Broader Conception?
--Broader Conception?--
Returning to the subject mentioned in the first blog "Hypothetical
Cosmos, Preliminary Analyses" under the heading "Possible Tests", and
again dealt with at the beginning of the second blog, "Further
Considerations", we examine objects on, or near the surface described by
the region of space at a distance of about 2516.25 ly.secant(d), from
the Sun, d the declination of the 'points' on the surface (ly= light
years). This follows from the supposition that in an S.G.I. (Special
Galilean Invariant) Universe a light wave departing a light source would
be expected to depart the source with the Maxwellian velocity c (1.),
relative to the observer who later observes it, if both the source and
observer are in inertial frames of reference. Therefore it is in
principle possible in such a universe for two different light waves to
depart the same light source simultaneously with two distinct velocities
relative to that source. These two light waves traveling at different
velocities relative to the source could become widely separated in space
over time, the one which departed with a lesser velocity falling
significantly behind the one which departed with the greater velocity.
One could then imagine a single observer whose velocity changes so
that the first light wave is traveling at velocity c relative to this
observer, when it reaches the observer and is thus perceived as a light
wave by the observer at that time. Assume then that this observer later decreases
its removal velocity from the source in time for the later arrival of
the second, slower (relative to the source) moving light wave so that
this second light wave arrives at the observer's location when it is now
moving also at velocity c relative to the observer as a result of the
observer's now altered velocity; then this second light wave would also
be perceivable by the observer. Since the two light waves would have
departed the source at (essentially) the same moment, if the observer
were to record the image formed by each of the two light waves, said
observer might then possess two essentially identical images of the same
object
(Assuming each light wave possessed enough information to reveal a complete image of the object)
which would have emanated from it at the same moment but been received at two separate later moments in time.
If the observer were on an Earth-like planet in such a universe, the
difference in the observer's velocities might result from a difference
in the location of the planet in its orbit about its star. At one point
in the planets motion around its star it might be moving with component
velocity equal to its orbital velocity away from a light source behind
it in its line of motion; and a half orbit later it would be moving with
component velocity of equal magnitude in the direction of the light
source, at which time it might pick up a light wave which had departed
relative to the light source with component velocity equal to twice the
observer's planet's orbital velocity less than the first light wave had
departed from it, and which therefore would have been traveling for a
longer period of time, because it was moving slower than the first
light wave. Given that the Earth-like planet would have the same period
of one year to its orbit as the Earth has, and that the speed of light
was the same as in our universe; then a light source on the surface
described by the expression mentioned earlier (i.e. 2516 ly.sec d)
distant from the planet's star, would in such a universe emit light
waves essentially simultaneously that would reach the observing planet 6
months apart, that could then be observed there because each of the two
light waves should arrive with velocity c relative to the observing
planet. Assuming that the calendar of the observing planet were also
identical with that used by astronomers on earth, and that star charts
were composed in the same way, the days to first observe the light from a
celestial object on or near the surface defined above, which would have
been emitted essentially simultaneously by such an object on, or very
near the surface, and which could be perceivable six months apart would
be given by the formula: D=A/4m-10, with D referring to the day of the
year; January 1 being day 1, and A being the right ascension of the
object (m=minutes of right ascension).
To try to determine, for example, if we are living in an S.G.I.
Universe, we might imagine examining a given celestial object judged to
be near the above defined surface for at least about one year. In
particular we compare the image of the object as observed on the day
denoted by D with that observed on the day denoted approximately by
D+182, thus falling exactly 6 months after the observation made on the
day denoted by D. We are particularly interested in objects with some
degree of cyclic behavior or variability.
If our universe is an S.G.I. Universe then the two images should show
the object in a similar phase (We assume the phase change is in general
slow enough, requiring at least days or weeks to complete).
Observations during the intervening 6 months between D and D+182 should
be expected to show relatively little variation as if one were (almost)
looking at a kind of frozen moment (Since light waves reaching the Earth
during this period would have left the object at almost the same
time). Observations during the subsequent six months between D+182 back
to D should show greater variation. Of course it is important that our
estimates of nearby celestial distances such as are involved here be
sufficiently accurate to be able to determine which objects actually do
lie near the surface defined above.
Based on currently published estimates, objects lying near the
previously delineated surface include 24 Com Bernice in the
constellation Coma Bernice (2.) to be observed first at approximately D=
June 29; and then compared with observations made six months later on
about December 29. We note that 24 Com Bernice is listed as a
"suspected variable star" with its observed magnitude varying between
6.55 and 6.59 (2.). Another object is HD 36591 in Orion which is also
listed as a "suspected variable star" (3.) to be observed on about March
14 and compared with observations made about six months later around
September 14. Also in this group are Carot-29 in Ophiuchus (4.) to be
observed on D= September 28, and again on March 28. This star is
described as having a transiting planet (4.). Also Kepler-67 in Cygnus
(5.) is by current estimates close to the surface earlier defined. It
should be observed on D= October 12 and again on about April 12 six
months later. Kepler-67 is said to have "one planet slightly smaller
than Neptune" (5.).
Continuing our examination of the strangeness of the theory of
Relativity from the last blogs, we note that a popular book on the
subject makes the statement that since light possesses energy, then
because of the mass-energy equation it should also have mass and
therefore it should be effected by gravity in a way similar to other
masses (6.). "Because light has energy and is subject to the effects of
gravity it will, in Albert Einstein's theory, tend to fall towards the
earth like any other object." (7.). However the same book on another
page states that gravity slows down light, as it slows down time " ...
there are two related and mutually reinforcing effects. One is the
influence of gravity on time, which is equivalent to slowing down light
and making the space around the Sun into a lens that bends the ray of
light." (8.), so that light will travel more slowly near a gravitating
mass; a statement for which the book even claims to present experimental
evidence (9.).
We notice that
this supposition that light travels more slowly in the vicinity of a
gravitating mass should, it seems, mean that a light wave while
"falling" into such a mass would be expected to, it would appear, be
decelerating as it as it falls. If I throw a ball or any other mass
upwards away from the Earth's center of gravity it will, in my
experience, inevitably decelerate as it moves away from the Earth's
center of gravity; and then accelerate as it comes back down again. And
yet light which in this book is said to "fall toward the earth, like
any other object." (7.), should it seems, if the other statement in the
book is true, be apparently doing the opposite, accelerating as it
travels up from the Earth, while light from a distant star while
"falling" to the Earth should be slowing down or decelerating as it
comes down! Light then does not appear to "fall" (just) "like any other
object" if this second statement in the book is also true!
An ordinary object as I understand it will curve around a gravitating
mass as it moves past the mass since the force of gravity is pulling on
it, or in other words, is accelerating it towards the mass. Shouldn't
light then, if it is decelerating rather than accelerating as it
approaches a gravitating mass rather be deflected away from the mass
than to curve around it? Yet the same book also suggests that (as has
been observed, and as was predicted by Newton), light does curve around
objects, and is not deflected by them! I am sure some of the
relativists who may be reading this are likely to have some nice
explanation of how this happens, but I don't know if any will bother to
inform this blog of it!
Some
sources speaking of a clock running faster in a weaker gravitational
field (10.) seem to overlook that it is the height in a gravitational
field that, according to General Relativity, results in a clock running
faster. Thus a clock standing 3,000,000 kilometers above the surface of
the Sun should experience a slightly weaker gravitational force than
one resting on the surface of the Earth, but would non-the-less be
predicted to lose more than 12 seconds per year compared to the clock on
the Earth's surface according to G.R. It would of course be
interesting if such an experiment were one day carried out to see if the
results would accord with the expectation.
The equation cited in an earlier blog which is given by Feynman, essentially t'=t/(1+aL/cc) (11.)
which he derives for the time dilation at a distance L between two
clocks registering time t' and t respectively moving forwards along the line of
acceleration a in an accelerating rocket, (12), can be used to generate the equation t'=t/(1+Gm/rcc), (13.)
with G=(The universal gravitational constant) (14.), t=(The time
elapsed on a clock removed from the gravity field), and t'=(The time
elapsed on a clock residing a distance r from the center of gravity
generated by the mass m), which is itself approximate to the
Schwartzschild solution (15.) to the principle equation of General
Relativity. (16.) It can be derived conceptually from the presumed time
dilation resulting from acceleration, and then transferred to the case
of gravity via the "equivalence" principle, by substituting Newton's
equation for gravitation: g=Gm/rr, (17.) g the force of gravity
experienced at distance r from the center of gravity (m,G as before)
into the Feynman equation t'=t/(1+aL/cc), by letting a=g, and L=r.
Feynman admits that this last equation is only approximate. The exact
equation which one would expect to hold can apparently be derived from
Special Relativity. It does not make such an obvious reference to the
presumed behavior of light under acceleration, and curiously comes out
to be t'=t/(1+aL/cc.ln2), (ln= The natural logarithm), (18.) which is of
course approximate to the Feynman equation for most purposes as
ln2=0.6931... is approximately equal to 1.
While the derivation of this last equation perhaps
shows a connection between Special and General Relativity, one not
uncommonly has the sense of Relativity being pieced together out of
somewhat dissimilar ideas that have been merged in an effort to make a
whole. The speed of light is constant in inertial frames of reference;
in accelerated frames of reference its velocity can not be measured
because of the lack of a common time measure, so it need not be constant
here. According to the equivalence principle of Relativity, matter
curves a region of space making it in every way indistinguishable from
an accelerated reference frame; ... in which space is actually not
curved. In gravitational frames of reference the curvature of space
makes the idea of determining the speed of falling light seem wholly
implausible (Though as said earlier, according to someone outside the
gravity field it should be seen to be decreasing in velocity as it
falls).
In particular the idea of
time dilation during acceleration appears peculiar. A rocket of large
length coasting next to a similarly long rocket which is undergoing a
constant acceleration (but which has just become momentarily stationary relative to the first rocket), could have synchronized clocks on it, since such a
rocket is in an inertial frame of reference as it is not itself
accelerating. At the same time a clock at the front of the accelerating
rocket might in accordance with General Relativity be running at a
widely faster rate than a clock at the rear of the same rocket. Being
in essentially the same reference frame as the coasting rocket however,
the clocks at both ends of the accelerating rocket should remain almost
exactly synchronized with nearby clocks on the coasting rocket, which
themselves would remain synchronized with one another.
So this is similar to saying that one has two clocks running
at widely different rates on the accelerating rocket; the one in the
front is perhaps registering many seconds or even minutes in the
interval of one second being registered at the rear of the rocket; and
yet both of the two clocks are at the same time clearly running
essentially at the same rate as a third! (19.) A believer in Relativity
I know will not be bothered by these observations; has not Relativity
after all done away with conventional ideas about time and space? The
problem then remains how do we real beings of time and space maneuver
around all this?
And then
there is the observation that in order that individuals in the front of
the accelerating rocket agree with those in the rear about the final
velocity attained by the rocket; since they differ about how long the
rocket has been undergoing acceleration, that this implies that those in
the front of the rocket must therefore experience a lesser force of
acceleration proportional to the greater period of time which they
believe has elapsed in the course of accelerating; compared with those
in the rear of the rocket. Thus the rear of the rocket is experiencing a
constant acceleration greater than the front is. If a car moving
behind another on a road is experiencing a greater constant acceleration
than one in front of it, won't it eventually catch up with the car in
front of it and pass it by? (Leaving the rocket presumably turned
inside out as its rear passes its front!)
Not according to Relativity if the distance between the two is
great enough. (20.) This its equations predict. But what are mere
equations to us who live in a world with actual palpable laws we
experience? How do I know whether all this mathematics actually
describes something real? Are a few atomic clocks in airplanes and
satellites enough to convince me of this?
We should perhaps not forget that the detected values for the
time dilation are extremely small. While atomic clocks are in theory
extremely accurate measurers of time; they are also very complicated
devices; conceivably sensitive to minor changes in their environment.
It is essential that there be a feedback mechanism keeping the clock
resonating with the electronic transition frequency of the element being
used as a time setter, and that this resonance not diverge from that
frequency. (21) Can changes in the environment including gravity
changes or acceleration effect this fine tuning and cause the clock to
move slightly out of sink with the frequency of the element, giving the
impression that it is the element's frequency, rather than the resonance
of the clock itself which has changed?
As stated earlier (In "Preliminary Analyses"), in an S.G.I.
Universe in general, no time dilation would be expected; so one would
instead expect that all clocks in such a universe could remain
synchronized. Since one of our goals is to try to imagine an S.G.I.
Universe which would particularly closely correspond to what appears to
have been established about our universe, we call to mind what was
stated in the first blog under "Possible Difficulties", which is that
while such a universe is not predicted in general to have any kind of
time dilation at all, and the existence of a time dilation in a purely
inertial frame of reference would essentially be excluded; the S.G.I.
model does not preclude the possible existence of time dilation in, for
example, accelerated or gravitational reference frames. We also note
that while the theory of an S.G.I. Universe does not apparently
necessitate or even call for the existence of some kind of "curved
space" caused by the presence of matter as in G.R.; it does not itself
exclude the possible existence of something like that.
The existence of such a time dilation in such a universe might then
yield evidence to scientists in such a universe of the sort of results
to experiments similar to those conducted in our universe which are
cited as evidence for General Relativity; but would then possibly not
suffice to distinguish whether the given universe was Special
Relativistic or Special Galilean Invariant. The possible time dilation
that might exist in such a universe in the presence of reference frames
undergoing an acceleration, and which are not purely under the influence
of gravity is less clear. Some accredited scientists have apparently
hypothesized the existence of subtle violations to the equivalence
principle of General Relativity (22.), which would otherwise eliminate
any measurable distinction between gravitational and accelerated
reference frames (23.).
At this
point we recall what was said earlier when we compared the gravity
field on the Earth with the gravity field about 3,000,000 kilometers
above the Sun's surface. As far as I know there is only one type of
constant acceleration; similar to the seat of the chair I am sitting in
accelerating with a certain constant acceleration a which can be
measured in meters per second squared in some direction of space.
Relativity as far as I understand it yields one coherent formula for
this acceleration and the presumed associated time dilation at a height L
above (or below) this point; that which was given earlier:
t'=t/(1+aL/cc.ln2). Against this we place the gravity field on the
surface of the Earth which we compare with that just a little bit less
than 3,000,000 kilometers above the surface of the Sun. In both
locations we should experience an identical force of gravity of just
under 10 meters per second squared. In both locations, according to the
equivalence principle of Relativity, there should be no way to
distinguish if our eyes are closed whether we are in a gravity field or
are in a rocket accelerating at almost 10 meters per second squared, so
according to this equivalence principle both gravitational reference
frames are in every way equivalent to an acceleration of about a=10m/ss;
(m-meters, s-seconds), (23.) yet the time dilation at the specified
location above the Sun is almost 12 seconds per year greater than that
on Earth according to Relativity (And furthermore, the rate at which the
time dilation varies as we alter the distance r from the center of
gravity also varies between the two cases, since the mass m in the
Schwartzschild equation has a different value, in each case. In each
case it is also so that the rate at which the strength of the gravity
field varies with different values of r is different in the two gravity
fields. Compare this with the one relation which was given earlier for
both variations during uniform acceleration in accordance with
Relativity.)
So here again we
have two things (in this case two gravitational reference frames) which
in order to accord with Relativity can clearly not be equivalent to one
another; but which according to the equivalence principle of Relativity
are both equivalent to a third thing ("In every way equivalent"!), in
this case to acceleration at about 10m/ss! This reminds us of the
clocks on the accelerating rocket which both run off from one another,
while they both remain synchronized with a third clock (on the coasting
rocket). Note here that in both separate gravitational cases we are
looking at, the gravity on the surface of the Earth, and about 3,000,000
kilometers above the Sun, we are deriving the different values for the
time dilation in each case from the Schwartzschild solution to the
principle equation of Relativity (16.), by varying the key parameters m
and r. We are overlooking for the moment the many other possible
solutions to the principle equation of General Relativity; which are
obtained by substituting different possible values for the "Stress
Energy Tenser" (24.) into the left side of this principle equation;
which yield different types of gravity fields with different associated
time dilations on the right side of the equation; and which according to
some people may also allow for "repulsive" or "anti-gravity" fields.
All of these fields which are clearly different from one another and
thus "in-equivalent" should all none-the-less through the "equivalence
principle" be exactly equivalent and in every way indistinguishable from
acceleration with its one variable a, and its one possible time
dilation (In accordance with Special Relativity) given by
t'=t/(1+aL/cc.ln2). All these different types of gravity fields
certainly make Feynman's statement look problematic: "Using the
principle of equivalence we can figure out how much the speed of a clock
changes with height in a gravity field. We just work out the apparent
discrepancy between the two clocks in the accelerating rocket ship."
(25.).
Going back to the problem
of time dilation in an S.G.I. Universe, while within the confines of
the basic model of such a universe, in general, as said earlier no time
dilation would be expected; the possibility that by some fluke a time
dilation in certain situations, for example in a gravity field or during
acceleration, might exist in such a universe is not excluded (though
any time dilation in inertial reference frames would essentially be).
It is true that in an S.G.I. Universe without time dilation, light waves
emitted in a gravity field or during acceleration would have the
property that they would contract along their length as they move
through space, since the rear of the wave would be moving faster than
the front, which would have lost velocity in a gravity field by the time
the rear of the wave was emitted. The time intervening between the
reception of successive wave crests by an observer, however, would
remain equal to the time separating the emission of successive wave
crests at the source (if there were no time dilation), though the
wavelength L' of the light wave observed would be shifted to the red by
comparison with a light wave of wavelength L emitted in the absence of a
gravity field; by about L'=L{1+2Gmh/ccr(r+h)}^1/2, where h is the
vertical distance traveled by the wave which originates at an initial
distance r from the center of gravity. Taking h to infinity yields
L'=L(1+2Gm/rcc)^1/2. A light wave having once left a gravity field
would then maintain a constant length.
Equivalently under acceleration the rear of the light wave would be
expected to have to be emitted with the velocity added to it which the
accelerating observer would later gain between the reception of the
front, followed by the reception of the rear of the light wave; so that
each is traveling with velocity c relative to the observer when they
arrive at the observer's location. It is here that we come across what
appears to be a palpable problem. If we imagine the light wave instead
passing the observer in the accelerating rocket, and instead traveling
out of the front of the rocket and being picked up by another observer
(observers) who are ahead of the rocket but are themselves not
accelerating; we expect the light wave (waves) observed by them, since
these observers are not accelerating, should have a constant length and
should not be contracting as they travel. We then expect that the front
of the light wave picked up by the accelerating observer should form
the front of a light wave traveling with velocity c through one
reference frame, while the rear of that same light wave should form the
rear of another light wave, traveling with velocity c in a second,
slightly faster reference frame. We note that Maxwell's theory of
electromagnetism assumes that light waves propagate through space
"unchanged" (26.), (I am curious how the geometrical contortions light
waves must undergo to satisfy the theory of Relativity get around this
limitation!)
If, as seems
likely, the individual photons of the light wave that are seen by the
observer in the accelerated frame of reference, presumably being subject
to quantum mechanical laws, might turn out to be some kind of random
selection of the photons which had belonged to either the slower, or the
faster light wave, (or to one of intermediate velocity), that pertained
to one or another inertial reference frame, this could conceivably lead
to a somewhat haphazard perception, on the part of the observer. There
might in a given reception, be more photons from the "rear" light wave
or more from the "front"; or there might perhaps be (at random), an
excess or dearth of photons in comparison with reception in a simpler
inertial frame of reference.
If one then imagines
that this phenomena, the contraction of light waves along their length
during their journey through space during acceleration, or analogously
in a gravity field (27.) might somehow upset their integrity, or
compromise their nature, possibly scrambling or reducing the amount of
information they carry, or that can be perceived by an observer; this
might conceivably somehow interfere with the evolution of life (or of
some types of life) in such a universe, favoring for this outcome
universes in which some counter deformation of the light wave were
present. Making what appears to be the logical assumption that given
that: r''(t)= -Gm/rr, (28.) and that lim(r -> infinity) of r'(t)=u,
implies that then in general v=r'(t)=(uu+2Gm/r)^1/2, (See also
"Addendum" to this blog), we obtain for an S.G.I.Universe without time
dilation that a light wave emitted by a source in a gravity field which
is later observed by someone outside of the gravity field, and who
remains stationary with respect to the center of gravity would be
expected to travel with velocity v=(cc+2Gm/r)^1/2, at distance r from
the center of gravity, through the gravity field.
If we then introduce the idea of an S.G.I. Universe with a time
dilation in gravity fields, which would have the property that light
waves traveling between sources and observers in line with the center of
gravity would not lengthen or shorten as they travel, at least from the
perspective of someone removed from the gravity field (Though these
light waves might appear to be shifted to the red or blue by comparison
with light waves emitted in the absence of a gravity field, from the
perspective ether of someone observing the light waves from within the
gravity field, or after the light waves would have left this field), but
that there would be a time dilation (with concomitant red shift) given
by t1.v1=t2.v2, where v1 and v2 are the velocities of light waves if
measured at distance r1 and r2 from the center of gravity respectively,
that would have traveled in journeys in line with the center of gravity;
which would have departed from one of these two locations and been
received at the other. Thus either v1 or v2 would be equal to c,
depending upon whether the light wave were received at distance r1 or r2
respectively. If this seems to introduce a preferred frame of
reference because light waves in the gravity field would not seem to
contract according to someone outside the gravity field, but would
according to someone inside it; this would only be in the sense that
inertial frames of reference as would pertain to someone outside the
gravity field might be (perhaps slightly) preferred to gravitational
frames of reference. All inertial frames of reference would be
equivalent to one another; and all gravitational frames of reference
caused by an equal mass might also be equivalent. t1 and t2 are the
times elapsed on clocks at the locations r1 and r2 respectively. By
substitution into the preceding equation above we obtain that, in
general: t1=t2.{(1+2Gm/r2cc)/(1+2Gm/
We note here that a light wave "falling" into a gravity field that would be actually visible to someone in the gravity field, would appear to have accelerated as it fell in a manner similar to other falling objects, according to someone in the gravity field; but according to someone removed from the field the light wave would not have either accelerated or decelerated as it fell, but would instead have seemed to maintain a constant velocity. A slight fluctuation in the local gravity field might then conceivably cause a light wave to accelerate slightly if it involved an increase in the field. This could possibly have the result that a light wave reaching a detector might be one which would have departed its source at a slightly lesser velocity than another one would have prior to the increase according to a timing mechanism in the field. It would then possibly take a slightly longer time to arrive at the detector, causing it to move a little out of phase with a light wave moving perpendicular to it which would have been unaffected by the fluctuation. (A local reduction in the gravity field might cause the opposite effect of the light wave reaching the detector faster). Such a scenario might conceivably allow for the possibility of the detection of some kind of "gravity waves" (29.), if oscillations of masses produce waves in the universe in question in analogy with electromagnetism.
We note here that a light wave "falling" into a gravity field that would be actually visible to someone in the gravity field, would appear to have accelerated as it fell in a manner similar to other falling objects, according to someone in the gravity field; but according to someone removed from the field the light wave would not have either accelerated or decelerated as it fell, but would instead have seemed to maintain a constant velocity. A slight fluctuation in the local gravity field might then conceivably cause a light wave to accelerate slightly if it involved an increase in the field. This could possibly have the result that a light wave reaching a detector might be one which would have departed its source at a slightly lesser velocity than another one would have prior to the increase according to a timing mechanism in the field. It would then possibly take a slightly longer time to arrive at the detector, causing it to move a little out of phase with a light wave moving perpendicular to it which would have been unaffected by the fluctuation. (A local reduction in the gravity field might cause the opposite effect of the light wave reaching the detector faster). Such a scenario might conceivably allow for the possibility of the detection of some kind of "gravity waves" (29.), if oscillations of masses produce waves in the universe in question in analogy with electromagnetism.
The above equation for time dilation yields for most testable purposes
values almost exactly equal (though it gives values for t slightly
[infinitesimally] greater) to those given by the Schwartzschild equation
from G.R. and it conforms well nigh exactly with the time dilation
listed for that assumed to be caused by gravitational time dilation that
is effecting the G.P.S. satellites (as well as that predicted for the
other experiments of the General Relativistic time dilation that I've
heard of)! If we extend this notion further to also eliminate length
expansion or contraction of light waves (as seen by someone outside the
gravity field), relative to an object moving horizontally through a
gravity field like a moving airplane, or orbiting satellite, we must
consider that in an S.G.I. Universe the light waves reaching such an
object from a source within the gravity field would have to travel
slightly faster relative to the center of gravity than those reaching an
object stationary with respect to that center, since the velocities of
the light wave, and that of the moving observer would have to sum to c,
the Maxwellian speed of light according to its observer. This might
possibly reduce slightly the time dilation effecting the observer
approximately in accordance with the formula (See next to last formula
in Addendum to this blog): t1=t2.{(cc+uu+2Gm/r2)/(cc+2Gm/
We note that unlike in a Relativistic Universe, in which light waves moving in
opposite directions through a rotating Sagnac interferometer would move
at different velocities relative to the apparatus, and thus possibly go
out of phase when they rejoin before reaching the detector; it is true
that in an S.G.I. Universe one would expect both light waves moving in
opposite direction to travel with velocity c (relative to the apparatus
if the detector is also taking part in the rotation). However relative
to someone outside the apparatus who is not rotating, the two light
waves that are later picked up by the detector would be traveling at
different velocities. This, it would seem, should cause the two light
waves to experience a somewhat different centrifugal acceleration as
they move, possibly causing the faster moving light wave to follow a
slightly longer path to the detector than the slower light wave, which
might then none-the-less cause the two light waves to move out of phase
with one another.
So I hope that I have at least
convinced some of my readers that a hypothetical S.G.I. Universe of this
or a related sort might indeed very closely resemble our own universe,
to the point that it would possibly be worthy of individuals in such a
universe to at least conduct the necessary experiments and observations
needed to determine exactly what kind of universe they live in! In any
case, in looking for the truth, what ever ones preconceptions may be,
and whatever the evidence may eventually show, I think one does not want
to entirely overlook an alternate way of interpreting the already
available evidence, in the way that this concept of an S.G.I. Universe
has apparently been overlook; at least publicly, by the scientific
establishment. One at least, it seems to me, should want to address
this idea!
(1.) See "Hypothetical Cosmos, Preliminary Analyses", note
(2.). A Special Galilean Invariant Universe is one in which the
observed constancy of the speed of light results from a special case of
Galilean Invariance, where each individual light wave is restricted to a
single inertial reference frame, through which it travels with the
velocity c predicted by the theory of Maxwell. (See Maxwell, James
Clerk "Treatise on Electricity and Magnetism").
(2.) en.wikipedia.org/wiki/Lists_
(3.) Ibid "Orion".
(4.) Ibid "Ophiuchus".
(5.) Ibid "Cygnus".
(6.) Calder, Nigel "Einstein's Universe", pp.-39-40.
(7.) Ibid p.-45.
(8.) Ibid p.-53.
(9.) Ibid pp.- 58-59.
(11.) Feynman, Richard P. "Six Not So Easy Pieces" p.-134.
(12.) Ibid pp.-129-144.
(13.) Menzel, Donald H. "Fundamental Formulas of Physics" p.-217: "(20)".
(14.) Nelson, David "The Penguin Dictionary of Mathematics, Third Edition" p.-191.
(15.) Menzel p.-217: "(20)" is obtained through approximation to the Schwartzschild equation: "(19)".
(16.) " t'=t(1-2Gm/rcc)^1/2", is a portion of the Schwartzschild
solution to the principle equation of General Relativity, i.e.
"G(a,b)=-8[Pi]GT(a,b)/(c^4)", where G(a,b), and T(a,b) are 4 by 4
symmetric matrices called "tensors" because of the operations they
undergo. [(Pi)=3.141592...], See Menzel, pp.-46-51,210-217.
(17.) Nelson, pp.-3,191.
(18.) For the definition of the natural logarithm, see Ibid
pp.-260,261. a in this equation refers to the acceleration that would
be experienced by someone at the rear of the rocket, where the time t'
is found to elapse.
(19.) Note
that the accelerating rocket need not be undergoing any great
acceleration, as long as its length is great enough. For example, if
the length of the accelerating rocket is about 6 light years, then the
clock at the front of the rocket should be registering about 10 seconds
for every 1 second registered at the rear, while the acceleration
experienced at the rear of the rocket would be the same as the
acceleration due to gravity on the Earth's surface.
(20.) Since we have at'=a't, substituting into the equation
for the time dilation for an accelerated reference frame which we
posited earlier, and inverting, we get: "if the distance is greater
than or equal to cc(ln2)(a-a')/a.a' ", where a is the acceleration
experienced by the car in the rear, and a' that by the car in the
front. Then to accord with Relativity the car in the rear should never
overtake the car in the front, even if the car race goes on for all
eternity (as long as the acceleration experienced by both cars always
remains constant)!
(21.) en.wikipedia.org/wiki/Atomic_
(22.) See, for example en.wikipedia.org/wiki/Brans-
(24.) T(a,b) in "The principle equation of General Relativity". See (16.).
(25.) Feynman, p.-133.
(26.)
Haber-Schaim, U, Cross, J.B., Dodge, J.H., and Walter, J.A. "PSSC Physics, Forth Edition" p.-544.
(27.) We prefer, for the sake of
argument to replace the "equivalence principle" of Relativity with a
slightly weaker "analogy principle", by which gravitational frames of
reference would be seen as very similar to, and even difficult to
distinguish from, accelerated frames of reference; but we are inclined
to think that considering the two as entirely equivalent might be
jumping to conclusions.
(28.)
Again Newton's equation for gravitation [See note (17.)], with
g=r''(t). For the theory and notation of calculus, see Anton, Howard
"Calculus", or an equivalent text.
(29.) See, en.wikipedia.org/wiki/Gravitational_wave
(c= The Maxwellian speed of light.)
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