror time as calculated by the spaceman and the earthman. For the spacemen the calculation is extremely simple:

           Mirror to mirror time = (distance between mirrors)/(speed of light)
              (spacemen see)         = d/c

        For the earthmen the photon diagonal distance between mirrors is the hypotenuse of a right triangle. Since the photon always moves (as measured by everyone) at the speed of light (c), the distance it moves along the diagonal in time delta t is {c x delta t}. The base of the triangle is the distance covered by the spacecraft while the photon is traveling between mirror {v x delta t}. The height of the triangle is the distance between the mirrors (d). The earthmen calculate the time it takes the photon to diagonally travel between mirrors {delta t} by using Pythagorean theorem, known since Greek times, which is the square of the hypotenuse is equal to the sum of the squares of the other two sides. The result (see the notes below for the details) is

       Mirror to mirror time = (d/c)  x  (1/sq root [1 - (v/c)^2])
           (earthmen see)
                                              =  Mirror to mirror time  x  (1/sq root [1 - (v/c)^2])
                                                        (spacement see)

        As the spacecraft speed gets close to the speed of light, the term (1/sq root {1 - (v/c)^2}) begins to grow larger than 1. At 90% of the speed of light it is 2.29 and at 99% the speed of light 7.08. The earthmen measure the period of the fast moving clock to be longer, i.e. the clock running slow, compared to the spacemen traveling with the clock. This phenomena is known as time dilation.

Light clock sketch

Length contraction
        Sitting on a high speed muon coming through the atmosphere (see next section) your explanation of why so many muons survive to hit the ground is that the atmosphere (& the whole earth)  looks flatted out like a pancake. It is contracted in the direction of motion. So what's the formula for contraction?  We know it already. Consider, if earthmen see the muon clock running slow by the factor n, then the spacemen must see the atmosphere thinned by exactly the same factor n. The factor for length contraction (in the direction of motion) has to be just the inverse of factor for time dilation. Inverse because the increased period of the high speed muon clock translates to a decreased length of the atmosphere as seen by muon.

                    Length            =       Length       x  sq root [1 - (v/c)^2]
              (spacemen see)      (earthmen see)

How to remember time dilation and length contraction
        The math of Einstein's special theory of relativity is very simple, a little trigonometry and simple algebra. The concept of simultaneous events is tricky conceptually. But hardest (for me) to keep straight always has always been: who is seeing what time dilate, who is seeing what length contract, etc.

Trick --- Remember excess muons from cosmic rays survive to hit the ground. From earth's viewpoint the reason is that the muons are decaying more slowly than normal (time dilation), but from a muon spaceman's point of view the reason is that the atmosphere looks thin  (length contraction).

        Muons are created high up in the atmosphere (10 miles) by cosmic rays (high speed protons) hitting air molecules, and then travel downward through the atmosphere at near the speed of light . Muons have a short lifetime of about 2 usec (half life) as measured in the lab. At the speed of light it takes about 1 usec to travel 1,000 ft (300 m). In 1941 an experiment was conducted where atmospheric muons were counted at the top of a 6,000 ft mountain (Mt Washington) and at sea level. In the 6 usec it takes the muons to travel 6,000 feet wouldn't you expect that only (1/2)^3 =  1/8 of them would survive?  Well, that's not what the data shows. The 1941 experiment found the sea level count was 72% of the mountain top count. Somehow most of the muons survive the 6,000 foot, 6 usec trip, almost six times (72%/12.5%) more than would naively be expected.
        Our explanation from the ground is that we expect to see things happening slowly on a fast moving particle. because we see their clocks running slow. We say their time is dilated.  Therefore we expect it will take longer than normal (i.e. what we measure in the lab) for a high speed muon to decay. That's our explanation as to why so many muons hit the ground.

        Let's consider the situation from the muon's point of view. Imagine you're sitting on the muon. What do you see?

        On the muon you can consider yourself not moving and that the earth is rushing up at you at nearly the speed of light. When you measure the muons' half life (in your muon lab), it looks normal to you at 2 usec. Yet you too see you have a good chance of hitting the ground.

        Sitting on the muon your explanation for hitting the ground is that the earth looks flattened into a pancake. Side to side normal, but front to back contracted, i.e. you see it contracted (only) along the direction in which you see it moving.  Looking from the muon the thickness of the atmosphere (from creation of the muon to hitting the ground) it's much less than 10 miles, Mt Washington looks to guys on muons to be less than 1,000 ft high. So from the muon point of view, the reason a lot of muons survive to hit the ground with 2 usec half lives is that the atmosphere looks thin, it's length has been contracted.

        Let's apply what we've learned to a space ship that goes zooming by the earth. From the earth we see the people moving around on the space ship slowly and the space ship is shortened front to back.

Here's two references on excess muons hitting the ground.


Time dilation and length contraction -- in particle physics (update 4/11)
        It turns out that special relativity applies to subatomic particle collisions too. The memory picture here is like the earth/muon picture except the high speed muon is replaced by a high speed electron, and the flattened earth/atmosphere is replaced by a flatten proton.

        I recently came across a striking picture from particle physics that allows visualization of simultaneous length contraction and time dilation. It is in a new technical biography about Feynman called, 'Quantum Man'. The SLAC particle accelerator (still operating) accelerates electrons to very high speed down a two mile straight path and collides them with protons. Feynman realized that length contraction and time dilation view of the proton, as seen by a high speed incoming electron, drastically simplified calculations of the interaction of the electron with (possible) proton components, then known as partons, soon to be called quarks.

        The incoming electron sees the proton flattened to a pancake, and if the proton has charged subcomponents (partons), their motion would be seen to be nearly frozen by time dilation, thus allowing the partons to be treated as independent. The proton pancake shape is because the electron sees length contraction along the direction of motion. Within the pancake shaped proton, in a plane perpendicular to the direction of electron motion, any subcomponents the proton has would be seen moving very slowly due to time dilation.

Proton as seen by high speed SLAC electron
        High speed (SLAC) electrons 'see' (and collide with) ---- pancake shaped protons that each contain three barely moving charged subcomponents (quarks).
        The 'scaling' rule Feynman derived for this collision was that the electron/proton interaction depended on the momentum of the proton's three constituent quarks and their electric charge squared (p298). This explained a regularity seen in the SLAC data and was one of the first indications that protons had an internal structure.

Time dilation and length contraction --- Is that all there is to it?
        Unfortunately, no. Just applying time dilation and length contraction sometimes gives the right answer, and sometimes it doesn't! This is what makes understanding special relativity so difficult. You think you understand, and then the next problem you try to solve, you can't.

        Here is one of the keys --- When it comes to time, what you see depends on two effects: time dilation and the doppler effect. If you are not moving (or can consider that your not moving), then you always 'know'  that any clock on a moving (at constant speed) body is running slow, in other words, time dilation always applies. The doppler effect, however, only applies sometimes, it applies when you are directly observing frequency or period of light from (or to?) a moving body. So if you observe the effect of a slowed clock indirectly, like how long (on average) does it take a muon to decay, then time dilation is (often) all that is needed. But if (by some means) you 'look at' or 'track' how a moving clock is beating, you need to apply both time dilation and doppler effect to solve the problem.

A simple way to figure what you see when a moving body is emitting light

                        a) Time dilation affects when, or in what rate, the light is emitted
                        b) Doppler affects how the emitted light is seen. (The light can be visualized in
                                    space traveling at c and the observer is traveling toward or away from
                                    the light at some fraction of c.)

        So if the emitted light is in the form of light pulses (or light frequency) controlled by the clock on the moving body, then you apply time dilation to slow the clock and figure what effect this has on when or how the light is emitted, then you use the doppler effect to see how the emitted light (its period or frequency) is affected by its velocity relative to the non-moving observer.

Doppler effect
         Does the speed of a light source have any effect on (or how your see) the emitted light?  The answer is yes. The speed of the emitted light is not affected (a key relativity concept), but you do see the frequency (or period) and energy of the emitted light to be affected by the speed of its source. This frequency/period/energy change in the light due to the relative velocity between the source of the light and the observer is known as the doppler effect.  The doppler effect is used by astronomers to measure (line of sight) velocity of astronomical objects relative to the earth. This is done by measuring how much the frequency of prominent spectral absorption lines are shifted relative to the same lines measured in the lab.

        Period is the inverse of frequency, so if a moving light source is putting out light in the form of pulses, then the doppler effect causes the measured time between pulses to change, decreasing when the source is moving toward the observer, and increasing when the source is moving away. Note doppler is a separate effect from time dilation and is a stronger effect than time dilation. Doppler is a first order effect {(1+/-v/c) or 1/(1 +/- v/c)} whereas time dilation is a second order effect {sqrt(1 - (v/c)^2)}.

        In summary, the timing of light pulses controlled by a clock on a moving source are seen by (inertial) observers to have two sources of variation: relativistic time dilation and doppler effect. To find the timing seen by the (inertial) observers multiply the the local moving clock timing by both the time dilation multiplier and doppler effect multiplier.

                1) relativistic time dilation  multiplier --- always slows a moving clock, increasing
                            the time between clock beats by the factor (1/sqrt{1 - (v/c)^2}.
                2) doppler effect multiplier ---  changes the time between clock beats by the factor
                            (1 +/- v /c), so depending on the sign of v the time between clock beats can
                            increase or decrease.

       Note the doppler effect is very asymmetrical. For high speed particles, as v approaches c, (1 + v/c) approaches (1 + 1) = 2, but (1-v/c) approaches (1-1) = 0. So if a fast moving body is moving away from the observer, time between its emitted light pules can (at most) be seen to double. On the other hand when a fast moving body is moving inward toward the observer, time between its emitted light pulses can be seen to shrink nearly to zero, so that the emitted light pulses arrive very close together.

Introduction to the twin paradox problem
        I have read about the twin paradox special relativity problem in two books and researched it online. This problem in relativity is over a hundred years old. It was referred to by Einstein in his original 1905 paper, and the formulation involving twins and rockets ships is due to Langevin from 1911. Here is the full version of the problem:

        One twin gets on a rocket ship and travels to a nearby star at near the speed of light and immediately returns to earth at the same high speed, while the other twin remains on earth. Both twins are able to track the clock (time) of the other because they each flash a light based on the beats of their local clock. When the traveling twin returns home and the twins are reunited, the traveling twin is found to be (much) younger than the earth twin, as both twins expected. Explain.
        Most explanations of this problem that you see are either incomplete, saying the count of light pulses (somehow) work out, or are based on time dilation (tricky), or use 'world line' analysis. World line analysis is a powerful and general tool for studying special relativity problems, but if you aren't familiar with it (and I am not), it's not very explanatory.

        Most newcomer approach this problem by applying time dilation (moving clocks go slower), and then realize that each twin thinks the other is moving and hence aging more slowly, hence the paradox. In fact most solutions to the problem agree that on the outbound leg this is exactly true, the paradox is alive and well. This is a pretty tricky problem in special relativity. A good way to approach this problem is to diagram out how the spaceship moves and light flashes travel in space. From earth perspective this is not too difficult, it's harder to do from the spaceship point of view.

My explanation of the twin paradox problem
        Here's my analysis of the twin paradox problem. I came up with this analyis on my own, and despite a lot of research into the problem I have yet to see anything like it. My analysis is based on length contraction, not time dilation. I think this is a much more intuitive and easy to understand approach than the usual one that explains the paradox in terms of time dilation.

         The perception of paradox, referred to as the twin paradox (sometimes called the 'clock paradox') is caused by the error of assuming that relativity implies that only relative motion between objects should be considered in determining clock rates. The result of this error is the prediction that upon return to Earth, each twin sees the other as younger -- which is clearly impossible. (Wikipedia)

One twin travels in a spaceship to a star 32 light-years away (as seen by earth) at speed is 0.995c and returns immediately at the same speed. The other twin stays on earth. They flash light beams at each other once a second (local time). The Lorentz dilation/length contraction factor for 0.995c is 1/sqrt (1 -.995^2) = 10. When the traveling twin returns to earth he has aged 6.4 years, whereas the earth twin has aged 64 years.
        Overview --- Earth starts firing light flashes into space once a second when the spaceship leaves. Relatively few (only 1 in 200) of the light flashes reach the ship on the outbound leg since the ship is traveling almost as fast as the light flashes. On the inbound leg the light flashes are all lined up in a row (in space) as the ship heads into them. The key to my explanation is to visualize how the light pulses exist in space from earth's point of view, then to employ length contraction to show that the arrangement in space of the light flashes looks the same to the twin in the ship except squeezed together (length contracted) by a factor of 10. Finally the doppler effect (as seen from the spaceship) must be factored in.

        Inbound leg --- When the ship turns around and starts back to earth most (99.5%) of the light flashes have still not having reached the ship and are lined up in a row in space separated in distance by 1 light second (as seen by earth). However, the twin in the spaceship sees the world outside the spaceship as length contracted by a factor of 10, so he sees the light flashes are only separated in distance by 0.1 light second. Furthermore, since he is traveling toward each light flash at almost the same speed as the light flash is traveling toward him, the ship and the light meet (about) halfway at a distance of 0.05 light second (as seen on the spaceship). This is the doppler effect. The result is that on the inbound leg the spaceship runs into another of the row of light flashes each 1/20 of a second. So in the 3.2 years (as seen from the spaceship) it takes the space ship to leave the star and return to earth, the ship counts (about) 20 x 3.2 yr = 64 years worth of earth light pulses.

        Outbound leg --- The spaceship sees the light flashes behind the ship as separated in distance by 0.1 light second, which is 1/10 of the distance separation as seen from earth. Since the spaceship is only slower than the light by a factor of 1/200, the time between light flashes reaching the space ship is stretched out 200 times from 0.1 second to 20 seconds (as seen on the spaceship). Therefore the ship counts (about) 1/20 x 3.2 yr = 2 months worth of earth light flashes on the outbound leg.

Travel to a star 32 light years away
        A excellent relativity example is a spaceship traveling from earth to a star and back at a constant speed with the spaceship flashing a beacon at a regular rate. By thinking about the trip from the perspective of earthmen and spacemen the time dilation/length contraction duality is illustrated. How the earthmen see the flashing beacon is affected by time dilation and the doppler effect. The sketch (below) shows how the doppler effect both stretches (outleg) and contracts (in leg) the flash timing as seen by earthmen, allowing the doppler effect to be disentangled from time dilation.

        By having one twin ride on the spaceship while the other twin remains on earth, it sets the stage to explain the classic relativity twin paradox.  At the end of the trip one twin has aged less than the other. Why, and which one is younger? The twin paradox is classic case where time dilation and length contraction fail (or seem to fail).

        The specifics are these --- One twin gets on a spaceship and travels to a star 32 light years away at (about) 86.6% speed of light, then immediately turns around and comes back to earth at the same speed. A speed of 0.866 c gives us a nice round number for relativistic time dilation & length contraction, since sqrt{1-(v/c)^2} = 0.5. We will assume the time it takes the spaceship to accelerate and decelerate is negligible, because there is no theoretical limit on acceleration (unlike velocity!), all you need do is pour on the power. The spaceship's position vs time (as seen from earth's perspective) is a triangle, as shown in the sketch below.

        Since the spaceship is going reasonably close to the speed of light (0.866 c), from the earthmen's perspective the total travel time to a star at 32 light years and back is a just a little more than 64 light years, or 73.9 years (= 2 x 32 years/0.866). To the spacemen zipping along relative to earth at 0.866 c, the distance to the star looks to them to be contracted by a factor of two (= 1/sqrt{1-.866^2}) to 16 light years. So the spacemen see the total travel time to a star at 16 light years and back is a just a little more than 32 light years, or 36.95 years ( = 2 x 16 years/0.866), and of course, 36.95 years is how much the spacemen expect to  age during the journey.

        Before the trip starts, the twins agree that the both the spaceship and earth will both fire a brief light flash every five years (local time). The earthmen 'know' (due to time dilation) that the spacemen are aging at only half the rate of earthmen, so from earth's perspective the spaceship will flash only every ten years, and that the doppler effect, since the spaceship is at different distances from earth at each flash, will strongly affect how the flashes are seen on earth.  The sketch (below) shows the path of all the light flashes (from earth's perspective). The pencil diagonals (w/down arrows) are the light flashes fired from the spaceship and the red diagonals (w/up arrows) are the light flashes fired from earth.

Summarizing, the parameters:

Total trip (out and back) Earthmen see Spacemen see
distance 64 light years 32 light years
speed 0.866 c 0.866 c
Time dilation/Length contraction
sqrt{1-(v/c)^2} = sqrt{1-.866^2}
0.5 0.5
travel time 73.9 years 36.95 years
Doppler effect (for ship flashes)
Out leg
In leg

(1-.866c/c) =0.134
Doppler effect (for earth flashes)
Out leg
In leg

1/(1-.866c/c) =7.46
speed of spaceship
relative to earth
64 lyr/73.9 yr
= .866 c
32 lyr/36.95 yr
 = .866 c
spaceship flash rate
(@ 5 yr local time)
(not including doppler effect)
5 years x (1/0.5) =
10 years
5 years
spaceship flash rate
(@ 5 yr local time)
(including doppler effect)
Out leg
In leg
10 years x 1.866 = 18.66
10 years x 0.134 = 1.34
 5 years
# of spaceship flashes (sent/received) 7 7

        An important point is this --- earthmen and spacemen always agree on what the relative velocity is between them. They disagree on the travel time and the travel distance, but they agree that the relative velocity between the spaceship and the earth is either +0.866 c or -0.866 c. The reason, of course, is that for the spaceman both distance and time are shortened by the same (Lorentz) factor leaving velocity unchanged.

Check on doppler effect (for earth flashes)
        On the out leg the it takes a long time for the light flashes sent from earth to reach the ship because the ship is racing away from earth at nearly the speed of light. The out leg doppler effect (from earth perspecive) is easily calculated: equate the distance of the ship from earth and the distance of the first light flast (sent 5 years after the ship).

                                   Distance of ship from earth = 0.866c x earth time
                                   Distance of first light flash from earth = c x (earth time - 5 yr)

equate and solve for earth time
                                  0.866c x earth time = c x (earth time - 5 yr)
                                  earth time x (1 - 0.866) = 5 yr
                                  earth time = 1/(1 - 0.866) x 5 yr
                                  Doppler effect (out leg) = 1/(1 - 0.866)
                                                                            = 7.46

        The in leg doppler calulation, where the ship is heading into the flashes from earth, is sum the ship travel distance and flash travel distance to equal to the 5 light years distance between earth light flashes.
                                flash travel distance + ship travel distance = 5 light years
                               c x earth time +0.866c x earth time = 5 yr x c
                                (1 + 0.866) earth time = 5 yr
                                earth time = 1/(1+0.866) x 5 yr
                                  Doppler effect (in leg) = 1/(1 + 0.866)
                                                                          = 0.536

View of clock based light flashes from earth's perspective
        Below is a sketch I did showing how clock based light flashes (spaced 5 years local time) sent from the space ship and sent from earth appear from earth's perspective.

        On the outbound leg doppler causes the time between pulses to be stretched out and on the inbound leg squeezed together, but notice the asymmetry. The spaceship sees the earth flash rate (red) increase when the ship reverses direction at the middle of its journey (at the star) and stay high all the way home, but earth sees the spaceship flash rate (pencil) increase only near the end of the journey. The spaceship counts 14 earth flashes, so the space twin knows 70 years have passed on earth, whereas earth counts only 7 flashes, so the earth twin knows only 35 years have passed on the spaceship.

trip to a star --- view from earth of spaceship's uniformly spaced light flashes

        From the spacemen's perspective they are sending a flash out every five years, for a total number of flashes of seven on what they see as a 36.95 year (total) journey. The earthmen 'know' (due to time dilation) that the spaceship clock is running only only half as fast as the earth clock, so five years to a spaceman is ten years to an earthman. Hence earthmen 'know' that (as measured by earth time) the spaceship will flash every ten years for a total of seven flashes in a 73.9 year (total) journey.

        However, the spaceship is at a different distance from earth each time it flashes, hence the travel time of each light flash to earth is different. As can be seen in the sketch above, the three flashes sent on the 'out leg' arrive on earth separated by 18.66 years, whereas the four flashes sent on the 'in leg' arrive on earth separated by only 1.34 years. On the 'out leg'  the spaceship and light are going in the opposite directions, so the doppler multiplier is 1.866 = (1 + .866), stretching 10 years to 18.66 years. On the 'in leg'  the spaceship and light are going in the same direction (spaceship traveling only a little slower than the light), so the doppler multiplier is 0.134 = (1 - .866), shrinking 10 years to 1.34 years.

Twin paradox
         When the twins meet again on earth after the journey, how have they aged?  The official 'right' answer is that the twin who stayed on earth has aged 73.9 years, while the twin who traveled on the spaceship aged 1/2 this, or 36.95 years.

        Is this not just a simple case of time dilation and length contraction?  It seems so. Does not the earth bound twin see the clocks on the fast moving spaceship running slow, so of course he is not surprised that he finds his spaceship twin has aged less? Also the time dilation equation has only v^2 in it, so the earth twin see the spaceship's clocks running slow on both the outbound and inbound legs of the journey, shouldn't he? Correspondingly the spaceship twin see the universe whipping past, so for him the distance to the star looks contracted. He sees the distance to the star as 16 light years, not 32 light years. Hence, he is not surprised that during the journey he has aged only aged a little more than 32 years (36.95 years), because for him this is how long the trip seemed to take.

Where is the paradox?
       So where's the paradox? Well, is not the spaceship twin perfectly free to consider that his spaceship did not move and the earth was moving? He would see the earth zipping away and back. Should not the spaceman then see earth bound clocks running slow and see his earth bound twin aging more slowly than him? It seems so. Whoa, there's a contradiction here! Both twins seem able to argue the other twin was the one who moved, and hence the other twin aged more slowly! No wonder it's called the twin paradox.

        Many books explain (?) the twin paradox by saying the situation of the twins is not the same. They say the spaceship twin decelerated to zero and then back up to speed. So what? Doesn't the spaceship twin see the earth do the same thing? And all the lorentz equations have only the speed squared in them. What difference does a speed reversal make?  Well, it's more than the speed reversal ...

        Above we asked, "Is not the spaceship twin perfectly free to consider that his spaceship did not move?" We implied the answer was yes, but if you think about it, the answer is no. Four times during the journey the space twin has sensed (& can measure) that his spaceship has accelerated or decelerated, so the space twin knows he has moved. This is a key issue in understanding the nature of time and space (see time and sapce below).
        Is there is still a puzzle here?  Maybe. This is not a case of uniform motion, yet time dilation and length contraction can be applied to get the right answer. What I want to know is when can you apply time dilation/length contraction, and when can't you? I have ordered a book by David Merimen, who is reputed to be the best at explaining Special Relativity. We'll see what he has to say. I know one way the twin paradox is explained is to detailed it out. This can be done by having the space ship fire a light flash at a regular rate based on the spaceship clock, and then figuring out how those light flashes are seen on earth. Another step up in complexity used to solve special relativity problems is to graph 'world lines', which are space and time diagrams.
Update --- Is there is still a puzzle here? Well, not really. Relativistic time dilation and length contraction multipliers always seem to apply. Sometimes they alone will give the right answers, for example to travel times and distances. But when an observer in one frame tries to directly observe the clock or actions? (via light of course) in a moving frame, the doppler effect comes into play and the doppler multiplier must also be used.
How is a moving flashing beacon seen?
        The spaceship trip to a star 32 light years away at 0.866 c and return can be understood much better if we add light beacons, on earth and on the spaceship, that flash at a regular rate controlled by the local clocks. In this way both spacemen and earthmen can observe the other guys clock. Let's assume that beacons flash when the local clocks click off five years, so earth is going to flash 14 times during the journey and the spacemen, who see the journey as only half as long, are going to flash their beacon 7 times.

Case 1 -- earthmen's view
        The earthmen 'know'  that the spacemen's clocks are running 2 times (= 1/sqrt{1-.866^2}) slower than earth clocks, so (from an earth perspective) the spacemen are only going to flash every 10 years. So do the earthmen expect to see a flash (arrive on earth) every 10 years? Answer is no! The reason, and here's where it gets interesting, is that earthmen know the flashing spaceship is moving, so the doppler effect must also be considered.

        The doppler effect, which is usually associated with red and blue frequency shifts in the absorption line in the spectrum of stars and galaxies, also affects the time between light flashes. From earth's perspective the spaceship will move in the 10 years between flashes emitted by the spaceship away from the earth by 8.66 light years (0.866 x 10 light years). Therefore the time as seen on earth between light flashes from the spaceship is the time between light flashes (emitted from the spaceship) plus the extra travel time to earth since the spaceship is further away. The general formula for time between pulses as seen on earth is

                         local clock period x  time dilation factor  x  (moving away) doppler effect
                         5 year  x  (1/sqrt{1 - (v/c)^2})   x   [1 + v/c]
                                                          v = 0.866 c

                         5 year x 2 x [1 + 0.866] = 18.66 years

Note on the outward leg of the journey the emitted time between pulses (10 years) is seen on earth to be nearly doubled (18.66 years) because the doppler effect, which is the moving away of the spaceship between pulses.

        On the outward leg the spacemen send 3 flashes during what they see as a 18.475 year trip to the star. The combined effect of time dilation (affecting the emitting rate) and doppler (affecting how the time between pulses is seen) causes the time elapsed on earth to receive these three pulses to be almost 56 years! On the return leg the doppler effect works to reduce the time between pulses. And when the spaceship is traveling fairly close to the speed of light and in the same direction of the light it emits, it nearly nearly keeps up with its light flashes. The formula has the same time dilation term, but the sign of v in the doppler term is reversed (v = -0.866 c).

                         5 year  x  2  x [1 + (-0.866c/c)] = 10 years (1 -  0.866)
                                                                            = 10 years (0.134)
                                                                            = 1.34 years

        Notice the huge effect the sign reversal makes. Whereas it took almost 56 years on earth to receive the three pulses sent on the outbound leg, the four pulses sent on the inbound leg arrive squished together, all four pulses arriving in (about) four years just slightly ahead of the spaceship arriving on earth. Notice also that on the inbound leg to the earthmen, who are looking at the spaceship clock via the flashes, if they forget to take the doppler effect into account, will incorrectly assume that the spaceship clock does not 'look' slowed down, i.e. that the spaceship time is not dilated.

        The dopper effect is a first order effect that is sign dependent, going as [1 +/-( v/c)], whereas the relativistic time dilation effect is second order effect that is not sign dependent going as 1/sqrt{1-(v/c)^2}.  The general rule for an observed clock is

        When the source is coming toward the the observer, the (apparent) speeding up of the moving clock due to the dopper effect (seen as a crowding of pulses) is always stronger than the (real) slowing down of the clock due to relativistic time dilation (seen as a spreading out of pulses).
How is it spacemen measure c for the speed of light?
        An interesting question not often addressed (and which can be very confusing) is this --- Can earthmen explain how it is that when spacemen measure the speed of light (in their moving spaceship lab) they get the same value as earthmen. And, of course, they better get the same value because this is a key principle of relativity?

        Let's extend our spaceship example --- Before the journey started the spacemen and earthmen set up similar light speed measuring apartus in their labs and get the same results. Using a ruler they measured out 100 ft  and with a (local) clock that has 1 count/nsec they time how long it takes a light pulse to cross the measured distance. In the spaceship the measured course is laid out so tha the light travels in the same direction that the spaceship is traveling. On earth before the journey started both labs find it takes 100 counts of the clock for light to travel the 100 ft course, which is a speed of 1nsec/foot (eq to 3 x 10^8 m/sec).

        During the journey when the spaceship is traveling (with respect to earth) at 0.866 c the spacemen repeat the experiment and get the same result: 100 counts. Of course, to the spacemen this is not surprising, since to them their spacelab looks exactly like it did on earth. When they check the measured distance with their ruler it still measures 100 ft, and as far as they can tell their (local) clock is running just like it did on earth.

        How do the earthmen 'explain' why the spacemen in their spacelab, moving at 0.866 c with respect to earth, still measure the same speed of light (100 counts). From the earthmen's point of view the moving spacelab is not the same as it was on earth. One, due to lenght contraction, the spaceship has squashed front to back, so the measured course is really only 50 feet, not 100 feet as it was originally. Two, due to time dilation, the spaceship clock has slowed down (relative to its speed on earth) such that it is running only half as fast. In the time it takes the spaceship clock to count 100 beats the earth clock makes 200 beats.

        Since velocity is delta distance divided by delta time,  earthmen observing (in their mind's eye) the experiment on the moving spaceship say the distance is down by two and the time is up by two, so the speed is different by four!! Yikes, I always get this result. Where is the error here??

Einstein in his 20's
        In 1902 Henri Poincare (who is known to all math students) wrote a book listing three big unsolved problems in physics: photoelectric effect, brownian motion, and luminiferous ether. Einstein read the book in 1904 (at age 24) and within one year solved all three problems. Einstein, who had failed to get his doctorate, solved the problems working alone while employed full time in an (engineering) job and supporting a wife and child.. At age 25 he had four papers published in a leading German physics journal (in March, April, May and June of 1905). These four papers, each now a classic, solved all three problems laid out by Poincare.  (Ref:  Einstein 1905: The Standard of Greatness, by John S. Rigden)

        1) The brownian motion problem was that small pollen grains suspended in water could be seen under a microscope to constantly zig zag around. No one had any explanation for this motion. At the time little to nothing was known about atoms, how big were they, or even if they existed.

         Einstein's brownian motion paper showed how atoms were responsible for the jiggly motion and showed how this motion could be used to calculate the (approximate) size of atoms. He won a Noble prize for his work on brownian motion.

        2) The photoelectric problem was that high frequency ultraviolet light was known to eject electrons from a metal plate, but lower frequency (visible) light did not do this regardless of how bright the source was. This could not be explained by then current wave theory of light.

        Einstein's photoelectric paper showed that the experimental result could be explained if light was quantified (now called a photon) and that the energy in each quantum of light was proportional to the frequency of the light.  This later became one of the core concepts in the development (by others) of quantum physics.

        3) At the time all known waves were vibrations of some substance (sound is a vibration of air or water). When evidence piled up that light was a wave (it refracted and reflected), it was generally accepted that there must exist an as yet undetected substance (known as 'the ether') that was everywhere (including in space) and that light was a vibration of the ether. The ether also nicely explained why the speed of light was independent of the speed of the source of the light. If it was assumed that the ether was fixed (unmoving) and that planets and stars moved through it, then of course, light moved at a speed relative to the fixed ether, not the source.

        One difficulty with the ether was no one could come up with a reasonable model for the ether (people mostly built mechanical models with springs and weights in those days). The properties of the ether were very hard to explain, because on the one hand the ether needed to be very stiff to support the very frequencies and high speed of light, yet on the other hand it had to be incredible thin and tenuous so that planets and binary stars could move easily through it. By the mid 1880's an even more serious problem with the ether arose. A series of experiments was done to measure the slight variations in the speed of light expected as the earth traveled through the ether around the sun (famous Michelson-Morley experiments), but they found the speed of light was always the same regardless of orientation of the apparatus or the season.

        Einstein's ether paper (completely) detailed his new theory of special relativity. It showed that if the speed of light was accepted as a constant (consistent with experiments), then the concept of time as absolute (proposed by Newton and almost universally accepted) must be wrong. It showed how time and length must vary when seen by moving observers, and the theory predicted that the variations in time and space were exactly what Lorentz had earlier proposed (ad hoc) to explain away the experiments that had failed to find a variation in the speed of light. The theory proposed an entirely new and revolutionary view of space and time in which the the speed of light measured by all (inertial) observers would be the same without need for an ether. In other words, there was no ether.

        In Sept of the same year (1905) Einstein published a three page supplement to the relativity paper that derived one of the most famous equations in physics (E= mc^2). This equation showed that mass carried a fixed and very large amount of energy and was (in some sense) equivalent to energy. (Ohanian in his book Einstein's Mistakes says Einstein proof is not general that it is restricted to low speeds, but it was proved in general by others a few years later.)

        To the non-scientist it was sometime stated as, mass is frozen energy. Ohanian calls mass congealed energy and contrasts it with ordinary energy that is free to move around and be transformed. He makes an analogy with water/ice. Ice in ice sheets on earth is congrealed water, whereas water in the oceans is free to move around.

Nature of time and space
            (from the book Life of the Cosmos by Less Smolin)

        Galileo and Newton both well understood that you cannot determine if you are at rest or moving at a constant speed, but you can determine, without reference to anything external, if you are rotating or accelerating.  Smolin says of this, "the distinction between uniform motion and accelerating motion is absolutely central to our understanding of motion"  and that this is a "fundamental discovery, and the deepest mystery, in the whole history of physical science" (p 230)

        Newton formulated his laws of motion by explicitly assuming that space and time were absolutely fixed. Many, like Leibniz, thought this was crazy, that you could only determine if you were moving by making reference to other parts of the universe, that you needed to construct equations of motion using relative motion.. But Newton, no dummy he, saw that absolute space and time, even if philosophically flawed, was a good starting point for formulating equations of motions because the equations were greatly simplified. Example, equations like (Force = mass x acceleration) couldn't be much simpler.

        In the 19th century the concepts of an observer & frame of reference were invented. This allowed Newton's equations of motion to be used without reference to absolute space and time. You could use Newton's equation and say you were moving with respect to a frame of reference. Frames of reference that were fixed or moving at constant speeds to each other became a special case and were called inertial frames. In these frames an observer is unable to determine if his frame is moving or not, so he is allowed to assume it is at rest and that other inertial frames are moving.

        A puzzling think about Maxwell's equations (formulated 1860's) was that they gave the speed of light without reference to any frame. Forty years later Einstein took this as a starting point and developed special relativity as an explanation of this puzzle.

        Einstein's special relativity applies only to inertial frames and is concerned with how observers in one inertial frame, which they can assume is at rest,  see motion in another inertial frame, which they assert is moving.

        Smolin continues -- "The relativity of motion is both a fact and a mystery." He says the key question to ask to penetrate the mystery is "What is the cause of the distinction between velocity (which you cannot feel) and acceleration (which you can feel)." (p 230) Ernest Mach had a (possible) answer.

Is Special Relativity hard to understand?
        There is no question (in my mind) that getting a really good understanding of special relativity is difficult. Consider the following:

History --- A. P. French in his book on relativity says that in in 1957-1959 the Twin Pradox was "raging controversy" in physics. And he adds that in a book of (paper) reprints on special relativity in 1963 by the American Institue of Physics 9 of the 17 papers reprinted were devoted to the twin paradox problem.  Note this is (about) 50 years after the twin paradox/clock problem was first mentioned by Einstein!!

       A classic example of the difficulty of special relativity is Herbert Dingle (1890 – 1978). He was an English astronomer,  President of the Royal Astronomical Society, and in 1922 authored a book on relativity. Yet in his later years he became convinced that special relativity was wrong and wrote a book about it. He was not nuts, he just (according to others) got tripped up on some subtlies of special relativity.  More details can be found searching Dingle and Wikipedia.

        * Simpler relativity problems generally involve just two object. Problems with three moving objects are a lot trickier.

        * There is the conceptual difficulty of keeping straight the difference between what you 'see' from what you 'know' when looking at a moving object. When you look at a moving clock, you are seeing a real effect (time dilation) through the distorting (lens) of the doppler effect. Some reference talk in terms of how the moving clock appears (including the doppler effect), while others separate out the doppler effect to talk about how the moving clock is (really) slowed, even if it may appear to be running fast. And to add to the confusion the doppler distortion equation comes in four flavors, depending on whose viewpoint is taken and whether relative motion is moving away or together: (1-v/c), (1+v/c), 1/(1-v/c), 1/(1+v/c).

        * Combo equation --- You think you have the doppler effect all figured out, and then you come upon a reference that uses the relativistic doppler effect. What the hell is that?  It turns out this is a combo equation. Instead of an analysis using (time dilation x doppler), relativistic doppler combines the time dilation equation and the Doppler equation into a new (combo) equation. For example:

                doppler  x      time dilation      =  relativistic doppler
            1/(1 - v/c)  x sqrt {1 - ( v/c)^2}  =  sqrt{(1 + v/c)/(1 - v/c)}
                   sqrt {1 - ( v/c)^2} = (1 -v/c)  x sqrt{(1 + v/c)/(1 - v/c)}
                                                   = sqrt{[(1 + v/c)/(1 - v/c)]  x (1 - v/c)^2}
                                                   = sqrt{[(1 + v/c) x (1 - v/c)}
                                                   = sqrt{1 +v/c -v/c -(v/c)^2}
                                                   = sqrt{1 - (v/c)^2}

        * There is a four dimension (three space + time) formalism, called world line analysis, that apparently allows many relativity problems to be graphed.  While I have not studied it, getting the right answer this way may (or may not) help you to better understand relativity.

Paradoxes ---    When you read the explanation of various paradoxes, you find the analysis given is almost always different in different references. If special relativity is so simple, why is that?

        Some of the more tricky and/or more difficult to solve problems in special relativity have become famous. Some of the famous problems or paradoxes go back  to the early days of relativity, like the Twin Paradox problem, but others are relatively new, like the Bell's spaceship paradox (1976). Wikipedia has a list of these problems and paradoxes.

        Furthermore some of the published, standard, explanations of paradoxes are very hard to swallow. For example, the analysis of the twin paradox in the recent book E=Einstein has the spacemen seeing the clock on earth as running slower than the spaceship clock on both the outbound and inbound legs. Yet the spaceship twin is going to arrive home younger. What?? The hard to swallow part is that during the brief interval at the star while the spaceship decelerates and accelerates again the spaceship twin sees virtually all the aging of the earth bound twin happening during this brief time! (See my own much more common sense explanation of the twin paradox based on length contraction).

        If we lived at a time when astronauts, where going to stars at near the speed of light, I think it would be different. It would then be common knowledge that the astronauts would sense the journey time and distance very differently from those on earth.

Why is the universe not rotating?
        Almost every body we see in the universe has angular momentum and rotates. The earth rotates on its axis as does the sun, every planet, stars and black holes. The earth, planets, and every object in the solar system rotates around the sun. The solar system is embedded in an arm of a spiral galaxy and is rotating around the center of the galaxy. There's even a mechanism whereby objects like neutrons stars and black holes can be 'spun up' when rotating gas falls on them.

        But... the universe as a whole with respect to us does not appear to be rotating. Weird... As Smolin points out there is nothing in Newtonian mechanics that says it can't, and Einstein's general relativity doesn't prevent it either.  In fact in 1949 Godel (famous mathematician) found a rotating, static, universe was a solution to Einstein's general fields equation, the outward force of rotation counteracting the inward force of gravity. Godel argued that in such a universe you would know it was the universe rotating and not you if you did not feel dizzy!   From a Newtonian perspective of absolute time and space it must just be coincidence that the universe is not rotating. (p 231)

        Mach in the 1880's argued that when you sense rotation, it must be because you rotating with respect to something and that something is the universe. Seems pretty mushy to me, and it's not proven, but as Smolin points out this view point does provide a (sort of) explanation as to why the universe we see around us is not rotating.

Aside --- How can the rotation of the universe be measured? Note you cannot use galaxy red and blue shift doppler effects. Doppler only measures the component of velocity on the line of sight between the object and earth. Rotation (relative to earth) would be at 90 degrees to our line of sight, and thus would cause no doppler effect.

        A goggle search shows two methods. One is based on using measured planet positions (with respect to the background stars) in a Kepler/Newton planet rotation model. This model has a lot of correction factors, one of which is the rotation of the background stars. Within experimental error this term is zero. (my comment --- Does this make any sense? All the background stars are in our own galaxy. This method would only makes sense if planet positions are measured with respect to the background galaxies, which I bet is impossible.) The second method is based on analysis of the background microwave background. The argument seems to be 'If the universe were rotating, the rotational axis would single out a preferred direction in space', and there is no evidence of this.

Nature of simultaneous
Simple version
        Light bulb flashes in the center of a moving train car. People in the train car know (& can measure) the speed of light is the same in every direction, so since the light bulb is in the center of the car, they see the light hit the front and back of the car simultaneously.

        The people on the platform see that as the light is traveling the car moves forward a little, so they see that the light has less distance to travel to reach the back of the car and more distance to hit the front of the car. It's an experimental fact that the speed of light is not affected by the speed of its source, so unlike say a ball thrown in the car, the speed of light from the bulb is not given any push forward by the speed of the bulb. Therefore the people on the platform see the two events (i.e. light hitting the car ends) as not simultaneously.

Fancy version --  (frin Brian Greene's book, The Elegant Universe)
        Two warring kings sit at opposite ends of a long table ready to sign a peace agreement, but neither wants to sign before the other. To solve the problem a light bulb, not lit, is placed in the exact center of the table. The kings agree that they will each sign the instant they see the light bulb come on. The know it takes a finite time for the light (at 1 nsec/ft) to travel from the bulb to the end of the table, but since they are each the same number of feet from the bulb, they know they will be signing simultaneously.

        I forgot to tell you the table and signing ceremony is in a train car moving at a constant speed. Does it make any difference to the kings? Answer, no. As Galileo argued nearly 400 years ago, there is no test you can do to determine if you are at rest or moving at a constant speed. Everything will look exactly the same in the car to the kings whether the train is at rest or moving (through the station) at a constant speed.

        But to the kings subjects looking on from the station there is a problem. They don't see the signing as simultaneous. They see the king nearer the back end of the train signing first, then later the king nearer the front end signing. Why is this?

        During the time the light is traveling from the light bulb toward both ends of the table the train is moving forward. The light does not get any special 'push' forward from the speed of the train. One of the principles of relativity is that the speed of light everyone sees (measures) is totally unaffected by the speed of the body emitting the light. The kings' subjects on the platform see the light heading toward the back of the train has less distance to travel, because while the light is traveling back the end of the table moves forward a little. Similarly the light heading toward the front of the train has more distance to travel, because while the light is traveling forward the end of the table moves forward a little. Since the light is traveling in both directions at c, it reaches the back end of the table first because the distance it has to travel is less.

        Moral of the story  --- An event (signing) seen as simultaneous to one group (those in the moving train) is seen as not simultaneous to another group (those outside the train). In general observers moving relative to each other at constant speed will not agree on whether an event is simultaneous. This is a fundamental result with great consequences about the nature of time and space.
Magnetic fields are 'caused' by relativity!
        You often see this claim  --- Relativistic effects are only significant at high speeds. Wrong, wrong!!  Whoever says this doesn't understand special relativity. In the same sense that the curvature of space (general relativity) explains what gravity is, the motion of charges (special relativity) explains what magnetism is.

        In electrical engineering electric and magnetic fields are treated as fundamental entities. Both types of these fields exert forces on charged particles and store energy in the fields.  Electric fields are created by charges and are measured in volts/distance. Magnetic fields are created by currents and are measured in amps/distance. In a capacitor energy is stored in the electric field. In an inductor energy is stored in the magnetic field. Maxwell's equations describe how electric and magnetic fields are related to charges and currents. Physicists take a different view. Physicists regard the magnetic field as secondary to the electric field. The magnetic field can be derived from the electric field with the magnetic field amplitude (always) coming out to be electric field amplitude divided by the speed of light squared.

Magnetic force vs electric force
            Two electrons spaced at distance r feel an electric force outward (pushing them apart).

                                Felec = 1/(4 pi e0) x q^2/r^2
                                                    e0 = permittivity of free space

        If the same two electrons at distance (r) apart are moving sideways to me with velocity (v), they look to me like little currents (current is just moving charge), so they make a magnetic field. I can calculate the magnetic field (B) from the Biot-Savart law which physicist use to calculate the magnetic fields from a small length (dl) of current (i) of length (dl).. Current is just moving charge, so i=q/dt and idl = qdl/dt = qv. .

                                B = (u0/4 pi) x (idl)/r^2 = (u0/4 pi) x (qv)/r^2
                                                    u0 = permeability of free space

 From the general (vector) force equation (F = q (E + v cross B) magnetic force in this case is qvB and its direction to inward. (An electrical engineer can figure the direction from the right hand rule of currents and remembering that the charge of q is negative.)

                                Fmag = qv x B
                                Fmag = (u0/4 pi) x (qv)^2/r^2

But there is a relation ship between e0, u0 and the speed of light (c)

                                c= 1/(sq rt {e0 u0)
                                c^2 = 1/(e0 u0)
                                u0 = 1/(e0 c^2)

eliminating u0 from Fmag we get

                                Fmag = (u0/4 pi) x (qv)^2/r^2
                                          = (1/e0 c^2)/(4 pi) x (qv)^2/r^2
                                          = 1/(4 pi e0 c^2) x (qv)^2/r^2
                                          = 1/(4 pi e0) x q^2/r^2 x (v/c)^2

but the first expression is just Felect, so we have

                                Fmag = Felect x (v/c)^2

Thus the net force (Fnet), which is the repulsive force between two electrons moving at speed (v) relative to an observer, is the large electric repulsive force (Felect) minus the normally much smaller magnetic attractive force (Fmag).

                               Fnet = Felect - Fmag
                               Fnet = Felect x {1-(v/c)^2}

        As you can see above 'motion fudge factor' reducing the electric force {1-(v/c)^2} between electrons turns out to be just the square of the relativistic 'fudge factor' that appears in the special relativity length contraction. The inward magnetic force between the electrons only equals the outward electric force when the speed of the electrons (with respect to me) reaches the speed of light. So magnetic force, which arrises when charges move, sure looks like it's just a relativistic correction of electric force. And so it is. This point of view provides us with an explantion of the origin of the magnetic field and magnetic force.

        There's a lot of good stuff about special relativity at this link (Australian university)


Wikipedia comments on electromagnetism and relativity this way:

        "In particular, a phenomenon that appears purely electric to one observer may be purely magnetic to another, or more generally the relative contributions of electricity and magnetism are dependent on the frame of reference. Thus, special relativity "mixes" electricity and magnetism into a single, inseparable phenomenon called electromagnetism (analogously to how special relativity "mixes" space and time into spacetime). .. When both electricity and magnetism are taken into account, the resulting theory (electromagnetism) is fully consistent with special relativity." (Wikipedia, 'Magnetism')
Magnetic force between two parallel wires
        A textbook example for measuring magnetic force (as a function of current) is the case of two parallel wires carrying the same current in the same direction. There is an attractive force between the two wires. Normally in the textbooks the force inward is derived from the electromagnetic equations that relate force, current and distance.

        If we look deep at two current carrying wires parallel wires, where  at the currents are equal and in the same direction, what we see is this. Two rows of negative charge (free electrons in the metal wire) that are moving in parallel with the same (average) velocity and two rows of positive charge (the atoms of the metal wire that have lost an electron, called ions) that are not moving. Above we derived the force between moving electrons as a static electric repulsive force minus a velocity sensitive attractive magnetic force.

                               Fnet = Felect - Fmag
                               Fmag = 1/(4 pi e0) x q^2/r^2 x (v/c)^2

        There is a positive ion in each wire for each negative electron, so in the case of two parallel wires there is no static electric force between the wires (Felect = 0). The remaining force between the wires is the attractive, relativistic, magnetic force (Fmag), which varies as flow of charge (q x v) squared, or current squared.

Magnetic field as seen by a test charge outside a wire
        It is an amazing thing is that magnetic forces and fields are (at root) really just relativistic corrections of the electric field caused by the motion of electrons that make up a current. In the case of forces on a test charge outside a wire carrying a current the relativist nature of the magnetic field is a more obvious. The text below shows that the magnetic force on a test charge outside a wire is explained by invoking length contractionthat slightly unbalances the normally exquisitely balanced positive and negative forces of the electrons and copper ions in the wire.

        But the really incredible thing is that the speed of electrons in a wire (at a few amps), which is the cause of this relativistic effect, is less than 1 mm/sec!! This means, of course, that the resultant length contraction is unbelievably tiny, about one part in 10^23, since (v/c)^2 = (10^-3/3 x 10^8)^2.  What this tells us is that electric forces from bare (unshielded) charges must be astoundingly huge.

        Current in a copper wire is nothing more than a flow of loosely bound outer electrons of the copper atoms moving along with wire. The wire as a whole is electrically neutral. The freely moving negatively charged electrons are balanced by the non-moving positively charged atoms (ions) that have lost an electron to the current. A (test) electron near, but outside, a wire carrying a current feels no force if the test electron is not moving (because the wire is neutral). But if the outside test electron is moving (either as a free electron or as part of a current in another wire), it does feel a force. The force felt by the moving test electron is usually described as coming from the magnetic field created by the current in the wire {Force = q(v cross B)}

        If viewed at the charge level, however, thing look different. Let's assume the outside freely moving test electron is moving parallel to the moving electrons in the wire and at the same (average) speed. When the wire is viewed from (a frame attached to) the outside moving electron, the electrons in the wire are seen to be fixed and the positively charged ions moving backwards. Even though speeds are slow relativity arguments can be applied, and the spacing between the positively charged ions, as seen from the freely moving test electron, is length contracted.  The key is this: the outside test electron sees the positively charged ions more closely packed than the negatively charged moving electrons of the wire, so from the point of view of the outside moving test electron the wire has a net positive charge.  Since opposite charges attact, the outside test electron feels a force pulling it toward the wire.

        The (average) speed of moving electrons is a wire is very slow (< 1mm/sec), so on the one hand the relativistic length contraction is unbelievably tiny, but on the other hand the attractive/repulsive forces between bare charges is incredibly huge. One references summarized this example this way:

        This is why there are magnetic fields: their effects are a manifestation of the fact that charges exert electrostatic forces on other charges, and that special relativity makes current-carrying wires which are neutral in one frame appear charged in another frame.
How time dilation and length contraction allow everyone to measure the same speed
        We can now explain how earthmen and moving spacement both get the same result when measuring the speed of light, or any speed. The key is that speed is a ratio (distance/time). A spaceman can (from our point of view) measure distance wrong and measure time wrong, but still calculate (measure) correct speeds because the distance and time errors cancel each other. Of course, the formula for time dilation and length contraction have been constructed so that they will divide out when the ratio of speed is calculated.

        Consider, a spaceman uses his clock to count how many beats it takes a photon to travel across a test distance he has measured with his ruler.

        Spaceman sees --- Measures out a 1 foot test distance and counts 1,000 beats of his (local) 10^12 hz oscillator (period = 1 picosecend) clock. So spaceman calculates speed of light = 1 foot/1 nsec or 3 x 10^8 m/sec.

        Earthman sees (assume v/c = 0.866, so sq rt (1-(v/c)^2) = 0.5)  ---
            1) Travel distance is only 6 inches, because the ruler the spaceman used was contracted by a factor of 2. Since his distance is low, his calculated speed due to this 'error' (alone) will be low by a factor of 2, or in general by the length contraction factor {sq rt (1-(v/c)^2)}.

            2) Time dilation causes everything on the spaceship (except photons) to move slowly. So Earthmen see the oscillator running at half freq (period = 2 nsec). We agree with spacement there were 1,000 beats during the photon travel distance, earthmen measure the travel time at 1,000 x 2 nsec = 2 nsec. Yikes!!!

        Earthmen say moving 6 inches took 2 nsec or 1/4 ft/nsec . Where the hell is the error here!!

figuring from the muon case
        earthmen see muons travel down 6,000 ft mountain in 6 usec. Speed close to c To explain the observed high survival the time is assumed to be running slow on the muon, such that the travel time was only 0.7 usec
        muon men see mountain only 700 t high so it can be traversed in 0.7 usec. Speed close to c. Result only 30% decay.

 earthmen see muon men see
travel time down mountain 6 usec 0.7 usec
mountain height 6,000 ft 700 ft
relative earth/muon speed 6000 ft/6 usec =
1000 ft/usec
(close to c)
700 ft/0.7 usec =
1000 ft/usec
(close to c)
time for photon travel the distance 2 nsec (1000 beats of  period 2 psec osc) 1 nsec (1000 beats of  period 2 psec osc)
distance 2 feet/6 in
1 foot
measured speed of light 2 ft/2nsec=1 ft/nsec 1 ft/nsec


Mea speed across the shorten distance quickly. His speed is scaled up by the factor {1/, the inverse of the contraction of his ruler.

         Mistake #2 is that his measured time (from our point of view) is too short because his clock is beating too slow. Mistake #2 causes him to get a speed result which is also too high, since he finds it takes a short time for the photon to cross the measured distance.  Yikes, I don't understand this!!

        Earthman sees (looking at spaceman zooming by at 90% of speed of light ) --- Spaceman has measured out 5.24 inch test distance and counts 437 beats of the clock. Why???
(while we agree he counted 1,000 counts of his clock, his clock is running slow so that each beat takes 2 picoseconds. So his measured speed is 0.5 foot/2 nsec. Same problem!!)
        Spacemen and earthmen both agree --- clock on spaceship beats 1,000 times as photon crosses measured distance.

        Spacemen say --- Measured distance was 1 ft and since 1,000 beats of the (local) 10^12 hz oscillator (period = 1 picosecend) clock means travel time was 1 nsec. So spaceman measure speed of light = 1 foot/1 nsec.

        Earthmen say --- Distance was really 2.29 ft (length expansion???) and 1,000 beats of the slowed down 10^12 hz oscillator (period = 1 picosecend) clock means travel time was 2.29 nsec. So earthmen measure speed of light = 2.29 foot/2.29 nsec
How time dilation and length contraction allow everyone to measure the same speed (2nd try) (11/15/11)

        Principle --- Length contraction is (only) in the direction of travel
                              Everyone measures the same speed for light

        From earth we watch spacemen (in a glass space ship) going by as they measure the speed of light.

                * Inside the ship they orient a 1 ft ruler from front to back of the ship.  Inside they measure the travel time
                            of light along the ruler with their clock and get 1 nsec. (Visualize we can watch the space ship clock move
                            and we see it hits the 1 nsec mark as the light photon crosses the end of the ruler.)

                            -- Hence even though we measure the ruler as contracted and the travel time along the ruler as extended
                                we can 'understand' how the spacemen get 1 nsec/ft because we can see their clock hit 1 nsec when
                                the light crosses the end of their marked 1 ft ruler. In other words we observe the simultaneous event of
                                the light photon crossing the end of the ruler and the marking on their clock. We see the same spaceship
                                instrument readings as the spacemen.

               * From earth perspective we see events happening more slowly on the ship, so when we time the light travel
                            along the ruler using our earth clock we get more than 1 nsec, say 1.1 nsec. Thus the time dilation effect,
                            considered alone, would cause us to say that the speed of light measured on this moving platform
                            would give a speed that is too low.

                * Now we come to the crux of the matter. Does, or does not, length contraction cancel out time dilation?
                                -- We see the ruler (oriented along the length of travel) as contracted, so we measure it at less
                                            than one foot, say 0.9 ft. So measuring with our instruments it takes 1.1 nsec for light to travel 0.9 ft.
                                            Twice the error!

        Does this violate the principle, 'Everyone measures the same speed for light'? Maybe not. My little though experiment says that if we try to measure the speed of light with instruments in our frame as we observe the experiment in a moving frame, the relativistic effects do not cancel they add (really multiply). Also even if by some logic (with shrunk rulers), we do argue there is calculation, when the ruler is turned 90 degrees the cancellation will be gone, because time dilation will remain, whereas there is no contraction in direction perpendicular to the direction of motion.

        I bet the explanation is that there is a corollary:

                       'Instruments must be in the same frame as the experimental apparatus'

While I can't find this explicitly stated in a search, I suspect it is common sence. When it is stated an observer makes a measurment in his reference frame, it is implied that the instruments he uses are in his frame too.

Riff on '1/2' in energy formulas
        Energy can be stored in a lot of ways and a surprising number of the energy formulas look very similar. The table below shows four of many. A moving mass stores energy in motion, a compressed spring stores energy in material elastic deformation, a capacitor with a voltage stores energy in an electric field, an inductor with current stores energy in a magnetic field.

Energy formula Parameters Energy stored
1/2 m v^2 kinetic   mass,
1/2 k x^2 linear
spring constant,
spring displacement
1/2 C V^2 capacitor capacitance,
electric field
1/2 L I^2 inductor inductance,
magnetic field

        How come all these formula have a '1/2'?  Well let's start with zero energy in a system and then pulse the power (off, on, off) to add some energy. In mechanics power is the product of force x velocity, in electronics power is current x voltage. In the following examples one power term is held constant and the other power term ramps up linearly:

-- When you drop a body in the earth's gravitational field, the force is (nominally) constant and the velocity ramps up linearly.
-- When you compress a (linear) spring by squeezing it at a steady rate,  the force ramps up linearly.
-- When you add charge to a capacitor with a constant current source, the voltage on the capacitor ramps up linearly.
-- When you apply a constant voltage source across an inductor, the current ramps up linearly.
        Energy is the integral of power, so the area under the power pulse is energy added to the system. All these power pulses, being the product of a constant and a ramp that starts at zero and then steps back to zero, have the shape of a triangle. The area of a triangle is 1/2 times the peak power times the time of the ramp. This is where the 1/2 comes from!

        Here's it is from another point of view. Consider the body dropped in earth's gravitational field, which we assume constant. Since (E = force x distance), the kinetic energy a falling body just depends on how far it has fallen. The energy increases linearly with distance, and there is no '1/2' in the energy equation if written in terms of force and distance. Of course, as a function of time a falling body falls faster and faster, resulting in the distance it has moved going up as the square of time. The math is this: velocity increases linearly over time in a (constant) gravitation field, so speed plotted vs time is a triangle. In time t the speed is at, where a is the gravitational acceleration constant on earth (32ft/sec^2). The distance moved is the area under the velocity triangle. The triangle shape explains the '1/2' and since both the hor and vertical size of the triangle grow with time, this explains why the distance increases as time squared. (distance = 1/2 a t^2).

        From still another point of view --- The distance that a dropped body moves in a given time is its average speed x time. Since the speed increases linearly (from zero), its average speed (during the fall) must be equal to 1/2 of its present speed. There's the '1/2' again.

        The point of all this is that the '1/2' in the energy formula is related to the fact that (if power is finite) it takes time to move energy in or out of bodies. The flow of energy in/out over time is why there is a 1/2 in all the above square law energy formulas.

Why no '1/2' in Einstein's formula?
        Curiously, Einstein's famous formula E= m c^2 is similar to the energy formulas above except it's missing the '1/2'.  In all the other energy formulas the 1/2 arises during the process when the energy is added to the system. Einstein's formula is interpreted as the rest energy of mass. There is no charging process, the energy is always there. I think that's why there is no '1/2' in Einstein's formula.

Rewriting E = mc^2
            The link below is Einstein speaking and explaining that (E = mc^2) means mass and energy are (basically) the "same thing". If the important thing about (E = mc^2) is the proportionality between mass and energy, then notice that the equation E=mc^2 is written rather oddly. It always looks to me like an energy equation (sans the 1/2), yet it is really a linear, proportionality equation. What seems at first glance to be velocity square term (c^2) term is really the proportionality constant.

        The math books I was taught from always used the format below for a proportional, or straight line,  relationship.

                                y = ax
                                        x is independent variable
                                        y is dependent variable
                                        a is constant of proportionality

To get the E=mc^2 equation in this format all we need do is reverse the order of the terms on the right side of the equation.

         E = (c^2)m

Curiously, I have never seen E=mc^2 written as shown above, even though it is the standard format for a proportional relationship. In this format it is much more obvious, at least to me, that mass scaled by a huge proportionality constant is equivalent to energy.


Relativistic mass
        There is sometimes confusion surrounding the subject of mass in relativity.  This is because there are two separate uses of the term.  Sometimes people say "mass" when they mean "relativistic mass", mr but at other times they say "mass" when they mean 'invariant mass' or 'rest mass', m0.  These two meanings are not the same.  The invariant mass of a particle is independent of its velocity v, whereas relativistic mass increases with velocity and tends to infinity as the velocity approaches the speed of light c.  They can be defined as follows:
                       mr = E/c^2
                        m0 = sqrt(E^2/c^4 - p^2/c^2)

where E is energy, p is momentum and c is the speed of light in a vacuum.  The velocity dependent relation between the two is
                       mr = m0 /sqrt(1 - v^2/c^2)

        Einstein's famous equation E=mc^2  remains generally true for all observers only if the m  in the equation is considered to be relativistic mass (mr).
How an increase in kinetic energy of a relativistic particle increases its mass
        Let's think about the kinetic energy of one of these particles traveling close to the speed of light. Recall that in an earlier lecture we found the kinetic energy of an ordinary non-relativistic (i.e. slow moving) mass m was ½mv². The way we did that was by considering how much work we had to do to raise it through a certain height - we had to exert a force equal to its weight W to lift it through height h, the total work done, or energy expended, being force x distance, Wh. As it fell back down, the force of gravity, W, did an exactly equal amount of work Wh on the falling object, but this time the work went into accelerating the object, to give it kinetic energy. Since we know how fast falling objects pick up speed, we were able to conclude that the kinetic energy was ½mv².

        More generally, we could have accelerated the mass with any constant force F, and found the work done by the force (force x distance) to get it to speed v from a standing start. The kinetic energy of the mass, E = ½mv², is exactly equal to the work done by the force in bringing the mass up to that speed. (It can be shown in a similar way that if a force is applied to a particle already moving at speed u, say, and it is accelerated to speed v, the work necessary is ½mv² - ½mu².)

It is interesting to try to repeat the exercise for a particle moving very close to the speed of light, like the particles in the accelerators mentioned in the previous paragraph. Newton's Second Law, in the form

            Force = rate of change of momentum

is still true, but close to the speed of light the speed changes negligibly as the force continues to work -- instead, the mass increases! Therefore, we can write to an excellent approximation,

            Force = (rate of change of mass) x c

where as usual c is the speed of light. To get more specific, suppose we have a constant force F pushing a particle. At some instant, the particle has mass M, and speed extremely close to c. One second later, since the force is continuing to work on the particle, and thus increase its momentum from Newton's Second Law, the particle will have mass M + m say, where m is the increase in mass as a result of the work done by the force.

        What is the increase in the kinetic energy E of the particle during that one second period? By exact analogy with the non-relativistic case reviewed above, it is just the work done by the force during that period. Now, since the mass of the particle changes by m in one second, m is also the rate of change of mass. Therefore, from Newton's Second Law in the form

            Force = (rate of change of mass) x c,

we can write

            Force = mc.

The increase in kinetic energy E over the one second period is just the work done by the force,

            force x distance.

Since the particle is moving essentially at the speed of light, the distance the force acts over in the one-second period is just c meters (c = 3×10^8 m/sec). So the total work the force does in that second is

            force x distance = mc x c = mc²

Hence the relationship between the increase in mass of the relativistic particle and its increase in kinetic energy is:
            E = mc²

Mass increase is equal to the total work done divided by c²
        At lower speeds we can show how E=mass x c^2 includes kinetic energy by doing an expansion of the equation of  relativistic mass. Consider a mass M, moving at speed v, much less than the speed of light. Its kinetic energy is E =½Mv². Its relativistic mass is M/sqrt(1 - v²/c²), which we can write as M + m. What is m?  For small v/c we can make the following approximatations:

                     sqrt(1 - v²/c²)         =>     (1 - ½v²/c²)
                     1/(1 - ½v²/c²)         =>    (1 + ½v²/c²)

total (relativistic) mass at speed v is

                     M/sqrt(1 - v²/c²)   =>   M(1 + ½v²/c²) = M + ½Mv²/c² = M + m
                                              m  =  ½Mv²/c²

so Einstein's total energy = mass x c^2 becomes

                          Energy = (M +m)c² = Mc² +  ½Mv²

        By expanding the relativistic mass equation we have shown that for low (v/c) the total energy of a particle is just its rest energy (Mc²), where M = rest mass,  plus its kinetic energy (½Mv²).  We conclude that when a force does work accelerating a body to give it kinetic energy, the mass of the body increases by an amount equal to the total work done by the force, the energy transferred, divided by c².
Reversible example of E=m x c^2
        The helium atom has a nucleus which has two protons and two neutrons bound together very tightly by a strong nuclear attraction force. If sufficient outside force is applied, this can be separated into two "heavy hydrogen" (deuterium) nuclei, each of which has one proton and one neutron. A lot of outside energy has to be spent to achieve this separation. It is found that the total mass of the two heavy hydrogen nuclei is measurably (about half a percent) heavier than the original helium nucleus.

        This extra mass, multiplied by c², is just equal to the energy needed to split the helium nucleus in half.  Even more important, this energy can be recovered by letting the two heavy hydrogen nuclei collide and join to form a helium nucleus again. They are both electrically charged positive, so they repel each other, and must come together fairly fast to overcome this repulsion and get to the closeness where the much stronger nuclear attraction kicks in. This is, of course, fusion,  the basic power source of the hydrogen bomb, and of the sun.
Electric forces are huge!
        Since positive and negative charges are almost always in balance, or nearly in balance, we have little intuition of how huge electric forces really are. The example below of 1 million tons of repulsion is for the huge distance of 1 meter. Note that the force goes up inversely with the square of the distance, so at 1 cm distance the force would be 10^4 higher, or 10 billion tons!

Here's the math. Current is the flow of charge {I = delta q/delta time}. In the example above the bulb current is 1 amp  (1 amp x 120 volt = 120 watt). 1 amp = 1 coulomb/sec, so in the example above q = 1 coulomb.

            F = q^2/4 pi e0 r^2

                        f = newtons
                        q = coulombs
                        e0 = electric permitivity of vacuum (8.8 x 10^-12)
                        r = meters

            F = 1/4 pi x 8.8 x 10^-12 x 1
                = 1/110 x 10^-12
                = 9 x 10^-3 x 10^12
                = 9 x 10^9 newtons

To convert newtons to pounds of force think of a 1 Kg weight sitting on a table. A 1 kg in English units weighs 2.2 lbs. F = ma and acceleration of gravity on earth  is 9.8 m/sec^2 so

                    F = m x a =1 kg x 9.8 m/sec^2 = 9.8 newtons
                    9.8 newtons <==> 2.2 lbs force

                    F = 9 x 10^9 newtons x (2.2 lb/9.8 newton) x (1 ton/2,000 lbs)
                        = 10^6 tons (1 million tons)
Lorentz reference 'ether' frame (5/07/09)
        Physicist Hans Ohanian in his excellent new book 'Einstein's Mistakes' says that Hendrik Lorentz, whom Einstein revered and who had discovered the Lorentz transformations before Einstein in his study of electromagnetism, continued to believe until his death in 1928 that absolute Newtonian time existed. He believed that time relative to the ether was the absolute Newtonian time. Clocks moving relative to the ether measured "local time" and were subject to time dilation. Lorentz had explained in 1895 that Michelson-Morley experiment had failed to detect motion through the ether because of length contraction in one arm in the interferometer and time dilation. Einstein, and most physicist since, have concluded that the null results of experiments to measure the speed of the ether is best explained by concluding there is no ether.

        Lorentz lived more than two decades after Einstein's 1905 relativity paper and many years after it was well accepted. Lorentz agreed with Einstein that observers in inertial frames saw things equivalently and that they were unable to sense if they were stationary or moving relative to the ether. Poincare had written a paper on (special) relativity before Einstein. In fact, in the early the theory was generally known as Lorentz-Einstein relativity (sometimes Poincare name was included). Both Lorentz and Poincare continued to believe in the ether all their lives that frames are not relative, they just appear that way to observers moving with a constant relative velocity. Michelson believed in the ether until his death too.

        When I read about this in Ohanian's book, it gave me an idea. If Lorentz and other famous physicists continued to believe in the ether as an absolute reference frame, it must be that recasting relativity problems into a fixed reference frame does not introduce any mathematical error. So why not use a 'fixed' reference frame (mythical or not) in working relativity problems. Seems to be this viewpoint might be useful, and it might very well be simpler because the speed of light can be drawn as a constant in this reference frame. (I always find that the lack of a reference frame to be one of the confusing aspects of working special relativity problems.)

        Lorentz's belief in a reference ether frame (virtually) proves that working with a fixed ether frame has to be mathematically equivalent to the more conventional approach. Why not use this fact to get the right answers? As an engineer, this seems to me to be a sensible thing to do. For years engineers have successfully used this approach: one example being the use of the 'impulse' function in circuit analysis. My professors at MIT introducing it said that it may not be mathematically rigorous, but it was convenient and always appeared to give the right answer, so it was used extensively in circuit analysis. (Years later I believe the mathematicians did show it to be rigorous.)

Background microwave radiation -- new preferred reference frame
        Ohanian makes the point that a with the discovery of the microwave background radiation there is now a preferred reference frame. In other words everything is not relative!
Doppler effects from an ether viewpoint
        Below is a figure I drew showing light pulses traveling in the fixed (Lorentz) ether frame. Light travels in the ether about 1 nsec/foot (c). Each small box of the grid is one foot, so the light flashes (black arrow) drift rightward at c. The light source (black box on left) is assumed flash briefly every 5 nsec. Thus the source creates in the ether a pulse train (series of light flashes) that drift rightward at c, or 5 box (feet) in 5 nsec.

Stationary case
       At the top of the figure the source and receiver (say respectively the back and front of a space ship) are stationary in the ether. The pulse train in the ether is a series of light flashes spaced 5 feet (5 box) apart. The receiver sense a flash every 5 nsec, or exactly at the rate (frequency) as the sources is sending.

Blue shift at source
        In the lower part of the figure the source and receiver are both moving with respect to the ether rightward at (a constant speed) 0.2c, or 1 box/5 nsec. The light flashes in the ether are shown at 5 nsec intervals. Each light flash upon entering the ether moves at c, drifting rightward at 5 box/5nsec exactly as in the stationary case. Note, however, the spacing between the light flashes in the ether has changed. The spacing in the ether is now 4 feet not 5 feet as in the stationary case.

        Why is this? It's a doppler effect. The source moving in the same direction as the light (@ 0.2c) somewhat closes the gap to the previous flash. It travels 20% as far as the light, so the spacing of the light flashes in the ether is compressed from 5 feet to 4 feet. It's a doppler blue shift. In frequency terms the spacial frequency is higher. (If the source had been sending sinewaves, then in space the crests would be seen to be closer together, it's a higher spacial frequency.)

Red shift at receiver
       So does the moving receiver see a higher frequency (equivalently less time between pulses) that the source is sending? No. Why not. At the receiver there is a doppler red shift, just the the opposite of what happens at the source. The receiver is moving away from the incoming light flashes at 0.2c, so even though the light flashes in the ether are (actually) separated by four feet (equivalent to 4 nsec), the receiver is running away from the light at 0.2c, so it 'sees' a light flash every 5 nsec, which is exactly at the rate (frequency) the sources is sending.

        The amplitude of a doppler shift is proportional to the speed of the source or receiver through the ether. In this case they are both traveling at the same speed (0.2c). From the perspective of the (fixed absolute) ether frame in which light travels we see that what is 'really' happening: The blue doppler shift at the source is exactly cancelled by the red doppler shift at the receiver, so the receiver 'sees' the time between flashes to be exactly the same as the source (5 nsec)!
        For a receiver not moving (with respect to the ether) the pulses are received every 4 nsec, in other words he sees the pulse rate blue doppler shifted due to the inward motion of the source.

Gravitational red shift
       Well what's so interesting about that? Well perhaps at constant speed not much, but we can use the same viewpoint for an accelerating space ship. Now when the light arrives at the receiver on the front of the ship, sent from a source at the back of the ship, the receiver is now moving faster (with respect to the ether) than at the time the source fired the light pulse because the ship is accelerating.  We are now in position to see why light fired in the direction of the accelerating space ship is (gravitationally) red shifted.

        The red shift at the faster moving receiver (with respect to the ether) is larger than the blue shift at the source. The red and blue shift no longer cancel leaving a net red shift for light fired in the direction of acceleration. (For light fired in the direction opposite to acceleration there is a gravitational blue shift.)
General relativity test
        In terms of the earth this means earth's gravity causes light fired upward to redden and light coming into earth downward to shift toward blue. The gravity on the sun 'surface' is about x28 higher than earth, so Einstein proposed a test of relativity to measure the frequency of spectral element lines as seen in the solar spectrum and compare them with the frequency measured on earth. The sun's spectral lines are expected to be about 2 parts in a million lower in frequency due to the sun's higher gravity over the earth. This frequency shift (with a lot of complications) has been verified to tolerance of a few percent.

Clocks slowed by gravity
        But the receiver over time cannot just sense less flashes than are being sent, flashes just don't get disappear. So there must more to the explanation than just a doppler shift! Einstein in an early general relativity paper provided the answer:

        The clocks at the source and receiver run at different rates. Adjusting the clock at the receiver to run slower will provide more time to count pulses, and if done right, mirabile dictu the doppler effect is gone. [By the equivalence principle someone in a sealed box can't determine if the box is continuously accelerating or sitting on a planet surface in a gravitational field.] So above explains the gravitational red shift of light and why clocks in gravitational fields run slower (these are general relativity effects).
Physical explanation for length contraction
        Ohanian says what bothered many of Einstein's contemporaries was that relativity was totally mathematical. There was no mechanism proposed, hence no possible visualization, of how length contraction (or time dilation) worked. The math just said that for the speed of light to be constant to all observers the length of moving objects had to (in an unspecified way) contract in the direction of motion and moving clocks had to run slower. There was no explanation, zero, of how these effects occurred at a molecular level or any level.

        What's interesting is that Ohanian says there now is a quantum mechanical explanation for length contraction (& indirectly I think for time dilation). He says the calculation could not be done in the early years of relativity because it requires relativistic quantum mechanics. This became available in the form of Dirac's relativistic quantum mechanical equation in 1928.

        Ohanian says in 1941 W.F.G. Swann calculated that a bar quantum mechanically does contract while moving, but nobody paid any attention. In 1976 it was again calculated again by J.S. Bell of CERN and Bell urged this material be put in textbooks on relativity. Ohanian says no current textbook includes this material and very few even mention that "length contraction and time dilation have a physical explanation."  The only physical visualization mechanism found in textbooks is the light clock, but Ohanian criticism is that this is a very special case, for example, it only works when light bounced perpendicular to motion. Also the light clock provides no intuition as to how the shape or size of atom or molecules might change to to speed induced effects of the their electric fields.

Michelson - Morley Experiment in ether frame (5/09)
        The Michelson - Morley interferometer experiment was strongly featured in my freshman physics course, but I had long forgotten all the details remembering only the conclusion: the speed of light was the same in all orientations. The time to travel out and back in quadrature tubes was compared in various orientations relative to earth and the earth's motion around the sun and no variation was found, i.e. the interferometer stayed nulled regardless of how it was oriented on earth or in space.

19th century viewpoint
            * Light moves at a fixed speed (c) relative to the ether
            * Ether assumed to be fixed relative to the sun
            * Earth in its motion around sun is moving through the ether
                        (might be some partial entrainment)
            * Velocities add and subtract in a newtonian way
            * Light travel time goes down traveling with the ether flow,
                        goes up traveling against the ether flow, and always
                        the round trip time increases

        Below is a link to good animation of the Michelson-Morley experiment in the earth's frame with the ether flowing by. The simulation shows the flow of the ether (relative to earth) and can be adjusted. This is the world as viewed (before the experiment) by everyone including Michelson from the earth's frame.

        . If you adjust the speed of the ether in the simulation to up near c, an asymmetry in travel times is clearly evident. Light traveling in the same direction as the ether can only (in the limit) be doubled in speed (c + c) = 2c, but light traveling against the ether can be slowed (in the limit) to a stop (c - c) = 0. The result of this asymmetry is that the round trip travel time is always increased by the flow of the ether. The extra time it takes traveling against the ether is always (at least a little) more than than the time saved traveling with the ether.


        The mathematics for out and back in an 8 foot long tube moving with ether flow at v (relative to the ether) are as follows:

                Time (out & back) = 8 ft/(c+v) + 8ft/(c-v)

                                                = (8ft/c)[1/(1 + v/c) + 1/(1 - v/c)]

                                                = (16 ft/c)[1/(1 - (v/c)^2]

        When moving in the same direction as the ether, the doppler shifts going into and out of the ether don't quite cancel. There is a residual 2nd order term (v/c)^2 that slightly increases the time it takes for light to go out and back. The correction factor is 1/ [1 - (v/c)^2]. Note, no square root, it is not the usual Lorentz correction factor, it's the lorentz correction factor squared.

Sketch of light flashes out and back from eiher perspective
        The increase in transit time (out and back) is confirmed and can be visualized in a rough sketch I made in the Lorentz fixed ether frame. Here Earth is speeded up to 20% of speed of light ( v = 0.2c). The sketch shows that when light travels with the ether flow spacially it's doppler blue shifted, the spacing of the light flashes in the ether is 4 boxes. The reflecting mirror introduces a doppler red shift, so in the reverse direction the spacial separation of the light flashes in the ether is 6 boxes.

        The sketch shows graphically that round trip time for each light flash is 16.666 nsec, whereas it with no ether flow it would be 16.0 nsec, so round trip time is increased about 4% compared to no ether flow. More exactly the increase in travel time is by the factor 16.6666 nsec/16 nsec = 1.041666 = 1/0.96. Plugging v= 0.2c into the correction factor above:

                                [1/(1 - (v/c)^2] = [1/(1- (0.2)^2] = [1/(1 - 0.04)[ = 1/0.96  checks

Lorentz explanation
        Lorentz, so I had read, explained the failure of the Michelson Morley experiment by saying the tube aligned with the ether flow length contracted. This is the origin of the famous lorentz contraction. But this didn't seem right. The travel time increase in the tube aligned with the ether flow is 1/(1 - (v/c)^2), it's only partially corrected by a length correction of sqrt{1 - (v/c)^2}.

        For a while I thought the answer must be that Lorentz in 1895 had proposed a length contraction and a time dilation (slowing of the clock). With two lorentz corrction factors at work the effects of ether flow in the aligned tube [1/(1 - (v/c)^2)] would be cancelled. But I had missed something. A null Michelson Morley result just means the travel times in the two 90 degree tubes are the same.What I had missed is that the ether flow changes (increases) the round trip travel times in both tubes. The 90 degree tube is effectively a light clock! It's 'beating' is slowed (dilated) by travelling (in the ether) meaning it takes longer (by a lorentz correction factor) for the light to make a round trip.

90 degree tube is a light clock!
        Seen from the fixed ether frame the light flash in the 90 degree tube on the moving earth is traveling not only out and back but also forward with the earth. The light path (seen by an outside observer in the fixed either (sun) frame) is two diagonals whose base is (v x delta t). Thus with ether flow the round trip time in the 90 degree tube is increased by 1/sqrt{1 - (v/c)^2}. This is about half the [1/(1 - (v/c)^2)] increase in the tube aligned with the ether flow. Hence a lorentz length contraction [sqrt{1 - (v/c)^2}] of only the tube aligned with the ether flow does the trick. It cuts the extra travel time of the aligned tube (about) in half matching the travel time in the quadrature tube. The equations of Michelson Morley showing all this are here:


Bottom line on Michelson Morley
        * From an earth perspective there is no motion so length contraction (or time dilation). The null result of the experiment is proof light is not being conducted in an ether flowing through earth.

        * From an outsider in the fixed ether frame (here assumed to be attached to the sun) the earth is moving.

        The arm not aligned with the earth's travel around the sun acts like a light clock. It's light not only has to go out and back but also move perpendicularly through space, hence the round trip time in that arm is increased by the lorentz factor (1/sqrt{1 - (v/c)^2}).

        The arm aligned with the earth's motion around the sun has an increased in round trip time due to the doppler blue and red shifts going into and out of the ether that don't quite cancel. This results in an increase in round trip travel time of [1/(1 - (v/c)^2)], which is roughly two lorentz factors. However, this arm pointed in the direction of motion is length contracted, which reduces travel time by a lorentz factor. The combined result of these two terms in an increase in round trip time by the lorentz factor (1/sqrt{1 - (v/c)^2}).

        The increase in round trip time due to earth's motion is thus the same both arms which explains why the interferometer stays balanced with no fringe changes.

        Wikipedia says this about the length contraction in the Michelson Morley experiment article (5/14/09)
        The explanation (for the null result) was found in the Fitzgerald–Lorentz contraction. According to this physical law all objects physically contract along the line of motion (originally thought to be relative to the ether),  so while the light may indeed transit slower on that arm, it also ends up travelling a shorter distance that exactly cancels out the drift.
        It's poorly worded ('cancels out the drift'). Also I sees there are different opinions as whether an object 'physically' length contracts, or whether it just appears to contract by an outside observer who is not moving. I generally tend toward toward the latter view. But in the Michelson Morley experiment it certainly seems that saying it physically contracts is right, because the human observer sees no fringe changes.

Contraction due to earth's speed around sun
        The earth's speed around the sun is about 30 km/sec (3 x 10^4 m/sec), so (v/c)^2 = [(3 x 10^4/(3 x 10^8)]^2 = 10^-8. So the expected increase in round trip travel time in the arm aligned with the ether was about 1 part in 100 million, and in the quadrature tube about half this (1/2 part in 100 million). If the aligned tube length contracts, then it causes the the round trip times in both tubes to be the same (up about 1/2 part in 100 million from the case of no ether flow).