Einstein's Physics of Relativity 
    Cosmology is the direct derivative of the relativity theory developed 
by Einstein in the 1900s and 1910s. Yet home astronomers for the most part 
have no grounding in it. True, most never were treated to Einstein physics 
in their schooling. But they generally do not acquire on their own a 
tuition in Einstein physics any where as easily as they do in astronomy. 
For one thing there are no Einstein physics clubs or Einstein sections in 
astronomy clubs! 
    Efforts to learn about relativity often end in utter disgust. There 
are plenty of self-teaching books on relativity, yet so few are successful 
at giving the home astronomer a tuition sufficient to understand cosmology 
or even just astrophysics. 
    Yet for any furtherance in astronomy the home astronomer 
absolutely has to embrace Einstein physics. For without it the world 
is in deed quite incomprehensible and cosmology becomes essentially 
    Here we can not cover any great fraction of relativity. We do explore 
a few selected topics of immediate application to astronomy. In doing so 
we show some of the hazards home astronomers must watch for in their own 
learning efforts. 

Time and Space
    In ordinary physics and geometry, that of Newton and Euclid, space and 
time are absolute parameters. The regime of space and time for any one 
person is the same for all other persons. A coordinate of an event 
measured by a one observer is valid for all observers. All classical 
mechanics is founded on this absolute nature of space and time. 
    Under Einstein, on the other hand, time and space are not the same for 
all persons. They have different measurements depending on the motion of a 
one person relative to an other. Thus, 'relativity' refers to the relative 
-- and not absolute -- character of time and space among observers in 
motion. This is a concept so alien to the common sense that home 
astronomers often simply can not deal with it. In all their earthly 
existence and experience, there is nothing what so ever to shake their 
visceral instincts that time and space are somehow external to the motions 
of persons who measure them. 
    But it is surprisingly simple to show that time and space as 
measured by two moving observers is in fact and deed not the same. 
Einstein broke the earthly mindset first by combining space and time 
into a single unified construct of four dimensions. Three are the 
ordinary XYZ space coordinate system and the fourth is time T. So far 
so good. The next step trips most people: All four dimensions are at 
right angle to each other. In the 3D world it is easy to see that the 
dimensions are orthogonal. Adding a fourth dimension perpendicular to 
the other three is beyond the visualization powers of most folk. It is 
just not at all common sense that time is perpendicular to space! 
    It is proven in all challenges and tests of Einstein physics that 
there are four dimensions and they are all orthogonal among themselves. 
Some physicists claim that there are more than four dimensions in the 
universe, of which the XYZT set is only a 'slice' of the multidimensional 
world. But so far there is no generally accepted evidence for any more 
than four dimensions. 

    Einstein in combining time with space made both peers in the 
dimensions. Time is no longer a separate thing from space. We call this 
amalgam of time and space 'spacetime'. There is one niggling defect. The 
XYZ dimensions of space are measured in units of length, meters, but the T 
dimension is measured in units of time, seconds. This makes it hard to 
manipulate them as equals in the maths of 4D geometry. To remove this 
defect we convert the time axis into a new space axis by applying the 
factor c to the units of time. c is the speed of light 3E8m/s. So all time 
measures are now space measures with the factor 3E8m equaling one 
second. The time axis in many treatments is called the cT axis to 
emphasize the embedding of c into it. An event in spacetime has four 
coordinates: X, Y, Z, and cT. 
    It is entirely true that 'seeing' four dimensions is hardly easy. 
Worse, it is just about impossible to use drawings and diagrams on two 
dimensional paper to illustrate 4D spacetime. What we often do is drop two 
of the space dimensions and take a slice in 2D using the remaining space 
and the one time dimension. For instance a X-cT slice maps to an ordinary 
two dimensional depiction. 

Natural Gravity
    Einstein first developed a preliminary physics called the Special 
Theory of Relativity in 1905. This worked out the behavior of spacetime in 
a world devoid of gravity. To be fair, Special Relativity never prevails, 
being that the universe is everywhere permeated by gravity. However, we 
can attain to an gravity-free space to any degree of approximation we want 
by two means. 
    First we can confine our experiments to a tiny volume in the 
gravity field so the field is uniform thruout that volume. In this 
case the gradient of the field is small enough to ignore. In a room on 
the Earth the Sun's gravity field is completely uniform; we can not 
sense any tides in the room caused by the Sun. 
    In the second case we can remove ourselves far from the attrahent 
body. Then the gradient is so shallow that it can be ignored. In the solar 
system if we stay a few tenths AU from the Sun or a few dozen radii from a 
planet we are pretty much removed for the tidal effects of that central 
    There are, of course, many situations where Special Relativity just 
doesn't cut it. There are tides in Earth's oceans from the Moon's gravity. 
The belly of Io suffers tides from Jupiter's gravity. Saturn's rings came 
from the tidal bursting of a too-close satellite. Contact binary stars are 
wildly distorted by their mutual tides. And so on. In these cases, where 
the gradient of gravity can not be neglected, the laws of General 
Relativity, issued by Einstein in 1915, apply. 
   When we speak of gravity we mean only natural gravity, that induced by 
mass. It can not be an artificial or simulated gravity like that of a 
spinning spaceship. Altho the riders do stick to the inner circumference, 
as if held there by gravity, there is no gravitational gradient. By a 
similar reasoning the thrust of the rockets to press the rider onto the 
back of the ship does not substitute for natural gravity. In the 
spaceship, then, Special and not General Relativity rules. 

Inertial Platform or Frame 
    A person in a gravity-free region of the universe has only forces 
acting on him due to inertia. The rotation of the spaceship and the 
propulsion by the rockets are inertial forces, not gravity. When these 
forces are turned off (rockets or spinning stops) there now are no forces 
on the observer. He procedes in a straight line at constant speed. His 
velocity vector is constant in both size and direction. 
    The terms 'frame' or 'platform' derive from the common picture of an 
observer carried in a vessel or vehicle. This emphasizes that within this 
structure, the frame or platform, the observer is stational. Because the 
observer travels thru space, the most prevalent vehicle is a spaceship, of 
the Buck Rogers sort, but there is no need for any enclosing structure 
around the observer. So long as he moves in linear uniform motion thru a 
gravity-free territory, he is by himself an inertial observer. 
    Many works on relativity fail to carefully define the inertial 
frame. Some allow that a person in free-fall within gravity, like a 
satellite in Earth orbit to be an inertial observer. ALtho within the 
satellite the Earth's gravity field is essentially uniform, this is 
due only to the utterly tiny volume of space occupied by the 
satellite. Recall that the [un]tethered satellite deployed by the 
Shuttle in 1996 actually relied on the gradient or tidal effect of the 
Earth's gravity! 

Impossibility of Absolute Rest 
    On the Earth the ground is a natural foundation against which to 
measure motion. A person standing still on the ground is absolutely at 
rest while one in motion is really moving. Away from a ground things are 
not so simple. Galileo in the 1610s was first to show that without the 
external reference frame of the ground it is impossible to determine which 
of two inertial platforms is at rest and which is moving. 
    Each may perform various experiments to establish absolute motion or 
station without success. Each platform may declare itself stational and 
the other motional. Each measures all other inertial frames as being 
motional relative to itself. This is the source of the term 'relativity' 
to describe the Einstein physics. The new physics is based on the 
relativity -- not absolutivity -- of motion among inertial platforms. 
    The experiments a person can do in his frame rely on the laws of 
physics to work. The idea is to examine these laws and see how they differ 
among the frames. If there is a difference between the way things behave 
in a one frame as against an other, then that frame can declare itself 
absolutely at rest because of such-&-such phaenomena that occur in it and 
not in the others. This does not happen. For all inertial frames the way 
things happen is identicly the same. There is no difference among inertial 
frames and so no one can distinguish any peculiar one as the true place of 
rest against all the others. 
The Speed of Light
    For ages light was treated as a distinct physical phaenomenon, 
like sound and inertia. One early discovery was that it takes time for 
light to travel from its source to a receiver. Galileo was the first 
to seriously inquire after the speed of light by timing the flashes of 
lanterns across distant hilltops. He even gave a perfectly scientific 
result: Light travels excedingly fast or perhaps instantly. His 
failure to obtain a definite value was, of course, due to the very 
swift passage of light, the absurdly small distances over which it ran, 
and the patheticly crude means for measuring the travel time. 
    Roemer about 1660 derived the first actual speed for light by 
observing Jupiter's moons. Their occultations and eclipses were 
advanced or retarded in time as Jupiter was nearer or farther from the 
Earth. This he attributed to the finite speed of light; his value was 
still crude because the Earth-Jupiter distance was imperfectly known. 
    In the early 1800s Fizeau tried an indoor method using offset cogs 
spinning rapidly. A lightbeam passing between two teeth on one cog 
would hit the offset tooth on the other cog. However, if the cogs span 
rapidly enough the second one would rotate out of the way so the beam 
passed thru both wheels. The mechanics of the systems revealed the 
speed of light. 
    The person who obtained the best value, one essentially used to 
today, was Michelson about 1880. He used a racetrack of mirrors to split a 
lightbeam into two paths of almost equal length. The difference was a 
split wavelength. At one adjustment the beams joined exactly and the image 
so formed was bright and clear. At a tweaked off setting the image 
darkened due to interference between the peaks of the waves of the one 
beam and the valleys of those of the other. Again the mechanics of the 
apparatus gave the actual speed. For this he earned America's first Nobel 
Prize in physics. 

Maxwell's Laws 
    Physicists held to the notion that light was merely a wavelike 
thingy moving thru some medium. Its speed was merely an experimental 
quantity which happened to be 2.998E8m/s. All this changed in 1864. 
Then Maxwell developed his theory of electromagnetic waves and proved 
that light was simply one of these waves. It differed from others only 
in its wavelength. Simultaneously biologists were realizing that light 
is a response in the eye-brain of people to electromagnetic waves of a 
narrow waveband, which arose the response of vision while other 
wavebands do not. 
    But Maxwell came onto an other far reaching discovery. The 'speed 
of light' is NOT a haphazardly measured quantity; it is an artifice of 
nature. That is, its value is a simple arithmetic combination of other 
fundamental physical properties of nature. To convey the 
electromagnetic radiation (EMR) Maxwell found that empty space, 
vacuum, had the property to sustain both a magnetic and an electric 
field. The oscillation of these two fields in halfwave offset 
constitutes the EMR. The propagation of the wave is a function of 
these two fixed properties, the one epsilon0, the other mu0. These in turn 
are expressed in terms of already known electricity and magnetism. 
    epsilon0 = 8.854E-12C^2.s^2/(Kg.m^3) and mu0 = 4*piE-7Kg.m/C^2. C 
  (capital-C) is the couloumb, the unit of electric charge, equal to 
some large but definite number of electrons. Now these are innate 
constants of nature, not chance values found by experiment. Vacuum can 
have no other value for these parameters. From them the quantity c, 
the propagation speed of EMR in vacuo, was derived as 

    c = 1/(epsilon0 * mu0)^(1/2) 
      = 1/(8.854E-12C^2.s^2/(kg.m^3) * 4*piE-7kg.m/C^2)^(1/2) 
      = 1/(1.113E-17C^2.s^2.kg.n/(kg.m^3.C^2))^(1/2) 
      = 1/(1.113E-17s^2/m^2)^(1/2) 
      = (8.988E16m2/s2)^(1/2) 
      = 2.998E8m/s 

which is exactly, like exactly, the measured speed of light in vacuo. 

 | SPEED OF LIGHT               | 
 |                              | 
 | c = 1/(epsilon0 * mu0)^(1/2) | 
 |                              | 
 |   = 2.998E8m/s               | 

    In other words, this combination of innate physical parameters is 
itself a new physical parameter, which is what here to fore has been 
called 'speed of light'. 

Michelson-Morley Experiment 
                  When physicists caught onto Maxwell's laws they hit 
on an amazing prospect. Astronomy really never had a convincing proof 
of the Earth's motion thru space. True, Bradley in the mid1700s found 
the aberration of starlight and Foucault demonstrated his pendulum in 
the 1850s. But there was no 'absolute' proof of the Earth's motion. 
    Maxwell's discovery that the speed of light is a feature of empty 
space got the astronomers thinking. If we measure the speed of light 
parallel to the Earth's motion thru empty space and then antiparallel 
to it we should perceive a difference. This discrepancy is nothing 
more than the addition or subtraction of the Earth's speed in space. 
In this way we can measure in an absolute sense the real motion of the 
    Michelson & Morley set out to do this measurement in the 1880s. 
Their apparatus could have detected a speed as low as 500m/s for the 
Earth altho it was expected that 30Km/s would be the correct result. 
But however careful and diligent they were, Michelson & Morley were 
driven against the one damn result. The speed of light was the SAME no 
matter in what direction they measured it. It DID NOT add or subtract 
against the motion of the Earth!! 
    There were only two conclusions possible. Being that space was an 
absolute medium against which motion is banked off via the assessed 
discrepancies in the lightspeed, the Earth in fact stands still after 
all. Or something is very queer and weird about the speed of light. 
    The first conclusion was insane. For by every other account the 
Earth does move thru space. But the second was even more absurd! The 
speed of light does not obey the addition rules so natural and 
sensible in every other domain of physics. What can be the way out of this 

Laws of Physics 
    Einstein cleaned the mess by noting that in all inertial platforms 
the laws of physics are the same. In astronomy this is examplified by 
the laws of gravity. When first elaborated by Newton gravity could be 
applied only within the solar system. This region of the universe in 
Newton's day was the only place where mass existed. The nature of the 
stars was unknown at that time; they were generally treated as 
insubstantial points of light with no real body to them. 
    In the 1780s the binary star was discovered by Herschel. Here in 
very remote and scattered places thruout the universe were examples of 
gravity acting between two (or, in some cases, more) stars. The laws 
doped out from within the solar system worked out there, too. It was 
only then, about 100 years after Newton released his gravity theory, 
that his laws could truly be called universal.
    It is crucial to appreciate that in any inertial platform the laws 
of physics are the same. The value of gamma is not altered by being on 
a one or an other inertial platform. Now there is one very special law 
that is the same for all inertial observers and this is the cardinal 
feature of relativity that offends everyone's common sense. The speed 
of light is the same for all inertial observers. No matter what the 
velocity of a one frame is banked off of an other each frame measures 
the same value for c. 
    It is sometimes explained that the laws of physics and -- as a
separate issue -- also the speed of light are the same for inertial 
frames. The speed of EMR is itself a physical law, being the result of the 
magnetic permeability and electric conductivity of a vacuum. There is no 
reason to treat light like something apart from other physical laws. 
    Some works state that c is the same for all platforms. Not so. c 
is constant only for inertial observers. Observers immersed in gravity 
experience a speed of light less than c. Such cases are the domain of 
the General Theory of Relativity. Altho very minutely detected in the
usual astronomy applications, the decrease in c within gravity was not 
really appreciated until the blackhole was entertained in astronomy. 

The Tern 'Speed of Light'
    The term 'speed of light' is an albatross of history. Light was 
historicly considered like some sort of mechanical wave whose speed 
had to assayed by experiment. And it was determined by experiment. 
    When Maxwell issued his theory of electromagnetism he found that 
light is an EMR wave and its speed was a derivative of the electric 
and magnetic properties of space. In particular he found that the 
speed in a vacuum, empty space, was the same for all EMR waves. Light 
simply was EMR of a certain band of wavelengths (or frequency) that 
excited our eyes and brains into vision. 
    Hence, far from being a chance measurement, the speed of light c 
is one of the laws of physics. It ranks with, say, gamma, the Newton 
constant, as being the same for all inertial frames. It is unfortunate 
that a new term was not adopted for c, such as 'Maxwell's constant'. 
The continued use of the phrase 'speed of light' leads to the most 
ridiculous and unimaginable misapprehensions about relativity!

Comparing Clocks 
    We examine the behavior of time between two inertial observers. One is 
selected as the stational frame and the other is then the motional one. 
Because it is pretty much beyond most people to visualize 4D spacetime we 
allow the X-axis of both frames to be aligned and the motional frame to 
pass by along the X axis, so that the Y and Z space coordinates can be 
missed out. We have a 2D slice of this 4D world with the X and cT axes 
at right angle. Yes, it is not easy to think of time (even converted 
to a length measure) as having an angle against the space dimension, 
but it is very true that it is. 
    Each observer carries a clock. Each uses it to measure the behavior of 
time. The stational frame we call the s frame; motional, m. We select the 
motional frame as our target which we examine both from itself and from 
the stational frame. Which is say, we explore what happens to the clock of 
the m platform as experienced by both that same m platform and the s 
    Please be careful! Many books on relativity do not so neatly declare 
the target, the motion or station of the two observers, or the way the 
target is experienced! The notation is often ambiguous. The result is that 
after a few pages one loses all grasp of who's looking at whom and what's 
really going on. Hence, many home astronomers simply give up on relativity 
after a dutiful effort. 

The Scenario
    For convenience -- and only for convenience -- the s frame is the 
Earth and the m frame is the spaceship. It is utterly immaterial what the 
vehicles or vessels are. The m frame passes along the X axis, which is 
aligned the same for both frames. This way we can map the X-cT spacetime 
system onto an ordinary 2D system; Y and Z can be ignored. m moves with 
constant velocity (it is an inertial platform) V toward positive X, as 
measured by s. Which is to say, as m procedes, m acquires relative to s 
ever greater value of its X coordinate. 
    We note here a crucial fact because we need it later and because so 
many other treatments omit it. Because of the relativity of motion -- 
either s or m can stand for the stational frame! -- the V measured by s is 
the same as that measured by m, except for the signum. As experienced by m 
the s frame recedes toward negative X. 

Equation of Motion
    On the spaceship a pulse of light is measured for speed. It travels at 
a certain rate c[m/m]. The notation means 'speed of light in the m 
frame as experienced in the m frame'. This same pulse is experienced 
by the Earth as c[m/s], read as 'speed of light in the m (Earth) frame 
as experienced by the s (spaceship)frame'. 
    Because relativity compares events in the m and s frame, extreme 
care must be given to the notation 'Q[m/s]' and similar. The subscript 
is NOT a maths division. The example means 'the parameter Q prevailing 
in the M frame as experienced from the s frame'. This scheme of 
subscript is used thruout these articles about Einstein physics and 
its applications. Relaxing the attention will, WILL, derail your 
attempts to appreciate Einstein's relativity theory. 
    In the same manner the velocity of the spaceship as experienced by the 
Earth is V[m/s]. The same velocity as measured by the spaceship itself is 
V[m/m]. Note very well that we are talking about only the velocity of the 
m frame and not alternately that of m and s. So often this is mixed up in 
relativity works! 
    The clocks are constructed identicly and one is placed in the 
spaceship while the other remains on Earth. The clocks have marks for the 
time units, seconds, and we compare these units between the spaceship and 
the Earth. 

Units versus Total
    Be careful! Here we are comparing the unit of measure, the second, 
between the two frames. We could also compare the total interval, in 
seconds, from one point to an other between the two frames. We can 
measure a slat of wood and read off of the ruler '53cm'. The unit of 
length is the centimeter. We can also flip the ruler over and read 
'21 inch' (rounded. The unit is the oldstyle inch. Note very well that 
the inch is LARGER than the centimeter. One inch is about 2-1/2 cm. 
The length of the slat is SMALLER. 21 (inch) is about 4/10 of 53 (cm). 
    We work in these papers ONLY with the unit of measure, the second, 
and NOT the full duration in seconds. We so the Same for length,the 
unit of meter and NOT total meters. 
    Either method is valid but you must know which is being used. The 
equations resulting from each type of comparison look different, yet 
the conclusion in words is the same! Some relativity works don't fully 
lay out which method they are using, leading to ridiculous wheel 
spinning for the novice. 

Home and Target Frame
    An other pitfall is the choice of the home frame and target frame. We 
may deem either the motional or the stational frame as the basis of the 
comparison. Again, the books often are not clear which is compared to 
which. The equations end up looking quite different yet the conclusion 
in the text is the same! 
    While the choice of the home observer, the observer whose clock is the 
'good' one, is arbitrary, books are quite loose with the notation or 
symbols for this observer. You simply can not let the notation of one 
author lull you to think the same notation of an other author has the 
same meaning! Chances are quite against you that they do. 
    In particular it is just not a given that the stational observer is 
also the home observer. He may quite rightfully be the target 
observer, the fellow with the 'bad' clock. You really have to suss out 
what the instant author is saying. 

Spacetime interval 
    While we must allow that the m and s frames may record different 
unit times (and spaces) between them, they can -- and must -- agree on 
the one & same combined unit of space & time. That is, the distance in 
spacetime is the same for the two observers.
    With delL being the common spacetime unit, we have 

    delL^2 = (delX[m/m])^2 + (c[m/m] * delT[m/m])^2 
           = (delX[m/s])^2 + (c[m/s] * delT[m/s])^2         
           = (V[m/m] * delT[m/m])^2 + (c[m/m] * delT[m/m])^2 
           = (V[m/s] * delT[m/s])^2 + (c[m/s] * delT[m/s])^2 

    Why the square functions? Recall that we are dealing with two 
orthogonal axes X and cT. The distance unit delL is the sum of the 
distance units along the two orthogonal axes and this is obtained by 
nothing more than the Pythagoras theorem. Remember!, in 4D geometry cT is 
just one of the dimensions, just like X or Y or Z. With both observers 
having the same unit of spacetime interval 

    (V[m/m] * delT[m/m])^2 + (c[m/m) * delT[m/m])^2 
    = (V[m/s] * delT[m/s])^2 + (c[m/s) * delT[m/s])^2 

    V[m/s] is the speed of the spaceship measured by the Earth. V[m/m] is 
the spaceship speed measured by the spaceship itself. This latter is zero; 
the spaceship is not moving relative to its own self. Note quite carefully 
that we are not speaking of a V[s/m], which would be the speed of the 
Earth measured in the spaceship! In fact, V[s/m] = -V[m/s}. 

    V[m/m] = 0 

     (0 * delT[m/m])^2 + (c[m/m) * delT[m/m])^2 
     = (V[m/s] * delT[m/s])^2 + (c[m/s) * delT[m/s])^2  

     (c[m/m] * delT[m/m])^2 
    = (V[m/s] * delT[m/s])^2 + (c[m/s) * delT[m/s])^2 

Four Roads to Follow 
    You may ask why we write 'c[m/m]' and 'c[m/s]' as if lightspeed is 
different between the m and s frames. Isn't the speed of light the 
same for all observers? 
    Not yet. We can not assert this right now but we do eventually 
show that the two are the same and we replace them with poain 'c'. 
    To continue further we have four routes. One is the Newton path 
which has a pancosmic time for all observers regardless of their 
motion. The second is deliberately a wrong analysis to show that there 
really can be divergent times between the observers. The third is the 
correct formulation in Special Relativity. The fourth shows up one 
misconception about clocks when compared in their own frames. 

The Newton Road 
    Under Newton physics there is an independent time parameter thruout 
the universe that is the same for all observers. Thus delT[m/m] = 
delT[m/s] = delT, saying that the clock on the spaceship ticks the same 
size of second whether measured from the spaceship or from the Earth. 
Recalling that V[m/m] = 0 

    (c[m/m] * delT)^2 = (V[m/s] * delT)^2 + (c[m/s] * delT)^2 

     c[m/m]^2 = V[m/s]^2 + c[m/s]^2 

    This is the ordinary rule of addition of velocities. In this case 
there is nothing special about c. It is treated just like any other 
mechanical motion. It is largely the persistent use of the phrase 'speed 
of light' that causes so much angst among home astronomers trying to 
comprehend how this c can be the same for all inertial platforms. It in 
the Newton sense certainly behaves like any other speed. Perhaps if the 
name for c were something more innocent, such as 'Maxwell constant' or 
'Michelson-Morley number', it would be easier to vacate the Newton 
    This was the thinking behind the Michelson & Morley experiment. It was 
known from Maxwell's work that thru aether EMR (and light) travels at 
3E8m/s. If we on the Earth measure a different speed, the offset is 
due to the Earth's motion thru the aether. Ergo, we should have a 
means to measure motion in an absolute frame, that of the aether. 
    In the above equation c[m/m] would be c experienced on the Earth, the 
motional observer, by the same Earth. c[m/s] is c on the Earth as measured 
by the aether, which is the stational frame. This is the c in 
Maxwell's laws. V[m/s] is then the supposedly absolute speed of the 
Earth measured relative to the aether. 
    When the experiment -- and all others trying to upset it! -- 
failed to find any variation in c caused by the motion of the Earth, 
the scene was set for Einstein. 

Einstein's Mistake?
    We don't know if Einstein actually tried this second case. He could 
have and then rejected it as wrong. We go thru this second case to show 
that not every propounded idea works. 
    Einstein started with the fact that c is measured the same by all 
inertial observers. c[m/m] = c[m/s] = c. 

    (c[m/m] * delT[m/m])^2 
    = (V[m/s] * delT[m/s])^2 + (c[m/s] * delT[m/s])^2 

    (c * delT[m/m])^2 
      = (V[m/s] * delT[m/s])^2 + (c * delT[m/s])^2 

    (c * delT[m/m])^2
      = (V[m/s]^2 + c^2) * delT[m/s]^2 

      = ((V[m/s]^2 + c^2) * delT[m/s]^2) / c^2 
      = (V[m/s]^2 / c^2 + c^2 / c^2) * delT[m/s]^2 
      = (V[m/s]^ 2 /c^2 + 1) * delT[m/s]^2 
      = (1 + (V[m/s] / c)^2) * delT[m/s]^2 
    delT[m/m) = (1 + (V[m/s]/c)^2)^(1/2) * delT[m/s] 
    = pseudobeta * delT[m/s] 

    delT[m/s] = delT[m/m] / pseudobeta 

   This is one unexpected result! The time unit on the spaceship is 
experienced differently by the spaceship and by the Earth! Before we get 
too excited about this result, we must remind that this particular 
solution is false & wrong. That's why we call the 1+V/c thingy a 
pseudobeta. The correct formulation yields a very similar expression called 
beta. But we continue. 

Behavior of Time
    More from Einstein's wrong case. 
    V[m/s] is greater than zero, so 

    delT[m/s] = delT[m/m] / ((1 + (>0) / c)^2)^(1/2)) 
              = delT[m/m] / ((1 + (<c) / c)^2)^(1/2)) 
              = delT[m/m] / ((1 + (<1)^2)^(1/2)) 
              = delT[m/m] / (1 + <1)^(1/2) 
              = delT[m/m] / (>1)^(1/2) 
              = delT[m/m] / (>1) 
           = delT[m/m]*(<1)
           < delT[m/m]

The time unit, the second, on the spaceship as measured by the Earth 
is less than that same second as measured from the spaceship. Or Earth 
experiences the spaceship second to be less than what the spaceship 
itself experiences it. In other words, time on the spaceship as 
experienced by the Earth is speeded up compared to the time experienced on 
the spaceship itself! 
      This is not a trivial matter to grasp, even if it is really the 
false formulation. How can time be different simply because two 
observers are in relative motion? The answer is that in our ordinary 
every day life we never come onto motions so severe that the 
difference shows up. In deed, it is we who are 'culturally deficient' 
in the real world of relativity. It is, in the grand scheme of things, 
perfect common sense that unit of time has different sizes for 
platforms in motion among themselfs. 

Why is This Case Wrong?
    Look at the inequality 

    delT[m/s] = delT[m/m] / (1 + (>0) / c)^2)^(1/2) 
              < delT[m/m] 

    pseudobeta has no limit! It may be as large as we like for any 
value of V[m/s] we please. It is valid even for V[m/s] greater than c. 
In the extreme we can have the situation of a bullet travelling so 
fast that it arrives at its target before its own image! There is 
nothing special about c except for a leverage on how big pseudobeta is 
for a given V[m/s]. 
    But this behavior of V[m/s] and c is not in concord with what we in 
fact observe around us. While, yes, delT[m/m} <> delT[m/s] the sense of 
the inequality is opposite what we do experience and there are no bodies 
travelling faster than lightspeed. 
    In a broader principle, if the theory does not match the observed 
facts, it is invalid. And ir doesn't matter how good the maths are in 
the theory. 

Einstein's Solution 
    We make a seemingly trivial and minor adjustment in the delL^2

    delL^2 = (V[m/m] * delT[m/m])^2 - (c * delT[m/m])^2 
           = (V[m/s] * delT[m/s])^2 - (c * delT[m/s])^2 

 |                                                     | 
 | delL^2 = (V[m/m] * delT[m/m])^2 - (c*delT[m/m])^2   | 
 |        = (V[m/s] * delT[m/s])^2 - (c * delT[m/s])^2 | 

    See?, the plus sign is replaced by a minus sign. We go thru the 
derivation like before, noting that V[m/m] = 0. 

 delL^2 = -(c*delT[m/m])^2
        = (V[m/s]*delT[m/s])^2-(c*delT[m/s])^2

 -(c*delT[m/m])^2 = (V[m/s]*delT[m/s])^2-(c*delT[m/s])^2

 -delT[m/m]^2 = ((V[m/s]*delT[m/s})^2-(c*delT[m/s])^2)/c^2
              = ((V[m/s]^2-c^2)*delT[m/s]^2/c^2
              = ((V[m/s]^2/c^2)-(c^2/c^2))*delT[m/s]^2
              = ((V[m/s]/c)^2-1)*delT[m/s]^2

 delT[m/m]^2 = (1-(V[m/s]/c)^2)*delT[m/s]^2

 delT[m/m] = (1-(V[m/s]/c)^2)^(1/2)*delT[m/s]
           = beta*delT[m/s]

 delT[m/s] = delT[m/m]/beta

 |                                                |
 | delT[m/s] = delT[m/m]/((1-(V[m/s]/c)^2)^(1/2)) |
 |           = delT[m/m]/beta                     |

    This result at first looks like the 'wrong' case, except that we 
have the 1-(V/c)^2 in the stead of 1+(V/c)^2. But this the correct 
equation for the behavior of time between two inertial platforms 
carries some incredible features. 

    Because the factors V/c, (V/c)^2, 1-(V/c)^2, (1-(V/c)^2)^(1/2) 
occur so often in relativity work they are given special symbols. 
Alas, there is no general agreement on what symbols to use. beta is a 
very common choice but it is applied to any one of these factors 
noted, plus the inverse of the squareroot factor! In one book you may 
see delT[m/s] = delT[m/m]/beta and in an other you read delT[m/s] = 
delT[m/m]*beta. At first you may think one is using the inverse of the 
other's factor. But they may BOTH be using the SAME factor, but NAMING 
it differently. The one's 'beta' is the other's '1/beta'. 
    gamma is also used when beta is applied to one factor. Again, 
there is no general standard. You really do have to read carefully and 
note well just what the instant author's symbols mean. 
    In this book we use 'beta' for (1-(V/c)^2)^(1/2); gamma is not at 
all used here. 

Time Dilation
    With V[m/s] less than c the spaceship's unit of time compared on the
Earth is greater than that compared on the spaceship. Its second is 
longer, greater, more than Earth's. Phrased in terms of total elapsed 
time, the spaceship's clock as experienced by Earth runs slower and shows
fewer accumulated seconds than what the spaceship itself experiences.

    delT[m/s] = delT[m/m] / ((1 - (V[m/s] / c)^2)^(1/2)) 
              = delT[m/m] / ((1 - (<c / c)^2)^(1/2)) 
              = delT[m/m] / ((1 - (<1)^2)^(1/2)) 
              = delT[m/m] / (1 - (<1))^(1/2)) 
              = delT[m/m]/((<1)^(1/2)) 
              = delT[m/m] / (<1) 
              > delT[m/m] 

    This is in full concord with the real world. Time on the motional
platform does run slower than on the stational platform. This is a 
manifestation of a new and higher order of cosmic common sense. Time 
'should' be different between two inertial frames. It is the special 
case when V[m/s] is quite small so beta is very near to one that delT[m/m] 
= delT[m/s]. In all our cotidian life we encounter only such small speeds,
fooling us to believe that time is independent of the observer's motion.

 The Barrier of c
    As V[m/s] increases from zero (V[m/s]/c)^2 increases from zero also.
1-(V[m/s]/c)^2 decreases from one. As long as this is still greater than
zero, the squareroot is defined. When V[m/s] = c, (V[m/s]/c)^2 = 1 and 1-
(V[m/s]/c)^2 = 0. Up to this point the squareroot is taken of a number
between one and zero and is a real number. However, once V[m/s] > c the
argument of the squareroot becomes negative and the squareroot itself 
is not defined by real numbers. 
    c is, therefore, not merely some very high speed but a natural and 
real barrier against arbitrarily fast relative motion. It is a barrage or 
boundary of speed. Einstein was a bit more careful by showing that no mass
or energy can travel faster than light relative to an inertial platform.
    It is possible to make illusions travel vastly faster, like a
laser beam emitted from Earth and sweeping across Jupiter. The spot of 
light can easily excede lightspeed across the Jovian ground. However, 
it is not possible to convey from one part of Jupiter to an other by 
this beam any mass or energy. Other examples include sliding shadows, 
spinning Moire patterns, and abstract mechanisms. When confronted with a 
supposed violation of the lightspeed limit, ask if the proposed exception
can carry any mass or energy from the one place to the other.

The Fourth Case
    The statement is commonly made that the spaceship clock is somehow
felt in the spaceship to be running slower than on the Earth.
    Not so.
    The divergence of time occurs only when a one clock is measured by 
different inertial frames. Within any one inertial frame there is no 
slowing of time at all. In fact, unless a one observer compares his 
clock with an other, he feels time passing by at the same rate as any 
other inertial observer feels. 
    This sounds quite contradictory from all we did above to show that
time diverges between inertial observers! We start with the basic 
equation of spacetime unit interval and compare the spaceship's time 
on the spaceship with Earth's time on the Earth. 

    (V[m/m] * delT[m/m])^2 - (c * delT[m/m])^2 
      = (V[s/s] * delT[s/s])^2 - (c * delT[s/s])^2 

In each frame the velocity relative to its own self is zero, 

    (0) - (c * delT[m/m])^2
       = (0) - (c *delT[s/s])^2 

    -(c * delT[m/m])^2 = -(c * delT[s/s])^2 

    -delT[m/m]^2 = -delT[s/s]^2 

    delT[m/m]^2 = delT[s/s]^2 

    delT[m/m] = delT[s/s] 

 |                       |
 | delT[m/m] = delT[s/s] |

    The time dilation shows up only when we compare unit time intervals
between inertial observers. No matter what the speed of an inertial 
observer relative to an other, each feels its own time to elapse at the 
same normal and natural rate. 
    We can see the absurdity of the incorrect notion by simply noting 
that in this wide wide universe the Earth is in relative motion 
against some other place at some large fraction of lightspeed. Yet we 
do not sense within ourselfs any slowing of time. And we are moving at 
many fractions of lightspeed relative to many places. Which of these 
governs our rate of time? 
    It is simply not at all true that you can slow your own time by 
travelling at speeds near c with respect to a stational frame