Preliminary Analyses

                        Comments, questions, and criticisms in relation to the following are welcome, but may be met by the same:

                          --A Hypothetical Cosmological Model, Preliminary Analyses--

                        I have spent some time looking at a certain type of hypothetical "universe". As this conception is the result of a very simple and seemingly provocative idea, I don't understand why I can find no mention of it anywhere in whatever sources of information appear normally to be available to the general public on this or related topics.  I am inclined to imagine that others have examined this same question, however I can not understand why it is not apparently addressed somewhere in any of those places where an ordinary person looks for information on such a subject.  Certainly, less plausible seeming notions appear not uncommonly to be considered in one or another context.  Not knowing personally anyone who might be seen as an expert on cosmology, and finding that any inquiry made to others seems to be met with apparent disinterest or bafflement, I have finally been moved to attempt to write an exposition on the subject to the best of my level of comprehension, and of my ability to analyze with the capacity I have, and the knowledge I am able to garner from what information I can find on related subjects.  At last I have found attempts to contact 'peer revue journals' to try to have them look at this report to be unsuccessful.  Therefore I have decided to try to make this information directly available to as many others as possible who might have an interest in the same subject.  I view it as a kind of obligation, to make this information available to the public, even if it may seem that I am in some ways stating the obvious; I non-the-less think that there are reasons to make such statements.

                        If an expert decides that for some reason I am following a spurious lead in even attempting to address such a question, then I think that is partly to blame on the fact that such an obvious notion as I am here alluding to has previously, as far as I am able to tell, been left unaddressed by these same so called experts.

                       In this report I will not attempt to answer the question of whether the hypothetical universe I describe really could, or could not exist somewhere.  I will also not attempt to address the question as to whether, given that the later possibility might turn out in the affirmative, our universe could somehow be such a universe.  I have simply tried to follow the lead to wherever I might, at my level of understanding, seem to be able to take it.  Let the experts address the questions put forth if they think it is worth their while to do so.     

                                                                                  --    July 21, 2017

                                    --Special Galilean Invariance--
                     
                       An S.G.I. (Special Galilean Invariant) universe is a hypothetical universe model based essentially upon Newtonian or inertial frames of reference (1.).  All such frames of reference would be expected to obey the same 'flow of time' (1.), so that the clocks in one frame of reference would be expected to remain synchronized with those in reference frames in motion with respect to it.

                       Such a universe, however, would also be expected to have much in common with what is understood to obtain for our universe.  Thus, the speed of light in a vacuum  would be found by all observers in each inertial reference frame to have the same velocity c predicted by Maxwell's theory (2.) relative to them.  The famous 'mass energy relation', for E=(energy), m=(mass): E=mcc,(3.) would also be expected to hold in it.  As a result of this, the 'mass dilation equation' (4.) accepted for our universe might also be expected to hold (or to appear to hold), at least in some situations, as this equation can be shown to be a direct mathematical consequence of the 'mass energy equation' (5.).

                       The explanation for this state of affairs is that in an S.G.I. universe light, and in general electromagnetic waves would not be accredited with an absolute existence outside of the frame of reference in which they are observed, but would instead be viewed as being a particular attribute of the given reference frame in which they are detected.  Therefore in an S.G.I. universe a light (or e.m.) pulse would not be conceived of as an entity which would occupy a fixed location in space at a given moment in time for all observers as it is apparently viewed to be by most theories of our universe at present (though the given time flow providing the location for the 'moment' may be seen as varying between reference frames so that clocks in one such frame might be unsynchronizable with those in another).  Rather the light pulse would be conceived as an entity which might be present in different locations of space simultaneously; but only be observed by someone who's location and relative velocity would put them in the right time and place in a given frame of reference to which a representative of the light pulse would have reached traveling with velocity c through it from the source of the pulse.  Thus in such a universe, each separate frame of reference would posses its own complement of light (or e.m.) waves that would all travel with the constant speed c predicted by Maxwell's theory through the given reference frame.  The light (or e.m.) waves present in one frame of reference would not exist in the others (though near replicas of them typically would).

                     Therefore a light or e.m. wave emitted by a source in one frame of reference that is observed in another would not have to depart with velocity c relative to this source   (though it would have to travel with velocity c through the frame of reference in which it is observed).  A light source would thus produce light waves in a continuous range of velocities each of which might normally only be observed in the given reference frame through which the light wave would be moving with velocity c. 

                    Assuming this second reference frame were moving in such a way that a given observer in it would be traveling with precise velocity v in the direction exactly away from the light source; this would mean that the light wave seen by the observer would have departed the source, and moved relative to its reference frame with velocity c+v, (v here would have a positive sign when the motion is actually away from the source, and a negative sign if the motion were really towards the source).  This necessitates a space dilation (contraction) given by:  x'=cx/(c+v) ; where x is the distance initially measured to an object just before an observer accelerates to component velocity v, (6.) in the direction AWAY from the object (assume very rapid, near instantaneous acceleration, v can be

negative[in which case the motion would actually be towards the object]); and x' is the distance measured by the observer to the same object immediately after having accelerated.  Space would thus expand in the direction of increased velocity and contract behind it.

                   The new distance x' is measured exclusively within the frame of reference of the observer who has accelerated.  Within the frame of reference of the object observed the observer mentioned would have accelerated to velocity v from the point in that frame of reference in which the observer had been positioned prior to accelerating.  The observer having experienced an acceleration to velocity v, would then see the object appear to be moving AWAY with the velocity:  u=cv/(c+v).  Again, u would be negative if v were negative (in which case the magnitude of u would be greater than that of v).  An object passing one would seem to decrease its velocity by 2cvv/(cc-vv), as it goes by one with v its original velocity of approach; without itself experiencing any change in velocity.  This effect would, of course, be negligible if the relative velocities were much less than c.    

                   The conception of an S.G.I. universe in which each reference frame would have its own complement of light or electromagnetic waves which would not exist outside of the given reference frame results from the observation that electric and magnetic fields are caused by the relative distances to, and relative motions of, charged particles respectively; to an observer of these fields (7.).  An electromagnetic (for example, light) wave is in general conceived of as being nothing more than a set of changing electric and magnetic field values.  A change in velocity taking an observer from one reference frame to another would alter the magnitude of these distances (because of space dilation, see above), and of these motions relative to the observer; and so alter the electric and magnetic field values out of which the wave in the original reference frame was composed.  This would cause the original e.m. wave to vanish; and be replaced by a new wave comprised of the new values for the e. and m. fields (8.).  In order for this new wave to conform to the theoretical necessities of Maxwell's theory it would also have to travel with velocity c through the reference frame in which it exists and could be observed, and therefore could have its velocity measured (9.).

                  The suggestion that in an S.G.I. universe the 'mass energy equation' E=mcc, would be expected to hold results from the observation that the theory of Maxwell predicts that for a light (or e.m.) wave of momentum p and energy E the relation p=E/c, would hold (10.).  The theory of Newton on the other hand predicts that in general for a matter particle of mass m and velocity v the relation p=mv would hold (11.).  Then noting the often cited wave-particle duality (i.e., that light behaves sometimes as a wave and sometimes as a particle) one assumes for a light particle that p=mc.  Putting the two equations together one obtains the result p=mc=E/c, => E=mcc.  Thus this last equation can be seen as being a consequence of Maxwell's and Newton's theories alone, both of which would hold in an S.G.I. universe.

                  As a consequence of the 'mass energy equation' holding one would expect that the 'mass dilation equation' from Special Relativity (12.) might also hold, or at least appear to hold in some situations (5.).  In particular one might expect this equation to work in situations where acceleration is caused by external electric and magnetic fields being applied to charged particles, since it is derived from the ' mass energy equation' which we have just shown to be itself derivable from the equation yielding the momentum of an electromagnetic wave.  Therefore charged particles put through particle accelerators in an S.G.I. universe which use electric and magnetic fields to accelerate them might be expected to behave in a way that upholds this 'mass dilation equation'.   

                  In an S.G.I. universe this mass dilation is conceived of as being more of an apparent, than a fully real phenomena.  The particles are not to be viewed, in general, as actually gaining mass, but rather that a reduction in the force exerted by the electric and magnetic fields applied to the charged particles takes place as these particles increase their velocity moving through these fields.  Such a diminution in force might be seen as resulting from a combination of the kind of effects such as the space dilation experienced by the particles increasing the distance to e.m. sources in front of the particles and a 'red shift' (13.) of e.m. energies from sources the particles would be moving away from.

                  As the particles approach the speed of light, the distance in front of them to the e.m. sources would approach infinity, causing the force exerted upon them from these sources to fall to zero.  This might prevent the particles from exceeding the speed of light relative to the apparatus used to accelerate them.

                  The 'mass dilation' is not however to be considered to be otherwise a property intrinsic to the particles themselves.  The particle masses would not have increased according to someone in the same frame of reference as the particles, and without a time dilation (14.) one would not expect there to be an absolute barrier to prevent the particles from traveling faster than light.  Thus, for example, if the particles having approached near to the speed of light were to pass through a second accelerator that itself were moving relative to the observers along the path of the particles (or if the observers or observing apparatus were moving in a direction opposite to the particles motion, or both) a relative velocity greater than c might be detected in an S.G.I. universe.

                  Neither would one expect that a vehicle which makes use of an internal means of propulsion should be subject to an absolute limit to the velocity it might attain; in the way that a charged particle moving through a set of fields with velocity close to c might be.  Again for someone accelerating along with the vehicle in its reference frame, the mass of the vehicle would appear to remain constant, and since there would be no time dilation, and with the vehicle undergoing a constant acceleration a from its propulsion; after starting from rest its velocity v might be given at time t by the Newtonian relation:  v=at.  Therefore v>c might hold after t>c/a, (15.).  Furthermore, given that the acceleration were equal to the home planets gravity, one would expect a clock on the vehicle to remain essentially synchronized with one on the home planet.  Thus the vehicle might exceed the speed of light at about t>0.97 years if a were about equal to earth's gravitational acceleration.

                  Unlike the 'mass dilation' equation, the 'space' and 'time' dilation equations predicted by S.R. do not, to my knowledge, have any direct derivation from the 'mass energy equation', and are therefore not apparently implied by it.  They might thus be seen as having only an accidental resemblance to the former, the partial validity of which might not by itself lead one inevitably to conclude that the other two must necessarily also hold.

                  One characteristic of an S.G.I. universe would be that a radio signal sent out to a space probe from a planet which the space probe were moving away from with component velocity v, which was then relayed back to the planet by the space probe, would take less time to make the round trip than it would in a L. I. (Lorenz Invariant) (16.) universe.  This would be the case since such a signal would have to travel with velocity c+v relative to the planet to be detected by the probe.  The return signal would have to travel from the probe to the planet with velocity c to be detected there.  This might cause inhabitants of the planet to imagine that the probe had not traveled as far as expected and so had become 'delayed'.  A simple calculation suggests that the amount of 'delay' would be approximately:  Dx=xv/2(c+v), where x would be the expected distance traveled by the probe, and Dx would be the difference between the apparent and expected distances traveled (17.).

                  Another apparent attribute of an S.G.I. universe would be that a light (or e.m.) pulse sent out from one location in space in a given direction would over time become widely separated in space in different frames of reference, since the light (or e.m.) waves emitted in the pulse would be traveling at different velocities relative to one another.  The possibility might then exist that the total energy of the pulse could be reabsorbed in its entirety in different locations of space, thus generating a surplus greater than the input energy.  Some of the energy obtained could then be used to regenerate the pulse with the surplus to be utilized elsewhere.  This might, of course, in theory provide for a very cheap source of energy, that might be gained with out any apparent consumption of fuel taking place.

                  Such a scenario might even be possible without there needing to be any real violation of conservation of energy.  Given that the universe were of infinite volume, and noting that in such a universe energy can propagate at faster than light speeds, there might be an infinite supply of energy available in such a universe which could also be transported through it at essentially unlimited velocities.  As subtraction of any finite amount, no matter how large from an infinite amount leaves the latter value unchanged; the total energy of such a universe would still be 'conserved', after any finite amount of energy were deducted somewhere in the universe from the total!                                                    Of course, in dealing with unfamiliar physics some caution would be advised.  For example, the possibility might exist that through this kind of approach high energies could build up in a small region of space.  One also notes that in such a universe as described here, the possibility exists in theory that energies could propagate from some distant region of the cosmos at faster than light speeds.  One might not exclude the possibility that although these would in general remain in another frame of reference than the majority of local objects where the energies might have reached, and would thus not normally tend to effect them; that in some cases some kind of influence might take place.  Therefore reasonable caution might be advised with some kinds of experiments.

                  Similarly to what is considered to hold in our universe, when looking at an object that is at a distance of, say x light years away from one in an S.G.I. universe; one would be seeing the light that had departed from the object x years in the past.  Thus one would be seeing the object as it would have appeared x years in the past.  However, in an S.G.I. universe, light waves in different reference frames from ones own would be reaching ones location in space that would have left the object both earlier, as well as light waves that had left the object later, than x years ago.  These light waves might then in theory be picked up by one without one having to leave ones immediate location in space (for example, by allowing some sort of telescope-camera to briefly accelerate to high velocities towards, or away from the object to be observed while taking its picture; or conceivably by {somehow} deciphering the e. and m. fields to determine what their values might be in other reference frames than ones own, and then reconstructing the image that would be produced by the resulting light waves in these other reference frames).

                  Thus it might be possible, for those individuals living on a planet in such a universe to detect the light waves emanating from a star at a distance of, say 10 light years, and observe how the star appeared a mere one year ago, or still more recently, (or alternately how it appeared 100 years ago or later); without having to leave the immediate proximity of the planet.  Using satellites of some sort one might be able to photograph events that took place, and conditions that obtained on the planet during its ancient or early history by detecting the light waves in the frames of reference that an observer would be in who was approaching the planet at high velocities (i.e. many times the speed of light).  As one would possibly be able to observe the light waves present in a frame of reference moving very rapidly (many times the speed of light) away from some distant light source in the universe, one might be able thus to perceive how it would recently have appeared.

                  The determination of what kind of universe one lives in can not, of course, be made purely through theorization alone.  Only experimentation can give certainty on this. One can't even entirely exclude the possibility that somehow nature might be capricious, and produce a universe which seems to behave in a way which defies logic.  Some of the particulars of the L.I. model seem to many to be tantamount to this.  However, the general experience of nature's laws tends to point in the opposite direction; to the sense that at base the universe should be simple and logical, and at least in the case of Newton's or Maxwell's laws; even comprehensible to common humanity, without there being the need of an elite to interpret them!     

                                             --Possible Tests--  

                  A fairly simple approach to trying to determine, for example, if our universe might be an S.G.I. universe, would be to examine celestial objects; preferably ones which exhibit definite variable or periodic behavior; that would be very near the circumference of a circle centered on the sun, in the plane of earth's orbit about it, with a radius of very nearly 2516 light years (18.). (One would want to be certain that a candidate object had been correctly estimated to be at this distance).  An object 'on' this circle would preferably be observed for at least one year.  In particular one would want to study the image of such an object seen from the location of the earth at the exact time when it would be directly behind the earth's motion in orbit about the sun.  This would then have to be compared with an image of the same object seen from the earth six months later, when the earth in its orbit about the sun would be moving directly towards the object with its orbital velocity.   

                 If our universe were an S.G.I. universe, then the two images should closely match, showing the object to be in the same phase of its periodic behavior in each.  This would be the case because the light seen in each 'observation' six months apart would have left the object at approximately the same moment.

                 Observations of the object during the whole six month period between the two observations mentioned above would, in fact, be expected to show limited periodic behavior; as during the whole six month period the object would appear almost frozen in time.  Observations during the other six month period, on the other hand, would show much greater periodic behavior.        

                 Another possible test would be that indicated in the 'problem' definition below.

                                          --Possible Difficulties--

                While Special Galilean Invariance apparently ensures that the speed of light in a vacuum should always be measured to be equal to the constant c predicted by Maxwell's theory in any inertial frame of reference, the question is more complex in an accelerated reference frame as any measurement of the velocity of a light wave would require time to allow the wave to pass through the measurement apparatus.  During this time the apparatus has changed velocities and when the measurement is completed one would then be observing a different light wave than initially, which would have traveled from the source with a velocity differing by Dv from the first wave; Dv being the change in the component velocity of the reference frame in the direction of the light source in the course of the measurement, assuming the light source were outside the reference frame and not itself accelerating.  This might lead, for example, by some types of 'line of flight' measurements to discrepancies which would increase with the distance to the light source.

                 In practice however, using a Michelson type of set up (19.), one would be measuring the velocity of the light wave that traveled with velocity c relative to that of the end point of the path taken through the apparatus by the light pulse, since this is the point where it is detected.  In this case the acceleration, if not large, might make the path just slightly longer or shorter, distorting the result.  However, if all the measurements of this sort were made under a similar acceleration; that of the gravitational field of a single planet for example, then the distortion would be presumably identical in each case giving a constant value for the speed of light, that might be slightly different if measurements were taken in a weightless environment without any kind of acceleration.  The acceleration caused by the centrifugal force of the planets rotation or orbit about its star might lead to discrepancies between different measurements but one imagines that these would be very slight.

                 Use of such an apparatus in a significantly accelerated frame of reference might be difficult anyway since the acceleration undergone might cause the apertures or mirrors to move off the path of the light pulse altogether.  I don't myself know of any experiments run to measure the speed of light in a significantly accelerated frame of reference, so what would result from such experiments remains unclear.       

                 In an S.G.I. universe in an accelerated frame of reference, however, one would expect a light pulse emitted by an object in the reference frame to travel to another object in the same reference frame, in the direction of acceleration at an apparent velocity slightly greater than c (and slower than c in the opposite direction with a negative acceleration); namely at apparent velocity: ah/{(cc+2ah)^1/2-c}, measured in the reference frame, after having departed relative to its source with velocity: (cc+2ah)^1/2, with h=(distance between objects in direction of acceleration, measured prior to accelerating), a=(acceleration).  However this would not, one would expect, itself be the cause of any time dilation (though possibly of a space dilation), as it would presumably cause the pulses to arrive sooner (if they are sent in the direction of acceleration), however it would not be expected to effect the time measured between the pulses.  Therefore one should not, in general, anticipate that there would likely be any time dilation, even in an accelerated frame of reference, in an S.G.I. universe. 
                Interesting in this regard might be to measure the speed of light as it travels along a vertical (rather than horizontal) path in one direction in a gravity field (or in general, in an accelerated frame of reference) to see how this result would compare with the other direction, and with the horizontal path.  A precise measurement would, of course, be needed as the acceleration of gravity, say on earth accounts for a small change in velocity compared with the very large velocity of light.

                 Non-the-less the question is somewhat complicated, and the existence of some kind of General Relativistic (20.) principle in an S.G.I. universe to account for observations of time dilation in accelerated frames of reference, as well as the as equivalent construed gravitational frames of reference (21.); and which would also ensure that the constancy of the speed of light is upheld in such cases might not be impossible.  This however, does not seem to me to necessitate that Special Relativity with its more troubling temporal notions such as non-simultaneity (22.) would also have to hold, as the S.G.I. principle would have accounted for the constancy of the speed of light in the inertial frames of reference otherwise addressed by S.R.  As said elsewhere one notes in this regard that tests appearing to uphold the theory of relativity to my understanding have almost invariably dealt with accelerated frames of reference, and have not as far as I know looked at pure inertial reference frames.  It is to be expected that a formulation of General Relativity in an S.G.I. universe would differ with and make different predictions than such a formulation for a L.I. universe.  Therefore the exact expectations for the results of experiments of the type done to try to verify Special or General Relativity in our universe are unclear, as one would have to first determine the possible formulation that General Relativity might have in an S.G.I. universe. 

                                                   --Summery--

                  An S.G.I.(Special Galilean Invariant) universe could perhaps be most succinctly defined to be one in which two postulates hold:

                     1) Maxwell's theory of light (or electromagnetic phenomena) holds (or at least appears essentially to hold) with the consequences thereof (some having been mentioned above).

                     2) Each inertial frame of reference (1.) possesses its own set of light (or in general electromagnetic) waves, which all move with the constant speed c predicted by 1) through it. The light (or e.m.) waves found in one such frame of reference do not exist in the others (though near replicas of them typically would).    

                 Special Galilean Invariance denotes an extension of the idea of Classical Galilean Invariance to e.m. phenomena.       

                     'Problem' definition of an S.G.I. universe:   

                At time T a radio signal is sent out from a radio antenna atop a high tower.  At time T+d/(c+v), a radio on the ground, and a radio in a speeding car are both a distance d from the top of the tower.  The car is moving directly away from the antenna on the tower with velocity v at this time.  c=(the speed of light).

                         t1=(time that the radio on the ground receives the radio signal)
                         t2=(time that the radio in the speeding car receives the radio signal)

                                In an S.G.I. (Special Galilean Invariant) universe:

                                   t1=t2+dv/c(c+v)>t2 

                      Poetic definition of an S.G.I. universe:  

                              One in which,

                           If you leave your star with velocity v,

                           the light you see left with c+v,

                           If you change v, you exchange the light wave that you see!   

                                                   --Notes--

            (1.)            See Nelson, David "The Penguin Dictionary of Mathematics, Third Edition" pp.-170,171.
            (2.)            Ibid. pp.-273-274.  James Clerk Maxwell was a Scottish Physicist and mathematician who wrote "Treatise on Electricity and Magnetism" in 1873.       

            (3.)            Ibid. p-270

            (4.)            m=M(1-v v/cc)^-1/2,  M=(rest mass), m=(accelerated mass), ^ -'raised to',  See Loeb, Leonad B. "Fundamentals of Electricity and Magnetism" p.-426.  
            (5.)            i.e. See Feynman, Richard  "Six Not So Easy Pieces" pp.-68-71. 
            (6.)           The correct definition of 'component velocity' to be used here must be understood clearly.  More precise is:  x'=xc/{v.cosA+[cc-(v.sinA)^2]^1/2}, where an observer traveling with velocity v makes an angle A on crossing the path of a light wave emanating from the source, at a point in space where an observer stationary with respect to the light source would have seen the source at a distance x.  Note:  A=(pi)/2 yields x'=xc/(cc-vv)^1/2. (pi= 3.141592...).  Though the velocity v is at a right angle with the path of the light wave, the 'component velocity parallel' to the light source is apparently non-zero as x' is not equal to x.  This is true as the observer does not see the light wave crossing his or her path at a right angle, rather an oblique one.  A=0 (placing the observer's course on the path of the light wave) yields x'=xc/(c+v).  The light wave seen by the observer in these cases would have departed relative to its source with velocity:  u=v.cosA+[cc-(v.sinA)^2]^1/2.

            (7.)           See Haber-Schaim, U.  Cross, J.B.  Dodge, J.H.  Walter, J.A. "PSSC Physics Forth Edition" 'Electric force': p-412, Magnetic fields: pp.-491-500, Electromagnetic waves: pp.-522-551.

            (8.)           In actuality the observer would move through a continuous set of reference frames with a resulting continuous variation in the observed values of the component electric and magnetic field values while accelerating.  Thus the observer would be witnessing a continuously 'renewing' light (or e.m. wave) which would in a sense be accelerating along with the observer, giving the impression that it were keeping a constant velocity if the observer were not noting the acceleration.      

            (9.)           Haber-Schaim, p-544.  This is evident from the discussion on this page.

            (10.)         Ibid p.-581.  

            (11.)         Ibid p.-305.

            (12.)         One notes that the two "founding assumptions" of Special Relativity (See Einstein, Albert and Infeld, Leopold "The Evolution of Physics from Early Concepts to Relativity and Quanta", pp.-156,177) are also upheld in an S.G.I. universe.  However in this case, the "third principle" of 'transformations', "according to the classical transformation" does not have to be given up.  Instead the "time dilation in an inertial reference frame" and all the confusion and paradox it leads to is given up.  It comes to mind here that all of the experiments I've heard of which appear to confirm S.R. seem to deal with "accelerated reference frames" i.e. reference frames undergoing continuous acceleration; rather than "inertial reference frames", i.e. reference frames which may at some time have undergone acceleration with respect to another and then continue to move through inertia.  It is not clear to me if the predictions of S.R. have been as well tested in these cases.   

            (13.)         The red shift would be given by the 'classical equation':  DL=Lv/c, where v is the velocity with which the light source is moving away from one, L is the 'stationary' wave length, and DL is the change in the wavelength.    

            (14.)         One wonders if some of the 'longer lived particles' thought of as resulting from time dilation that emerge in particle accelerator experiments could in reality be particles newly produced from the high energy environments of the particle accelerator, the original particles having earlier decayed.    

            (15.)         A simple calculation suggests, that given that the vehicle undergoes a constant acceleration throughout, and that the means of propulsion is uniformly efficient across different thrust settings; that its mass m after the time t when the speed of light is passed, would conform to: m<Mexp(-1/k), where M is its initial mass, and k the fraction of its consumed fuel mass which is converted to energy in the form of thrust ('exp' is the exponential function, see Nelson pp.-133,154).  The loss in mass of the vehicle would be entirely a result of its consumption of fuel and would not affect the components of the vehicle or of its occupants otherwise.   

            (16.)         Lorenz Invariance denotes the set of symmetry transformations (first proposed by H.A. Lorenz) pertaining to the theory of Special Relativity.  See Isaacs, Alan "Dictionary of Physics"  p.-274.

            (17.)         I believe one or another discrepancy in our universe of this sort has been noted, as I recall, without any clear explanation having been offered at the time to account for it.

            (18.)        (cc-vv)w/4v=(2516.25... ) Light Years, w=(1 year), v=2(pi)r/w=(velocity of earth in orbit around the sun), r=(one astronomical unit), c=(speed of light in a vacuum), pi=3.141592...  .  This formula can reasonably be extended to regions of space off the ecliptic (plane of the earth's orbit around the sun) to:  (2516.25)secant(D) Light Years, D=(declination of the region of space).      

           (19.)         Haber-Schaim, pp.-84,85.

           (20.)         Feynman,  pp.-128-143.

           (21.)         Einstein, A. and Infeld, L. pp.-218-221.  

           (22.)         Non-simultaneity is the principle in S.R. which claims that if a space ship passes the earth at half the speed of light coming from the direction of Pluto at perihelion at the same moment that another space ship passes Pluto with the same velocity, someone on the first space ship could claim to have reached the earth about two and one third hours before the other space ship reached Pluto in their reference frame; if I did the calculation correctly (23.).  This apparently gives the crew of the second ship enough time to change their minds and go in another direction instead, to prevent what has already taken place from happening.  However, there is no way in an L.I. universe for communication to occur between the four points at faster than light speed to enable one to arrange to change history; without using either quantum tunneling or E.P.R. effects (24.) and these methods are undoubtedly highly frowned upon, so don't worry!

           (23.)         Einstein A., and Infeld, C.  pp.-178,179.  Gives the basic argument for this hypothetical state of affairs.     

           (24.)          See Greene, Brian "The Fabric of the Cosmos" pp.-99-115.           

                           

                            Could a single light pulse be picked up at either (or both) A and B?
                            (km-'kilometer', rpm-'revolutions per minute'; Four light sensors are attached to the                             wheel centered below point B, one is fixed two meters behind B along the path of                                 the light pulse reflected back to the Earth from the mirror on the Moon.)