In an effort to understand Einstein's relativity I have prepared the
numerical example which is presented below. My question is: Did I
get the math and the analysis right? I suggest that the question be
offered first to mathtalk-ga since he provided some of the calculus in
answer to an earlier question. I apologize for the poor quality of
the diagrams but Google Answers doesn't seem to permit the insertion
of better diagrams.
A Numerical Example of Einstein's Relativity
March 11, 2004
In his famous first published paper on Special Relativity in 1905
Einstein stated that his conclusions were derived from two basic
hypotheses. He later added that these hypotheses are only applicable
to domains ?in which no gravitational field exists.? The first basic
hypothesis was ?The laws by which the states of physical systems
undergo change are not affected whether these changes of state be
referred to one or the other of two systems of co-ordinates in uniform
translatory motion.? He later provided a General Relativity version of
the first hypothesis to the effect that all coordinate systems in
which ?readings? ?observed simultaneously on adjacent clocks (in
space) differ from each other by an indefinitely small amount? ?are
essentially equivalent for the formulation of the general laws of
nature.? He sometimes referred to such a system of coordinates as ?a
stationary system of coordinates.? Under this more general formulation
the laws of physics are the same in two such systems of coordinates
even if their relative motion is not uniformly transitory with respect
to each other.
His second basic hypothesis was ?Any ray of light moves in the
?stationary? system of co-ordinates with the determined velocity c,
whether the ray be emitted by a stationary or by a moving body.? That
second basic hypothesis can be analysed into two parts. It is possible
to conceive of a universe in which only the first part was valid, but
Einstein hypothesized that both parts were valid. One part specifies
that two rays of light which started off in a certain direction
side-by-side would continue to move side-by-side, even if the sources
of the lights were moving with respect to each other when the rays
were emitted. The other part specifies that any ray of light will have
a speed of c, approximately 300,000 kilometers per second in a vacuum,
when measured in a stationary system of coordinates even if that
system is moving with respect to the source of the light. The
statement requires that the two points between which the speed is
being measured must be stationary with respect to each other at the
time of measurement even if the source of the light and the target of
the light are moving with respect to each other. Any two points
stationary with respect to each other could provide the system of
coordinates for measuring the speed even if those two points were
moving with respect to other points.
Although the second basic hypothesiss was stated in terms of light the
hypothesis was considered to apply as well to all forms of
electro-magnetic radiation.
Some effects of those two basic hypotheses, as well of four other
hypotheses implicit in Special Relativity, are illustrated in the
numerical example described below. The effects discussed are those of
an electric force acting on one body and not on another upon
measurements taken on those bodies of the velocities and lengths of
the bodies and of the readings of clocks attached to them.
Suppose that there were at some time two spaceships, as depicted in
the diagram below, parallel to each other and at rest with respect to
each other out in the near-vacuum of space in a region with no
discernible net gravitational or electro/magnetic influences other
than one electric force specifically described below.
Diagram I
_______________________M'______________________
I______________________0:00_____________________I
_______________________M_______________________
I______________________0:00_____________________I
Each rectangle in the diagram represents a space ship. Any two points
on either ship are considered to provide a stationary system of
coordinates. Each ship has a mass of 100,000 kilograms (kg). The upper
ship has a net positive electric charge. The lower ship has no net
electric charge. Each ship has clocks at many points along its length
and next to each clock a camera recording continuously the reading of
that clock, the reading of the adjacent clock, if any, on the other
body, and the position of the clocks on the bodies relative to
distance markers on those bodies. The colons in the diagram indicate
the positions of selected clocks. The number to the right of a colon
indicates the reading of the clock at that point in nanoseconds (ns).
The number to the left indicates the distance in meters (m) to that
point as measured on that body from its midpoint, which on the upper
ship is marked by an M?above the colon and on the lower ship is marked
by an M above the colon.
When the cameras recorded the numbers shown in the diagram above all
the clocks on the ships were synchronized and read zero. Starting then
and continuing for 50 ns on the clocks of the lower ship a constant
positive electric force of 800 kg m/ns/ns was applied to the
positively-charged upper ship pushing it to the right. The force was
applied to all parts of the upper ship in proportion to the masses of
those parts. The magnitude of the force was measured with respect to
the lower ship. The force in this case had its origin in some body
other than those in the diagram and did not affect the lower ship.
Such a constant assumed force is convenient for an illustrative
example but would be very difficult to arrange in practice. The
distances from the emitting body to the upper ship at the times of
radiation would probably be constantly changing in practice. The
distances the radiation would have to travel after emission to reach
the upper ship would also be changing as the ship?s velocity varied.
And account would also have to taken of the way in which the
accelerating effect of an incoming electrical force is affected by the
motion of the body being affected.
As mentioned above in the discussion of the model, Special Relativity
hypothesizes that all light or electrical influence arriving at a
body, such as the upper ship, will always be measured on that body to
have the same velocity, regardless of the relative motion of that body
with respect to other bodies, so long as the velocity is measured on
that body by use of two points constituting a stationary system.
Nevertheless, according to Special Relativity, incoming forces
measured on a body, such as the upper ship, to be equal would be seen
to have different effects on that ship when it is moving at different
velocities with respect to the lower ship, as those effects are
measured on the lower ship. According to Special Relativity, a force F
of a certain size as measured on the lower ship would somehow give
acceleration A to the upper ship to a greater extent relative to the
lower ship when the upper ship was in motion away from the source of
the force with a lesser velocity V with respect to the lower ship than
when the upper ship had higher velocity. Under the rules of Special
Relativity the effect of a body?s velocity in reducing the
accelerating effect of an incoming force is not simply proportional to
the ratio of the body?s velocity to the velocity of the incoming
force. Rather, according to one implicit hypothesis the effect can be
expressed precisely in either of the following formulations, in which
H represents the mass of the body receiving a force and c has a value
of approximately 300,000 km/sec, which shall here be considered to .3
m/ns.
F = HxA/(1-V^2/c^2)^.5 or A = (1-V^2/c^2)^.5xF/H
As a result when a constant force is applied to an accelerating body
over equal increments of time smaller and smaller increases in
velocity are hypothesized to result as the body?s velocity increases.
If a body?s velocity approached the value of c then according to the
formulas its acceleration would approach zero and the resulting
velocity could never exceed c.
With the constant force and the values for c and H assumed above the
equation for acceleration becomes:
A = (1 ? V^2/.3^2^.5x800/100,000 = .008(1 ? V^2/.3^2)^.5
When the distance moved by the upper ship from its rest position
relative to the lower ship in any time T is considered to be a
function F(T) the equation for acceleration can be restated as:
F??(T) = .008(1 ? (F?(T))^2/3^2)^.5
By integrating from this formula it is possible to calculate that in
50 ns the movement of the upper ship as measured on the lower ship
was:
F(T) = 11.25(1 ? cos[2T/75]) = 8.6035 m.
This movement is less than the 10 m the movement would have been if
the velocity of the ship had not reduced the accelerating effect of
the incoming force.
At the end of the 50 nanoseconds the velocity of the upper ship would be:
F?(T) = .3sin(2T/75) = .2916 m/ns
At that time the velocity would have been .4 m/ns in the absence of
the effect of the velocity in reducing acceleration.
According to a second implicit hypothesis of Special Relativity a
body, in this case the upper ship, which had been at rest with its
clocks synchronized with the clocks on a reference body, in this case
the lower ship, would have the pace of time-keeping advance of its
clocks, as measured on the lower ship, lowered relative to the pace of
timekeeping advance of clocks on the reference lower ship when the
clocks on the ship had all been accelerated into motion relative to
the lower ship. At each velocity of the upper ship?s midpoint clock
relative to the reference body the pace of timekeeping advance of that
clock would be measured at (1-V^2/c^2).5 times the pace of timekeeping
advance of the clocks on the reference body, considered here to be one
ns per ns. Accordingly the pace of timekeeping advance of the upper
ship midpoint clock after 50 nanoseconds would be:
{1 ? (.3sin[2T/75])^2/.3^2}^.5 = .2353 ns/ns.
By integration it can also be determined that over the full 50
nanoseconds of the period of acceleration on the lower ship?s clocks
the total timekeeping advance of the upper ship?s midpoint clock would
be 36.4476 ns. On the basis of the calculations just described the
calculated position and reading of the upper ship midpoint clock
relative to the lower ship at the end of the 50 ns period on the lower
ship is shown in the following diagram. At that time the upper ship?s
clock was no longer synchronized with the clocks on the lower ship.
Diagram II
__________________________________M'__________
I_________________________________0:00_________I
_______________________M______________________
I______________________0:50_____8.60:50_________I
After the end of the period of acceleration the velocity of the upper
ship midpoint clock as measured from the lower ship would continue to
be the .2916 m/ns achieved at the end of the period, and the pace of
timekeeping advance of the upper ship clock as measured on the lower
ship would continue to be .2353 of that of the lower ship clocks. The
next diagram shows then what would be the positions of the upper
midpoint clock and the adjacent lower ship clocks an illustrative
28.6783 ns after the end of the period of acceleration as measured on
the ship.
Diagram III
_____________________________________M'______
I____________________________________0:43.20__I
____________M_______________________________
I___________0:78.68___8.60:78.68____16.97:78.68__I
The upper midpoint clock had moved a further 8.3626 m to the right
relative to the lower ship, and the upper clock reading had advanced a
further 6.7480 ns while the ship?s clocks advanced 28.6783 ns.
The two diagrams above illustrate a result of Special Relativity often referred
to as the Twin Paradox. If when the ships had been at rest with
respect to each other there had been on board the lower ship at its
midpoint a twin whose aging process proceeded like clockwork and at
the midpoint of the upper ship the other twin with the same aging
process then the lower twin would observe ? when later studying
pictures taken by the cameras on the lower ship - that over the
acceleration period the upper twin had aged only about three quarters
as much as he or she had. During periods after the end of acceleration
the upper twin would appear to have aged only about as quarter as much
as the lower twin.
Since in the case just discussed the force was applied to all parts of
the upper ship in proportion to the mass of the parts, the force would
have caused no physical stretching or contraction of the ship as
measured on that ship, but according to a third implicit hypothesis of
Special Relativity a body which is accelerated relative to a second
body would have all segments of its length, as measured on that second
body at the end of the period of acceleration, contracted to
(1-V2/c2).5 times the previous rest length of that segment, when V is
the velocity attained by all parts of the body relative to the second
body at the end of the acceleration period. The apparent contraction
would be toward the center of gravity of the contracting body. The
following diagram then shows what the distances from the upper ship
midpoint to two other illustrative upper ship points, A? and B?, were
after 50 ns on the lower ship clocks from two different viewpoints,
that is as measured on the upper ship and as measured on the lower
ship at the end of the period of acceleration..
Diagram IV
___________________________M'_________________
I_____-72.10:_______________0:36.45______72.10:__I
_________________M___________________________
I______-8.36:50___0:50____8.60:50________________I
_
The selected illustrative upper ship points A? and B? had both been
72.1041 m from the upper ship midpoint as measured from both ships
before the acceleration. After the acceleration the selected points
were still 72.1041 m from the upper ship midpoint as measured on the
upper ship but only 16.9661 m from point on the lower ship adjacent to
the upper ship midpoint as measured from the lower ship. During the
acceleration period as measured on the lower ship the A? point on the
upper ship moved 63.7415 m, from ?72.1041 m to -8.3626 m, that is
faster and further than did the upper ship midpoint, which moved only
from 0 to 8.6035 m. The B? point, as measured on the lower ship,
actually moved in the direction opposite to the upper midpoint?s
movement, that is from 72.1041 back to 25.5696 m..
The following diagram shows the different distance measurements
28.6783 ns after the end of the acceleration period. During this
period all points of the upper ship moved 8.3620 m to the right
relative to the lower ship at .2918 m/ns as measured on the lower ship
and experienced no further acceleration. During that time there was no
further contraction of the upper ship as measured from the lower ship.
Diagram V
________A'______________M'______________B'_____
I________________________0:43.20_____72.10:_____I
________M____________________________________
I_______0:78.68_______16.97:78.68_____33.92:78.68_I
The two diagrams just above did not show the readings of the clocks at
points A? and B? on the upper ship. To calculate these readings it is
necessary to take into account the fact that under the rules of
Special Relativity the apparent advance of accelerated clocks other
than those at the center of gravity of a body, as seen from an
un-accelerated body, is not governed by the hypothesis, used above for
the upper ship midpoint clock, that the advance of an accelerated
clock is determined solely by the velocity experienced during the
period of acceleration. For clocks not at the center of gravity on a
body accelerated from a rest position a fourth implicit hypothesis of
Special Relativity provides that at the end of the period of
acceleration the reading of a clock which is ahead of the center of
gravity will be behind the clock at the center of gravity by VxL/c^2
where V is the velocity of the body at the end of the acceleration and
L was the distance between the two clocks before the acceleration
began, all as measured on the reference body. By the same hypothesis
the reading of a clock behind the center of gravity will be ahead of
the center of gravity clock by VxL/c^2. The readings of the ship
clocks under this hypothesis at the end of the 50 ns acceleration
period are shown in Diagram VI below. While the upper ship midpoint
clock moved forward by 36.4476 ns during the acceleration period, as
measured from the lower ship, the point A? clock, which had presented
a zero reading at the same time the other clocks did, moved forward by
more, that is by 270.0649 ns, to a reading 233.6173 ns ahead of the
clock at the upper ship midpoint as measured on the lower ship. During
the acceleration the forward upper clock ended up 233.6173 behind the
upper middle clock at -197.1697. The reading of the upper ship forward
clock as seen from the lower ship actually moved backward, therefore,
during the acceleration period.
Diagram VI
______________________M'_____________________
I__-72.10:270.06________0:36.46____72.10:-197.17__I
____________M_______________________________
I___8.36:50__0:78.68__8.60:50______25.57:50______I
(As revealed by the diagram above, if a twin were at the A? point on
the upper ship, rather than at the upper midpoint as discussed
earlier, that twin would have aged more than a twin at the lower
midpoint over the acceleration period, but if the upper ship continued
to move at .2916 m/ns after the end of acceleration the upper twin
would quickly fall behind the lower one in age.)
As also shown above, the acceleration period, which was 50 ns on all
the lower clocks, appeared to have durations of various lengths of
time at different points on the upper ship as measured by the clocks
at those points.
The next diagram shows all the clocks after a further 28.6783 ns on
the lower ship clocks. During that period all the upper clocks
appeared to move 6.7480 ns at the same pace.
Diagram VII
__________A'___________M'__________B'________
I____-72.10:276.81_____0:43.20____72.10:-190.42__I
I_________M_________________________________
I________0:78.68__16.97:78.68____33.92:78.68____I
If a single force, rather than the distributed force assumed above,
had been applied to the upper ship, say to the selected point A?, then
it would have taken time for the effect of the force to travel by
electro-magnetic influence from particle to particle from point A? to
other points in the ship. The A? point of the ship would therefore
have moved sooner than a point ahead, thereby compressing the part of
the ship forward of point A?. And the rear part of the ship behind the
A? point would have been stretched. This compression and stretching
would last as long as the ship was being accelerated by the force. As
a result the positions shown in Diagram VI above for the various
points 50 ns after the distributed force had been begun to be applied
would not be applicable at the exact end of a 50 ns acceleration by a
single force applied to a point on the ship. But if the ships were
constructed out of material which made them totally resilient in the
sense that when one was held in place but compressed or stretched
there developed within the body a countervailing force which would,
sometime after the compression and stretching force had ended, return
the body to its original shape and size, if no other forces were then
affecting the body?s shape and size, then the compressing and
stretching time effects in the upper ship would be quickly offset
after the end of the acceleration. Diagram VII above of positions
after a total of 78.6783 ns on the lower ship could then properly
represent the configuration of the ships as seen from the lower ship
no matter where the force had been applied for 50 ns.
The second basic hypothesis of Special Relativity indicates that a
flash of light, or of other electro-magnetic radiation, passing
between two points of any body using a stationary reference frame
should be measured on that body to have a speed of c regardless of the
movement of that body with respect to other bodies and regardless of
the source of the flash. With the changes in the positions of points
on the upper ship and in the readings of the upper ship clocks, all as
seen from the lower ship, as illustrated above in accordance with the
hypotheses of Special Relativity, it can be calculated that a light
flash would be measured at the same speed on the upper ship as on the
lower ship even though the ships were moving with respect to each
other. For example, suppose a light flash from dead ahead of the
accelerated ship, that is from the right, reached the upper ship
midpoint M? and the adjacent 8.6035 m point of the lower ship at the
end of the period of acceleration when the clock at the upper point
read 36.4476ns and the adjacent lower clock read 50 ns, all as shown
in Diagram VI above and repeated in part in Diagram VIII..
Diagram VIII
________________________________M'___________
I________-72.10:270.06____________0:36.45_______I
______________________M______________________
I_________-8.36:50_____0:50____8.60:50__________I
On the assumption that the flash was moving at .3 m/ns relative to the
lower ship that flash would travel 8.6035 m from the lower 8.6035 m
point and reach the midpoint on the lower ship 28.6783 ns later when
the clock there read 78.6783 ns, as shown in Diagram IX below. When
the flash reached that point on the lower ship the ?72.1186 m point on
the upper ship would have moved, as measured from the lower ship with
a velocity of .2916 m/ns for 28.6783 ns for 8.3626 m to a position
adjacent to the lower ship midpoint. And the left clock on the upper
ship would, like the upper midpoint clock, have advanced by 6.7480 ns
to 276.8129 ns. The light flash would thus reach the lower ship
midpoint and the then adjacent ?72.1041 upper ship point on the same
occasion in the light?s travel. During its travel from the lower ship
point adjacent to the upper ship midpoint to the lower ship midpoint
the flash traveled 8.6035 m in 28.6783 ns as measured on the lower
ship, a measured light velocity of .3 m/ns. During the same segment of
its travel the flash traveled, according to measurements on the upper
ship, 72.1041 m from a point with a clock reading of 36.45 ns to a
point with a clock reading of 276.81 a duration of 240.36 ns,
therefore at a measured light velocity of .3 m/ns.
Diagram IX
___________________________________M'________
I__________________72.10:276.81_____0:43.20_____I
______________________M______________________
I_____________________0:78.67__________________I
Similar measurements on both ships of light approaching the ships from
other directions would, according to Special Relativity, also confirm
the constancy of light speed measurements regardless of the movement
of the measuring devices relative to each other.
If the force described above had pushed the upper ship in the opposite
direction relative to the lower ship, measurements taken on the lower
ship would have recorded the same change in velocity magnitude, the
same length contraction, and the same type of re-adjustment of the
clocks of the upper ship as detailed above.
If after the 50 ns of acceleration to the right described above for
the upper ship it had been slowed down to a position at rest next to a
position, not the original one, on the lower ship then the length of
the upper ship and the pace of its clocks would be returned to their
original magnitudes as measured on the lower ship. The upper ship
clocks would then appear to the lower ship to have the same reading as
each other, but that reading would remain behind the reading of the
lower ship clocks.
If the upper ship?s movement to the right were brought to a halt, the
ship were then accelerated back to the left and decelerated to a halt
in its original position next to the lower ship the upper ship?s
length and pace of clock advance would have returned to their original
magnitudes, but the return journey would have put the readings of the
upper ship clocks even further behind the clocks of the lower ship.
Pictures taken of the data at any specific pair of adjacent points on
the ships would always be the same whether the pictures were taken by
a camera on the lower ship or by a camera on the upper ship. But the
effects of a force on a ship are different depending upon whether the
force was applied to that body or to the other one. A number of these
differences have already been pointed out for sets of data
photographed when all the clocks on the lower ship had equal readings.
Differences are also illustrated in diagrams below when the effects of
the accelerating force are measured on the basis of equal readings on
the clocks of the upper ship, which was the object of the force. (The
differential effects of acceleration depend upon which ship received
the force and not upon whether any particular observer could feel the
force. A force may affect an observer and a ship differently or,
alternatively, a force may affect a ship and an observer equally so
that the observer is not aware of the force. The latter result may
happen, for example, when an observer is inside a ship falling freely
under the influence of gravity.)
Diagram X below presents two different sets of data. The top part of
the diagram shows the configuration of points and clock readings when
all the clocks on the upper ship showed readings of 270.0649 ns and
the lower clocks showed various readings. At the time of these
readings acceleration was just ending for upper ship point A?, for
which the same data were displayed in earlier diagrams. Acceleration
had ended earlier for the upper mid-point and all other points to the
right of the A? point but they had continued to move to the right. At
the end of the acceleration period the upper ship midpoint had a clock
reading of 36.4476 ns. Later by 233.6173 ns when that clock had
reached a reading of 270.0649 ns it had moved a further 214.4553 m to
the right on the lower ship from the lower ship 8.6035 m point to a
lower ship distance of 223.0508 m from the lower ship midpoint as
measured on the lower ship. The bottom portion of the diagram shows
the data sets 6.7463 ns later on all the upper clocks, when they had
readings of 276.81 ns, and 28.68 ns later for the lower clocks.
Diagram X
When All Upper Clocks Read 270.06 ns:
___________A'_____________________M'__________
I______-72.10:270.06_______________0:270.06_____I
__________________M_________________________
I_______8.36:50____0:50_______276.81:1042.84____I
When All Upper Clocks Read 276.81 ns:
______A'____________________________M'________
I_-72.10:276.81______________________0:276.81___I
_______________M_____________________________
I__-8.36:78.68___0:78.68_________285.98:1071.53__I
These diagrams illustrate the fact that under the hypotheses of
Special Relativity there is no reciprocal slowing of clocks. The
clocks of the upper ship, which was accelerated, run slower than the
clocks of the lower ship after acceleration whether viewed from the
upper ship or the lower ship.
As measured on the upper ship at 270.06 ns all the upper clocks were
well ahead of the readings of the adjacent lower ship clocks. This
condition arose because of the large increase in the reading of clocks
at the rear of the upper ship during acceleration. But 6.75 ns later
on the upper clocks the upper clocks were ahead by a lesser amount as
the slower advance of all the upper clocks had begun to have its
effect. At a later time the upper clocks would all fall behind the
lower clocks even when measurements were made with equal upper ship
clock readings.
In summary the illustration described above shows that, according to
the hypotheses of Special Relativity, when in a vacuum one of two
bodies which have been at rest with respect to each other with
synchronized clocks is accelerated apart for a period by a force then,
measurements by measuring rods and simultaneous clocks on the
un-accelerated body will indicate that at the end of the acceleration
period the length of the accelerated body has been reduced, the
reading of the clock at the center of gravity of the accelerated body
is behind the reading of the clocks on the measuring body, clocks
behind the clocks at the center of gravity of the accelerated body
have readings ahead of the reading of the clock at that ship?s center
of gravity, clocks ahead of the clock at the center of gravity of the
other ship will have readings behind the clock at the center of
gravity, and the readings of the clocks on the accelerated body will
all be advancing at a common pace slower than the pace of the clocks
on the measuring body. The illustration also indicates that as a
result of the hypotheses of Special Relativity both bodies would
measure the same speed of any light approaching them regardless of the
velocity of the source of the light.
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