A Relativity Primer

v1.1.0 / 01 sep 17 / greg goebel

* In 1905, Albert Einstein (1879:1955) published his "Special Theory of Relativity", a simple and elegant hypothesis that made a number of remarkable assertions: nothing could exceed the speed of light; clocks in a moving object slow down; the length of a moving object shrinks; and the mass of a moving object increases. These assertions have been confirmed by observations and experiment, and are now generally accepted. Einstein used his theory of Special Relativity as a springboard for a much more comprehensive "Theory of General Relativity", which is also now generally accepted. This document provides a brief introduction to Special and General Relativity.



* Special Relativity was derived from two principles. The first was "Galilean Relativity", the simple principle established by Galileo Galilei (1564:1642) that there was no way to distinguish a state of uniform motion from a state of rest. The laws of motion appear the same to observers in a cube moving uniformly as they do if the cube were sitting still.

This simple, almost intuitive concept leads to the relativistic "equivalence principle": if two objects are moving relative to each other, it is just as valid to regard the first as stationary and the second as moving, as it is the reverse. For example, suppose two friends, Alice and Bob, are both in their own glass cubes moving relative to each other in empty space. Alice will believe she is standing still and Bob is moving; Bob will believe he is standing still and Alice is moving.

The second principle arose from considerations of the observed behavior of light. This principle was articulated by Albert Einstein as a postulate in response to a very difficult problem in physics. What Einstein said was that the speed of light is an absolute constant. Why he said this, and why it was such an astonishing thing to say, requires a little explanation.

* In the later half of the 19th century, the Scots physicist James Clerk Maxwell (1831:1879) devised a set of equations describing electricity and magnetism that suggested that electromagnetic energy could be transmitted in the form of waves. These electromagnetic waves were quickly identified with light and invisible forms of electromagnetic radiation, such as radio waves.

In classical physics, a wave is a disturbance through a propagating medium. For example, a sound wave consists of variations in pressure propagating through the air, while a water wave consists of variations in height of water propagating across a body of water. If electromagnetic radiation was in the form of waves, then by classical thinking it had to be a disturbance of some sort of propagation medium. The only problem was that light propagated through free space where nothing of substance could be detected. Physicists therefore suggested the existence of an "luminiferous (light-bearing) ether" as the propagation medium.

Although the ether was invisible and undetectable, it was thought to fill the entire Universe, with the planets and stars moving through it unimpeded. That implied that the ether established an "absolute frame of reference" for the motion of objects through it. The ether was universal and fixed, and so the motions of heavenly bodies could in principle be measured relative to it.

By this time, the speed of light had been determined to be very close to 300,000,000 meters per second. The question quickly arose: just how fast is the Earth moving relative to the ether? Experiments were conducted to determine the velocity of the "ether wind". The most famous of these experiments was conducted by the brilliant American experimental physicist Albert A. Michelson (1852:1931) and his colleague Edward W. Morley (1838:1923), who had established a solid reputation as an experimentalist in determining the abundances of various elements.

* The "Michelson-Morley experiment" was an early application of an "optical interferometer". If two waves propagate through the same medium, they can constructively or destructively interfere with each other. For example, if two water waves of the same frequency propagate over water, they could be out of phase and cancel out to calm the water, or add up to make the waves higher. Two beams of light can similarly interfere with each other, leading to constructive interference, resulting in brightness; or destructive interference, resulting in darkness.

An optical interferometer works by splitting a beam of light with a half-silvered mirror, running the split beams through two paths or "arms" at a right angle to each other, reflecting the two beams back together and summing them in an eyepiece, then observing the interference effects between them, manifested as a pattern of alternating light and dark "fringes". The split light beams can be "bounced" back and forth in the arms to increase the effective path length. Shifts in light propagation through the two paths are easily detected. Optical interferometry is extremely sensitive, and is now often used an as element in highly accurate sensor and instrument systems. The illustration below shows a simplified Michelson interferometer:

Michelson-Morley interferometer

The Michelson-Morley interferometer was set up in the basement of a stone building and was built on top of a heavy marble slab to ensure stability and reduce vibration. The slab could be rotated around the vertical axis on a thin film of mercury. A light beam was split, reflected along two paths at right angles to each other, and then summed again to generate interference effects. If the reflected light beam was in the direction of the ether wind, the velocity of the ether wind would be subtracted from the velocity of light in one direction, then added in the other; there would be no net effect on the round-trip time. However, if the reflected light beam was crossways to the ether wind, then the effect of the ether wind would be to increase the effective path length, slowing the beam down, in each direction.

The idea behind the Michelson-Morley apparatus was to observe the interference effects when the apparatus was in one orientation, and then observe their change when the apparatus was rotated. This would cyclically shift the position of the interference fringes in the eyepiece as per the changes in path length of the light beams in the two arms. The problem was that, when the experiment was actually performed, there was no perceptible shift; Earth didn't seem to have any appreciable speed relative to the ether wind. The speed of light seemed to be the same along both arms of the interferometer, no matter what the orientation of the device was.

There was no perceptible effect between readings 12 hours apart, when the spin of the Earth changed the direction of the interferometer's movement relative to the Earth's orbit, or readings six months apart, when the Earth's movement along its orbit had reverse relative to local space. Refined experiments gave the same results. Something seemed to be wrong, but nobody could figure out exactly what; no matter what was done, the experiment always gave the same result.

* In 1905, Albert Einstein published a paper that gave an explanation for what was happening, using a concept that became known as "Special Relativity". The paper considered a very simple but bewildering postulate and examined its implications. Einstein's postulate, mentioned above, was that the speed of light is an absolute constant, an invariant. Any beam of light from any source, no matter how fast its relative motion, will be measured to have exactly the same velocity of about 300,000,000 meters per second by any observer. This is why the Michelson-Morley experiment gave negative results.

That sounds like an unremarkable thing to say at first, until the implications are considered. Supposed Alice is flying towards Bob on Earth in a starship at half the speed of light, and then fires off a flash bulb. By classical physics, the flash would reach Bob at 1.5 times the speed of light. No, said Einstein, it still comes at Bob at the speed of light, no more or less.

Similarly, if Alice's starship is moving away from Bob at half the speed of light, classical physics says that the flash will move toward Bob at half the speed of light -- but according to Einstein, it still comes at Bob at the speed of light. Of course, the speed of the flash as seen from the frame of reference of Alice's starship is the speed of light as well, and Alice sees the flash propagate away from the ship in all directions at that speed.

By classical physics, that is completely absurd. If Alice was riding in a the back of a truck at 80 KPH and shot a pebble with a slingshot in the forward direction at a velocity of 50 KPH, then ignoring wind resistance the pebble's velocity as seen by Bob at the side of the road would be 80+50 = 130 KPH. If Alice shot the pebble in the backward direction, Bob would see it as flying at 80-50 = 30 KPH.

To emphasize just how crazy an idea this is, suppose Alice flies right by Bob at half the speed of light, and Bob fires off a flash bulb when she passes by. The light flies away from Bob at what he sees as the speed of light in all directions; but it also flies away from Alice at what she sees as the speed of light in all directions, even though she is moving at half the speed of light relative to Bob. That sounds completely illogical.

However, what Einstein pointed out was that it was classical physics that was illogical as far as the propagation of light was concerned. If the speed of light were relative to the motion of a specific frame of reference, there could be a frame of reference where light would stand still or go backwards. If Alice lived in such a frame of reference, she wouldn't be able to see herself in a mirror. Incidentally, Einstein always insisted that it was this line of reasoning that led him to Special Relativity, not the negative results of the Michelson-Morley experiment; he couldn't recollect if he had even heard of the experiment when he started writing the paper.

What Einstein did was extend the concept of Galilean Relativity to the observed behavior of light. Galileo observed that there was no way to tell any difference between an object at rest and an object in uniform motion, and according to Einstein this was also true of light. Alice cannot determine if she is in motion or at rest by measuring the speed of light relative to her. The equivalence principle says it will always be the same. Nobody has ever observed light traveling at any less or more than 300,000,000 meters per second in a vacuum. There was no ether wind.

* The notion that the speed of light was an absolute constant led to a number of counterintuitive implications. The first was that nothing could go faster than the speed of light. Suppose Alice is on a starship and emits a flash of light. By the equivalence principle, it propagates away from Alice at the speed of light in all directions, and obviously precedes her in the direction of her motion. Suppose Bob on Earth sees that flash. The light of the flash has necessarily arrived before Alice has, and since the flash is moving at the speed of light, she must have been moving more slowly than that.

There were other interesting implications: time slows down in a moving object, and a moving object shrinks in the direction of its motion.



* To see why time slows down in a moving object, suppose that Bob is watching Alice flying across his line of sight in her glass cube. Suppose Alice is carrying in her cube a "clock" consisting of a pair of mirrors oriented at a vertical right angle to Bob's line of sight, with the "ticks" of the clock defined by the time it takes a light pulse to bounce back and forth between the mirrors:

relativistic time dilation example

This clock keeps perfect time on the cube. The light pulse moves back and forth between the mirrors at a constant speed, and the distance between the mirrors remains constant as well. If we designate the speed of light as C and the distance between the mirrors as D, then the tick time T is given by:

   T = 2 * D / C 

However, if Bob is watching the operation of this clock from Earth as Alice's cube flies by at a velocity V, given as a fraction of the speed of light, Cf, then the mirrors are moving as the pulse flies between them. This makes the path length longer for the light pulse and increases the tick time as measured by Bob. If this longer measured time is given by Tm and the velocity of Alice's cube is given by V, then using a little simple geometry -- the Pythagorean theorem -- shows that Bob sees Alice's clock slowing down by the factor:

   1 / SQRT( 1 - Cf^2 )

This is the first consequence of Special Relativity: "time dilation". A moving clock slows down. As the velocity of Alice's cube approaches that of light, the clock tick time lengthens toward infinity:

   Cf       factor
   _______  ______

   0.1        1.01
   0.5        1.15
   0.8        1.67
   0.9        2.29
   0.95       3.20
   0.99       7.09
   0.999     22.37
   0.9999    70.71
   0.99999  223.61
   _______  ______

If something could reach the speed of light, time would stop.

Before Einstein, physicists had always assumed that time was an absolute. No matter where Alice and Bob were, if they both had accurate clocks, their clocks would keep the same time, and if they went their separate ways, no matter what they did, when they got back together their clocks would still be in agreement. This is, from the point of view of our daily lives, absolutely common sense. It is also wrong, though it takes moving around at velocities outside of our direct experience to make the effect noticeable.

* A consideration of "time dilation" leads immediately to another revelation. Suppose Alice is flying in a starship at half the speed of light to a star system ten light-years away. Alice's journey as seen by Bob on Earth would take 20 years. However, Bob also determines that the starship's clock is running more slowly, and in principle Bob could count all the ticks of Alice's clock from departure from Earth to arrival at the remote star system to prove that in fact time has run more slowly for Alice on the starship -- time dilation is not an optical illusion. This means that Alice has traversed the distance D in a time Tm, which is shorter than T as given by the formula we have already derived:

   Tm = T / SQRT( 1 - Cf^2 )

By the equivalence principle, the velocity of the external Universe as seen by Alice on the starship is the same as the velocity of the starship as seen by Bob on Earth. Since the velocity is the same from either point of view but the flight time is shorter, that means that the distance Dm from the Earth to the distant star system as seen by Alice has to be shorter by the same factor:

   Dm = D * SQRT( 1 - Cf^2 )

Of course, the equivalence principle also implies that Bob sees the starship as being shorter in the direction of its motion by this same factor. This is the second consequence of Special Relativity: "length contraction". A moving object becomes shorter in the direction of its motion; at the speed of light, its length would go to zero, another big hint that the speed of light is an absolute barrier.

* This discussion demonstrates that time and space are not constants. This is a necessary consequence of the fact that the speed of light is a constant. The Michelson-Morley experiment was conducted on the basis that space and time were constants, and so the speed of light had to vary. Einstein turned that logic around, showing that if the speed of light were a constant, then space and time had to vary.



* Along with time dilation and length contraction, Einstein also realized that a constant speed of light meant that events that were simultaneous in one frame of reference were not necessarily simultaneous in another.

Let's go back to the scenario in which Alice is flying across Bob's line of sight at great velocity in a glass cube. This time, however, let's suppose that as Alice flies by, she fires off a flashbulb positioned in the center of the cube, which sends a light pulse out towards detectors on the front and back of the cube.

relativistic simultaneity

To Alice in the cube, the pulse has to move an equal distance to hit the front and back, and so the pulse hits both detectors simultaneously. To Bob, however, as the pulse moves backward, the back of the cube moves toward it, and as the pulse moves forward, the front of the cube moves away from it. The light from the pulse strikes the detector in the back before it strikes the detector in front. To Alice, the detectors pick up the pulse at the same time. To Bob, the rear detector picks up the pulse before the front detector. If Alice has mounted clocks on the front and rear faces of the cube and synchronized them to her satisfaction, Bob will see the clock on the rear run ahead of the clock in the front.

Of course, Bob will see Alice's cube as length-contracted, but this is of no great importance in this example. Length contraction is linear: the front half of the cube contracts by exactly the same amount as the rear half, and so the middle of the cube stays where it is.

Incidentally, it is obvious from this scenario that no matter how fast Alice is moving relative to Bob, he will see the pulse hit the walls after Alice fires off the flashbulb. One of the implications of Special Relativity is that causality is preserved in all frames of reference: if events with a causal relationship to each other occur in the sequence A-B-C-D in one frame of reference, then they will be observed to happen in the sequence A-B-C-D in every other possible frame of reference. One of the conceptual difficulties with faster-than-light travel is that it implies there will be a frame of reference where this causal sequence doesn't hold -- B happens before A -- violating the law of cause and effect, something that's never been seen to happen.

* What Einstein was saying, in short, was that simultaneity is as relative as length and time. The relativity of simultaneity helps resolve a number of apparent contradictions posed by Special Relativity.

Imagine Alice is flying a starship of length L. If Alice's starship is moving at Cf, then it is length-contracted by the factor:

    Lc1 = L * SQRT( 1 - Cf^2 )

For example, if L is 100 meters and Cf is 0.8 C, then Lc1 is 60 meters.

Now let's imagine that the starship passes through a tube or tunnel floating in space, at rest relative to Bob, whose length Bob sees as exactly Lc1, or 60 meters in this case. Bob will be able to observe that at one instant the starship will fit exactly into the tunnel. However, from the point of view of Alice on the starship, it is the tunnel that is length-contracted, by the factor:

   Lc2 = Lc1 * SQRT( 1 - Cf^2 )  

       = L * ( 1 - Cf^2 )

If L = 100 and Cf = 0.8 C, then Lc2 = 36 meters. There's no way Alice's starship will fit in the tunnel, is there? Reduced to simple terms, say using measuring sticks, if Bob has a short measuring stick and Alice has a long measuring stick, then from Bob's point of view length contraction can make the lengths of the two sticks equal -- but from Alice's point of view, it makes them even more unequal.

Tunnel Paradox

From a simple consideration of length contraction, we have just painted ourselves into a corner, but there's a way out, through one of the other basic elements of Special Relativity: relativistic simultaneity. Bob can see both ends of Alice's starship fit inside the tunnel while Alice doesn't, because as far as he is concerned time in the rear of her starship is running ahead of time in the front.

Let's imagine that Alice has a clock at each end of her starship, and that she has synchronized the clocks. Alice, who perceives her starship at rest and the tunnel in motion, sees the Lc2 = 36 meter tunnel approaching her L = 100 meter starship at Cf = 0.8 C. Here's what Alice sees:

Alice will measure a short time interval of Td = T2-T1 from the time the nose of her starship leaves the tunnel to the time the rear of her starship passes into the tunnel. Of course, we know that Bob sees the nose of Alice's starship reach the end of the tunnel at exactly the same instant that the rear of her starship passes into the tunnel.

Bob can see Alice's entire starship in the tunnel at one time because the clock in the rear her starship is running ahead of the clock in the front. The short time interval of Td that Alice measures will give the advance of the clock in the rear as seen by Bob. Doing the algebra, the front of the tunnel will reach the rear of her starship after a time interval of:

   Td = L * Cf / C

If L = 100 and Cf = 0.8 C, then Td = 267 nanoseconds (nsec), or in other words Bob sees the count on the clock at the rear of Alice's starship running 267 nsec ahead of a clock at the nose, while Alice sees the clocks as synchronized. This is a worthwhile little formula to remember: it provides a key for unlocking a number of seeming paradoxes.

There really is no paradox here. When the nose of Alice's starship reaches the end of the tunnel, both Bob and Alice agree on that event. However, Bob sees the rear of the starship pass into the tunnel at what amounts to 267 nsec in the future for Alice. Alice will not see the rear of her starship pass into the tunnel until a (very) short time Td later.

Tunnel Paradox resolution

The faster Alice's starship goes, the shorter it becomes, and the more time advances in the rear of her starship. A cartoonish way to visualize length contraction is to imagine a centipede in which each successive set of legs is marching a bit ahead of the legs in front of it, forcing the centipede to pile up on itself and become shorter.

There are many relativistic paradoxes similar to the "Tunnel Paradox", as it's sometimes known, but they're almost always resolved by an understanding of relativistic simultaneity. However, another paradox deserves a closer inspection -- the most famous paradox in relativity, the "Twin Paradox".



* Suppose Alice and Bob are twins. Alice flies to the stars in a starship moving at a good percentage of the speed of light ("relativistic" speed) while Bob stays at home. When Alice comes back to Earth, she will be younger than Bob. For example, suppose that Alice and Bob are both 20 years old when Alice sets out in a starship to travel to the planet Minbar, 10 light-years away, at half the speed of light. She spends a negligible amount of time at Minbar, and then comes back to Earth at half the speed of light. Her time-dilation factor would be:

   1 / SQRT( 1 - Cf^2 )  

     =  1 / SQRT( 1 - 0.5^2 ) 

        = 1.15 

Her trip would take 40 years as far as Bob back on Earth was concerned, but only 40/1.15 = 34.6 years as far as Alice was concerned. Bob would be 60 years old; Alice would not quite be 55 years old.

The Twin Paradox certainly seems contradictory, since Alice's starship can be just as validly regarded as being at rest and the Earth as being in motion. This leads to the first objection: if Bob sees Alice's clock as running slow and Alice sees Bob's clock as running slow, then how can Alice age less than Bob?

The answer is that the two scenarios, though balanced in the general sense, are not mirror images of each other. To Bob back on Earth, Alice's starship is moving at half the speed of light, and only her starship is length-contracted. To Alice, the rest of the Universe is moving at half the speed of light, and the rest of the Universe is length-contracted: the distance to Minbar is only 10/1.15 = 8.66 light-years.

This is the critical point of the Twin Paradox. Alice is taking a trip through a Universe that appears length-contracted to her, and so a trip at half the speed of light takes a shorter time. In fact, it was the necessity to ensure that the Alice and Bob's clock could balance out that was used to introduce the notion of length contraction in the first place. Alice is out of context with Bob's Universe; to her, lengths in that Universe have shrunk, so in compensation her clock ticks have expanded.

* That still doesn't show how Alice and Bob can see each other's clock running slow and come up with different times. In fact, Alice and Bob do not always see the other's clock running slow. This is because of the fact that although time dilation is not an optical illusion, optical illusions are also involved in the twin paradox.

The phenomenon of relativistic time dilation was derived in an earlier section in the case of a starship moving across our line of sight, in which the clock appears to run more slowly. However, that does not mean that the clock will be seen to run slower in all circumstances, which is a common misconception about Special Relativity. If Alice's starship is moving toward Bob, her clock will seem to run faster than his. If Alice's starship is moving away from Bob, her clock will seem to run even more slowly than it would if her starship were running across his line of sight.

Discussion of this phenomenon requires an understanding of the "Doppler shift", which was well understood by classical physicists at the beginning of the 20th century. It is the change in pitch caused by the motion of an object. If a train approaching at high speed blows a whistle, the pitch of the whistle is higher than it would be if the train were at rest. Similarly, if the train is moving away, the pitch of the whistle is lower. If the time of the period of the whistle's wavelength when the train is at rest is T and the period when the train is moving at velocity V is Tm, then if the speed of sound is given by S, the ratio of the change in period due to the Doppler shift is given by:

   Tm / T = ( 1 - V/S )

-- if the train is approaching. If the train is moving away, the ratio is given by:

   Tm / T = ( 1 + V/S )

The classical Doppler shift can be redefined to apply to light with a minor change in variable definitions:

   Tm / T = ( 1 - Cf )      ! if approaching
   Tm / T = ( 1 + Cf )      ! if receding

The classic Doppler shift is not correct at relativistic speeds, since the time dilation factor SQRT(1 -Cf^2) applies and has to be also multiplied in. This gives:

   Tm / T = SQRT(( 1 - Cf ) / ( 1 + Cf ))

For a receding starship, the relativistic Doppler shift is:

   Tm / T = SQRT(( 1 + Cf ) / ( 1 - Cf ))

If we have a clock that emits a pulse of light on every tick, the pulse / tick period will decrease if the clock is approaching and increase if it is receding. As Alice and Bob are moving apart, they each see each other's clock running very slow -- but when they are coming together, they see the clock running very fast.

Here's the trick: Alice reaches Minbar after 17.3 years, and then turns around. The instant she does so, as far as she's concerned, Bob's clock stops running slow and starts running fast. As far as Bob is concerned, Alice turns around at Minbar after 20 years -- but since he's 10 light-years away, he won't see Alice's clock running fast until 30 years after she left Earth. Alice sees Bob's clock running slow half the time and fast half the time; doing the arithmetic shows she will count 40 years on his clock. Bob sees Alice's clock running slow 75% of the time and running fast 25% of the time; doing the arithmetic shows he will count 34.6 years on her clock.

* A graph known as a "spacetime diagram" or "Minkowski diagram" -- after Hermann Minkowski (1864:1909), a Russian expatriate who had been one of Einstein's instructors at the Swiss Federal Institute of Technology (ETH in its French acronym) in Zurich -- can be used to help visualize the scenario. It's just a graph with time on the vertical axis and space on the horizontal axis, with the time and space defined for one observer's frame of reference, in this case the Earth's.

Twin Paradox & Minkowski diagrams

If the time axis is measured in years, the space axis is measured in light-years, and the two axes have equal increments, then light always travels at a 45 degree path on the diagram. An object at rest in the Earth's frame of reference follows a vertical path, since it passes through time but not space, while an object moving closer and closer to the speed of light approaches the 45 degree angle. Minkowski diagrams are often used in texts on relativistic physics.

* As proven above, measurements by Alice and Bob of each other's clock ticks show that time passes more slowly for her -- she ages less -- than Bob. However, the confusing factor remains that both know for a fact that each other's clock is running slow.

During the trip to Minbar, Alice, by her own clock, determines that 17.3 years have passed. However, she knows that, in spite of the fact that "blueshifting" makes the clock on Minbar appear to run fast, it is actually running slow due to time dilation by the factor of 1.15, and so it only counts 17.3/1.15 = 15 years during the course of her outbound journey. Since the clocks on Earth and Minbar are synchronized in their frame of reference using coded signals, then when Alice departed from Earth the clock on Minbar read Tz. When she arrived at Minbar, it read Tz + 20, not Tz + 15.

The answer should be apparent, though maybe not obvious, given an understanding of relativistic simultaneity. Notice the qualification that the clocks on Earth and Minbar are synchronized in their own frame of reference. They are not synchronized in Alice's frame of reference, with the difference in synchronization is given by, as before:

   Td = L * Cf / C

If L is specified in light-years and Td in years, this simplifies to:

   Td = L * Cf 

Given that L is ten light-years and Cf is 0.5 C, that means that as far as Alice is concerned the clock in Minbar is running 5 years ahead through the entire outbound journey. It times through 15 years during her trip, and ends up at a value of 20 years.

By the same logic, while Alice spends 34.6 years on her interstellar round trip, she knows that Bob's clock is running slow by a factor of 1.15 and so only counts through 34.6/1.15 = 30 years. However, Alice got five years out of sync with the rest of the local Universe, and when she turns around, she gets out of sync with it by another five years, for a total of ten years, which means that when she arrives back home Bob's clock has counted 40 years since her departure.

Yes, Alice and Bob do see each other's clocks running slow, but there's a synchronization issue, somewhat along the lines of the Tunnel Paradox. By accelerating to 0.5 C, Alice effectively advances the time she sees on Bob's clocks, so the end result is that time passes less swiftly for her than it does for Bob.



* All the concepts discussed so far in this document were more or less laid out in Einstein's 1905 paper. In 1906, almost as an afterthought, he followed up with another paper that discussed mass and energy in the context of relativistic physics.

As Einstein discovered, not only did time and length change with velocity, mass did as well. Suppose Alice's starship zips by Earth at half the speed of light, to be observed by Bob on Earth. At closest approach, the starship and the Earth observer are separated by a distance D. Bob is feeling rude and hostile, and decides to shoot a cannonball of mass M at Alice so that it will hit the side of her starship at closest approach.

Bob fires the cannonball. We assume it travels at a low (non-relativistic) constant velocity V as measured by him. The flight time of the cannonball is given by:

   T = D / V

Flipping this around gives the velocity as, of course:

   V = D / T

Alice observes the same distance D as Bob, since it's at a right angle to her line of motion and no length contraction occurs. Alice unsurprisingly gets annoyed at being fired on, and so fires a cannonball of the same mass M and velocity V, as measured in her frame of reference, to hit the Bob's cannonball in an absolutely precise head-on collision at D/2, and bounce it back at the Earth observer in an elastic collision.

In principle, both cannonballs hit each other at the low velocity V and bounce away at the same low velocity V. The starship's relativistic velocity Cf, imparted to the cannonball, does not directly affect the collision, since it takes place at a perfect right angle to the direction of the starship's motion. However, there's an indirect effect. Both Bob and Alice realize after they fire their cannonballs their clocks are running slow relative to each other. Since the distance D is not affected by their relative velocity, but the flight time is longer, both see the other's cannonball as moving at a velocity Vm given by:

   Vm = V * SQRT( 1 - Cf^2 )

This means that each should expect that their cannonball is moving faster than the other (at velocity V instead of Vm) and so will have greater momentum. Bob thinks that Alice will see her cannonball bouncing back faster after the collision, while his cannonball bounces back slower; Alice naturally thinks the reverse. The problem is that they can't both have it that way, and the only logical conclusion is that the momenta will in fact cancel out.

Since momentum is simply the product of mass times velocity, and since Bob and Alice see a different and lower velocity of the other's cannonball, then the only way out of the contradiction is to assume that the mass increases by the same factor. That is, Bob sees that Alice's cannonball has increased from its "rest mass" or "invariant mass" to a larger "relativistic mass" by the factor:

   Mm = M / SQRT( 1 - Cf^2 )

Alice sees Bob's cannonball as having increased in mass by the same amount. In other words, mass increases as an object approaches the speed of light. By the way, this formula implies that as speed of light is approached, the mass increases towards infinity. Since force is equal to mass times acceleration, the amount of force required to get further acceleration also increases towards infinity, and the amount of energy required increases towards infinity as well. Incidentally, strictly speaking it isn't the mass that increases, it's just the momentum -- but that's fine print that doesn't fit into a short document, and so mass it remains here, as a workable simplification.

relativistic mass increase

* An analysis of relativistic mass increase leads to an interesting result. Intuitively, the fact that our starship's mass increases as more energy is pumped into it suggests a deep connection between mass and energy. In fact, an analysis of the energetics of a relativistic object leads to:

   energy = mass * speed_of_light^2

-- which is just Einstein's famous equation "E=MC2". Mass-energy equivalence leads to an extension of the law of conservation of energy, in which mass-energy is conserved, with the two forms converted from one to the other under specific circumstances. The idea was not actually entirely new; Newton had speculated that light could be interconvertible with matter, and E=MC2 can be easily derived from Maxwell's equations if someone knows where to start. However, Einstein was the first to clearly articulate the principle.



* Nobody can study relativity without having the suspicion that it's all trickery, but observations do back up theory. The first observation of course was the Michelson-Morley experiment, which proved there was no ether wind. More modern experiments have failed to find any ether wind either, and no more plausible or economical theory than Special Relativity has been devised to account for that fact. The speed of light is a constant, and nobody has been able to measure any variation from that speed.

Time dilation has been confirmed in the case of certain unstable elementary particles. When moving at relativistic speeds, their decay time increases over the value at rest by the factor:

   1 / SQRT( 1 - Cf^2 )

Similarly, observations confirm that particles such as electrons moving at relativistic speeds are more massive than they are when at rest, by the same factor. Mass-energy equivalence has been frighteningly demonstrated by fusion weapons.

One particularly significant proof is the relationship between magnetism and electricity. A magnetic field is caused by a moving current and exists at a right angle to that current. In fact, there actually is no such thing as a distinct magnetic force, it's just a relativistic manifestation of electric forces.

Imagine a loop of wire carrying a current. If the current is large enough and the wire is limp enough, the moving current will set up a magnetic repulsion that stretches the loop into a neat circle. In principle, the electric charge of the wire should be neutral overall, but if electrons are moving through the loop, then electrons on one side of the loop will see electrons in motion on the other side of the loop. This relative motion leads to a length contraction of the electron stream on the other side of the loop that is greater than the length contraction of the metal matrix it is flowing through. This means that the other side of the loop appears to have a net negative charge and repels the electrons on the first side of the loop. The length contraction is very small, but electric forces are very powerful, and only a small length contraction is enough to set up a tangible force.



* One thing that is apparent is that Special Relativity's modifications to traditional concepts of space and time are not arbitrary. They follow specific rules and have to add up according to them. Under these rules, the three spatial dimensions are modified in a way that interacts with modifications in time. Put another way, Special Relativity deals with the properties of four-dimensional "spacetime", while classical physics deals with three-dimensional space and time as independent physical properties.

It was Minkowski who suggested the notion of spacetime, introducing the Minkowski diagrams shown previously, in a lecture delivered in 1908 and published in 1909 shortly after his death. The idea had certainly been implicit in Einstein's 1905 paper, but Einstein had taken a more strictly mathematical approach. It would take him some time to appreciate Minkowski's idea.

What led Einstein to this appreciation was his attempt to reconcile gravity with Special Relativity. Special Relativity ignores the effects of gravity, and that means that the Earth or any other object with an appreciable mass and gravitational effect can only be considered a frame of reference by ignoring gravity and simply assuming everything is glued to its surface. In 1907, Einstein began to work on fixing the omission of gravity.

The first thing he did was extend his relativistic equivalence principle to scenarios in which gravity is involved. Suppose Alice is in a cube with no windows, drifting in free space, weightlessly floating around inside the cube; and then suppose Alice's cube is being pulled down from space by the force of gravity towards the surface of a planet. Since she is being accelerated at the same rate as the cube, she will still be floating around weightlessly and will have no perception that she is moving. She will be in "free fall".

In broader terms, Alice will observe exactly the same physical laws inside her cube no matter if it is floating free or being drawn down in a gravitational field. On the other side of the coin, if Alice's windowless cube is being accelerated through free space at 9.81 meters per second, everything will appear to her just as if the cube were sitting motionless on the surface of the Earth.

Einstein saw the scenario in more commonplace terms, thinking of somebody falling off of a roof. One of the first implications that Einstein pulled from this concept was that time would slow down in a gravity field. Suppose Bob is at the front of a long spaceship and Alice is at the rear. Suppose further that they send light pulses to each other every second, by their own individual clocks. If the spaceship is not accelerating, Bob will receive one pulse from Alice every second, and Alice will receive one pulse from Bob every second.

However, if the spaceship is accelerating, say at 9.81 meters per second per second, Bob will receive Alice's light pulse on an interval longer than one second, and Alice will receive Bob's light pulse on an interval shorter than one second. By the equivalence principle, there is no way to distinguish between this situation and the long spaceship sitting on the Earth, in a gravity field with an acceleration of 9.81 meters per second per second. Alice's clock runs slower than Bob's. One of the implications of this is that light would also be "redshifted" coming out of a gravity well, just as if it were undergoing a Doppler shift after being emitted by a source moving away from an observer.

* Einstein then dropped the matter for a time, returning to it in 1911. In his earlier work on the subject, he had ignored the issue of tidal forces. Tidal forces arise because gravity falls off with distance, and so there is a differential gravitational force on an extended object that sets up a stress in it from front to back, and also tends to squeeze it in towards the middle.

Einstein had previously assumed gravity worked on all parts of an object in its field uniformly. This is close enough to the truth with Alice in a cube in the gravity field of a normal planet, but not true at all for a very large object in the gravity field of a planet, or for a small object like Alice's cube in a close vicinity of a huge, very dense mass. If Alice was in her windowless cube and was being pulled towards such a mass, she could tell she was actually in a gravity field because she would be stretched and squeezed, warning her to quickly get her cube out of the potentially deadly fall towards the object.

How could this be reconciled with the equivalence principle? After about a year's work, he finally adopted Minkowski's notion of spacetime. Einstein worked out that a warpage of spacetime could deal with the matter of tides, or in more general terms that gravity was actually a warpage of spacetime caused by a mass.

Suppose Alice and Bob are in their cubes and have been flying unpowered through deep space a good distance apart with no relative motion; during their flight, they follow precisely parallel lines. Suppose, then, that they come up on a planet, with Bob flying along a line that would take him along one side of the planet, and Alice flying along a line that would take her along the other side of the planet. What happens then is that they both are pulled toward the planet and their paths begin to converge. According to Einstein, the flight of Alice and Bob is continuing on path, it's just that the spacetime path has been distorted by the mass to bring the two paths closer together. They are following the shortest path or "geodesic" through spacetime. Tidal effects become noticeable when the spacetime curvature is large relative to the object in question.

In most elementary discussions of General Relativity, the analogy is made with a rubber sheet on which heavy balls are placed. The balls sink into the sheet, distorting it in a way analogous to the way masses reside in a distortion of spacetime in their vicinity. It is important to remember that this distortion is not caused by gravity; it is gravity. The paths of objects moving across the sheet will curve along with the distortions.

warped spacetime

In fact, light itself will follow curved paths through this distorted spacetime. This also demonstrated the equivalence with accelerated motion, since a light beam flashed across an accelerating starship would curve across the starship's direction of motion. Other clauses of General Relativity state that, as described above, clocks will slow down in a gravitational field and that light will be redshifted coming out of a strong gravitational field.

* The description of General Relativity given above gives a highly simplified view of an extremely difficult theory, one which Einstein himself said made Special Relativity look like "child's play". Sir Arthur Stanley Eddington (1882:1944), a prominent physicist who was a pioneer in understanding stellar evolution and one of General Relativity's early backers, was asked once by a reporter if it were true that only three people in the world understood the theory. Eddington simply stared off into space in an abstracted fashion. When asked what the trouble was, he replied: "I'm trying to think of who the third person might be."



* General Relativity's analysis of the behavior of gravity differs from that of Newton's classical theory in a few subtle respects. General Relativity was used to account for small discrepancies in the orbit of Mercury that could not be explained by classical theory. The bending of starlight predicted by General Relativity was also confirmed during an eclipse of the Sun in 1919 -- though later analysis of the photographs taken showed the errors in the measurements were of the same magnitude as the effect being measured.

Later, more reliable estimates definitely proved the predictions of General Relativity. The bending of the images of distant galaxies around large intermediate galaxies to create "double" images is very well established in the present day, and in fact has been used as a tool in deep-space surveys. Carefully conducted lab experiments also confirm relativistic effects. In recent precision experiments conducted by the US National Institute of Standards & Technology (NIST), a precision clock was raised 33 centimeters (about a foot) and measured to be running slower by a rate of about a 90-billionth of a second in 79 years. In another part of the experiment, an aluminum ion being used as a clocking element was put into cyclical motion using an electric field, with the "clock" becoming slower once it was in motion.

* One interesting prediction of General Relativity is that if a massive object becomes sufficiently dense, spacetime will curve around it and light will no longer be able to escape from it. The object would become a "black hole" in space, the mass collapsing in on itself forever to form a "singularity". Such objects could result from the explosive collapse of very large stars. Einstein himself was reluctant to believe that they were possible, even writing a paper in 1939 to prove they couldn't exist.

A few decades later, physicists returned to the matter and demonstrated that black holes may well exist after all. Since then, a number of star systems have been discovered where one of the stars is so massive and compact that theory hasn't got a better explanation than that it is a black hole. Supermassive black holes with the mass of millions or even hundreds of millions of Suns are believed to commonly reside at the center of galaxies.

a hole in space

Black holes have very interesting properties. Suppose Alice sends a robot named Robby on a spaceship to investigate a black hole. As he descends deeper into the gravity well of the black hole, his clock will slow down compared to Alice's clock, and any light emitted by his ship will be increasingly redshifted. He will also suffer from increasing tidal forces that will threaten to rip him from head to foot; these tidal forces will be more pronounced for a small black hole than a large one, since the gravitational gradient is steeper for a small hole than a big one.

At a certain critical distance from the center of the black hole, light will no longer be able to escape, and so if Robby the robot goes below that critical distance, neither will he. This critical distance is known as the "Schwarzchild radius" -- after the physicist Karl Schwarzchild (1873:1916), who came up with the notion of a black hole in 1915 -- and the "surface" defined by this radius is known as the "event horizon". Once Robby falls below the event horizon, there is no way for him to escape or even communicate with the rest of the Universe, since any signals he tries to send back up will not be able to get out of the plummeting gravity well. As far as Alice was concerned, poor Robby disappeared from view and will never re-emerge.



* In 1999 I released the first revision of the document ELEMENTARY RELATIVISTIC PHYSICS, which ultimately evolved into a fairly elaborate and lengthy description of the subject. I was pleased with the document, but then I got to wondering if it wasn't too eye-glazing for a lot of readers. As a result, I decided to render it down into a one-chapter primer for those who were interested in the subject but didn't want to do a lot of work. Those who have read this primer and want to learn more can go on to the parent document for further details. Since all this document amounts to is a heavily abridged version of the parent document, no sources are cited here.

* As concerns copyrights and permissions for this document, all illustrations and images credited to me are public domain. I reserve all rights to my writings. However, if anyone does want to make use of my writings, just contact me, and we can chat about it. I'm lenient in giving permissions, usually on the basis of being properly credited.

* Revision history:

   v1.0.0 / 01 may 07 
   v1.0.1 / 01 jan 08 / Review & polish.
   v1.0.2 / 01 sep 08 / Review & polish.
   v1.0.3 / 01 nov 09 / Updates of in Michelson-Morley experiment.
   v1.0.4 / 01 feb 10 / Review & polish.
   v1.0.5 / 01 mar 10 / Follow-on update to v1.0.4.
   v1.0.6 / 01 jun 11 / Review & polish.
   v1.0.7 / 01 mar 12 / Clarification on relativistic mass / momentum.
   v1.0.8 / 01 dec 13 / Review & polish.
   v1.0.9 / 01 oct 15 / Review & polish.
   v1.1.0 / 01 sep 17 / Review & polish.