This chapter expands on those ideas into more difficult areas, and in a less complete fashion. It concludes with a broad description of General Relativity, a subject whose mathematical complexity rules out discussion in detail in this document.
A second relativistic paradox concerns the relativistic behavior of three independently moving objects. The twin paradox concerns the motion of only two, with an Earth observer and a spaceship crew watching each other moving at the exact same relative velocity. Add another object with a motion of its own, and things become more complicated.
The Earth Federation receives a message from an outpost four light years away, saying that they have spotted a spaceship approaching them on a flight path towards Earth and moving at half the speed of light. The crew of the outpost identifies the ship as being a type built by "Daleks", an evil alien race determined to exterminate other species.
Since the signal from the outpost took four years to reach Earth, that means that the Dalek attack vessel has flown two more light years at half the speed of light, and is now only two light years from Earth. However, the Earth Federation manages to quickly modify a starship already under construction as an interceptor, and launches it at half the speed of light to stop the Dalek vessel.
Two years in Earth time after the interceptor is launched, the Dalek vessel has traveled one more light year and the interceptor has traveled one light year, closing the gap between them. The interceptor hits the Dalek vessel at a tremendous relative speed, destroying it and saving the Earth.
This can be shown with a little simple reasoning. The Dalek vessel is observing the Earth and can see the launch of the interceptor as a bright flash. From the point of view of an Earth observer, the flash reaches the Dalek ship 4/3rds of a year in Earth time after launch of the interceptor, during which time the Dalek vessel has traveled 2/3rds of a light year closer to Earth.
However, the interceptor has only traveled 2/3rds of a light year from Earth, and so the Dalek vessel and the interceptor are still 2/3rds of a light year apart. The flash emitted from the launch of the interceptor has preceded the interceptor from Earth, and since the Daleks know the flash moves at the speed of light, obviously the Daleks do not see the interceptor as coming at them at the speed of light.
Let's assume the following variables. First, let's define velocities in terms of fraction of lightspeed to simplify calculations:
Next, we want to determine how time appears to change for the different observers. Assuming that the interceptor emits a pulse of light, or a "clock tick", every second in its own frame of reference, then we can define the tick times seen in other frames of reference as:
The interceptor emits a pulse every second in its own frame of reference, but
in that interval it moves toward the Dalek ship. If the ship was stationary,
a Dalek would see the tick as being shorter than one second, due to the
relativistic Doppler shift. The length of this short tick is given by:
Ti = SQRT(( 1 - Cfi ) / ( 1 + Cfi ))
Now, from the point of view of the crew of an outpost, the distance between
the light pulses as they travel through space is:
Ti * C
Td * Vd = Td * C * Cfd
Td * C
This means that:
Ti * C = Td * C + Td * C * Cfd = C * Td * ( 1 + Cfd )
Td = Ti / ( 1 + Cfd )
Tid = Td * SQRT( 1 - Cfd^2 )
Since the interceptor is emitting a pulse every second, Tid gives the
relativistic Doppler shift of the tick signals coming from the interceptor:
Tid = SQRT(( 1 - Cfid ) / ( 1 + Cfid ))
The rest is simple algebra. We now have to get rid of Tid, Td, and Ti and
express Cfid in terms of Cfi and Cfd. First, let's rearrange this expression
to give Cfid in terms of Tid:
Tid^2 = ( 1 - Cfid ) / ( 1 + Cfid )
Tid^2 * ( 1 + Cfid ) = 1 - Cfid
Tid^2 + Tid^2 * Cfid = 1 - Cfid
Tid^2 * Cfid + Cfid = 1 - Tid^2
Cfid * ( Tid^2 + 1 ) = ( 1 - Tid^2 )
Cfid = ( 1 - Tid^2 ) / ( 1 + Tid^2 )
In either case, the relative velocity of the two spacecraft is not greater than the speed of light. Even if both were moving towards each other at 90% of the speed of light relative to an observer on an outpost, their relative velocity would still be only 99.4% of the speed of light.
Tid = Td * SQRT( 1 - Cfd^2 )
Td = Ti / ( 1 + Cfd )
Ti = SQRT(( 1 - Cfi ) / ( 1 + Cfi ))
Tid^2 = Td^2 * ( 1 - Cfd^2 )
Td^2 = Ti^2 / ( 1 + Cfd )^2
Ti^2 = ( 1 - Cfi ) / ( 1 + Cfi )
Tid^2 = ( 1 - Cfd^2 ) * ( 1 - Cfi ) / (( 1 + Cfd )^2 * ( 1 + Cfi ))
= ( 1 - Cfd ) * ( 1 - Cfi ) / (( 1 + Cfd ) * ( 1 + Cfi ))
= ( 1 - Cfi - Cfd + Cfi * Cfd ) / ( 1 + Cfd + Cfi + Cfd * Cfi )
Cfid = ( 1 - Tid^2 ) / ( 1 + Tid^2 )
1 - ( 1 - Cfi - Cfd + Cfi * Cfd ) / ( 1 + Cfd + Cfi + Cfi * Cfd )
= -----------------------------------------------------------------
1 + ( 1 - Cfi - Cfd + Cfi * Cfd ) / ( 1 + Cfd + Cfi + Cfi * Cfd )
( 1 + Cfd + Cfi + Cfi * Cfd ) - ( 1 - Cfi - Cfd + Cfi * Cfd )
= -----------------------------------------------------------------
( 1 + Cfd + Cfi + Cfi * Cfd ) + ( 1 - Cfi - Cfd + Cfi * Cfd )
2 * ( Cfi + Cfd )
= -----------------
2 + 2 * Cfi * Cfd
= ( Cfi + Cfd ) / ( 1 + Cfi * Cfd )
Cf12 = ( Cf1 + Cf2 ) / ( 1 + Cf1 * Cf2 )
Cf12 = ( Cf1 - Cf2 ) / ( 1 - Cf1 * Cf2 )
( 0.8 - 0.5 ) / ( 1 - 0.8 * 0.5 ) = 0.50
One thing that is apparent at this point 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 dealt with three-dimensional space and time as independent physical properties.
Spacetime is a concept that can be and often is belabored because it sounds mysterious. Special Relativity is in fact mysterious in many ways, but these mysteries can be explored using the basic rules and physics, and making too much of exotic terminology tends to obscure the details rather than make them clearer.
Suppose Alice's spaceship zips by Earth at half the speed of light in front of Bob on Earth. At closest approach, the spaceship 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 spaceship 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
V = D / T
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. (They haven't been able to afford photon torpedoes yet.)
In principle, both cannonballs hit each other at the low velocity V and bounce away at the same low velocity V. The spaceship's relativistic velocity Cf, imparted to the cannonball, does not directly affect the collision, as it takes place at a perfect right angle to the direction of the spaceship'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 )
Since momentum is simply the produce of mass times velocity, and since each
sees 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 in mass by
the factor:
Mm = M / SQRT( 1 - Cf^2 )
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.
In fact, a detailed analysis of the energetics of a relativistic object, too
complicated to be repeated here, leads to Einstein's most famous equation:
energy = mass * speed_of_light^2
However, from the frame of reference of Alice, they observe the Universe moving closer to the speed of light, with the entire Universe undergoing a relativistic mass increase. Since the rest mass of the Universe is vastly greater than that of the spaceship, the absolute mass increase of the Universe is proportionally vastly greater as well.
This might seem to be a contradiction. The energy pumped into accelerating the spaceship leads to an increase in the spaceship's mass, but the entire Universe can be perceived by Alice to grow in mass by the same proportion. Where did the energy come from to create the vastly greater relativistic mass increase of the rest of the Universe?
Suppose that we're standing off a constant distance from the hulk in our
mothership and launch the missile. It accelerates to a velocity of 100
kilometers per second, and collides with the hulk, destroying it. The
classical expression for kinetic energy gives the energy of the impact as:
(1/2) * missile_mass * missile_velocity^2
(1/2) * 1,000 * 100,000^2 = 5,000,000,000,000 joules
From the point of view of Galilean Relativity, this impact could be just as easily described as the missile standing still and being struck by the hulk moving at 100 kilometers per second. However, the kinetic energy of the hulk moving at 100 kilometers per second is 10,000 times greater than that of the missile. Where did all this kinetic energy come from?
This question proves to be silly, if a simple fact from classical physics is remembered: a force causes a change in motion. The energy expended by the missile causes a change in the motion of the missile, not that of the spaceship hulk. The fact that from the missile's point of view, the hulk has 10,000 times more kinetic energy is simply a matter of perspective. And conservation of momentum ensures that the collision won't release any more energy than possessed by the missile -- it's annihilated, but the hulk is only barely slowed down,
You have not in any way expended any energy on the mass, you have simply used (or in the strict physical sense, released, as you descended) energy to change your position relative to it, and the relative potential energy of the mass due to your change in position is far greater than the energy you released to change your position.
This argument is constructed strictly in terms of simple classical physics, but it also applies to relativistic physics, if the fact is understood that relativistic mass increase is an aspect of an increase in kinetic energy.
Yes, if you are flying through the Universe at an increasing fraction of the speed of light, from your frame of reference the entire Universe gains mass. However, the energy you are expending only changes your own motion, not that of the rest of the Universe, and all you are doing is changing your energetic "perspective" relative to the rest of the Universe. As the twin paradox showed, the equivalence between different frames of reference does not imply a level balance.
Suppose, as one such example put it, two objects of equal rest mass are approaching each other at half the speed of light in an Earth observer's frame of reference. They perform an inelastic collision and stop dead. Momentum is conserved in both classical and relativistic terms.
Now suppose a spaceship is moving alongside one object. The object appears to be motionless, while the other object is approaching at 80% of the speed of light. After impact, the combined mass remains in the frame of reference of the Earth, and so is moving away from the spaceship at 50% of the speed of light.
By classical terms, the momenta do not add up in the spaceship's frame of reference, and so the relativistic mass increase must be invoked.
Unfortunately, I found that just plugging the revised numbers into the example didn't work, either. To the Earth observer, when the objects were moving together at half the speed of light each, their mass was increased by 15%. What I had assumed was that when the two objects hit each other in the Earth observer's frame of reference, they returned to their rest mass. However, in the spaceship's frame of reference, the math wouldn't work out unless I assumed that the combined mass was also 15% greater than the rest mass.
This made a certain amount of sense, because the mass couldn't simply vanish. But what would actually happen to it?
The mass increase of the two objects was equivalent to the energy pumped into them to get them to half the speed of light. Bring them to an abrupt halt, and they do return to their rest mass, but they have to shed that energy to do it. You do not want to be anywhere near the fireball that would result from the instantaneous conversion of 15% of any appreciable amount of mass into energy.
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.
Time dilation has been confirmed in the case of certain unstable elementary particles. When moving at relativistic speeds, their decay time increases over their decay time when 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.
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 positively-charged 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 electrostatic forces are very powerful, and only a small length contraction is enough to set up a tangible force.
There really isn't much choice. We don't actually perceive objects moving at high velocity in our daily environment. They're either too small, or too big and far away. If objects big enough to be seen were to pass by close enough to be observed on Earth, they would be too fast to be perceived, and in any case would destroy everything in their path as their shock wave passed through the atmosphere.
Computer graphics simulations have been designed to allow visualization of the outside world as it would appear if we were moving through it at relativistic velocities. Such programs can be tricky to write, since at such high velocities more strange things happen than we have considered. For example, light takes longer to reach us from the more remote parts of an object than from the nearer parts. This is of no consequence at low velocities, but at relativistic velocities, we will see the more distant parts lagging behind the nearer parts along the line of motion, distorting their image.
An English physicist named Eric Sheldon wrote a computer program named STELLA that shows what the Universe would look like as we accelerated through it at one gee acceleration indefinitely. This may not seem like a great acceleration, but it would be enough to push our spaceship so close to the speed of light that time dilation would allow us to traverse across the Universe some 18 billion light years in only 23 years of our time.
Some of the things that we would see on this journey as we pushed closer and closer to the speed of light would be expected. We would see stars in front of us become bluer due to the Doppler shift, until they vanished into the ultraviolet. Those behind us would become redder until they were lost into the infrared.
Less intuitively, as velocity increases, stars in front of the ship appear to crowd closer and closer to a point directly ahead of the ship. Furthermore, time lags mean that stars that would have been observed behind the ship start to migrate into the forward field of view. Similarly, a dark region appears directly behind the ship.
With increasing speed, the dark region grows as the visible Universe crowds itself in front of the ship, until at extremely high relativistic velocities it crowds itself into a small area in front of the ship and then disappears into darkness as the starlight shifts into the ultraviolet. Eventually, the low-level microwave background radiation that pervades the Universe will be shifted into the visible range and will appear as a blue dot in front of the ship.
This distinction sounds bizarre, but it is simple. Mass is defined in
classical physics as the ratio of force to acceleration of an object:
mass = force / acceleration
gravitational_force = constant * mass1 * mass2 / distance^2
electrostatic_force = constant * charge1 * charge2 / distance^2
With gravity, in contrast, both the (gravitational) force and the acceleration depend on the mass. Einstein that this was no coincidence, that acceleration and gravity were actually the same thing. There was absolutely no way to determine the difference between being in a box accelerating at 9.81 meters per second and the same box sitting on the surface of the Earth.
Einstein merged the concepts of four-dimensional spacetime that he had devised for Special Relativity with the principle of equivalence of acceleration and gravity. He noted that accelerated motion of a mass caused a distortion of spacetime. Very well, then, a mass was associated with a distortion of spacetime as well.
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.
One of the implications of this was that light will follow curved paths through this distorted space. This also demonstrated the equivalence with accelerated motion, since a light beam flashed across an accelerating spaceship would curve away from the spaceship's direction of motion. Other clauses of General Relativity suggest that light will be redshifted coming out of a strong gravitational field, and that clocks will slow down in a gravitational field.
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, and was one of the first proofs of the concept.
The bending of starlight predicted by General Relativity was also quickly confirmed during an eclipse of the Sun, though there are questions about the quality of these measurements. That's an academic concern, however, because the bending of the images of distant galaxies around large intermediate galaxies to create "double" images is very well known in the present day, and in fact has been used as a tool in deep-space surveys.
General Relativity also leads to a few predictions that could not have been considered with classical theory. It suggests that the collapse of massive objects could set up a distortion of spacetime that propagates over great distances, and could in principle be detected. Work on gravitational wave detectors has proceeded in a slow fashion and with uncertain results over the past few decades, but it is likely that workable gravitational wave detection systems will be available in the near future.
Another prediction of General Relativity is that if a massive object became 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. Such objects could result from the explosive collapse of very large stars, and various multiple star systems have been discovered where one of the stars is so massive and compact that theory cannot describe it as anything but 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. Absolutely proving that black holes exist is very difficult, since they emit no light and cannot be observed directly, but continued observations and theoretical studies are making their reality steadily more convincing.
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."
Anybody with much familiarity with physics knows that Mr. Wright's question has an answer, but it isn't easy to understand, and even when understood, it is hard to explain to anyone else. Most people are likely happier not trying to understand relativity.
The math isn't complicated, at least not for Special Relativity, but it requires a grasp of different ways of thinking. Studying it leads to the suspicion that somebody's trying to trick you, but the logic hangs together, and in fact the classical view of things has illogical features that can't be resolved.
This short document only traces a path through the forest. Playing with the large number of puzzle problems available in formal texts on the subject would likely give those sufficiently interested a better conceptual grasp of the matter, and I suspect eventually that a fair number of computer videos illustrating the appearance of the Universe near lightspeed would help the rest of us who are somewhat lazier.
However, relativity still reminds me of an old Warner Brothers cartoon in which Foghorn Leghorn, that loudmouthed fool of a rooster, is trying to play with a kid chicken who would really rather be studying mathematics and physics.
As best I recall it, Foghorn persuades the kid to play hide-and-go-seek, and while the kid is counting, Foghorn sneaks through the barnyard and hides in a bin. "Heh!Heh!Heh! He'll nevah find me in hyar!"
The kid finishes counting, does some computations on a piece of paper, walks out to the middle of the barnyard, checks his calculations, draws a cross on the ground, goes and brings back a shovel, pushes it in the ground, leverages it and ... POP! Foghorn appears out of the ground on the shovel blade.
"WHO?! WHAT?! WHERE?! WHEN?! HOW?! WHY?!" The kid shows him the calculations and Foghorn backs up, waving his hands. Foghorn goes over to the bin and puts his hand on the lid, then says: "Nah, bettah not check. Ah might actually still BE in thar!"
I actually didn't do more than skim a few sources to write the bulk of this document. I originally tried to understand relativity back when I was about 20 or so in the early 1970s. I didn't really understand much then, except for the vital principle of the constancy of the velocity of light, from which all else flows.
I went back to it in the mid-1980s and created a set of hand-written notes. They were taken very directly from the textbook I was using, and though I did acquire an understanding of time dilation, for the other parts I only managed to get a grasp of mechanics of the math.
This third pass seems to have completed the circle, and I feel I have acquired a reasonable basic understanding of the essential elements of special relativity. It was a terrible headache to write, however. The initial editions had a few sections that I looked at later, and had to conclude were so muddled I had to say to myself: "What was I THINKING?!" I deleted them in v1.2.
As far as General Relativity goes, though, I recall poking around with tensor calculus when I was in college and backed off in dread. I think I've learned as much about General Relativity as I'm going to.
I think many textbooks tend to make relativity harder than it needs to be. One of the major things I learned in this pass was something I hadn't even considered before, what I called the "three-way paradox" in this document. I started trying to understand relativistic mass increase and found I was missing something, and had to backtrack to understand the three-way paradox.
This turned out to be the hardest part of the job, and it took me most of a Sunday to nail it down. The matter is discussed in all the texts, but they approach it by taking the relativistic equations for space and differentiating them with respect to time. This is mathematically elegant, but it gave me no visualization of what was actually happening. I chose instead to derive it from first principles using geometry and algebra.
v1.0 / 01 jul 99 / gvg
v1.1 / 01 aug 99 / gvg / Tidying and refinements.
v1.2 / 01 jan 01 / gvg / Cut out a few muddled sections.
v1.0.3 / 01 may 02 / gvg / Minor update.