Chapter 1: The Relative Response Time Hypothesis

 

A mainstream physicist who wishes to remain anonymous, I will refer to him as A, told me an interesting science fiction story about Special Relativity (SR).  This story caught-on and has been re-told by many different writers and in many different ways.  I will refer to these stories as SR stories.  One of these SR stories is posted on the internet at this Jasper and Zoe site.  Zoe was a girl who raced up and down the road at outrageous speeds back when she was a young woman and Jasper was a guy that was always studying what Zoe was doing from his veranda by the road.  Jasper did not have any computer models to simulate what was happening with Zoe and her car; but he did the best that he could with the tools that were available at the time.  Using mathematics and other tools in his tool box, Jasper had determined that the clock in Zoe’s car runs at a slower speed than his clock as she raced up and down the road.  Jasper also concluded that the clock in Zoe’s car runs precisely (1 - (v/c) ²)^1/2 times slower than his own clock where v is Zoe’s speed and c is the speed of light.  Jasper determined that Zoe’s car shrinks in length as she raced up and down the road compared to the length he measured when Zoe’s car was parked in her driveway. He determined that Zoe’s car shrinks by (1 - (v/c) ²)^1/2 times the length of the car when it is parked in her driveway, but strangely, it does not shrink in width.

 

Being the curious Information Technology (IT) measurement guy that I am, I had this irresistible urge to see if the concepts expressed in Jasper’s story as well as in other SR stories could be used to measure the response time components (or transmission delay times) between two objects moving at high speeds.  IT measurement guys have become somewhat skilled (or is it artistic) with the tools in their box such as mathematics, mathematical modeling, and Microsoft Excel. However, after trying to apply the fundamental concepts expressed in the SR stories to build a “relative response time” model it became obvious that SR had some questionable aspects.  It then occurred to me that a more generalized model was needed that could not only simulate and test the SR set of assumptions but could also simulate and test alternative sets of assumptions.  I thought that it wouldn't be cool to suggest that SR theory should be embraced as an unalterable truth for building a relative response time model without a check-out of its assumptions first.

 

I couldn't get this idea out of my mind and I had an irresistible urge to create such a generalized model.  My alter ego thought it was very naïve for someone who did not have a PhD in theoretical physics to try to tackle such a problem.  Therefore, I tried to fight off this urge many times.  But, I kept feeling guilty about pointing my finger at 'that mainstream physics community hiding behind the trees' because a response time measurement problem is an IT performance measurement problem.  In fact, it would not be unreasonable to take the contrarian view that since it forces us to apply free speculation then it is possible that an artist from the IT community with What If simulation modeling skills and an outside-the-box science fiction writer’s propensity for free speculation might be able to develop such a generalized model before a mainstream physicists could.  Mainstream physicists are biased and would assume that SR theory is an unalterable truth and would not be the kind of persons you would hire to check-out the credibility of SR theory.  Also, the assumption that the velocity of light as measured by any observer is limited by the value c (approximately 299,792.458 km/sec) would appear to be a queuing theory problem not a physics problem.  I have asked some PhD Physicists if it is possible that relativistic phenomena caused by relative velocities could be a queuing problem because light traveling on a moving path will approach the limit c as the velocity of the path approaches the limit c.  You may be surprised to learn that their answers were something to the effect that "I don't know anything (or enough) about queuing theory to answer your question" and that was the end of the conversations.  Physicists are not well educated in advanced queuing systems or in the application of queuing theory in the development of models for predicting the behavior of queuing systems.  Therefore, they also do not have the appropriate experienced or qualifications to do research on the mechanics of complex systems that involve queuing phenomena and require the application of queuing theory.      

Then I studied Jasper’s SR story about the mechanics of light moving up the street and down the street as Zoe raced back and forth as well as the mechanics of light moving over to one side of Zoe’s car and back to the other side.  It then occurred to me that the problem that Jasper had with his measurements was the same problem that Michelson and Morley had when they tried to detect the time difference between light traveling on the paths between mirrors in a moving ether stream going "upstream then back downstream" and light going "across to a shore and back" that was described in this MM Experiment link.  Then I decided to try working outside of the SR box.  So, I started to build a visual emulation model of the movements of light and the movements of the paths between the mirrors in the MM experiment using Microsoft Excel, Visual Basic and my knowledge of queuing problems and my "Keep it Simple Stupid" (KISS) philosophy for solving queuing problems.

 

During the intervening years of experimentation and testing, a new Relative Response Time (RRT) Hypothesis was derived.  The premise for the RRT Hypothesis includes the Principle of Relativity, the principle that the velocity of light along a path between two points is always measured as c by any observer, that is, path length / time = c, and Jasper’s measurements that the length of Zoe's car contracts as her velocity increases but the width of her car does not change as her velocity changes but remains the same width at any relative velocity.  In SR theory T_o/T’ = l = 1 / (1 - (v/c)²)^1/2 and when a signal travels along a path that is pointed in the direction of motion L_o/L = 1 / (1 - (v/c)²)^1/2.  The factor 1 / (1 - (v/c)²)^1/2 is the Fitzgerald-Lorentz Factor and will be denoted by the symbol l.  Since T_o/T' = l and L_o/L = l, then the relative space-time relationship T_o/T’ = L_o/L exists in SR theory when the signal travels on a path pointed in its direction of motion.  This relative space-time relationship indicates that the traveler’s time (T’) is directly proportional to the observer’s time (T_o) and inversely proportional to l (i.e. T' = T_o/ l).  In SR Theory, a moving horizontal signal path of length (L) as measured by a stationary observer (the Observer) contracts to its length at rest (L_o) divided by l when the path is pointed in the direction of motion (i.e. L = L_o/ l).  In addition, SR implies that T₀ is the same as Elapsed Time (ET_a) in seconds for the Observer at point 'a' and T' is the same as Elapsed Time (ET_b) in seconds prime for a traveler (the Traveler) at point 'b', therefore, ET_b = ET_a/ l when the signal travels in the direction of motion of it’s path.  In SR Theory, a signal sent from point 'b' at the back of a moving train to point 'd' at a fixed distance toward the front of the moving train will have a Response Time component (RT_bd) that is longer than a signal sent from the front to the back (RT_db).  In SR Theory, ET_a/ l = (RT_bd + RT_db) = ET_b but RT_bd must be longer than RT_db even though the distance between 'b' and 'd' has not changed. 

 

In the RRT hypothesis ET_a/ l = (RT_bd + RT_db) as it is in SR Theory (see send simulation in Video_1 and receive simulation in Video_2 for point 'a', 'b', 'd', orientations).  However, the one-way response times RT_bd and RT_db are equal when the distance between 'b' and 'd' is fixed (i.e. does not change).  According to the RRT hypothesis, response time (RT) depends upon Service Time (ST) and Queuing Time (QT) or Quantum Utilization (QU).  ET_bd = RT_bd = ST_bd + QT_bd  = ST_bd /(1 - QU_bd) and ET_db =  RT_db = ST_db + QT_db = ST_db /(1 - QU_db).  QU is dependent upon Path Velocity (PV) and c where QU = (PV/c) and PV is defined as the length of a photon's path in the Observer's coordinates minus the length of the same photon's path in the Traveler's inertial frame divided by Response Time (RT) according to the equations PV_bd = (PL_ae - PL_bd) / RT_bd and PV_db = (PL_ea - PL_db) / RT_db; where PL_ae and PL_ea are the distances traveled by a photon in the Observer's coordinates during the Elapsed Time ET_ae = RT_bd and ET_fb = RT_db respectively.  ET_fb is the elapsed time for a return of a signal sent from an Observer at point f; however, the length PL_fb cannot be used in the PV_db function.  The path (PL_fib) from point f image to point b must be used instead of the path from point f to b (see receive simulations in Video_f for point 'e', 'f' and 'f image' orientations and a discussion of the RRT Hypothesis).  Since the receive signal is a reflection of the send signal, the receive signal must appear to originate from the image of point f as described in Video_f.  A strict definition of path velocity is PV_bd =(PL_ae - PL_bd) / RT_bd for a send simulation because PL_ae and PL_bd are shared paths but for a receive simulation PV_db = (PL_fib - PL_db) / RT_db.  However, since PL_fib = PL_ea = (c*ET_fb) the equation PV_db = (PL_ea - PL_db) / RT_db and PV_db = ((c*ET_fb) - PL_db) / RT_db can be used as well as the equation PV_db = (PL_jib - PL_db) / RT_db to calculate the path velocity of a receive transmission.  PV/c must be less than 1.0 (or 100%) and as PV/c approaches 100% Queuing Time (QT) approaches infinity and Service Time (ST) becomes very small relative to QT.  PV is the same as the velocity (v) of the Traveler when the path of a photon moving from points 'b' and 'd' is pointed in the direction of the movement of the Traveler.  Therefore, QU_bd = v/c and QU_db = v/c. 

The concept of utilization as applied to a moving photon can be most easily understood if we follow the KISS principle and use the most simple explanation.  Quantum Utilization (QU) is zero (or 0%) when a photon can move unconstrained on the path because its velocity (v) is zero with respects to the Observer.  QU = 1 = 100% when a photon is completely constrained and cannot move along the path because its velocity (v) has reached the limit c with respects to the Observer.  The initial two equations (1) and (2) below that express QU_bd (utilization of a send signal) and QU_db (utilization of a receive signal) as a function of path lengths are as follows.  When the Traveler is at rest then PL_bd = Lo = PL_ae; therefore, (1) QU_bd = 1 -  (PL_bd/ PL_ae) = 1 - 1 = 0 (or 0%).  When v = c then PL_bd = 0 and PL_ae= c*ET_fb; therefore,  (2) QU_db = 1 -  (PL_bd/ PL_ae) = 1 - 0/ (c*ET_fb) =  1 - 0 = 1  (or 100%).   Equations (1) and (2) are summarized on line (a1) in the "Quantum Utilization (QU) Postulate" section below where the utilization of a send signal (QU_bd) and a receive signal (QU_db) are expressed as a function of path lengths only.  The line (a1) expressions are pivotal because they provide an interface between the geometry and mechanics of signal transmission and the queuing theory equations that express the behavior of a specific kind of queuing system known as the ubiquitous M/M/+1 queuing system.  The M/M/+1 queuing system is appropriate because the single paths with moving lengths PL_bd and PL_db along which photons must travel are single servers with random arrivals and constant Service Time per arrival.  The "Quantum Utilization (QU) Postulate" includes additional QU equations in lines a2 through a7 that have been derived through the mapping of QU terms into M/M/+1 queuing system equations.   In the RRT Hypothesis, ST_bd = PL_bd / c based upon the premise that ST_bd / PL_bd = c.  This means that ST_bd is the amount of time that the Observer would calculate as the time for a signal to travel the contracted path length (L) if it must travel at the velocity of light (c) along the moving path, but it cannot if the path is moving unless this Service Time measurement is equal to PL_bd/c. Therefore, the speed barrier of about 300,000 km/sec causes Queuing Time (QT_bd) that accounts for the difference between RT_bd and ST_bd (i.e. QT_bd = RT_bd – ST_bd). 

In the RRT hypothesis, the send Response Time for a traveler at 'b' (RT_bd) is directly proportional to ST_bd and inversely proportional to 1 - QU_bd for a signals traveling at any angle relative to the direction of motion as a natural consequence of the speed barrier c and the premise that PL_bd/ST_bd = c.  Also, the receive Response Time for a traveler at 'b' (RT_db) is directly proportional to ST_db and inversely proportional to 1 - QU_db for a signals traveling at any angle relative to the direction of motion as a natural consequence of the speed barrier c and the premise that QU is 100% when a signal’s path travels at velocity c. In the general equation for the traveler's response time (RT) as measured by the observer, ST replaces T', (1 – QU) replaces l and PV with its slope replaces L_o and L when the signal travels in any direction. The relative space-time relationship RT’/ST = L_o/L can be derived from the general response time equation and it replaces the SR space-time relationship T_o/T’ = L_o/L = l. The terms RT’/ST and L_o/L are not equivalent to l, nevertheless, L_o/L is a conversion factor as l pretends to be for converting RT in seconds to the traveler’s response time (RT’) in sec’ and 1/(RT’/ST) is a conversion factor as l pretends to be for converting L in km to the traveler’s signal path length L’ (same as L_o) in km’.  To lend credibility to these new conversion factors, they can be used to determine L’ and RT’ respectively when the signal travels on a path pointed at any angle to its direction of motion.  The single simple general response time equation RT = ST / (1 – U) not only replaces the equations that express time dilation and length contraction as a function of velocity and the speed of light, it replaces SR’s relative space-time relationship.  To lend additional credibility to this general response time equation, it does not require the clock synchronization fudge factor for simulation of What If scenarios but it does require this fudge factor for calibrating simulations that use SR assumptions.

 

If we compare the M/M/+M® Model equation RT = ST / (1 – U) with the SR equation T = T’/(1 - (v/c)²)^1/2 when applied to the vertical arm (same as signal path) of the MM Experiment then T = RT (in seconds) while the value of T’ (in sec’) = ST (in sec).  Since the equation T = T’/(1 - (v/c)²)^1/2 (same as T = T’ * l) was derived from a mechanical model of light transmission in one direction (one-way trip) along a path that is perpendicular to the direction of the path’s motion, then this equation is valid only for the vertical arm in the MM experiment.  However, T = RT = ET = ST / (1 – U) and ET = RT’ * l; but T is not = T’ * l except for an arm that points perpendicular to the direction of it’s motion or for the round trip time for light to travel from the back of the arm to the front and back.  This is evident because as a global variable, T’ does not contain the same value as the global variable RT’ except for an arm that points perpendicular to the direction of it’s motion or for a round trip.  RT’ is measured by the traveler’s clock that appears to run slower than the Observer’s clock according to the “observer”.  From the observer’s point of view, the traveler uses a slower clock and a shortened meter stick; therefore, to distinguish from the observer’s measurements we will designate measurement units taken by the traveler’s measurement devices as prime units.  When the arm is perpendicular to the direction of it’s motion ((1 – U) * l) = 1, then T/ l = ST/((1 – U) * l) = ST = T’. Consequently, it is easy to see that the “time” variable T’ has the same meaning as Service Time (ST) measured by the observer in seconds and cannot have the same meaning as the traveler’s Response Time (RT’) that is measured in sec’.  Also, the fact that SR uses the equation T’ = (T/ l) - (v*L_o/c²) for a one-way upstream send along a path pointed in it’s direction of motion is further evidence that as a global statement T’ is not = T/ l, therefore, as a global statement T is not = T’ * l.  The M/M/+M® Model general equation T = RT = ET = ST / (1 – U) = RT’ * l.sec/sec’ and the Phi Mode equation RT’ = ST*(L_o/L).sec’/sec are valid global statements that describes light transmission along a path that is pointed in any direction relative to it’s direction of motion.  In other words, the Special Relativity equation T = T’ * l is indeed a Special case statement while the M/M/+M® Model general equation RT = ET = ST / (1 – U) = RT’ * l.sec/sec’ and the Phi Mode statement RT’ = ST * (L_o/L).sec’/sec are global statements.

 

This simple unifying general equation RT = ET = ST / (1 – U) = RT’ * l was derived from the Computer Performance Measurement (CPM) box of tools.  This equation is the M/M/1 Response Time Equation that is a surprisingly ubiquitous general equation used in queuing models to predict response time components in computer networks.  The M/M/+M® Model is based upon queuing theory, therefore, it is not a new theory in itself; it is simply another CPM predictive model that is based upon queuing theory.  The ubiquitous Response Time Equation expresses Response Time (RT) as a function of Service Time (ST) and Utilization (U) as follows: RT = ST/(1 - U); where Utilization is a measurement of the capacity utilization of a single server (expressed as a percentage from 0% to 100%) and Service Time is the amount of time that the server is busy serving transactions.  Response Time, in this simple case, is the amount of time that passes between the times that the transaction enters a queue to wait for the server and the time that the transaction is completed by the server.  ST and QU will not have the same specific meanings in the RRT Hypothesis that ST and U have in classical queuing theory models; however, the classical Response Time functions RT = ST/(1 - U) does have the same general meaning and the same relationship as independent variables in the RRT Response Time functions RT_bd = ST_bd/(1 - QU_bd) and RT_db = ST_db/(1 - QU_db) .  The M/M/+M® Model can be used to simulate or predict the behavior of the MM Experiment as applied in V1R0 where point 'b' and 'd' are "bound" by an equation that defines path length (L) as a function such as the SR assumption that L = L_o/ l.  The RRT Hypothesis applies the SR length contraction assumption that L_o/L = l in SR Mode and it applies the RRT Hypothesis derivation that L_o/L = 1 / (l *(1 – QU)) in Phi Mode (see derivation of equation (5b) from (5a) following).  In summary, equation (5b) is fundamentally derived from the relationship L_o/RT_bd' = L/ST_bd = c where c is expressed in km'/sec' and km/sec respectively.  Since L_o = PL_aD and L = PL_bd for a send simulation, then PL_aD/RT_bd' = PL_bd/ST_bd = c and RT_bd' = ST_bd*(L_o/L) = ST_bd*(PL_aD/ PL_bd).  In the RRT hypothesis  RT_bd/RT_bd' where RT_bd' is a function ST, L_o and L.  This RRT l function is general and it applies to all transmission path (PL_bd or PL_db) slopes while the SRT l function can only be derived and logically applied to a geometry where the slope of the transmission path (PL_bd or PL_db) is perpendicular to the direction of its motion.  In Phi Mode the RT’ functions: RT’ = ST / (l *(1 – QU)) = ST*(L_o/L) = RT/l apply universally while the only RT’ function in SR Theory: RT’ = RT/l where l is defined only for the special case where the transmission is perpendicular to the direction of its path's motion.  Therefore, this SRT function is a non-sequitur for all other path slopes. 
  

The M/M/1 equation from queuing theory is applied in the following Relative Response Time (RRT) Hypothesis:

M/M/1 response time equation expressed as M/M/+M® Relative Response Time (RRT) Hypothesis:

Simply stated, the RRT Hypothesis is that the ratio (l) of the response time of a send transmission measured by the Traveler's clock (RT_bd') and the response time of the send transmission measured by the Observer's clock is l = RT_bd/RT_bd'.  The RRT Hypothesis is derived from the premise that Lo / RT_bd' = PL_aD / RT_bd' = c where Lo and PL_aD are prime units and where L / ST_bd = PL_bd / ST_bd = c.  Therefore, RT_bd' = ST_bd*(Lo/L) = STbd*(PL_aD/ PL_bd) and l = RT_bd/RT_bd' = RT_bd/ST_db*(Lo/L) = RT_bd/ST_db*(PL_aD/ PL_db).  Since the send photon also travels on the Observer's path that has a Path Length (PL_ae) then RT_bd = PL_ae/c.  The same premise applies to a receive transmission except the subscripts db are substituted for bd and the Observer's kinematical path discussed above and in Video_f has a Path Length (PL_fib) and RT_bd = PL_fib/c where PL_fib = PL_fa.

 The derivation of the RRT Hypothesis from a mapping of relative response times into a M/M/1 queuing system follows:
       
Observer’s measurements (units in a “stationary” frame are in seconds, meters, and transactions e.g. sec, km, trans – there is one transaction for each completed response from point D):

M/M/1 response time equation from queuing theory: 

(1) Response Time = Service Time / (1 - Utilization): RT = ST / (1 - U)

(1a) RT_b = RT_bd + RT_db = (ST_bd / (1 - QU_bd)) + (ST_db / (1 - QU_db))
Where RT_b is the response time at point 'b', ST_bd and ST_db is the Service Time (ST) measured by the Observer’s clock while a signal travels the distance from 'b' to 'd' and from 'd' back to 'b' respectively (i.e. ST = PL/c). QU_bd and QU_db are Quantum Utilization (see a1 through a7 below) for the send and receive signals respectively. Double letter subscripts reference a path length (e.g. PL_bd is the Path Length of a signal that moves from point 'b' to point 'd').

 

 

(1a) Expressed in other forms:

(1b) RT_b = RT_bd + RT_db = (ST_bd + QT_bd) + (ST_db + QT_db)

Where ST_bd = (RT_bd * (1 – QU_bd)) and ST_db = (RT_db * (1 – QU_db)) and

(b1) QT_b = QT_bd + QT_db = (QU_bd * RT_bd) + (QU_db * RT_db)

Where Q Time (QT_b) is the amount of time that is lost by a clock at 'b' during the time that a signal travels from 'b' to 'd' and back to 'b' and QT increases geometrically as QU increases.

(b2) QT_b = (QU_bd * ST_bd / (1 - QU_bd)) + (QU_db * ST_db / (1 - QU_db))

By applying Little's Law (applies to Phi Mode only):

(1c) RT_b =RT_bd + RT_db= PL_ae/c + PL_fib/c
Where PL_ae is the path between the beginning of the send signal at the Observer’s point 'a' and his point 'e' where the send signal reached its destination and where PL_fib is the path between the point ‘f image' and the end of the receive transmission at the Observer’s point 'b'.

When points ‘b’ and ‘d’ are moving in tandem (on parallel lines at the same velocity and direction) as in the M/M/+M® Model Version 1 (V1) Phi Mode then:

(1d) RT_b = RT _bd + RT _db = (L_o* l)/c + (L_o* l)/c = 2(L_o* l)/c

When the path is parallel to the direction of path motion in V1 Phi Mode:

(1e) RT_b = RT_bd + RT_db = PL_bd /(c – v) + PL_db /(c – v)

Quantum Utilization (QU) Postulate:

(a1) QU_bd = 1 -  (PL_bd/ PL_ae) and  QU_db = 1 -  (PL_db/ PL_ea )  
Since RT_bd* c = PL_ae and RT_db* c = PL_ea then:
(a2) QU_bd = (PL_ae - PL_bd) / (RT_bd* c) and QU_db = (PL_ea - PL_db) / (RT_db*c)
QU_bd and QU_db in (a2) can also be expressed as: 
(a2) QU_bd = ((PL_ae - PL_bd) / RT_bd) / c and QU_db = ((PL_ea - PL_db) / RT_db) / c

QU_bd and QU_db in (a2) can also be expressed as: 
(a3) QU_bd = (PV_bd / c) and QU_db = (PV_db / c), where PV is Path Velocity:

(a4) PV_bd = (PL_ae - PL_bd) / RT_bd and PV_db = (PL_ea - PL_db) / RT_db or PV_db = (PL_fib - PL_db) / RT_db

When the path is parallel to the direction of path motion:

(a5) PV_bd = v and PV_db = v

Since Arrival Rate (AR) = PV and Service Rate (SR) = c, then:

(a6) QU_bd = (AR_bd / SR_bd) and QU_db = (AR_db / SR_db)

When the path is parallel to the direction of path motion:

(a7) QU_bd = (v / c) and QU_db = (v / c);  therefore,  (v / c) = (AR_bd / SR_bd) = (AR_db / SR_db)

The Relative Response Time Hypothesis is derived as follows:

Response Time at 'b' (RT_b) as measured by clock at 'a' (for all path slopes in all Modes):

Since: (1a) RT_b = RT_bd + RT_db = (ST_bd / (1 - QU_bd)) + (ST_db / (1 - QU_db))

and ST_bd = ET_ae (1 - QU_bd)) and ST_db = ET_fb (1 - QU_db)), then:

(1f) RT_bd = (ST_bd / (1 - QU_bd)) = ET_ae (1 - QU_bd)) / (1 - QU_bd) = ET_ae

and RT_db = (ST_db / (1 - QU_db)) = ET_fb (1 - QU_db)) / (1 - QU_db) = ET_fb

Since ET_fb = ET_fa then: (1f) RT_b = RT_bd + RT_db = ET_ae+ ET_fa = ET_a

Response Time at 'b' (RT_b) as measured by clock at 'a' expressed as a function of ST and Path Lengths:

Since: (1a) RT_b = RT_bd + RT_db = (ST_bd / (1 - QU_bd)) + (ST_db / (1 - QU_db))

and (a1) QU_bd = (1 - (PL_bd / PL_ae)) and QU_db = (1 - (PL_db / PL_ea)) or QU_db = (1 - (PL_db / PL_fib)), then:
(3) RT_b = (ST_bd / (1 - (1 - PL_bd / PL_ae))) + (ST_db / (1 - (1 - PL_bd / PL_fib)))
therefore: (3) RT_b = RT_bd + RT_db = (ST_bd / (PL_bd / PL_ae)) + (ST_db / (PL_bd / PL_fib))

Since PL_aAa = PL_bd when 'b' is not in motion relative to 'a', then:

(4) RT_a = ST_a / (1 - QU_a) and RT_a = ST_a / (1 - (1 - PL_aD / PL_aD))
therefore: (4) RT_a = ST_a

Response Time at point 'a' (RT_a) as a function of send and receive components according to Little’s Law:

(4a) RT_a = RT_aD + RT_Da = (PL_aD / c) + (PL_Da / c)

Elaspsed Time at point 'a' as a function of Throughput at point 'a' (X_a) and RT_a according to Little's Law:

(4b) ET_a = X_a * RT_a

Traveler’s Measurements (units in a “moving” frame are prime units e.g. sec’ & km’, transactions are trans - there is one transaction for each completed response from ‘b’):

Relative Response Time Hypothesis (5 through 9) - Response Time as measured by the degraded clock at 'b' (RT_b’):

By applying conversion factors in bold text to observer’s measurements:

Response Time (RT) Conversion Factors and Time Dilation Equations:

For all modes when the signal path has slope = 90^o with respect to its direction of motion:

(5) RT_b’ = RT_bd’ + RT_db’ = (RT_bd/( l.sec/sec’)) + (RT_db/( l.sec/sec’))

For Phi mode when the signal path has any slope with respect to its direction of motion:

Since RT_bd=(ST_bd/(1 - QU_bd)) and RT_db=(ST_db/(1 - QU_db)), then:

(5a) RT_b’= RT_bd’+RT_db’ = ((ST_bd/(1 - QU_bd))/( l.sec/sec’)) + ((ST_bd/(1 - QU_bd))/( l.sec/sec’))

For Phi Mode (& SR Mode when the signal path has 90^o slope with respect to its direction of motion):

Since 1/(l*(1 – QU_bd)) = 1/( l *(1-(1- (PL_bd/PL_ae)) = 1/(( l *PL_bd)/PL_ae) and

1/( l *(1 – QU_db)) = 1/( l *(1-(1- (PL_db/PL_ea)) = 1/(( l *PL_db)/PL_ea) and

since PL_ae= PL_aD * l and PL_ea= PL_aD * l then

1/(( l *PL_bd)/( PL_aD* l)) = PL_aD/PL_bd and 1/(( l *PL_db)/( PL_aD* l)) = PL_aD/PL_db

Therefore, (5a) becomes:

(5b) RT_b’= RT_bd’+RT_db’=ST_bd* (PL_aD /PL_bd).sec’/sec + ST_db* (PL_aD /PL_db).sec’/sec

Expressed in directionless form: RT’ = ST * (L_o / L).sec’/sec

SR Mode requires an Out-of-sync calibration ±(v*L_o/c²) when slope equal 0^o, therefore:

(5c) RT_b’= (RT_bd’- (v*L_o/c²).sec’/sec) + (RT_db’ + (v*L_o/c²).sec’/sec) where L_o = PL_aD

Path Length (PL) Conversion Factors and Length Contraction Equations:

Since the value of L_o = PL_aD = PL_bd’ = PL_db’ then:

(6) PL_bd’ = PL_aD.km’/km

(7) PL_db’ = PL_aD.km’/km

For Phi Mode when the signal path has any slope with respect to its direction of motion:

Since (1d) RT_bd = (L_o* l)/c and RT_db = (L_o* l)/c and L_o = PL_bd’ = PL_db’ then:

(8) PL_bd’ = RT_bd*c / l.km/km’ and PL_db’ = RT_db*c / l.km/km’

Also, combining (5b) with (6) and (7) then:

(8a) PL_aD/PL_bd = (ST_bd/RT_bd’).sec’/sec = (PL_bd’/PL_bd).km/km’ therefore:

PL_bd’ = PL_bd * (ST_bd/RT_bd’).sec’/sec.km’/km

(8b) PL_aD/PL_db = (ST_db/RT_db’).sec’/sec = (PL_db’/PL_db).km/km’ therefore:

PL_db’ = PL_db * (ST_db/RT_db’).sec’/sec.km’/km

Since PL_bd’/RT_bd’ = c, PL_db’/ RT_db’ = c, PL_bd/ST_bd= c and PL_db/ST_db= c then :

(8c) PL_bd’ = c * RT_bd’ and PL_db’ = c * RT_db’

(8d) PL_bd = c * ST_bd and PL_db = c * ST_db

For Phi Mode when the signal path has slope = 0 with respect to its direction of motion:

Since from (1e) PL_bd = RT_bd*(c – v) and PL_db = RT_db*(c – v) then when combined with (8):

(9a) PL_bd’ = PL_bd'* c / ((c – v)* l.km/km’) and PL_db’ = PL_db* c / ((c – v)* l.km/km’)

(9b) PL_bd’ = PL_bd/((1 – (v/c))* l.km/km’) and PL_db’ = PL_db/((1 – (v/c))* l.km/km’)

For SR Mode when the signal path has slope = 0 with respect to its direction of motion:

(9c) PL_bd’ = PL_bd* l.km’/km and PL_db’ = PL_db* l.km’/km

For Phi Mode when the signal path has any slope with respect to its direction of motion:

(9d) PL_bd’ = PL_bd /((1 – QU_bd)* l.km/km’) and PL_db’ = PL_db /((1 – QU_db)* l.km/km’)

Throughput (X) Conversion Factor:

(10) X_{b =} X_bd/ l + X_db/ l = X_a / l

Where X_a is Throughput at point 'a'; X_bd and X_db are Throughput components for send and receive components for point 'b' respectively. Throughput is measured as transactions (count).

By applying Little’s Law:

(11) RT_b’ = RT_bd’ + RT_db’ = (PL_bd’ / c) + (PL_db’/ c)

If you find the above mathematics hard to follow then do not try to analyze the RRT Hypothesis or the MM+M Response Time Model output shown in Figure 1 before you have completed the science / fiction stories in Part I and Part II.

 

Chapter 2. Simulation of a Send Transmission Delay.

 

Are you rested and ready for more of this story or is your alter ego asking you “Is this guy kidding me?” or "What planet is this guy from?" or other such remarks that are irrelevant to the debate about Special Relativity?  Well, don’t worry about what your alter ego says.  This story began with a simple innocent question and scientific curiosity about whether SR could be used to measure response time between bodies moving at high speeds.  With this question and the given premise of the Principle of Relativity, the results of the MM Experiment, and a few simple concepts from SR theory, the rest of the story unfolds itself.  All you need to do is just fasten your seat belt, sit back, and relax because we are going to take a fast and exciting journey with two characters I will call A and B through space and time.

 

After I completed my work of creating the RRT Hypothesis and the MM+M Model, I decided to walk up the road to see A and tell him about the RRT Hypothesis and the MM+M Model.  When I saw A with the sun shining brightly behind his head, I said, “good morning A!  How are you?” A said “OK I guess!  What brings you up here so early in the morning?”  I replied, A, I have been thinking about Jasper’s SR story and other SR stories that I have been reading and I have some questions about his assumptions.  In fact, I told him, there are some questions about Jasper’s story that have been bugging me so much that I have had an irresistible urge to get to the bottom of this for whatever reason.  A, I exclaimed,  I would like to show you a new version or vision of the mechanics of light and see what you think!  Then, I commented that it would not have been possible for me to have a different version of the SR story if he had not had enough guts to stick his neck out and tell me the SR story in the first place.  Therefore, I think you should be the first to hear my questions and arguments and I think you may want to be a messenger of a story with a different perspective of SR.  Also, I want to show you some pictures from the Model to see if you agree that there is a bigger and more creative work of art in the nature of light propagation.

 

He then invited me to sit with him in his solarium so I could tell him my Computer Performance Measurement (CPM) version of SR.  We sat down and relaxed for a moment and then I began.  A, imagine that you are a photographer working in a solarium in outer space at point 'a' with the sun and other massive bodies far enough behind you that there are no significant gravitational effects near you or ahead of you.

Then I showed Figure 1 and said:  Imagine that you have a communications satellite that you call delta (D) and it is 150,000 km directly above you at point 'D'.  You setup your camera to take a snapshot of yourself and at the instant the snapshot is taken a spacecraft at point ‘b’ passes beside you at a velocity of 2.59807E+05 km / sec. (same as c * cos 30^o) with a young woman inside whom I will call B.  B also has a communications satellite moving with her that is 150,000 km directly above her at point 'd'.  Let’s assume that your camera is setup to broadcast its snapshots wirelessly in all directions immediately after the snapshot is taken.  Let’s assume that both you and B start your stop watches at the tine the snapshot is taken.  Figure 1 shows where B would be when your snapshot arrived at your satellite when your stopwatch read 0.5 seconds.  Let’s also assume you are stationary with respects to (wrt) about 1 million stars in your neighborhood of the milky way galaxy and you will be called the Observer at point ‘a’ who will study the transmitted snapshot as it travels between points ‘a’, 'D', ‘b’, and ‘d’.     

 

In figure 1, you can see that the yellow photon has moved from you at point 'a' to point 'D' at your satellite and then back to you at point 'a' and it can serve as a photon clock that measures your time.  In figure 1, where point 'D' is 150,000 km from point 'a', your photon clock has measured 0.5000 sec when your yellow photon reaches point 'D'.  You can also see that when B’s blue photon that has moved from her at point 'b' to point 'd' at her satellite and then back to her at point 'b' then the bouncing photon can serve as her photon clock that measures her time.  In the figure 1 where point 'd' is 150,000 km from point 'b', B’s photon clock has measured 0.2500 sec’ when her blue photon is half way to point 'd'.  When B’s satellite is above her and perpendicular to her direction of motion, her stop watch and her photon clock always measure the same time value just as your stop watch and your photon clock always measure the same time value.

 

As we study your transmissions and B’s transmissions, we notice that B’s signal had traveled only half-way to her satellite and her response time is simultaneous to yours in your seconds (sec).  However, when B uses her stop watch to measure transmission time (RT_bd’) she reads 0.2500 sec’, only half of your transmission time (RT_a) measurement of 0.5000 sec because her photon clock’s blue photon has traveled only one-half the distance that your photon clock’s yellow photon traveled.  That is, according to her meter stick she measures the distance traveled by her signal (PL_bd’) as 75,000 km’, only half the distance (PL_aD) or 150,000 km that your signal traveled.

 

Figure 2  shows where B would be when the snapshot first arrived at her satellite at point ‘d’.  At this time the Observer receives his first response (i.e. one end-to-end transmission or one completed transaction) at point 'a' assuming that his satellite did not have any internal processing delay.  That is, his response time is composed of send transmission delays and a receive transmission delays only and his response time can be computed as (4a) RT_a = RT_aD + RT_Da = (PL_aD / c) + (PL_Da / c) = (150000 km / 300000km/sec) + (1500 km / 300000km/sec) = 1.0000 sec.  At this time, 1.0 sec has passed since he started his first transmission and he gets his response back with a response time of 1 second, (sec).  However, B has her transaction only one-half complete in the same second (sec).  Therefore, B measures her response time (RT_bd’) as 0.5 sec’ by her stop watch.  The SR equation T’ = T/(l.sec/sec’) = (1.0 sec/(2 sec/sec’) = 0.5 sec’ and the RRT equation (5) RT_b‘ = RT_bd’ + RT_db’ = (RT_bd/( l.sec/sec’)) + (RT_db/( l.sec/sec’)) = (1.0 sec/(2 sec/sec’) = 0.5 sec’, give the same answers for T’ and RT_bd’ respectively for B’s vertical path to d.  The ratio l = T/T’ = 1.0 sec / 0.5 sec’ = 2 sec/sec’ and RT/ST = 1.0 sec / 0.5 sec = 2, agree with respect to the values but not with respect to the measurement units.  In the M/M/+M Model as well as in SR Theory, T/T’ is the conversion factor l and is constant for a given velocity v regardless of the slope of a path with respect to its direction of motion.  In Phi Mode RT/ST is not a conversion factor, it is only a ratio that will vary with the slope of a path with respect to its direction of motion.  The SR Model and the MM+M Model agree about the values in the results for the vertical arm in the MM Experiment.  The MM+M Model will use a convention that the Observer measures the traveler’s distances in meter’s (e.g. km) and measurements taken or derived by the Traveler will be in meters prime (e.g. km’).  The Observer must multiply his measurement L_o (same as PL_aD) by the conversion factor (.km’/km) in order to derive the path length PL_bd’ that would be seen by the traveler, that is, (6) PL_bd’ = (PL_aD (.km’/km)) = 150,000 km’.  The conversion factor (.km’/km) is derived from the Principle of Relativity premise that the traveler must always see the length of the path from point 'b' to point 'd' as being the same length regardless of travel velocity.  Also, the MM+M Model will use a convention that time measurements taken by the Observer are in seconds (sec) and time measurements taken by the traveler will be in seconds prime (sec’). The Observer must divide his measurement RT_bd by the conversion factor (l.sec/sec’) in order to derive the response time RT_bd’ that would be measured by the traveler, that is, (5) RT_bd’ = RT_bd / (l.sec’/sec) = 1.0 sec /(2.sec’/sec) = 0.5 sec’.

 

A, lets assume that everyone in your company and all of your customers share the same inertial frame except for B, that is, they all share your space-time coordinates.  In this case everyone except B can use the same clocks and measuring sticks.  When B uses her clock or her measuring stick, she actually measures her time in seconds and she measures distance in meters.  Therefore, it is not really necessary to convert to sec' and km' except to denote that B is using a clock that runs slower and a measuring stick that is shorter except when it is pointed perpendicular to her direction of motion.  From the performance measurement perspective, B does not see a slowdown in her response time as she increases her velocity.  However, the Computer Performance Analysts in the Observer’s inertial frame will notice that B’s transmissions are being degraded because it is undergoing some queuing that can be measured as (1b) QT_bd = RT_bd – ST_bd; QT_bd = 1.000 sec – 0.5000 sec; QT_bd = 0.5000 second.  B, however, does not see any queuing so she must use Little’s Law to calculate her send transmission time as (11) RT_bd’= PL_bd’ / c = 150000 km’ / 300000 km’ / sec’ = 0.5000 sec’.  She does not know she is using a different clock and she thinks her send transmission completed in 0.5000 seconds.  The Computer Performance Analyst in your company would be more interested in B's productivity than her response time or the rate at which she aged because her productivity would get the attention of the production manager and he/she would come to the Computer Performance Analyst to get an explanation of why B’s productivity and the productivity of the electro-mechanical platforms that travel with her have dropped.

 

Figure 3 shows B with her satellite configured at point 'd' that is now directly ahead of her in her direction of travel as she passes over your solarium.  In this configuration she will sees the snapshot being taken at the same instant that your camera takes the snapshot.  However, in figure 3, her satellite is 75,000 km directly in front of her by according to the SR story while your satellite is still 150,000 km directly above you.  B’s satellite is now closer to her according to your measurements as the Observer due to length contraction.  As the Observer you see the horizontal signal path contracted from 150,000 km to 75,000 km according to the SR equation L = L₀(1 - (v/c)²)^1/2 = 150,000 km * 0.5 = 75,000 km for the horizontal arm.  An essential characteristic of models used by computer capacity planners is that the models must be able to simulate various “What If” scenarios or changes in systems characteristics, configurations and workloads.  The M/M/+M® Model has the ability to simulate various “What If” scenarios for changes in path lengths due to length contraction.  These changes in path lengths may be observed as well as hypothetical "What If" changes that occur due to changes in velocity or other effects such as gravitational forces.  Therefore, the M/M/+M Model can simulate the SR assumption that L = L₀ (1 - (v/c)²)^1/2.  Also, the M/M/+M Model can apply the Out-of-sync calibration that is necessary in SR Mode modeling.  SR Mode simulations are simulations that use SR Theory assumptions.

 

Figure 3 and Figure 4 are M/M/+M Model SR Mode outputs that represent the horizontal arm in the MM Experiment where the send signal goes in the direction of motion or up the imaginary ether stream.  The horizontal arm (PL_bd) does shrink relative to PL_aD as the velocity of point 'b' and 'd' increases as described in the above discussion on length contraction (see www link above).  Now let’s take a look at the M/M/+M Model’s SR Mode output in figure 4 and compare that output to the SR theory results for the same network configuration.  The SR equation L = L₀ * (1 - (v/c)²)^1/2 = 75,000 km for the horizontal arm yields the same results as the M/M/+M Model in SR Mode, i.e. PL_bd = 75,000 km, as shown in figure 4 due to the SR Mode assumption discussed above.  The M/M/+M Model’s SR Mode response time equation (5) RT_bd’ = RT_bd /(l.sec/sec’) = 1.8660 sec / (2.sec/sec’) = 0.933 sec’ agrees with the SR theory calculation T’ = T(1 - (v/c)²)^1/2 = 1.866 sec * 0.5 sec’/sec = 0.933 sec’ for B’s horizontal path.  However, when B uses Little’s Law to calculate her response time as (11) RT_bd’ = PL_bd’ / c = 150,000 km’/300,000 km’/sec’, she gets the same answer RT_bd’ = 0.5 sec’ as shown in the M/M/+M Model’s SR Mode output in figure 4.

 

The equations in the foregoing paragraph agree with one another; but they do not agree with the SR theory equation T’ = PLbd’ / c = 150,000 km’ / c = 0.5 sec’.  We will call this disagreement the clock synchronization paradox.  However, the Concept of Simultaneity argues that when B has a clock at her satellite in front of her the Observer thinks it runs Out-of-sync with respect to a stop watch that she is wearing and the Out-of-sync Time = (v * PL_aD) / c² =(2.598E+05 km/sec * 150,000 km’) / (300,000 km’/sec’)² = 0.433 sec’ (see out-of-sync for derivation of Out-of-sync Time and its units (sec')).  This Concept of Simultaneity accounts for the difference (0.5 sec’ + 0.433 sec’ = 0.933 sec’).  We now see that the M/M/+M Model running in SR Mode can simulate the “What If” assumption that PL_bd shrinks from length L₀ at rest to length L in motion as described by the SR equation L = L₀ (1 - v/c)²)^1/2 and that B’s send response time component is predicted to degrade from 1.8660 sec to 0.9330 sec’ as predicted by SR theory.  However, B will read the send transmission time as 0.5000 sec’ (0.9330 sec’ – 0.4330 sec’) by her stop watch.  Therefore, equation (5) RT_bd’ = 0.9330 sec’ must be calibrated to 0.5000 sec’ by subtracting 0.433 sec’ of Out-of-sync Time as required by SR theory assumptions.

 

Chapter 3. Simulation of a Receive Transmission Delay.

 

A, let’s now take a look at the return signal from B’s satellite when her satellite is directly above her. Figure 2 shows where B’s satellite was when her blue photon or send signal arrived at her satellite.  This same blue photon or return signal starts at point 'e' as shown in Figure 5.  Assuming that your returning signal is instantly rebroadcasted without any delay when it reaches point ‘a’, then figure 5 shows where B’s transmission will be located when your retransmission reaches point ‘D’.  Figure 5 also shows a red photon.  The red photon is riding on B’s first return transmission signal and it represents your reflection from B’s satellite, as we will see more clearly in Figure 7.

 

Figure 6 shows the configuration when B’s return signal has completed.  This occurred two seconds after the start of the Observer’s first transaction.  The two seconds is equal to the accumulated response time or elapsed time for two completed transactions to the Observer’s satellite (i.e. two end-to-end transmissions of the snapshot).  At this time his response at point 'a' would have completed assuming that his satellite did not have any internal processing delay and his Elapsed Time (ET_a) can be computed as Throughput X_a times average Response Time (RT_a), i.e. (4b) X_a * RT_a = 2 * RT_a, where RT_a is computes as (4a) RT_a = RT_aD + RT_Da = (PL_aD / c) + (PL_Da / c) = (150000 km / 300000km/sec) + (150000 km / 300000km/sec) = 1.0000 sec.  Therefore, the Elapsed Time (ET_a) has been two seconds since the Observer started his first transaction.  According to B, this occurred 1 sec’ after the start of her first send signal.  B has only one transaction (i.e. one end-to-end transmission of the snapshot) completed during the process where the Observer measures that he has completed two transactions.  B measures her response time (RT_bd’) as 1.0 sec’ by her stop watch.  The SR equation T’ = T(1- (v/c)²)^1/2 = 2 sec * 0.5 sec’/sec = 1.0 sec’ and the M/M/+M Model SR Mode equation, (5) RT_b’= (RT_bd/(l.sec/sec’) + (RT_db/( l.sec/sec’) = 2*(1.0 sec/2.sec/sec’) = 1.0 sec’, give the same values for T’ and RT_b’ respectively for B’s vertical send signal to 'd' and return to 'b'.  Also, the ratios L / L₀ = PL_bd / PL_ae = 0.5 are in agreement for the vertical arm of the MM Experiment.

 

Now A, what do you as the Observer think that you and B will see if we imagine that both of you have telescopic eyes and the signals coming from B’s satellite is reflected as from a mirror like that of the MM Experiment?  Can you tell me what you will see in the mirror when it was at point 'f' as shown in Figure 7?  Then A said, “I see the mirror that appears to be at point 'f' 300,000 km from me and I see my image that appears to be 300,000 km behind the mirror”.  Thanks A, now can you see the time that is on your stop watch in the reflection and the time that is on the clock attached to the mirror?  Then A said, “Yes, the clock attached to the mirror reads 1 second that is at the time when B’s satellite was at point 'f' and my stop watch in the image reads 0 sec that is the time I started my first transaction to my satellite two seconds ago.  ” Then A added, “I must also see an image of B’s blue eye at the point image beside my brown eye as it were at time = 0,”  OK, I said, now can you tell me what you think B will see if a mirror is placed at point 'j' like in the MM Experiment as shown in figure 7?  Then A said, “From my perspective as the Observer, I think she will see the mirror 150,000 km’ from her at point 'j' and she will see an image of her blue eye beside my brown eye as they were at time = 0 and these images will appear to be 150,000 km’ behind the mirror and directly above her at a 90^o angle in accordance with the SR assumption of a completely symmetrical arrangement as well as with the physics of mirrors and the Principle of Relativity”.  Thanks for your opinion A, now can you see the time that is on her stop watch in the reflection and on the clock attached to her satellite at the mirror?  Then A said, “Yes, her stop watch in the reflection reads 0 sec’ that was the time of the first send to her satellite and the clock on her satellite at the mirror reads 0.5 sec’ that was the time the first send transmission reached her satellite according to the SR assumption of a completely symmetrical arrangement.”

 

We have now completed our analysis of the send and receive transmissions to B's satellite for the case where it is directly above her and perpendicular to her direction of travel.  In this special case, we see that the results of the SR Model and the MM+M Model's SR Mode appear to agree in all respects from the perspective of  the Observer’s measurements including what is seen in the mirrors as well as with respects to B's end-to-end response time.  The SR Model and the M/M/+M Model's SR Mode simulation of the vertical signal path both give identical results for B's send and receive transmission delays as well as her total end-to-end response time to her satellite that the Observer thinks she sees at 150,000 km' directly above her (at a 90º slope to her direction of motion).   

 

Now, let’s take a look at what happens in the case of a signal returning from the B’s satellite when it is directly in front of her in her direction of motion.  Figure 4 shows where B’s satellite was when her blue photon or send signal arrived at her satellite.  According to SR Theory, this same blue photon starts its return at point 'e' as shown in figure 4.  Figure 8 shows where B and her satellite would be when the Observer’s second transaction ends and his clock’s photon has returned to point 'a' for the second time.  Figure 8 also shows the configuration when B’s return signal has arrived and her photon returns to her at point 'b' for the first time according to SR theory.  This is the configuration two seconds after the start of the Observer’s first transmission.  The two seconds is equal to the accumulated response time (RTa) for two completed transactions (i.e. two complete end-to-end transmissions to his satellite and back to point ‘a’).  At this time the Observer’s second response at point 'a' would have completed assuming that his satellite did not have any internal processing delay and his response time for each end-to-end transaction can be computed as (4a) RT_a = RT_aD + RT_Da = (PL_aD/c) + (PL_Da/c) = (150,000 km / 300,000km/sec) + (150,000 km / 300,000km/sec) = 1.0000 sec.  At this time, two seconds have passed since the Observer started his first transaction and his total accumulated response time or total Elapsed Time (ET_a) is 2 seconds, (sec).  According to B, this occurred 1 sec’ after the start of the first send transmission to her satellite.  B has only one transaction completed in the two seconds that the Observer has two transactions completed.  B measures her total end-to-end response time (RT_b’) as 1.0 sec’ by her stop watch.  The SR equation T’ = T/( l.sec/sec’) = 2 sec/(2.sec/sec’) = 1.0 sec’ and equation (5) applied for SR Mode (5c) RT_b’ = ((RT_bd/( l.sec/sec’))-((v*Lo/c).sec’/sec)) + ((RT_db/( l.sec’/sec)) + ((v*Lo/c).sec’/sec)) = (1.8660 sec/(2.sec’/sec) - 0.433 sec’) + (0.1340 sec / (2.sec’/sec )) + 0.433 sec’) = (0.933 sec’ – 0.433 sec’) + (0.067 sec’ + 0.433 sec’) = 0.5 sec’ + 0.5 sec’ = 1.0 sec’, give the same values for T’ and RT_b’ respectively for B’s send signal to 'd' and return to 'b' that represent end-to-end response time values when the signal transmission path is pointed in its direction of motion.

 

If the simulation of light stories were ended here an M/M/+M Model critic might say: “So far you have not shown that the M/M/+M Model makes any significant improvement in anybody's range of vision or understanding of light mechanics beyond that of SR theory.  You have only raised some superfluous questions about which measurements should be expressed in prime units and it is clear that the Concept of Simultaneity is beyond your ken”.  The critic may conclude that this is simply a “science” story about the M/M/+M Model in SR Mode that reliably gives the same answers that the SR Model gives.  Others may tacitly agree with this criticism if they overlook the fact that the M/M/+M Model in SR Mode will simulate the behavior of the light transmission at any angle of the MM Experiment's arm with respect to the direction of motion and if they overlook the differences in the mechanics of the SR Model and the M/M/+M Model in SR Mode.  These differences mean that both models cannot be scientifically valid despite the fact that both are reliable; just as both the Copernican Model of a Sun-Centered solar system and the Earth-Centered model of the universe supported by Aristotle and Ptolemy are not both scientifically valid despite the fact that Ptolemy's Earth-Centered model was a more reliable predictor of some planets' location than the original Sun-Centered model.  Also, there are several other reasons why many, who can see with their own eyes, will not agree with this criticism.  One reason is that the M/M/+M Model is user friendly and can be used to do-the-math for you and get the same results that the three SR equations would get for the MM Experiment.  A more compelling reason is that SR Theory does not support the physics of mirrors and has no logical explanation to account for the physics of moving mirrors.  Finally, SR Theory has not been able to accurately account for starlight aberration that is known to be a relativistic phenomenon caused by the velocity of the earth in orbit around the sun.  This is evident because the NOVAS software developed by the U. S. Naval Observatory that is widely used by astronomers for precise predictions of star locations does not use the 1905 aberration equation that is based upon SR Theory.  NOVAS uses a more precise aberration equation that will be discussed in “Relativistic Stellar Aberration & the Bearing estimator (Best) Equation” to be posted at this web site.  SR’s problems with accounting for reflections and aberrations are enough to shake one’s belief that SR is a science story.  The most compelling reason is that the M/M/+M Model will produce significantly different results than the SR Model when executed in Phi Mode and these Phi Mode results are more consistent with the Principle of Relativity as well as the Physics of Mirrors and is more accurate than the 1905 aberration equation for predicting stellar aberration.  Phi Mode can be used to reliably predict starlight or signal aberration due to the velocity of an observer.  For these compelling reasons, I invite you to continue this check-out of SR theory and tell me what you can see in the reflections from B’s satellite in figure 9a.

 

A, could you as the Observer tell me what you think B will see with her telescopic blue eye if a mirror is attached to her satellite that is now in front of her as it was for the horizontal arm in the MM Experiment and Figure 8 and Figure 9a?  Then A made the following statements: “B sees the returning photon that reflected from point 'e' shown in figures 8 and 9a.  According to SR theory, this photon was the blue photon that B will see after it moves from the mirror’s reflecting surface at point 'e' as shown in figure 9a.  Point 'e' is the point where B’s satellite was when she was at point 'i'.  After the returning photon is reflected from the mirror, it is seen by B’s blue eye at point 'b'.  At this point B sees her reflection that appears to be 75,000 km’ behind the mirror and the mirror appears to be 75,000 km’ from her.  Also, when she adds the distance from point 'b' to the mirror to the distance from the mirror to her satellite at point 'd' she will calculate that the distance from point 'b' to point 'd' that is 150,000 km’.

 

Thanks A. Now, do you remember when you were analyzing what B saw in Figure 7 when the mirror appeared directly over her head at a 90^o angle to her direction of motion from your perspective?  In this analysis you concluded that B sees a mirror attached to her satellite and that the satellite as well as the mirror was 150,000 km’ above her in accordance with the Principle of Relativity.  Therefore, how can she see her satellite with the attached mirror at only 75,000 km’ in front of her and detached from the actual location of her satellite 150,000 km that is in front of her as shown in figure 9a?  Since she must see the mirror 150,000 km’ in front of her and she must see her reflection 150,000 km’ behind the mirror, then she must see the image of her blue eye beside an image of your brown eye as they were at time = 0 or 1 second in the past.  This means that she must see her point image directly in front of her on the solid blue circle that is the moving front of the original transmission from ‘a’ at time = 0 as she would if she were not moving.  Doesn’t figure 9a violate the Principle of Relativity?  Furthermore, B will see the returning photon travel 75,000 km’ from point 'f' to point 'b' in about 0.067 sec’ (1.0 sec’ – 0.933 sec’) and she would measure its speed at about 1,119,403 km’ / sec’ or more than three times the speed of light.  Since the Principle of Relativity is a premise we should determine what B would see in the mirror when it is at the proper distance from point 'b'.  Now A, what do you think B will see in Figure 9b?

 

A looked at figure 9b and said: “If we assume a wave front according to SR theory that has the shape of an ellipsoid of revolution about an x’ axis = (c * t)/l = 300,000 km in length with a radius = 150,000 km centered on point ‘b’ at x ˜ 519,615 km, then B would see her satellite with the attached mirror at point 'j' instead of at point 'e'.  In this case, she will measure the distance traveled from the mirror to ‘b’ as 150,000 km’ instead of the 75,000 km to point ‘d’ at x ˜ 594,615 km.  She would see her image beside A’s image at 150,000 km’ behind the mirror instead of 75,000 km behind the mirror at x ˜ 594,615 + 75,000 km ˜ 669,615 km where the ellipsoid wave front ends at t = 2 seconds at the moment her photon completes its return trip.”  This would be a plausible explanation only if you believe that the ellipsoid wave front started at x = 0, t = 0 and ended at x ˜ 669,615 km at t = 2 seconds approximately 69,615 km ahead of the spherical wave front originating from point a at t = 0 and ending at x = 600,000 km at t = 2 seconds.  B’s image must appear to occur on the expanding wave front of the original transmission from ‘a’ as it would occur according to the physics of mirrors laws if she were not moving and as it did occur in figure 7; otherwise, it will violate the speed limit c according to the Observer’s measurements.  Therefore, doesn’t figure 9b violate either the Physics of Mirrors laws and the Principle of Relativity or the speed limit c law?  Let’s review the “Physics of Stationary Mirrors” and see if we can develop a “Physics of Moving Mirrors” hypothesis to resolve these issues of whether SR violates the Principle of Relativity with respect to what is seen in mirrors from the viewpoints of travelers and observers.

 

In the case of the vertical path shown in figure 7, the solid blue upstream send signal is 150,000 km directly behind the mirror as its reflected blue photon is 150,000 km directly in front of the mirror.  Therefore, figure 7 does not violate the physics of mirrors laws.  Let’s look at Figure 7a and review the “Physics of Stationary Mirrors” laws to determine what these laws are and what constitutes a violation of these laws.  Figure 7a shows a configuration with the Observer’s telescopic brown eye, A, at x=0 km, y=200,000 km, z=0.01 km and shows B’s telescopic blue eye at x=o km, y=200,000 km, z=0 km (both shown as a gray eye) at point ‘a’.  The blue eye has a blue photon at its center and the brown eye has a red photon at its center (shown as a purple photon).  While at rest, the Observer turns on a red light and B turns on a blue light both at the same time t = 0 sec.  After traveling for 0.5 sec these two lights reached point 'D' where A’s satellite at z=0.01 km and B’s satellite at z=0 km are side by side at x-y coordinates x=0 km, y=350,000 km.  A’s red light is reflected back with red photons from a red mirror attached to ‘b’ at stationary point ‘D’.  B’s blue light is reflected back from a blue mirror attached to ‘b’ with blue photons.  These two mirrors are side-by-side and are shown as a purple mirror.  These respective reflected photon’s then strike the eyes of their respective “owner” at time t = 1.0 sec. as shown in figure 7a.  Let’s take due consideration of the law that the boundary of the original signal sent by B at t=0 as well as by A, must advance from t=0 at a velocity c to their point image in the same elapsed time taken for the signal to reach their respective satellites at ‘D’ and ‘d’ and then return to the sender.  That is, the boundary of the Observer’s initial red send signal from ‘a’ passes ‘d’ where it is reflected back to his brown eye but also must continue on to reach his image of point 'a' at the same elapsed time that his reflection reaches his eye at point ‘a’.  Also, B’s initial blue send signal from ‘b’ passes ‘d’ where it is reflected back to her blue eye but also must continue on to reach her point ‘b’ image at the same elapsed time that her reflection reaches her eye at ‘b’.

 

In figure 7, the point image of B’s blue eye always appears to be at the opposite edge of the gray region that spreads out from the center point at ‘d’ with the attached mirror.  The gray region is the cross-section of a bi-convex lens shaped wave front of the return transmission that is made up by joining two spherical sections.  The cross-section looks like two circular sections joined at a perpendicular bisector through the center of the shape, similar to the Greek letter Phi (f), in the Observer’s coordinates.  The behavior of this bi-convex lens shape is analogous to the gray circle of the reflection that spreads out from the center point at ‘d’ with its attached mirrors when points ‘b’ and ‘d’ are at rest as described in figure 7a.  However, the manifestation of a moving light zone or field is compressed according to the Observer and appears to have the bi-convex lens shape instead of a spherical or circular shape.  This point is illustrated more clearly in Figure 7b where the red-light zone of the Downstream Send Reflection and the blue/green-light zone of the Upstream Send Reflection overlap to form the gray zone that looks like the Greek letter Phi (f), therefore, we will call this gray area a Phi-light Zone.  The Phi-light zone is the embodiment of a new “Physics of Moving Mirrors” hypothesis and this zone will be centered on a remote reflection point such as point ‘d’ that is remote from the Traveler at point ‘b’.  The height of the Phi-light zone (perpendicular to the direction of motion) is always equal to 2Lo and its width (in the direction of motion) is always 2L when the leading edge of the Phi-light zone reaches point ‘b’.  Figure 7b shows two Phi-light Zones (gray areas).  The gray Phi-light Zone at the right represents the traveler’s Phi-light Zone according to the observer at 'a'.  The Observer’s telescopic brown eye at ‘a’ would see the mirror attached to B’s satellite at point ‘e’ at ETa = (l * 2 * Lo)/c = 2 sec at which time his telescopic brown eye sees his image beside an image of B in the mirror attached to point ‘d’ and these two images appear to be at point ‘n’ 300,000 km behind B’s mirror at point ‘e’.  However, B does not think her Phi-light Zone is flattened to a cross-section that looks like the cross-section of a biconvex lens.  She thinks her Light Zone looks like a sphere with a radius of length Lo.  Therefore, the perimeter the wave front from her satellite is described by the very faint gold circle (cross-section of her spherical wave front) with a center at point ‘d’ in figure 7b.  The faint gold Light Zone is the leading edge of the reflection from point ‘d’ at the time it reaches her at point ‘b’; that is, it is the return from ‘d’ to ‘b’.  The gray Phi-light Zone at the left would be the Phi-light Zone from the Observer’s satellite according to the SR assumption that B would see a Phi-light Zone centered at ‘D’ that is reciprocal and completely symmetrical to the one seen by the Observer centered at 'd' as a consequence of the theory that there can be no preferred inertial frame.  However, in a three part video series (Video_4 , Video_5 , and Video_6 ) at the "M/M/+M® Model, Version 1, Tutorial" page  the Traveler’s view from the center of her “Bi-convex Inertial Frame” (BIF) will show that her observations are reciprocal but not completely symmetrical.  The BIF also has the shape of a bi-convex lens but it will be referred to as the Traveler’s BIF instead of as a Phi-light Zone in order to distinguish it as being emitted from point ‘b’ instead of from a remote point in her inertial frame.

 

In figure 7b, point ‘b’ is at the perimeter of the gray Phi-Light Zone that is centered at point ‘D’.  According to the SR assumption of a completely symmetrical arrangement, B's telescopic blue eye at ‘b’ would see the green mirror attached to an image of your satellite at point ‘e’ at T’ = T/l = 1 sec’ at which time her telescopic blue eye could also see A’s image in the green mirror beside her image and these two images appear to be 300,000 km behind the green mirror.  However, it will be shown that B would see the first reflection from point ‘D’ at her time t=1 sec' coming from A's Database centered within a Phi-Light zone that is larger than the gray Phi-Light zone shown in figure 7b.  Video_2 will describe what the Traveler will see in the case where the Traveler’s satellite is traveling above her at an angle ˜ 90º wrt her direction of motion and will also address two major issues concerning SR assumptions about what the Traveler will see as follows.  The Traveler will see the Observer's SIF as a Phi-light zone that will be called the Observer's BIF.  The Observer's BIF will be described in the three part video series (Video_4 through Video_6).   

 

First, is the SR assumption that B’s telescopic blue eye sees an "image" of her mirror attached to her satellite at point ‘e’ as a “natural consequence” of a completely symmetrical arrangement because there cannot be preferred inertial frame.  This questionable SR assumption has lead to the unfortunate natural consequence that many physicists refer to as the “twin’s paradox”.  That is, the consequence that “a completely symmetrical” situation dictates that the traveler as well as the Observer both think that the other ages slower.  Perhaps the most convincing argument against Einstein's relativity argument about symmetry is the argument that SR's predictions of stellar aberration are in error.  According to an article on Stellar Aberration and Einstein's Relativity by Paul Marmet, and others (see references 2 thru 5 at the Marmet link) observations of stellar aberration do not agree with Einstein's argument of a completely symmetrical arrangement.  Consequently, we are forced to conclude that Einstein's principle of complete symmetry is in error and the accepted system of coordinates with respect to the solar system’s barycenter (center of solar system’s mass) should be considered to be a preferred or higher order system of coordinates and not a completely symmetric system of coordinates.  Also, the predictions of starlight aberration based upon the SR (1905) aberration equation do not agree with "Bradley Constant of Aberration" at velocities greater than the Earth’s orbital velocity and do not agree with more recently developed aberration equations that have been found to be more precise than the SR (1905) aberration equation by the U. S. Naval Observatory.

Second, is the assumption that B thinks the Observer’s clock runs slower than her own clock because there is no preferred inertial frame!  Is it not possible to imagine a scenario where B realizes that the clock at 'a' runs faster than her clock?  “What If” B is a 3^rd millennium woman with an M/M/+M® model and it tells her that she is the Traveler when she goes from point 'a' to point 'e' in outer space just as she is aware that she is the Traveler when she goes from New York to Los Angeles in an airplane?  “What If” she will not think that a “stationary” Observer at point ‘a’ is the Traveler but instead she measures 'a' to be a distance D_{a’ b} = RT_b' * v = (0.5 sec’ * 259,807.6211 km’/sec’) = 129,903.8106 km’ behind her when her send signal completes and will have been "dragged" to point 'g' in figure 7b that is a distance of 1 sec’ * 259,807.6211 km’/sec’ = 259,807.6211 km’ behind her when the receive signal returns to her, just as one would expect; instead of 519,615.2423 km as predicted by SR theory?  "What If" when she receives the signal from A's satellite she will think that his satellite has been "dragged" to point 'e' in figure 7b that is directly above point 'g' (the new location of point 'a'); then wouldn't she measure the distance to the new location of point ‘D’ as 300,000 km’, as one would expect; rather than the 600,000 km to point ‘D’ on the Observer’s y-axis?  “What If” she is not as a child who thinks it’s the trees that are moving instead of the car and “What If” she knows that she is moving even though point 'a' does seem to be moving away from her at velocity v' = 259,807.6211 km’/sec’?  “What If” she does not think that her clock has slowed but she knows that the clock at 'a' has speed up even though her measurements indicate that the clock at point ‘a’ is running slower than her own clock?  "What If" she is also aware that this does not change her measurement of her velocity or the measurement of her velocity by the Observer because her measurements indicate that point 'a' and 'b' are a distance D_a’b = 129,903.8106 km’ apart when her send signal completes while the measurements of the Observer indicate that 'a' and 'b' are D_ab = RT_a * v = (1.0 sec * 259,807.6211 km/sec) = 259,807.6211 km apart when her send signal completes?  “What If” her measurements do not indicate that her meter stick has contracted but her measurements indicate that 'a' has a meter stick that has contracted and that he must measures the distance 'a' to 'b' as twice the distance she measures?  "What If" she knows that she can use her meter stick to measure the dimensions of her space craft and the distance to her satellite as well as to measure the distance she has traveled from point 'a' to point 'b'?  Isn’t it conceivable that 3^rd millennium Homo sapiens with advanced modeling tools will be able to calculate their age as well as the ages of travelers in other inertial frames as long as their velocities v relative to a common or "absolute" starting point is known?  Also, wouldn’t you expect that all of their calculations about their velocities v, v’ and c would agree?  “What If” twins at point ‘a’ in a remote area of deep space synchronize their clocks and then one twin travels to a distant planet and returns to point ‘a’; then will both twins agree that the traveling twin’s clock has lost time?  Isn’t it reasonable to expect that both twins agree that the “traveling” twin’s clock has lost time simply because he moved some significant distance away from home and then back home at a significantly greater average speed wrt home than the “stay-at-home” twin has moved and not because he accelerated twice and reverse accelerated twice?  In a simulation of the twins "What If" scenario, there would be an absolute inertial starting point that is the same as the point of return with respects to both inertial frames but there would not be a completely symmetrical arrangement, the "stay at home" twin could think the traveling twin has "dragged" the stationary frame behind him on the trip away from home and reversed the "frame dragging" on the return to home trip.  Therefore, in a simulation of this "What If" scenario, the Twin’s Paradox would not be a natural consequence of a completely symmetrical concept of relativity.  Instead, this SR concept of symmetry would not be considered a necessity of thought and as a consequence it would not continue to achieve such authority over us that we forget its earthly origins.  The "What If" scenario described in this paragraph will be simulated by the M/M/+M® Model in Video_2.

 

In addition to SR's stellar aberration and twin’s paradox problems let’s call the questions about the spooky mirror illustrated in figure 9a the spooky-mirror paradox because the physics of mirror’s laws are not obeyed in SR theory as described in the example of the mirror that should appear to be attached at a distance of 150,000 km’ at the end of the MM Experiment’s horizontal arm.  Finally, if B sees the photon as shown in figure 9b then its image must travel 150,000 km’ from point 'd' to her point image in 0.067 sec’ or at more than seven times the speed of light in order to obey the “Physics of Moving Mirrors” laws.  Let’s call the questions about the speed of the photon returning from point ‘e’ the speeding-photon paradox.

 

A then exclaimed: “I can agree that these paradoxes and questions do warrant some explanations!”  Then I replied, yes, I agree and I would like to see you again soon to discuss the M/M/+M Phi Mode model that is a slightly modified SR Mode model that can resolve these paradoxes and questions about Special Relativity.  Very well, A replied, your model interests me; you can come back tomorrow afternoon if you would like and I will give you my undivided attention for an hour or two.

The discussion of the Phi Mode model can be accessed at the link below:

PART II: Simulation of Signal Transmission in Phi Mode