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
impossible.
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.
Spacetime
-------
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
body.
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
Earth!
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
mess?
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
platform.
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
mindset.
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
delT[m/m]^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
equation.
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
+-----------------------------------------------------+
| SPACETIME UNIT INTERVAL BETWEEN TWO INERTIAL FRAMES |
| |
| 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
+------------------------------------------------+
| TIME RELATION BETWEEN INERTIAL FRAMES |
| |
| 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.
beta
--
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]
+-----------------------+
| TIME EQUALITY WITHIN |
| INERTIAL PLATFORMS |
| |
| 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