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From: JE02%UTEP.BITNET@CUNYVM.CUNY.EDU
Newsgroups: sci.physics
Subject: Simple physical view of Relativity
Message-ID: <1302@sri-arpa.ARPA>
Date: 6 Mar 89 22:34:00 GMT
Lines: 1091
From: GALEN M KAIP <JE02%UTEP.BITNET@CUNYVM.CUNY.EDU>
Subject: Paradoxes of Special Relativity (all) resolved
A shift in perspective in Special Relativity provides a much
simpler and more visualizable physical model that gets rid of all the
well known 'paradoxes'.
This paper, entitled "A View of Relativity Without Paradoxes"
contains quick (relatively) reviews of what was in two previous
papers put out on USENET in April & May of 1988 but also adds about
a half dozen 'new' concepts which corroborate the previous ideas.
If after reading this paper you wish to contact me, please do
so by mail or telephone. I have had difficulty with contacts through
the computer network. You will find my address and phone number
at the end of this paper.
This paper is Copyright 1988 but permission is granted to
quote freely provided that FULL CREDIT is given the author including
name, address, and telephone number.
A VIEW OF RELATIVITY WITHOUT PARADOXES
Copyright 1988 by
Dr. Sherwood R. Kaip
PURPOSE
capsule: THE PURPOSE OF THIS PAPER IS TO SHOW, IN SEVERAL
DIFFERENT WAYS, THAT SPECIAL RELATIVITY, LOOKED AT
DIFFERENTLY, REPRESENTS A MATHEMATICAL TRANSFORMATION OF A
SIMPLE PHYSICAL INTERPRETATION OF OUR UNIVERSE, BOTH IN THE
LARGE AND IN THE SMALL, WHICH DOES NOT HAVE PARADOXES.
The purpose of this paper is to show that Special Relativity is
a physically impossible mathematical transformation of a much
simpler and more awesome physical reality.
This is exactly analogous to the 'discovery' you made as a kid
when you found out that Greenland isn't really approximately the
size of the United States. It just appears that way on many maps
because a Mercator projection is a mathematical transformation of
the physical 'reality' of the earth being a nearly round sphere.
Another exact analogy is the 'change' from pre-Copernican
astronomy to our current view of the solar system. What Copernicus
really said in effect was, 'Hey, fellas, there's nothing much wrong
with your equations but take your equations of motion of the bodies
of the solar system relative to the earth and rewrite these
equations of motion relative to the sun instead and you will discover
that the planets travel almost in a circle (ellipse) around the sun,
the moon around the earth, and the earth rotates on its axis every 24
hours.'
Special Relativity is the equivalent of the Mercator projection
(which can be navigated upon mathematically but not physically (in
higher latitudes you are travelling over mostly empty space) and
pre-Copernican astronomy (try to work out the centripetal forces!).
SUMMARY
capsule: SPECIAL RELATIVITY WAS AND COULD ONLY BE DERIVED
FROM A REALITY IN WHICH ELECTROMAGNETIC RADIATION MOVES
AWAY FROM ITS SOURCE AT A CONSTANT SPEED. THIS IS SHOWN IN
DETAIL IN SEVERAL WAYS.
Special Relativity not only mathematically transforms time
(dilation) and length (contraction) but, what apparently has been
little if at all recognized, also transforms velocity (contraction).
These transformations are only in the direction of relative motion
and furthermore are RECIPROCAL. Time is dilated in one frame and
length contracted in the other, but you can reverse the roles of
which is the 'fixed' and which the 'moving' frame (by the 1st
relativity principle), and thus which frame changes time and which
changes length, at your whim. (Talk about having it 'both ways'!)
However if you carefully follow the relativistic path of the
light travelling transversely and longitudinally in both reference
frames as described below, you will realize that time dilates,
length contracts, AND VELOCITY CONTRACTS because someone
ASSUMED that light travels the same velocity in all reference
frames. If you don't distort length, time, or VELOCITY of the
reference frames relative to each other, the light is exactly where
it should be if it is travelling 300,000,000 meters per second
RELATIVE TO THE SOURCE FROM WHICH IT WAS EMITTED. This is true
of the light in each reference frame separately and of the light
generated in one reference frame moving through the other. Even if
you leave the relativistic numbers in place, all locations in the
moving frame correspond to the points where they need to 'be' in the
fixed frame at various proper times if all the electromagnetic
radiation is traveling RELATIVE TO ITS SOURCE at the speed c.
In other words, ASSUMING light travels the same speed, c, in
all reference frames is either unnecessary or erroneous, depending
on how you wish to look at it. If you don't distort time or length
AND you recognize that the relative velocity between reference
frames is gamma (g in this paper) times the relativistic velocity (u
in this paper), then everything works out very simply in terms of
electromagnetic radiation leaving its source at speed c. This paper
shows that in several ways.
HISTORY OF THE IDEA
capsule: THE BASIC UNDERSTANDING WAS EXPRESSED IN A PAPER OF
1980. THE EFFECTS ON RELATIVISTIC MECHANICS WAS
SUBSEQUENTLY EXAMINED AND EXPLAINED IN A PAPER OF 1983. IN
1988 ANOTHER PAPER WAS WRITTEN COVERING THE BASIC
UNDERSTANDING FROM ANOTHER VIEWPOINT. THIS AND THE FIRST
PAPER WERE PUT ON USENET UNDER SCI.PHYSICS. THIS CURRENT
PAPER INCLUDES PROOF OF THE IDEAS FROM NEW MATERIAL.
Others have questioned Special Relativity and I have heard of
one person (there may be plenty of others, I don't know) who
supports this emission theory idea. But my history with this dates
back to a paper I wrote and registered the copyright in 1980 entitled
"THE ROLE OF VELOCITY IN SPECIAL RELATIVITY" where I examined
the light in a moving frame emitted perpendicular to the relative
motion (and reflected back down) and followed that light in the
fixed frame. This is the way Special Relativity is usually taught.
From examination of this I recognized the described emission
theory. This article was put out on the Usenet Bulletin Board under
SCI.PHYSICS in late April of 1988.
In 1983 I registered the copyright on an article entitled
"RELATIONSHIP BETWEEN SPECIAL RELATIVITY AND MODIFIED
NEWTONIAN MECHANICS" which examines the consequences on
mechanics of recognizing that Special Relativity is really a
mathematical distortion of the above emission theory. The 1983
paper covers completely the areas of relativistic momentum, kinetic
and total energy, and rest mass changes and will only make sense
when the relationship between Special Relativity and this theory is
understood, as shown in this paper. The implications are
considerable. Since the 1983 paper explains the consequences on
mechanics well and is available from the author (as described at the
end of this paper), no material from that source is included here.
In addition to the 1980 article put on Usenet in April 1988, I
wrote and immediately put on Usenet SCI.PHYSICS in early May 1988
another article which examines the same ideas from the point of
view of a light pulse emitted in the moving frame and its
relationship to the fixed frame but this pulse travels in the
direction of relative motion of the two reference frames. The same
conclusions were reached. I will now rather compactly summarize
both of these papers and their results and in addition show the same
conclusions by examining time and by examining the relativistic
transform equations directly. (If part of this material 'travels'
rather fast, getting copies of the 2 papers from someone who took
them off Usenet or from the author may make things easier since
these papers spread the earlier material out in somewhat more
detail than will be done here.)
Some of the following is a compression from the above papers.
The newer material is in sections below entitled: SIMULTANEITY;
TRY IT IN REVERSE; "GRANDMA WHAT STRANGE VARIABLES YOU
HAVE!"; PROPER TIME; THE TRANSFORM EQUATIONS; WHERE IT'S AT; A
CLOSER LOOK AT VELOCITY; and TRANSFORM EQUATION COMPARISONS
.
SETTING UP
capsule: THESE SIMPLE TO DRAW FIGURES, OFTEN USED TO
DEMONSTRATE RELATIVITY, ARE NEARLY NECESSARY TO UNDERSTAND
WHAT IS REALLY GOING ON.
To reasonably understand what is going on, you will need two
diagrams. They can be drawn on a sheet of 8.5 by 11 paper as
follows. 8 inches from the top of the page, draw a horizontal line 3
inches across approximately centered between the sides. Label the
left end LA (for Los Angeles), the right end NY (for New York), and
the exact middle of the line KC (for Kansas City). Consider the
length of the line 3000 miles in the earth's fixed, non-prime
reference frame. About 1 millimeter below this line draw another 3
inch line (parallel). Label the left end tail (for tail of the space ship
in the moving, primed reference frame), the right end nose (for nose
of the space ship), and label the middle of the space ship M. If this
line is given a little thickness, it may make visualization of it as
the moving space ship easier. The space ship and the line from LA to
NY should actually be at the same vertical level but then they
couldn't be recognized individually.
Draw from the midpoint of the earth fixed reference line (at
KC) a perpendicular dashed line rising 7.5 inches. "Cap" this line at
the top with a centered horizontal 0.25 inch line which may be
labeled "mirror". Also mark dots on the vertical dashed line at 4.0,
3.0, 2.0 1.5, 1.0, and 0.3 inches above the mid-point (KC). Draw
dashed lines from both the LA and NY ends of the horizontal earth
reference line to the top of the vertical line (middle of "mirror") and
to each of the other 6 dots on the vertical line. This should give the
effect of a series of superimposed isosceles triangles, all with a
common 3 inch base. From the left end (tail) of the space ship, draw
an upward perpendicular 7.5 inch solid line also capped with a
horizontal 0.25 inch "mirror". This "mirror" is attached to the tail
of the space ship by the structure represented by the line. This
whole configuration may be called Figure 1 and will be referred to
as such.
In the remaining space at the bottom of the paper, reproduce
the "frontmost" triangle (smallest) from Figure 1 along with the
rocket ship line and all the labels. Put in a vertical line upward for
0.3" (inch) from the tail of the rocket ship (just like with Figure 1
but this time only 0.3" instead of 7.5"). There should also be a 0.3"
dashed line up from KC just as in Figure 1 if you haven't done this
already. Now 'cap' both of these vertical lines with 0.25" centered
"mirrors". This should look like a 3.0" wide triangle only 0.3" high
with another line 3.0" wide just below it and a line rising 0.3" from
the latter's left end (tail).
Now draw a vertical dashed line 0.3" long downward from the
middle of the space ship, M, and a solid line down from NY 0.3" and
'cap' the bottom ends of these with centered 0.25" "mirrors". Draw
dashed lines from the nose and the tail of the space ship to the
bottom of the dashed line below the space ship's middle, M, so as to
complete a downward pointing triangle just like the upward pointing
one. In addition to the mirrors below the space ship and below NY,
put in a vertical mirror 'facing' the tail of the space ship (so it will
reflect light from the tail back to the tail) located 0.1 of the
distance (0.3") from the tail to the nose of the space ship and
another vertical mirror 'facing' NY and located 0.1 of the way
between NY and LA (located 300 miles west of NY, i.e., 0.3"). This is
Fig. 2 and will be referred to as such.
I also suggest making a machine copy of the 18 equations
located further on in the paper so they will be easy to check when
they are referred to. This avoids losing the train of thought while
hunting for them.
SPECIAL RELATIVITY
capsule: THIS SECTION CONTAINS A QUICK REVIEW OF THE
PRINCIPLES OF RELATIVITY. A LIGHT BEAM FROM THE MOVING
REFERENCE FRAME IS SENT PERPENDICULARLY TO ITS MIRROR AND
REFLECTED BACK TO ITS SOURCE. THERE IS NO LENGTH CONTRACTION
OR TIME DILATION BECAUSE OF NO RELATIVE MOTION IN THIS
DIRECTION. THE EQUATIONS OF RELATIVITY ARE THE DIRECT RESULT
OF ASSUMING THE LIGHT IS TRAVELING THE SAME SPEED ALONG ITS
PATH IN EACH REFERENCE FRAME. TIME DILATION, LENGTH
CONTRACTION, AND THE RELATIVE VELOCITY OF THE MOVING
REFERENCE FRAME ALWAYS BEING LESS THAN THE SPEED OF LIGHT
FOLLOW LOGICALLY.
Use Fig. 1. IN THIS PAPER THE SPEED OF LIGHT, c, WILL BE
ASSUMED TO BE EXACTLY 186,000 MILES PER SECOND. THIS
APPROXIMATION AFFECTS THE CONCEPTS INVOLVED IN NO
SIGNIFICANT WAY. THROUGHOUT THIS PAPER PRIMED QUANTITIES
SHALL ALWAYS REFER TO THE MOVING FRAME. The function
(1-((u/c)**2))**(-1/2) is usually called gamma and is being
represented in this paper by "g". That is, take the square of the ratio
u/c, subtract it from 1, divide this result into 1 and take the square
root of that. I will use ** to indicate raising to a power. Thus,
7**(-1/3) equals the reciprocal of the cube root of 7.
The space ship is at rest in the x' frame where x'=0 at the tail,
distance in the x' frame increases to the right, and the whole x'
frame is moving rapidly to the right. In the earth reference frame
we have dug a trench between LA and NY to get a straight line and
have stopped all rotation so that we have two inertial frames, with
the earth's frame considered fixed, and x=0 at LA, x=1500 miles at
KC, and x=3000 miles at NY. These distances were measured in the
fixed earth frame.
The space ship also measured its length in its own reference
frame (in which it is not moving, although its reference frame is
moving with respect to the earth's) and found the length of the space
ship from the tail to the nose to be 3,000 miles. (Of course this
will not be the length in the fixed earth reference frame because of
length contraction.) When the zeroes of the two reference frames
coincide, both t and t' = 0 at LA and the tail of the space ship,
respectively. (t will be different at other locations at t'=0 as can be
seen from transform equations 2 and 8. The 12 transform equations
are listed in a later section titled THE TRANSFORM EQUATIONS.) At
this instant the space ship fires off a powerful light from its tail
which burns a spot at LA and this light also travels straight up to
the mirror above the space ship where it reflects and arrives back
at the tail of the space ship just as the tail is at NY, where the light
also burns a spot in NY. The light in the moving frame traveled 7500
miles up and the same down in its own reference frame and
therefore took a proper time of t'=80,645 micro seconds. However
the light traveled 15,297 miles along the path of the two
hypotenuses (don't give me a hard time about the plural of this word)
in the fixed frame, and since light travels 186,000 miles per second,
c, in all reference frames, the time is t=82,242 micro seconds in
earth frame for the same event--the light leaving the tail and LA,
traveling to the mirror (at this point, over KC), and returning to NY
and the tail. Therefore by this reasoning, the tail of the rocket
traveled 3000 miles in 82,242 micro seconds so the relativistic
velocity equals 36,477.7 miles per second or 0.196116c. It can be
noted that there is some length contraction in the moving frame
because, according to transform equation 1, at x=3000 when t'=0,
x'=2,941.7 miles. The standard relativistic transform equations 5,
6, 7, and 8 are derived from these concepts.
Use the same methods of Special Relativity and have other
identical space ships go by at increasing speeds such that the light
at the tail burns LA on rising and NY on descending but the light
travels progressively less vertical distance in both frames as shown
by the fact that each successive faster space ship's mirror is lower
than the last. This will result in a series of velocities. The new
height of the mirror with each faster space ship is represented on
your drawing by the lower points above KC. The first space ship
mirror was at 7500 miles above LA, KC, NY --AND the space ship.
Please notice that earth observers can also tell PROPER time
by firing a pulse up and back to a mirror the same distance above LA
as the distance above the space ship since there is no relativistic
length contraction or time dilation in the perpendicular direction
where there is no relative motion.
The new relativistic velocities between the fixed and moving
reference frames, represented by the distance the tail of the space
ship traveled, 3000 miles, divided by the time the light took to
travel, at speed c, the hypotenuses (LA to above KC and back down to
NY in each instance) with the mirrors 4000, 3000, 2000, 1500,
1000, and 300 miles above are 0.351123c, 0.447214c, 0.6000c,
0.707107c, 0.832050c, and 0.980581c, respectively. The relative
velocity of the moving reference frame in all these instances is less
than c.
It will still be less than c even if the space ship is going so
fast that the the light burns LA, hits the mirror, and returns to burn
the tail and NY 3000 miles away, and the mirror is only 3 miles
above the space ship (and LA and the rest of the earth's fixed
reference frame). In this instance proper time t'=32.258 micro
seconds--the same as it would be for an LA light up 3 miles and
back down. Using exactly 186,000 miles per second for c, the
relative velocity between the moving and fixed reference frame is
u=185,999.6 miles per second. This is correct according to Special
Relativity. (Aren't you getting just a little suspicious that
something isn't quite right?)
SIMULTANEITY
capsule: "EVENTS SIMULTANEOUS IN ONE REFERENCE FRAME ARE NOT
SIMULTANEOUS IN ANY OTHER REFERENCE FRAME MOVING WITH
RESPECT TO THE FIRST." IN FACT, A CHAIN OF EVENTS WITH THE
SAME INITIAL CAUSE WILL OCCUR FIRST AT ONE END AND ALSO FIRST
AT THE OTHER END, AS WELL AS SIMULTANEOUSLY. READ ON.
Suppose that LA and NY both simultaneously fire powerful
omni-directional light pulses at the space ship as the tail is at LA.
This leaves burn marks on the tail and nose of the space ship. But
these burns cause explosions of ammunition magazines at the tail
and the nose of the space ship. The resulting space ship explosions
cause radio transmitters in LA and NY to fire an extra
electromagnetic pulse.
All of these event have to occur at t'=0 because, according to
transform equations #1 and 3, this is the only t' at which the tail is
at LA and the nose is at NY. After t'=0, by transform equation 5, the
tail (x'=0) will be at a location x>0 and the nose will be at a location
x>3000. (Remember that, because of length contraction, the value of
x' for the nose does not equal 3000 miles.)
We have stated that the earth frame light pulses that caused
this chain reaction were simultaneous in earth reference frame.
Therefore at a later time the wave fronts are converging toward KC
from the east and west and will arrive at KC at the same time. At
this current moment however the pulse from the west is at about
Dodge City, KS, and the pulse from the east is near Hannibal, MO and
in the meantime the space ship's middle, M, has moved east from KC
and just encountered the pulse from the east at Hannibal and will
get the one from the west, now at Dodge City, a while after the
pulse later passes Hannibal and the space ship has moved still
further east. Since light is assumed to move the same speed in all
reference frames, the relativistic explanation is that the firing of
the light pulses was not simultaneous in the space ship's reference
frame. The math can be carried out to show that in the space ship's
reference frame, the NY light pulse was fired before the other pulse
in LA. "Events simultaneous in one reference frame are not
simultaneous in any other reference frame moving with respect to
the first," to quote from a textbook.
However the only proper time that the tail was at LA and the
nose was at NY was at t'=0 (q.v. transform equation #3) and it is
reasonable to suppose that the light from ammo explosions in the
nose and the tail of the space ship at the space ship's proper time
t'=0 would reach the middle, M, simultaneously in the space ship's
reference frame. (If you wish to say that because of what was said
above the events at NY occurred before those at LA, then I simply
postulate that at proper time t'=0 the magazines in the tail and nose
of the space ship exploded for reasons of their own and immediately
caused the extra pulses from the LA and NY transmitters.) When the
light waves from the simultaneous explosions at the tail and nose of
the space ship are getting close to the middle, M (which they will
reach simultaneously in the space ship's reference frame), the space
ship will have moved eastward some distance so that the explosion
light wave from the tail will reach KC before the explosion light
wave from the nose. Therefore the earth observer, BECAUSE LIGHT
IS ASSUMED TO TRAVEL THE SAME SPEED IN ALL REFERENCE FRAMES,
will have to say that the events at the tail and LA occurred before
the events at the nose and NY. Meantime, the extra radio pulses
generated at LA and NY by the space ship ammo magazine explosions
have to arrive simultaneously at KC since they were generated in
earth reference frame at proper time t'=0. (More will be said about
proper time shortly. For now, if you are looking for a non-proper
time, t, when x'=0 is at x=0, then this occurs when t=0 at
x=0--anywhere else t will not equal 0. Two other significant
situations when t=0 occur at x=3000 when x' equals a number well
in front of the nose (by transform equation 7), and when x= 3000/g
(i.e., 3000/gamma), the contracted location of the nose in the
moving reference frame. In both of the latter instances, t' will be
negative, by transform equation 11, meaning that the events we are
talking about haven't even occurred yet!)
Now we have the paradox that of 3 sets of events that
occurred only at the proper time instant, t'=0, when the tail was at
LA and the nose was at NY (use the transform equation #3 to verify
this), each set of events was either simultaneous, occurred first at
the nose and NY, or occurred first at the tail and LA. It would seem,
if one did not insist on assuming that light travels the same speed
in all reference frames, that qualitatively the light is being
'dragged' along through one reference frame by the reference frame
in which it is not moving. It remains to be seen if there is a
quantitative relationship to this 'dragging' (there is). Why else
would the middle, M, of the space ship run into the original light at
Hannibal, MO. Why would the light from the tail of the space ship
reach KC before it reached its own middle, M.
ODDITIES MADE OBVIOUS AT VERY HIGH SPEED
capsule: USING A HIGH SPEED EXAMPLE, ALL POINTS IN THE MOVING
REFERENCE FRAME TRAVEL 3000 MILES ALONG THE FIXED REFERENCE
FRAME DURING A PROPER TIME INTERVAL OF 3,225.8 MICRO-SECONDS
WHILE THE LIGHT MEASURING THE PROPER TIME ONLY TRAVELS 600
MILES.
Use Fig. 2. The relativistic speed, u, between the reference
frames is 182,388 miles per second (i.e., c times 5 divided by the
square root of 26) or u=0.980581c. Fig. 2 is constructed just like
Fig. 1's smallest triangle where the mirror is 300 miles above the
space ship's tail plus there is also a triangle below. There should
also be 6 mirrors on your drawing. There should be one 'below' NY,
one 'above' the space ship's tail (above LA), one below the middle of
the space ship, one above KC, one shown 0.3 inches east of the space
ship's tail, and one 0.3 inches west of NY.
The procedure is the same as with Fig. 1. When the tail of the
space ship is at LA and the nose is at NY, t'=0. At LA t also equals 0.
As before, when the pulse of light leaves the tail of the space ship
it travels 300 miles straight up to the mirror and then back 300
miles, with the pulse burning LA when it starts and landing on NY
and the tail when it comes down. This is an omni-directional pulse
so that the pulse travels eastward in the space ship to a point 0.1
the length of the space ship where the mirror is located, then
reflects back to the tail.
The space ship measures its length as 3000 miles in its
reference frame. However due to length contraction in the moving
frame, from the viewpoint of the earth fixed frame the space ship is
actually 3000/g or 588.3484 miles long. This is shown by
transform equation 1 where t'=0, and x=3000. Therefore the x'
location of the nose opposite NY is x'=588.3484 and the x' of the
mirror is 0.1 of that or 58.8348. In the space ship's reference frame
its light traveled up 300 miles and eastward 300 miles to the 2
mirrors, then reflected downward and back toward the tail of the
space ship for a total of 600 miles, landing on the tail of the space
ship as the tail reached NY. (Again, because of length contraction, x'
will not show up as 3000 and 300 but as 588.3484 and 58.8348
miles.) However in the earth reference frame the space ship's light
travelled along the hypotenuse up to the mirror above KC and then
back down the other hypotenuse to NY.
The space ship light in its own reference frame generates
proper time (distance traveled divided by c) because the time is
measured at one point, the tail, in its reference frame and,
equivalently, the source of the light is not moving in the reference
frame in which it was generated. However proper time would also
be a light pulse at LA which was triggered by the light pulse of the
space ship's tail (x'=0) when it was at LA at t'=0. This light pulse
would also travel up 300 miles to a mirror above LA and would also
travel 300 miles eastward during the time the space ship pulse
went to above KC. It then would travel 300 miles back if reflected
or another 300 miles forward if not, for a total of 600 miles. This
pulse at LA also measures proper time, i.e., the light has traveled
600 miles at 186,000 miles per second in its own reference frame
and therefore has been traveling for 3,225.8 micro seconds when the
light has traveled a total of 600 miles. This proper time t' equals
3,225.8 if t' equalled 0 when the pulse started out. (NOTE THAT
WHILE THE LIGHT TRAVELLED 600 MILES THE TAIL OF THE SPACE SHIP
MOVED 3000 MILES FROM LA TO NY!)
But back to Relativity. Since by Special Relativity the space
ship light traveling in the earth reference frame must also be going
186,000 miles per second, and it has traveled 3,059.4117 miles in
going from LA to above KC and back down to NY, the length of the 2
hypotenuses, then 3,059.4117 miles divided by 186,000 miles per
second means that when the light pulse lands on NY, the time
assigned to this event in earth reference frame, t, is t=16,448.45
micro seconds. Since the tail of the space ship traveled x=3000
miles during the fixed frame time t=16,448.45 micro seconds, the
velocity of the reference frame must equal x/t = 182,388 miles per
second or 0.98058c. (The accurate number is c times 5 divided by
the square root of 26.)
By transform equation 1 when t'=0 and x=0 at x'=0, x=3000 is
'at' x'=588.3484. When the earth people in NY time the nose to tail
passage of the space ship using a light clock in their reference
frame (proper time), they will find that the space ship has passed in
3,225.8 micro seconds (their light up 300 miles to a mirror and back
down), and, since the space ship is 588.3484 miles long, the space
ship's reference frame must be moving by at speed u=182,388 miles
per second according to Special Relativity. Or you could say that
since the relative speed of the reference frames is u=182,388 miles
per second, the tail to nose length of the space ship must be
588.3484 miles.
TRY IT IN REVERSE
capsule: LENGTH CONTRACTION OCCURS IN THE MOVING REFERENCE
FRAME BUT SINCE EITHER FRAME CAN BE THE MOVING FRAME, BY THE
FIRST RELATIVITY PRINCIPLE, THE PHYSICAL REALITY OF WHAT
CONTRACTS IS DEPENDENT ON CHOICE.
Utilizing the exact analogy, with the light pulse emitted
'below' NY, and using the space ship as the fixed frame and the earth
as the moving frame you will get the same results in reverse. The
space ship people will see a light starting at their nose, going to a
point 300 miles 'below' their middle, and arriving back at their tail,
burning it. The space ship people know: that their space ship is
3000 miles long and that therefore the distance between NY and LA,
by transform equation 1, is x'=588.3484 miles; that the earth
reference frame is moving by at u=182,388 miles per second as
determined by the NY light starting at the nose, going at 186,000
miles per second to 300 miles below the middle, arriving back at the
tail after having traveled 3,059.4117 miles while NY traveled the
3,000 mile length of the space ship; that the time in the now fixed
space ship reference frame is 3000 miles divided by 182,338 miles
per second equals 16,448.45 micro seconds; and that the time it
took for LA to NY to pass by their tail is 3,225.8 micro seconds as
measured by a light sent 'down' 300 miles and back in the space ship
reference frame.
Some people have said that when you reverse which reference
frame is fixed and which moving, the distance from LA to NY is still
3000 miles but this is a length contraction from a space ship length
of 15,297.1 miles (g*3000). Going back to the space ship's frame
being the moving reference frame, if you use transform equation 9
and set t=0, then x'=0 when x=0 and x'=15,297.1 miles (g*3000)
when x=3000. This would seem to corroborate the idea. However, in
the first instance t'=0 but x'=15,297.1 when opposite x=3000 occurs
at t'= - 80,645 micro seconds. This is 80,645 micro seconds BEFORE
the tail of the space ship is at LA and the nose at NY. The measuring
events haven't even happened yet! Put another way, x'=15,297.1 is
opposite x=3000 at t'= - 80,645 micro seconds when x'=0 is at x= -
75,000 miles or 75,000 miles WEST of LA.
If the space ship and its reference frame were to slow down
greatly so that relative velocity between the reference frames was
slight, x'=0 would be at LA when x' equals approximately 3000 was
at NY. The un-contracted length of the space ship is 3000 miles.
Thus the length contraction occurs in the moving reference
frame, but either reference frame can be the moving frame by the
first relativity principle. The physical reality changes according to
the mathematician's choice. How clever!
"GRANDMA, WHAT STRANGE VARIABLES YOU HAVE!"
capsule: x' AND t ARE NOT THE SAME KIND OF ANIMALS AS x AND t'.
One important thing to note is that the space and time
variables x and t' are properties of only their own reference frames
whereas the variables x' and t are functions of a relationship
between two reference frames, namely the relative velocity. To
better illustrate, if you include other reference frames containing
an identical space ship at rest within and each reference frame
moving by the fixed frame at a different speed, the contracted
length, x', and dilated time, t, are different for each relative
velocity and obviously are not really properties of events occurring
in a single reference frame. Thus, is t really a time or x' really a
length? This point will be brought up again.
PROPER TIME
capsule: THE SAME PROPER TIME INTERVAL CAN BE GENERATED IN
ANY OTHER REFERENCE FRAME MOVING WITH RESPECT TO THE FIRST
BY USING THE ELECTRO-MAGNETIC PULSE OF THE FIRST REFERENCE
FRAME TO START AT THAT LOCATION IN THE SECOND REFERENCE
FRAME ANOTHER ELECTRO-MAGNETIC PULSE FROM A SOURCE NOT
MOVING WITH RESPECT TO THE SECOND FRAME. MEASURING THE
DISTANCES COVERED BY EACH PULSE IN ITS OWN REFERENCE FRAME IN
WHICH THE SOURCE IS NOT MOVING GIVES THE CHANGE IN PROPER
TIME SINCE THE SECOND PULSE WAS STARTED.
Proper time can only be measured directly in a reference
frame where the source of the light is at rest in the frame. Any
movement of the light source results in time dilation and length
contraction according to Special Relativity. (I would simply say
that relative motion has to be taken into account to determine a
'proper time'.) However, proper time IS available in both the 'fixed'
and 'moving' (which is which is your choice) reference frames.
Events which occur at the same place (i.e., at points, one each from
the moving and fixed reference frames, that at that instant are
'touching') can trigger proper clocks in each reference frame
consisting of light pulses sent perpendicular to the relative motion,
as is usually done in the explanatory example used to derive Special
Relativity (Figures 1 & 2).
We originally gave the proper time, t', to the moving frame by
sending a light pulse from the tail, up to its mirror which is
traveling along with its reference frame (remember, the space ship
and its mirror(s) are not moving with respect to each other in the
space ship's reference frame) and back to the tail. But it is totally
reasonable to have the simultaneous initiation of a pulse at a point
in the 'fixed' earth frame, e.g., when the space ship's tail is at LA at
t'=0 (and t=0), to travel up to a mirror the same distance above LA
as the mirror is above the tail and return to LA. That pulse, if
omni-directional, will also travel horizontally the same total
distance that it traveled up and back. This is proper time also
because it is being measured at the same place, LA in its reference
frame. But notice that proper time is occurring in both the 'moving'
and 'fixed' frames.
So while the space ship's proper time pulse travels up from its
tail and LA to a point 300 miles above its tail and KC in 1,612.9
micro seconds and back down to its tail and NY in another 1,612.9
micro seconds, for a total distance of 600 perpendicular miles in
3,225.8 micro seconds in both reference frames, the space ship
pulse's triggering of a proper time omni-directional pulse at LA will
result in this pulse going up and eastward 300 miles in 1,612.9
micro seconds and another 300 miles (back, if reflected) in the
succeeding 1,612.9 micro seconds, for a total of 600 miles in
3,225.8 micro seconds. The tail of the space ship has traveled 3000
miles across the earth during a proper time interval, t', of 3,225.8
micro seconds during which earth light only travelled 600 miles
from LA, A PROPER TIME INTERVAL WHICH IS THE SAME IN BOTH
REFERENCE FRAMES!
If you can recognize that 'proper time' is the same in BOTH
reference frames, then you realize that in this example the moving
reference frame of the space ship traveled 3000 miles in 3,225.8
micro seconds or 930,000 miles per proper time second, WHICH
PROPER TIME IS THE SAME WHETHER MEASURED IN THE MOVING OR
FIXED REFERENCE FRAME. This is five times the 'speed of light' as
measured in all reference frames in which the electro-magnetic
source is not moving with respect to the frame.
In other words, the relative velocity of these reference frames
is g*u as measured by a distance covered in the fixed frame by a
single moving frame point during a proper time interval which can
be determined in ANY reference frame using an electro-magnetic
source not moving within that reference frame. The same proper
time interval can be generated in any other reference frame moving
with respect to the first by using the electro-magnetic pulse of the
first reference frame to start an electro-magnetic pulse from a
source not moving with respect to the second frame and measuring
the distances covered by each pulse in its own reference frame. The
distances divided by c equal the proper time intervals being 'ticked'
off.
Now the situation should be getting obvious. Only BECAUSE
light and other electro-magnetic radiation leaves its source (in a
vacuum) at a speed, c, equal to approximately 300,000,000 meters
per second, is it possible to ASSUME that light travels at the same
speed, c, in ALL reference frames and obtain the mathematical
transformation known as the theory of Special Relativity in which
length is 'contracted' by g in the 'moving' reference frame
(whichever frame you choose), time (but not proper time) is 'dilated'
in the other 'fixed' reference frame, and the relative velocity
between the two reference frames (the distance covered by a point
in the moving frame divided by the 'proper' time interval available
from ANY reference frame) is 'reduced' by the factor g, resulting in a
relativistic velocity, u, which is always less than 1.
Originally scientists had difficulty with the 'ether' theory in
which all electro-magnetic radiation traveled at a constant speed,
c, in relation to one frame of reference, the 'lumeniferous ether'.
When Special Relativity was found to work experimentally, it was
assumed that Special Relativity represented reality much better. It
simply was not recognized that Special Relativity was a
mathematical transformation (distortion) of something much
simpler: a 'universe' in which electro-magnetic radiation leaves its
source at the speed c.
v is what I am calling this velocity, equal to un-contracted
fixed frame distance traversed by moving frame points (or vice
versa) divided by the proper time interval. Since v=ug, as can also
be seen by differentiating transform equation 5 with respect to t',
using the usual definition of gamma (g) in terms of relativistic
velocity u, quoted earlier in this paper, you will find that g also
equals (1+(v**2/c**2))**(1/2). Since this quantity is always
greater than v/c, u/c is always less than 1. (Velocities which are
'transformed' using the relativistic addition of velocities formula
(u1+u2) / (1 + u1*u2/(c*c)) are discussed fully in my 1983 paper.
Do not try to apply them to anything in this paper.)
THE TRANSFORM EQUATIONS
capsule: THE FULL SET OF RELATIVISTIC TRANSFORM EQUATIONS ARE
DERIVED AND LISTED TOGETHER WITH THE GALLILEAN EQUATIONS AND
THREE OTHERS.
Equations 1 through 12, following, are the Special Relativity
transform equations and their permutations. The Gallilean
transform equations are 13 through 15. 16 through 18 are three
other equations that will prove useful. g refers to gamma, defined
in the usual way in the section SPECIAL RELATIVITY.
1. x' = x/g - gut'/g or gx' = x - gut'
2. t = t'/g + gux/g(c**2)
3. t' = (x - gx')/gu
4. x = (gt - t')*(c**2)/gu
5. x = gx' + gut'
6. t' = gt - gux/(c**2)
7. x' = gx - gut
8. t = gt' + gux'/(c**2)
9. x = x'/g + ut
10. x' = (t - gt')*(c**2)/gu
11. t' = t/g - ux'/(c**2) or gt' = t - ugx'/(c**2)
12. t = (gx - x')/gu
13. x = X + vT
14. X = x - vT
15. T = (x - X)/v
16, v = gu
17. X = gx'
18. T = t'
All of the Special Relativity transform equations can be
derived from equations 1 & 2 although this is not the way they are
usually presented. (They can also all be derived from any other pair
that includes positions and times from both reference frames.)
Equations 1 & 2 find contracted length, x', and dilated time, t, in
terms of un-contracted length, x, and proper (un-dilated) time, t'. t'
is proper because it is a time interval measured at one point, the
tail of the space ship, in its own reference frame. Length is proper
in the rest frame--length contraction in Special Relativity occurs in
the moving frame. Both x and t' are variables definable in their own
reference frame without reference to another reference frame. They
are both definable even if you don't know the relative velocity, if
there is one, between the reference frames. And both variables are
the same as they would be from a non-relativistic or 'Gallilean'
viewpoint.
Equations 3 through 6 are derived by rearranging equations 1 &
2. Equations 7 & 8 are obtained by substituting transform equation
6 into equation 1 and transform equation 5 into equation 2. The
rest of the 12 equations are rearrangements of equations 7 and 8.
Equations 5 through 8 are the usual equations which are derived as
the transform equations of Special Relativity, but as you can see,
these can all be derived from equations 1 & 2. Equations 1 & 2 are in
essence relativistic definitions of the contracted length in the
moving frame and the dilated time in the fixed frame, caused by
relative motion, in terms of the variables which are not affected by
relative motion. Qv. section "GRANDMA, WHAT STRANGE VARIABLES
YOU HAVE!" above.
Equations 13, 14, & 15 are the conventional Gallilean
transform equations for relative motion in the x direction. Of
course there is only one 'time', denoted by T (capital), in these
equations.
WHERE IT'S AT
capsule: ALL PARTS OF THE MOVING FRAME ARE RELATIVISTICALLY
ALWAYS AT POINTS OF THE FIXED FRAME WHERE THEY WOULD ALSO BE
IF LENGTHS IN THE MOVING FRAME ARE NOT CONTRACTED, THE
RELATIVE VELOCITY BETWEEN THE REFERENCE FRAMES IS g*u
(RATHER THAN u), THE TIME USED IS t' (THE SAME PROPER TIME
AVAILABLE IN ALL REFERENCE FRAMES), AND ELECTRO-MAGNETIC
RADIATION TRAVELS AT SPEED c AWAY FROM ITS SOURCE.
At this time go back to our example for Fig. 2. Recognizing
that proper time occurs in both reference frames, as explained in
the section PROPER TIME, above, let us consider what locations from
the moving frame are at what locations in the fixed frame as this
proper time progresses. Starting at t'=0 and x'=0 (the location of
the tail), x'=58.8348 (the location of the mirror), and x'=588.3484
(the location of the nose), we find that the corresponding locations
in the fixed frame at this time are x=0, 300, and 3000 miles,
respectively. (The location of the nose at x'=588.3484 is actually
the result of applying transform equation 1 to find out what point of
the rocket, x', occurs at NY at the proper time, t'=0, when the tail is
at LA.) Notice that in spite of the fact that we claim length
contraction, the part touching NY which we call the nose occurs
3000 miles east of LA and the part we are calling the mirror 0.1 the
length of the space ship is at 300 miles east of LA just as the other
mirrors are 300 miles above LA. At proper time t'=1,612.9, using
transform equation #5, and the three values of x' used previously,
the tail (x'=0) is at x=1500 miles at KC, the mirror (x'=58.8348) is
at x=1800 or 300 miles east of KC, and the nose (x'=588.3484) is at
x=4500 miles. All parts of the space ship have moved 1500 miles
east across the earth.
Thus although we speak of a length contraction of the space
ship in the moving reference frame, a contraction which the space
ship people will not acknowledge, other than to be agreeable with
relativity, the points on the space ship corresponding to the tail, the
mirror, and the nose still represent a distance of 3000 miles on the
earth with the mirror at 300 miles east of the tail. At t'=3,225.8,
the tail (x'=0) is at x=3000, the mirror (x'=58.8348) is at x=3300,
300 miles east of NY and the tail, and the nose (x'=588.3484) is at
x=6000 miles, i.e., 3000 miles east of NY where the tail is.
Again keep in mind that this proper time we're using is a
proper time available in either the moving or the fixed reference
frame. The proper light pulse in earth reference frame has gone 300
miles up and 300 miles east followed by returning down to LA and
returning from the mirror east of LA or, if not reflected, going
another 300 miles east of that. The same thing has occurred with
the light pulse in the moving reference frame, except that because
of length contraction we insist on calling it 58.8348 miles in the
direction of motion instead of 300, although it is 300 miles
vertically. Keep in mind however that the space ship acts in earth
frame like it is 3000 miles long, the people on the space ship would
measure 3000 miles in their reference frame, and it is our
DECLARATION that has caused it to be only 588.3484 miles long and
the mirror which is 0.1 the length from the tail to be only 58.8348
miles from the tail. This DECLARATION that there was length
contraction in the moving frame occurs because we ASSUMED that
the light travelled the same speed in all reference frames.
Remember also that the contraction would occur in the OTHER
reference frame if we ARBITRARILY CHOSE to reverse which was the
moving and which was the fixed reference frames.
A CLOSER LOOK AT VELOCITY
capsule: IT IS POSSIBLE TO DEFINE A VELOCITY BASED ON THE
DISTANCE ANY POINT OF THE MOVING FRAME HAS MOVED THROUGH
THE FIXED FRAME COMPARED WITH HOW FAR LIGHT OF THE FIXED
FRAME HAS MOVED DURING THE INTERVAL. IT IS c TIMES THE RATIO
OF THE FIRST DISTANCE TO THE SECOND DISTANCE. THIS VELOCITY
WILL EQUAL g*u AND CAN BE VERY LARGE.
Consider equation 16. This defines a relative velocity between
the moving and fixed frames, v=gu, as the (un-contracted) distance
in the fixed frame which is traversed by ANY point of the moving
frame during (divided by) the proper time, t', of that traverse.
(Differentiate transform equation 5 with respect to t' or
differentiate gx' (transform equation 1, 2nd form) with respect to
t'.) This proper time interval is available from EITHER the fixed or
moving frame (or any other frame) as described above in the section
'PROPER TIME'. Although not the relativistic definition of velocity,
this is certainly an acceptable definition. In fact if you think about
it, it is the most reasonable one. In our space ship example it would
go as follows. The LA and NY observers have arranged to set off
light signals whenever the tail of a space ship passes by. Both do
this. The LA observer will receive the NY signal 19,354.8 micro
seconds after the NY observer sent his signal, 16,129.0 micro
seconds of which was the time for the NY signal to reach LA. Thus
the NY signal was sent 3,225.8 micro seconds after the LA signal,
the tail of the space ship traversed the 3000 miles from LA to NY in
3,225.8 micro seconds, and therefore it is extremely reasonable to
state that the relative speed, v, of the moving reference frame (in
which the space ship is at rest) is 930,000 miles per second, 5
times c, and equal to g*u.
Conversely, while the space ship's tail has traveled from LA to
NY (at which time the NY observer will send his signal) in 3,225.8
micro seconds after the LA signal is sent, the LA signal will have
moved 600 miles, or one fifth of the distance to NY. Therefore, the
NY observer will only have to wait another 12,903.2 micro seconds
after sending his signal until the LA signal arrives.
Please note that this is only a matter of viewpoint. I have not
changed any of the relativistic numbers. I have simply made
comparisons of other numbers that are correct relativistically. I am
using the x and t' that Special Relativity uses. But I use 'physical
properties' (x and t') that are obtainable in one reference frame and
do not depend on a relationship with another (or many other)
reference frames as do contracted length and dilated time.
If you find the last three paragraphs confusing because you
still believe 'nothing can go faster than the speed of light', apply the
principles therein to all the examples shown in Figure 1, starting
with the slowest relative speed between the reference frames
where the mirror is 7,500 miles above the space ship and proceeding
through each successively higher reference frame relative speed
exemplified by the lower height of the mirror required to reflect the
light back down on NY, the last being the example just above. Notice
that in all cases this 'conventional' velocity, v, equals g*u. This
velocity has nothing to do with time dilation or length contraction
since we are only talking about the location in earth reference
frame of one point, the tail of the space ship from the moving
reference frame, and proper time as measured in the fixed earth
reference frame by non-moving light sources at LA and NY. But it
should also be obvious that for the space ship's light to land back on
its tail and NY it will have to have velocity equal to the vector sum
of c in the up and down direction plus v=gu in the horizontal
direction.
This is what has been said throughout this paper and is the
point of it. Electro-magnetic radiation travels at speed c from its
source. Only because it does THIS can you 'ASSUME' light travels the
same speed in all reference frames and come up with the values of
length contraction, time dilation, and all relative speeds less than c
that you get in Special Relativity. For example, try applying the
assumptions of Special Relativity to an 'ether' theory and you won't
get the equations of Special Relativity.
TRANSFORM EQUATION COMPARISONS
capsule: USING EQUATIONS 16-18 SHOWS THAT THE RELATIVISTIC
TRANSFORM EQUATIONS ARE EQUIVALENT TO THE GALLILEAN
TRANSFORMS.
Let us look at equation 17, X=gx'. I used capital X in the
Gallilean transform equations to denote position in the moving
frame because in the conventional equations there is no length
contraction and therefore using X (capitalized) keeps it separated
from x' of the moving frame in relativity where there is length
contraction. Equation 17 removes the length contraction of the
moving frame.
Equation 18, T=t'. There is only one time in the Gallilean
transform equations. I have capitalized it, and it is the same as
proper time, which is t' in the relativistic transform equations. It
can be obtained by 'ticking off' the distance light travels (time
equals distance traveled divided by c) from a source not moving with
respect to the reference frame in which it is being measured. (If
you want the time measured 'at the same place' you can send the
light half the distance and reflect it back to the source with a
mirror also not moving in the reference frame.) Equation 18
expresses this.
Transform equations 13, 14, & 15 are the standard Gallilean
transforms except that the distance in the moving frame has been
denoted by a capital X rather than x'; the time, which is equal to the
proper time, has been given a capital T so that there will be no
difficulty knowing whether or not one is referring to relativity; and
v equals un-contracted distance traversed divided by the proper
time interval. (Proper time intervals are available in both the
moving and fixed reference frames as described in the section
PROPER TIME. Un-contracted locations are obtained by multiplying
the contracted locations by g (gamma), i.e., equation 17.) In the
relativistic formulas, whenever gu is seen together, this is the
equivalent of the non-relativistic velocity, v, shown in equation 16,
where the relative velocity of the reference frames is the
un-contracted distance traversed by any point of one reference
frame across the other divided by the proper time interval to
traverse that distance. In plain English, you are comparing how fast
something is moving through your reference frame compared with
how fast YOUR light (source not moving with respect to you) is
moving in a vacuum relative to you (speed c, of course). Isn't that
what you really wanted to know: how much distance is being
traversed in your reference frame compared with a known standard?
Equations 1 & 2 are essentially definitions of the quantities x'
and t, respectively, in terms of the familiar quantities t' and x
which are the same as in the Gallilean transformations. Equation 1,
as shown in its second form, directly becomes Gallilean equation 14
when 16, 17, and 18 are taken into account. The relativistic
equation 1 is really the Gallilean equation 14 in disguise because
someone insisted on ASSUMING that light travels the same speed in
all reference frames instead of recognizing that without this
assumption the radiation and objects are all where they must be and
when they must be there if the light travels at speed c away from
its source.
I chose to derive the transform equations from equations 1 & 2
because, as noted earlier, the independent variables x and t' are
properties of only their own reference frames. They have the same
meaning as in the Gallilean transform equations and do not change
even if there is not another reference frame around. However the
dependent variables x' and t do not even have any non-Gallilean
meaning unless there is relative motion. More than that, they take
on as many different values as there are different reference frame
motions to compare. x' and t, supposedly a location and a time, are
functions of a relationship between two reference frames, namely
the relative velocity. For example, without any physical change in a
reference frame, the dependent variables x' and t would change if a
3rd reference frame moving at a different relative velocity were
used. Again the question, is t really a time or x' really a length?
Relativistic velocity, u, equals changes in x, a physical
property not dependent on a relationship between reference frames,
divided by changes in a time, t, equal to proper time times g (at the
origin of the moving reference frame), which IS dependent on the
relationship. u also equals changes in x', a physical property in a
reference frame divided by a relationship between reference frames
(x/g), divided by proper time, a property definable in any reference
frame without reference to any other. In both cases relativistic
velocity u in a sense has a g (gamma) in its denominator. Without
that g, it becomes conventional velocity v=u*g, as is the point of
this paper.
Equation 3 shows proper time is just the same as what has
always been meant by time when using the conventional Gallilean
transformations if you recognize that the space ship is really 3000
miles long, as they would measure it in their own reference frame.
The gx' gets rid of the length contraction of the moving reference
frame. Equation 3 converts directly into Gallilean equation 15 by
use of equations 16, 17, and 18.
Equation 5. If you take the derivative of x with respect to t',
you will get what we have been saying all along, namely that the
relative velocity between the reference frames, utilizing the proper
time (the same in all reference frames), equals gu=v. This will give
the location of the space ship's tail (x'=0) in the earth frame at all
proper times. Of course, if you are not at the tail, you must add in
the distance you are along the space ship. But notice that this
distance you must add in is the un-contracted distance gx' which
equals X (capital). You can now see that equation 5 is exactly the
same as equation 13, a Gallilean transform equation, when you make
the allowances from equations16, 17, & 18.
Equation 8 shows that at the tail (x'=0) the time, t, will be g
times the proper time. This represents the time dilation. It also
demonstrates that in relativity the two times, relativistic, t, and
proper, t', although related to each other in a simple straightforward
way (known as time dilation) at the origin (x'=0) of the moving
reference frame, depend also on position everywhere else. Equation
11 is the same equation rearranged.
Equation 9 shows that the moving reference frame (x'=0)
travels the relativistic velocity times the dilated time past the
fixed reference frame. Equation 9, for x'=0, can also be written x =
gu * t/g which again is Gallilean equation 13 when X (capital) = 0.
The distance traveled by the origin of the moving frame over the
fixed frame is the conventional velocity v=gu times the proper
'un-dilated' time t/g (but only at x'=0=X (capital) because t varies
greatly with position).
Equations 1 and 5 show that at any time t', every point in one
reference frame will be opposite a unique point in the other
reference frame. (This is also true of holding t constant, but the
correspondence is strange.) But equations 4 & 10 show that because
of the nature of time in Relativity (since I have no difficulty with
proper time, I assume non-proper time is the culprit), if both proper
and non-proper time are specified, this will determine only one
point in each reference frame and these will be in contact with each
other. No other points from the two reference frames will be in
contact at these particular two times! How's that for a paradox--or
at least confusion.
TRANSFORMING RELATIVITY
capsule: SPECIAL RELATIVITY AND THIS 'EMISSION' THEORY ARE
TRANSFORMS OF EACH OTHER. ONLY ONE IS PHYSICALLY SENSIBLE.
In the 12 transformation equations, given the relative velocity
of the reference frames and any two of the four variables x, x', t, t',
the value of the other two is determined. With this information you
can convert to the conventional. You don't need the contracted
length, x', and you certainly don't need the dilated time, t, or a
contracted velocity either for that matter. But if you have the
conventional and care to do so, you can convert it into the
relativistic, just as you could convert the current equations of
motion of the bodies of the solar system into equations of motion
with respect to the earth ('with the earth as the center'). Both are
mathematically correct but one makes rational sense and one does
not. To convert to Relativity, you need g which equals
(1+((v/c)**2))**(1/2). This v must be un-contracted distance
divided by proper time. Then use equations 16-18 and the transform
equations. Special Relativity and the Gallilean transform equations
are transforms of each other. The Gallilean equations are used with
the understanding that light travels at speed c from its source in a
vacuum.
RECAP
The 'ether' theory was in error. Relativity did not fit the
'ether' theory just as pre-Copernican astronomy equations would not
have fit a 'flat earth' theory. Relativity works because it is based
on a correct idea. But Special Relativity is not that correct idea; it
is a mathematical transformation of that correct idea. The correct
idea is that electro-magnetic radiation in a vacuum travels at a
constant speed, c, relative to its source and Special relativity is a
mathematical transformation of that idea just as pre-Copernican
astronomical equations were equations of motions relative to the
earth instead of the much simpler and more sensible (for purposes
of looking at the solar system) equations of motions relative to the
sun.
You can write the equations of motion of the heavenly bodies
relative to where you are standing on earth. This 'works' but the
results were unintelligible to many generations. Or you can change
your viewpoint and write the equations of motion of the heavenly
bodies relative to the sun and make more sense of the whole thing.
(This assumes the other stars are 'fixed'. To understand galactic
motion and inter-galactic motion, other reference frames would be
preferable.)
If you assume that light travels the same speed in all
reference frames, you will come up with the theory of Special
Relativity. If you find (except for paradoxes) that this 'works'
physically, then you can also determine, as was done in this paper,
that electro-magnetic radiation travels in a vacuum at speed c
(186,000 miles per second approximately) RELATIVE TO ITS SOURCE
by simply dropping the 'same speed in all reference frames'
ASSUMPTION. YOU ARE SIMPLY CHANGING YOUR VIEWPOINT. All
physical objects and light will have the same physical relationships
as with Special Relativity but different (and much more common and
understandable) values will be given to SOME of the distances,
times, and the relative velocity between reference frames involved.
All space and time variables can be related to their own individual
reference frame. No knowledge of other reference frames moving
relative to theirs is needed. None of the space and time variables
are a function of a velocity. But by knowing the relative velocity of
reference frames, variables in different reference frames can be
related to each other. Because I am talking only about how we are
looking at the same 'data', if Special Relativity is 'valid', so is this
emission theory. But this emission theory does not have paradoxes.
P.S.: There is nothing here to say that light could not be influenced
by something--gravity, for instance. All I am saying is that Special
Relativity and this emission theory are mathematically equivalent
(can be transformed one into the other) but that only the emission
theory makes PHYSICAL sense.
EPILOGUE
It is a principle of physics that momentum (in the absence of
outside forces) and total energy (called mass-energy in Relativity)
are conserved. Both are functions of velocity. But this emission
theory and Special Relativity use different values for the relative
velocity of the reference frames.
For information on how this is resolved with momentum and
energy in relativistic mechanics, see the paper, "RELATIONSHIP
BETWEEN SPECIAL RELATIVITY AND MODIFIED NEWTONIAN
MECHANICS" available from the author. (I would appreciate $8 for
postage, production, handling, etc. In its current form it is 37 pages
long.)
Dr. Sherwood R. Kaip
1204 Turnberry
El Paso, TX 79912
U. S. America
Telephone (915) 833-2929
(PLEASE do not try to contact me directly through Usenet.
Because of certain factors, contact was missed last time and
probably would be missed again. Use the mails or telephone.)
END OF PAPER