Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six preeding millennium works are applied in the investigation of the relativisti behavior of ompound speeds involving both, the instantaneous speeds assoiated with onstant aeleration and the onstant speeds assoiated with uniform motion. he resulting findings of the investigation resolve the ontradition between the two opposing theories, millennium and speial relativity, on veloity omposition. In the proess, the validity of the millennium formula is onfirmed, new formulas are derived for the ompounded speeds of aeleration, and the speial relativity formula is shown to be inorret exept in speial ases involving aeleration and not uniform motion as stipulated in Einstein s theory.
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk. Introdution he purpose of this paper is to present the results of a study that examines the relationships between the priniples of onstant aeleration and the priniples of uniform motion. he researh involved the use of mathematial models employing formulas introdued in previous millennium relativity papers in addition to new formulas derived from them as needed. he study was made possible as a diret result of the distane formulas for stationary frame distanes traveled during onstant aeleration, developed in the previous most reent work, Relativisti Motion Perspetive. hese formulas together with other millennium formulas make it possible to orrelate the relationships of omposite onstant aeleration rates with those relating to the assoiated uniform motion. Sine all of these formulas were derived based on independent analyses, suh orrelation serves as a ross hek to verify the validity of these formulas and others used throughout the theory. Subsequently, it beomes possible to determine the validity of the millennium relativity veloity omposition formula and Einstein s veloity omposition formula. Suh theoretial distintion between the two different formulas is important beause the two formulas produe onfliting results and, as was pointed out in the millennium theory on veloity omposition, experimental proof may be very diffiult to obtain 3. In addition to learing up this issue and showing onlusively that the millennium formula is orret, new formulas are developed for aeleration involving the relationships of ompound instantaneous speeds, inluding formulas for aeleration omposition. It is then found that Einstein s formula for veloity omposition atually involves the instantaneous speeds resulting from onstant aeleration and not uniform motion speeds as stated in speial relativity. Furthermore, it is shown that even then, the formula produes orret results only under ertain speial onditions.. he Fundamental Relationships between Aeleration and niform Motion Illustrated in Figure are some of the important relationships between aeleration and uniform motion that will be exploited in this present work. In referring to Figure and the priniples to be presented throughout this work, it should be understood that all referenes to uniform motion are intended in the strit sense of that assoiated with inertial frames of referene. hat is, the stationary frame (SF) and all uniform motion frames (Fs) are understood to be inertial frames that are not involve in any form of aeleration. Additionally, all referenes to aeleration, unless otherwise speified, are intended as referenes to aeleration at a onstant rate. Referring now to Figure, the urved line designated A represents the rise in speed relative to a stationary frame of referene over SF time interval resulting from a onstant rate of aeleration, A., then represents the final instantaneous speed ahieved, and a, the SF distane traveled during that SF time interval. his same distane an also be represented as a uniform motion distane u, traveled at the uniform rate of speed, during the same SF time
interval. Sine it normally requires a higher attained speed to travel the same distane under aeleration that an be traveled at a uniform rate of speed for any given time interval, speed will normally be less than speed. (he qualifier, normally, is used here beause aording to the millennium theory, as approahes the speed of light,, approahes, and equals if speed is ahieved.) he final important onsideration onerning the relationships given in Figure, involve the dual nature of speed. Whereas speed is the instantaneous speed resulting from aeleration rate A, it also represents the speed of the uniform motion frame transitioned to during the period of aeleration. Sine this same speed, when viewed as a uniform motion speed, results in a greater distane traveled in the stationary frame during interval, a different designator, v is needed to distinguish this uniform motion distane from the distane ahieved by the aeleration. Although it may not be immediately obvious, when dealing with speeds and the resultant distanes traveled during a time interval, we are onfronted with a ondition that leads to great onfusion. For those who might argue otherwise, it will be shown later in this paper that even Einstein fell vitim to this onfusion. he basis for the onfusion involves the fat that with regard to aeleration and uniform motion, every distane an be defined by two different speeds, and onversely every speed defines two different distanes. his is shown in Figure, where the same distane, a, defined by speed, an also be defined by speed and is subsequently also shown as distane u. Conversely, Speed not only defines distane a, but also distane v. Even in Figure, this dual harateristi of speed is not limited to, but also applies to speed. Although not show in the Figure, this same speed an also represent the instantaneous speed of another rate of aeleration over time interval, and would then result in a shorter distane than u. And if that is not already onfusing enough, it should also be noted that speed has the same relationship with some higher rate of speed, based on some higher rate of aeleration, that speed has with speed relative to aeleration rate A. Obviously, we will not onern ourselves with issues suh as these if they are not relevant to the investigation. Even so, onsidering only the relationships shown in Figure with regard to a single rate of aeleration, one is onfronted with inreasing diffiulty of understanding as other rates of aeleration with similar relationships are introdued into the investigation. Proeeding with aution then, let us now refer to Figure. 3. he Speial Relationships between wo ifferent Rates of Constant Aeleration In Figure, the relationships shown in Figure are now applied additionally to a seond higher rate of aeleration, A that ours simultaneously with aeleration rate A during the same SF time interval. In this ase, the urved line designated A represents the rise in speed relative to the stationary frame of referene during whih speed is ahieved and the resulting SF distane, a traveled during SF time interval. u, then represents the same distane traveled at the uniform rate of speed,, while v is the distane traveled at the uniform rate of speed, during the same time interval. What is perhaps not immediately obvious in the illustration is the fat that in showing a uniform motion speed equal to instantaneous speed the illustration represents a speial ase. hat is, where aeleration rate A results in an ahieved speed with an assoiated uniform motion speed that has the same value as speed and therefore results in distanes, a and u respetively that equal distane, v is not usually to be expeted. With some thought, it should beome apparent that for any given rate of aeleration, A, there is only one rate of aeleration A that an give this result. hat is, if the rate of aeleration A is redued, the values of, a, and u will also be redued and, if 3
aeleration rate A was held at its original value, speed will then be less than speed, and distanes, a and u, though still equal to eah other, will be less than distane v. Conversely, if under the same onditions, the rate of aeleration, A is inreased, speed will be greater than speed, and distanes, a and u, though still equal to eah other, will be greater than distane, v. he speial ondition that leads to the results shown in Figure will be revealed in the next setion and overed in detail, along with two others, as the analysis progresses. 4. he Composite Relationships Involving the Speial Case In Figure 3, a third aeleration rate A is introdued. nlike the previous two, however, the new aeleration rate and its assoiated speeds,, u, and, are referened to the uniform motion frame defined by speed,, and not the stationary frame. his is the frame enlosed by the heavy dashed lines with the speed range indiated outside the upper right orner of the graph. Although this rendition is orret, it is a bit awkward for our purposes of making diret omparisons between aeleration related values of the different frames. hus, for onveniene purposes and to make the analysis easier to follow, the origin of this frame is shown in two different loations in Figure 4. Whereas it remains at its original loation with regard to uniform motion speed, it has been displaed to the left by a distane equal to a with regard to aeleration related values,, u, and. Referring now to Figure 4, sine the respetive SF aeleration related distanes, a and u are equal to the differene between distanes a and a, this third aeleration omponent interrelates all of the values assoiated with the two respetive aeleration rates A and A. Also of interest in the illustration is the distane v. his distane, that is equal to the differene between distanes v and v, is the distane traveled at a uniform rate of speed u during SF time interval. As an be seen in the proportionally onstruted graph, speed u is a lower speed than the instantaneous speed resulting from aeleration rate, A. Aordingly, the distane traveled at a uniform rate of speed equal to during SF interval, indiated by the heavy lined arrow in frame,, is a greater distane than distane v. In other words, whereas distanes v and v are the result of uniform motion speeds equal to the instantaneous speeds resulting from aeleration rates A and A respetively, distane v is the result of a uniform motion speed that is less than the instantaneous speed resulting from aeleration rate A. he final point of interest in Figure 4 is the fat that distanes a and u equal distane v. As will be shown during the analysis, the speial ase where these distanes are equal ours only when instantaneous speeds and are equal. It will also be shown that it is only when suh speial onditions are present (there are two others that have yet to be disussed) that Einstein s veloity omposition formula gives the orret result. Even more damaging to Einstein s theory is the fat that it will be additionally shown that even then, when one of three speial onditions is present, his formula deals with the instantaneous speeds of aeleration and not uniform motion speeds as stipulated in his theory. And finally, it will be shown that the millennium relativity formula for veloity omposition is the orret formula for uniform motion speeds as stipulated in the millennium relativity veloity omposition paper. Let us now introdue and derive as neessary the millennium formulas that were used in arriving at these onlusions. 5. he Aeleration Formulas Referring to Figure 5, as stated previously, the urved lines designated A and A represent the rise in speeds resulting from two different onstant rates of aeleration also 4
designated A and A respetively. Sine the time interval during whih the aelerations our is given, and sine both rates of aeleration are also given, and sine the respetive rises in the instantaneous speeds, and are plotted against the same stationary frame of referene, there an be little basis for onfusion or doubt as to the validity of this aspet of the analysis. he only thing one might question is the validity of the formulas used to plot the urves, and more importantly, arrive at the resulting instantaneous speeds, and, and the respetive distanes, a and a traveled in the stationary frame during the given time interval. Sine both sets of formulas, A () Instantaneous Speed, ( A ) and, A () Instantaneous Speed, ( A ) for speeds and, and, a (3) Aeleration istane for Speed and, a (4) Aeleration istane for Speed for the respetive distanes traveled in the stationary frame are diretly derived from formulas introdue in previous millennium papers 4 & 5, where they were shown to be in agreement with the evidene, it is strongly argued that the validity of these formulas has already been established beyond reasonable doubt. With that said, another previously derived formula 6 that will be used in the analysis, gives a, and a in terms of aeleration rates A, and A respetively, instead of the assoiated speeds, and, as shown below: a A (5) Aeleration istane for Aeleration Rate A ( A ) A a (6) Aeleration istane for Aeleration Rate A ( A ) o advane to the next step in the analysis, we need to onsider the redued time rate in the inertial frame transitioned to via aeleration rate, A. We an refer to this moving frame as 5
frame, sine its speed relative to the stationary frame is the same as the instantaneous speed ahieved by aeleration rate A during stationary frame time interval. sing the millennium relativity time transformation formula 7, (7) ime ransformation for Moving Frame, t v we an determine the time interval, t v for inertial frame that transpires during stationary frame time interval. hen, by using this shorter interval in a third version of the two previously given distane formulas 5 & 6, we an also define distane, a in Figure 5 in two different ways. his gives us, a t v (8) Aeleration istane for Speed, and, a tv ( A tv A (9) Aeleration istane for Aeleration Rate A ) for stationary frame distane, a. Modifiation of the original formulas in this manner is justified by the following rationale: Sine F, time interval, t v is used, the distanes are valid in F,. In aordane with the millennium theory, however, real distanes are unaffeted by time variane, thus these distanes, though they may be pereived differently beause of the differene in time, are also valid in the stationary frame. Moreover, when these two formulas are tested against other methods for finding distane a, the results are found to be idential. Again, using the time transformation formula, we an also define the time interval that must be used in the modifiation of yet another equation from a previous millennium paper 8, needed to determine the onstant rate of aeleration, A shown in Figure 5. his gives us, A (0) Constant Aeleration Rate for Ahieving Speed, tv for aeleration rate, A relative to inertial frame. o determine the rate of aeleration relative to inertial frame, we need to use another equation from a previous millennium paper 9, the formula for relative aeleration. hus, we have, A r A () Relative Aeleration Rate for Ahieving Speed, where A r is the relative rate of aeleration relative to inertial frame,. At this point, perhaps some larifiation is in order. First, let us define an instantaneous frame of referene as being the uniform motion frame that an aelerating objet is urrently in at any given instant in time. 6
We an then say that it is in regard to this frame of referene that the onstant rate of aeleration is a valid value. We an alternately refer to this frame as the aeleration frame. herefore, with regard to aeleration between two inertial referene frames, whereas the speed and time interval are given in terms of the frame that is being aelerated away from, the onstant rate of aeleration is given in terms of the aeleration frame. his means that in the ase of the stationary frame and inertial frame, the values of speed and time interval are given in terms that are valid in the stationary frame, but the value for the onstant rate of aeleration, A is a valid value in the stationary frame only at the start of the aeleration. At the end of the aeleration, it is a valid value in the visited frame, and not the stationary frame. uring the aeleration therefore, the valid value for the rate of aeleration in the stationary frame, is a relative value, A r that initially equals A at the start of aeleration, but then redues in value as the speed rises. Aordingly, in formula (0) above, speed and time interval t v are valid values in referene frame, and the value of the onstant aeleration rate, A, is valid in frame, at the end of the aeleration and not frame. In frame, it is the relative rate of aeleration, A r, given by formula () that is the valid value. hese relationships will be disussed in more detail as the analysis progresses. What is important to know for now is that the latter four formulas, (8, 9, 0, & ) along with all of the aeleration distanes traveled in the stationary frame provide us with one of two different methods that will be used for testing the results of the two different veloity omposition formulas. he seond method involves the uniform motion distanes traveled in the stationary frame during the same time interval at uniform rates of speed,,, and u. hese distanes are respetively designated v, v, and v in Figure 5. 6. he niform Motion Formulas Sine distanes v and v are based on speeds diretly related to the stationary frame, the formulas that define these distanes are quite simple and straightforward. Where and are uniform rates of speed relative to the stationary frame of referene and is the stationary frame time interval, the distanes traveled in the stationary frame are given by the formulas, and, v () niform Motion istane for Speed v (3) niform Motion istane for Speed respetively. Although the formula for distane v is also quite simple, the rationale behind it is a bit more ompliated. Our initial assumption might be that it is simply the produt of the uniform motion speed and the moving frame interval t v that ours simultaneously with stationary frame interval. hat, however, is not the ase, and fortunately very easy to prove. Sine speed is in fat referened to moving frame,, it is a orret assumption to assoiate it with moving frame time interval, t v. he produt, t v, however, does not define distane v. Consistent with the postulate that the laws of physis must be the same in all referene frames, the relationship of this uniform motion distane to moving frame distane a, must be the same as the relationship of uniform motion distane v (the produt, ) to stationary frame distane, a. he distane, t v is shown at the top of the graph in Figure 5. his is the distane indiated by the thik-lined arrow that extends horizontally to the right of the vertial 7
dashed line that defines the origin of moving frame. It an be seen quite readily in the proportionally onstruted graph that this distane is onsiderably longer than distane v. Quite simply, if u is a uniform motion speed based on moving frame time interval t v, it must be a lower speed than and also satisfy the equation, u v v. (4) Equation for efining Speed, u t v Moreover, sine by definition, v (5) Relationship of v to v, and v v v and therefore, (6) Relationship of v to v, and v v v v equation (4) an be restated as, v u (7) Formula for Moving Frame Speed, u tv resulting in, t (8) Formula for Stationary Frame istane, v v u v as a proper definition for distane v. his same distane, v, an also be defined as, v v u (9) Alternate Formula for istane, v where v u is the speed observed in the stationary frame. ( u is the speed observed in uniform motion frame,.) Given this relationship where speed rate is inversely proportional to time rate, we an state by way of equality that, t v u v u and derive the following relationships: (0) Equality of Speed and ime Ratios for u and v u v u u or, tv u vu () niform Motion Speed Relative to F, t v u tv vu or, tv vu u () niform Motion Speed Relative to SF 8
When viewing speeds, u, and as depited on the graph in Figure 5, it should be understood that the magnitudes of these speeds are shown in relation to inertial frame and not the stationary frame, as is the ase for all of the other speeds. 7. Methods of Problem Solving Given all of the relationships depited in Figure 5 there are a number of methods and many variations of those methods available for problem solving depending on whih variables are known and whih are yet to be determined. he variables referred to fall into eight main groups as shown below: ariable Groups. Aeleration Rates, A, A, and A. Instantaneous Speeds,,, and 3. Aeleration Related Speeds,,, and 4. niform Motion Speeds,,, and u 5. Aeleration istanes, a, a, and a 6. Aeleration Related istanes, u, u, and u 7. niform Motion istanes, v, v and v 8. SF ime Interval,, and F,, ime Interval, t v. If we know the SF time interval from group 8, and any two of the variables in one of the seven remaining groups, we should be able to solve for all of the other variables in all of the groups. On the other hand, if we know the F time interval t v, and not the SF time interval, we are more limited in whih other variables will be needed to solve for all of the variables. In either ase, it should be obvious that there are too many variations of problems resulting from whih variables are known and whih are not know to over in a reasonable length paper suh as intended here. Moreover, sine any suessful solution that gives the values of all of the variables is adequate for our purposes of validating the priniples of the millennium theory, there is really only a need to provide one suh solution. hat fat not withstanding, in the interest of thoroughness, the solution given will be overed in onsiderable detail. 8. Problem Solving Method for the Case where, A, and A are known Given, speed of light, SF time interval, and aeleration rates, A, and A, we an use formulas ( & ) to find instantaneous speeds, and. A () Instantaneous Speed, ( A ) 9
0 ( ) A A () Instantaneous Speed, Having found, and, we an then use formulas (3, 4, and 7) to solve for distanes, a, a, and F, time interval, t v. a (3) Aeleration istane for Speed a (4) Aeleration istane for Speed t v (7) ime ransformation for Moving Frame, We an then take formula (8) for distane, a, t v a (8) Aeleration istane for Speed, and solve it for to obtain, v a v a t t (3) Instantaneous Speed, And sine, a a a (4) Aeleration istane, a by way of substitution we arrive at, ) ( ) ( v a a v a a t t (5) Instantaneous Speed, where upon by further substitution using the right sides of formulas (7, 3 & 4) t v (7) ime ransformation for Moving Frame,
a (3) Aeleration istane for Speed a (4) Aeleration istane for Speed derive, (6) Instantaneous Speed, for the final version of the formula for instantaneous speed,. Although this is the orret formula for speed, it does not simplify easily and is a bit umbersome in its present form. Later a simpler formula that was atually used in the Mathad model will be derived based on the variable,. First, however, we will take a moment to derive four very important new formulas involving aeleration omposition. Given formulas, (5, 6, and 9) for the distanes traveled during aeleration, ( ) A A a (5) Aeleration istane for Aeleration Rate A ( ) A A a (6) Aeleration istane for Aeleration Rate A ) ( v v a t A t A (9) Aeleration istane for Aeleration Rate A we an take formula (5) and solve it for variable A to obtain, a a A (7) Formula for A using a for the stationary frame aeleration rate A. hen, given the proposition that, a a a (8) Relationship of a to a and a
we an substitute the right side of this latter formula (8) for the variable a in the preeding formula (7) to get, a a a a A (9) Formula for A using a, and a giving A in terms of distanes, a, and a. If we now substitute the right sides of formulas (6, and 9) for the variables, a, and a respetively in equation, (9) we obtain, ) ( ) ( ) ( ) ( v v v v t A t A A A t A t A A A A (30) Aeleration Composition (Rate) an important new formula for aeleration omposition that gives the rate of aeleration A based on aeleration rates A, and A. If we instead substitute the right sides of formulas (4, and 8) for the variables, a, and a in equation, (9) we obtain, t t A v v (3) Aeleration Composition (Rate) an alternate form of the aeleration omposition formula that uses instantaneous speeds, and instead of aeleration rates A, and A to find aeleration rate, A. If we now take formula (3) and solve it for, we an similarly derive aeleration omposition formulas that give the instantaneous speed, instead of aeleration rate A. hus, beginning with formula (3), a (3) Aeleration istane for Speed and solving for, we obtain, a a (3) Formula for using a whereupon substituting the right side of equation (8) in plae of a gives,
a a (33) Formula for using a, and a a a for determining in terms of a, and a. If we now substitute the right sides of formulas (6, and 9) for the variables, a, and a respetively in equation, (33) we obtain, A A tv ( ) ( ) A A tv (34) Aeleration Composition (Speed) A A tv ( ) ( ) A A tv another important new formula for aeleration omposition that gives the instantaneous speed, based on aeleration rates A, and A. If we instead substitute the right sides of formulas (4, and 8) for the variables, a, and a in equation, (33) we obtain, tv tv (35) Aeleration Composition (Speed) the alternate form of the aeleration omposition formula that uses instantaneous speeds, and instead of aeleration rates A, and A to find instantaneous speed,. It should be noted that the right side of time transformation formula (7) an be substituted in plae of t v in all of these four important new formulas for aeleration omposition thus reduing the number of variables to three. In addition, it should be understood that the same tehnique that was used to derive these four aeleration omposition formulas ould also be used to derived formulas for and. Getting bak to where we left off, sine we now have formula (6) for finding based on the values of and and not distane, a, we an use this value to find distane, a using formula (8). At this point then, besides the given values of, A, and A, we are able to determine the values of t v,,,, A, a, a, a, u, u, and u. his leaves only speeds,,,, u, and distanes, v, v, and v still to be found as we ontinue. Now, sine by definition, u (36) Formula for istane, u and also, u a 3
we an substitute the right sides of formulas (36) and (3) into the latter equation and obtain, that when solved for, gives, (37) Formula for Speed, where is a uniform rate of speed diretly related to an aeleration distane. his distintion is neessary beause although speeds,, and, are all uniform motion speeds, they are not diretly related to eah other in the manner that would normally be the ase involving veloity omposition. hat is, the value of is based on F time and not on F time as defined by veloity omposition theory. Continuing in the same manner for, we obtain, and also, u (38) Formula for istane, u u a whereby substituting the right sides of formulas (38 & 4) into the latter formula gives, that when solved for, results in, (39) Formula for Speed where is a uniform rate of speed diretly related to an aeleration distane. he same method an also be applied to to obtain, t (40) Formula for istane, u u v and also, u a whereupon substituting the right sides of formulas (40 & 8) into the latter formula gives, 4
t v t v that when solved for, gives, (4) Formula for Speed where is a uniform rate of speed diretly related to an aeleration distane. Speed, is of ourse the speed relative to F. If we wish to know what this speed is relative to the stationary frame, (the assoiated SF speed, u ) we simply use the equality priniple given earlier involving equations (0, & ). his gives us, that in turn gives, v t u (4) Equality of Speed and ime Ratios for and u u or, t v u (43) niform Motion Speed Relative to F, t v and, tv u or, tv u (44) niform Motion Speed Relative to SF for the assoiated uniform motion speed, u that would be measured in the stationary frame. If we now go bak to formula (4) and solve it for, we obtain, (45) Formula for based on for speed relative to F. his is the simpler formula for that was mentioned earlier when equation (6) was derived. Although we now have formulas for, u, and, we still do not have a formula that will give any of these values independent of the value of one of the others. However, sine by definition, u (46) Relationship of u to u and u u u where u is the SF distane traveled at speed during the same SF time interval that distanes u and u are traveled at speeds and u respetively, by way of substitution we obtain, u (Referene Eq s. 36, 38, 40, and 4) 5
that when simplified and solved for u gives, u (47) Relationship of u to and where u is a valid SF uniform motion speed require to travel SF distane u during SF time interval. At this point, the only speed not yet determined is uniform motion speed u. Sine this speed is dependent on distane v that is in turn dependent on distanes v, and v, let us first determine these three distanes. Sine we already know speeds, and, using formulas ( & 3) we quikly derive, and, v () niform Motion istane for Speed v (3) niform Motion istane for Speed for distanes v, and v. sing equation (6), this then gives us, (6) Relationship of v to v, and v v v v for the value of distane, v, that when used with formula (7) gives, v u (7) Formula for Moving Frame Speed, u tv for uniform motion speed, u relative to F. o find the value, v u of this speed relative to the stationary frame we simply repeat the proedure just used for speed u. hus, again using formulas (0 & ) we have, v t (0) Equality of Speed and ime Ratios u u v and therefore, tv vu u () niform Motion Speed Relative to SF giving us speed v u relative to the stationary frame. his ompletes all of the steps neessary to find the values that will be used in our final analysis. Prior to onduting the analysis, however, there is still one final order of business to take are of. It involves the relationship between distanes, a, and v, and is important enough to deserve larifiation before moving on to the final analysis. 6
9. he Composite Relationship between Aeleration and niform Motion Sine the relationship between distanes a and v is ruial to a proper understanding of the overall relationships between aeleration and uniform motion involving ompound speeds, the subjet will now be disussed in detail. Although this relationship was touhed on briefly at the beginning of the paper in the setions involving the speial relationship between omposite rates of aeleration, it is important enough to over again from a more fundamental perspetive. Speifially the issue to be addressed is the underlying ause for the normally expeted differene, or gap that appears between distanes a and v as shown in Figure 5. his gap is harateristi of the relativisti relationship involving two different rates of onstant aeleration suh as A and A and the uniform motion speed, that orresponds to the ahieved speed of A, the lesser of these two aeleration rates, during SF interval. It disappears only when the two omponent instantaneous speeds, and that orrespond to the ahieved speed of the greater of the two aeleration rates, A are equal. he rationale for this an be understood by examining the formulas that define onstant aeleration. It should be noted that all relativisti formulas, inluding the millennium formulas for onstant aeleration, redue to Newtonian formulas at low Newtonian speeds where time variane is not yet apparent. In other words, all relativisti formulas are simply lassial formulas that have been orreted for the effets of time variane that beomes signifiant only at very high (relativisti) speeds. With that in mind, we an ompare the relationship of the simpler Newtonian distane formula for onstant aeleration to the distane formula for uniform motion to understand what is happening. When the Newtonian formula for onstant aeleration is given in the form, a and ompared with the formula for uniform motion involving the same speed and time interval, u it an be seen that the distane traveled by an objet under onstant aeleration is exatly ½ that of the distane traveled by an objet at the same uniform motion speed ahieved during the aeleration. his means that if another objet were aelerating (relative to a uniform motion frame of speed ) at the same rate that the first objet is aelerating relative to the stationary frame to reah speed, the ombined aeleration distanes traveled by the two aelerating objets during the time it takes for the first objet to reah speed will (at Newtonian speeds) equal the distane of the uniform motion objet traveling at speed. In other words, when the rate of aeleration of the seond objet relative to frame, is equal to the rate of aeleration of the first objet relative to the stationary frame, the ahieved speed of the seond objet relative to frame will equal the ahieved speed of the first objet relative to the stationary frame. nder these onditions, where the speeds are not signifiant enough for time variane to be a fator, the ombined SF aeleration distanes traveled by the two aelerating objets during SF time interval, will be idential to the distane traveled by a uniform motion objet traveling at the speed attained by either of the two objets. Or, sine the speeds of the two aelerating objets are the same, we an simply say a uniform motion speed equal to the speed ahieved by 7
the first objet. If we relate this result to Figure 5, the aeleration distane, a, will equal the distane from uniform motion, v. Now, let us examine what will happen if either, the ommon aeleration rate of the two objets, or the time interval during whih the aelerations take plae is inreased to the extent that the ahieved speeds are a signifiant fration of light-speed. If equal aeleration rates are maintained as in the initial ase, at any point in time relative to the stationary frame, the faster moving seond objet will enounter the effets of time variane to a greater degree than the first objet. Sine speed is a funtion of time, and at any given instant, less referene frame time passes for the seond objet than for the first objet, the speed reahed by the seond objet relative to frame will be less than the speed reahed by the first objet relative to the stationary frame. In this ase, the distane traveled by the seond objet relative to frame, and thus the stationary frame, will be less than it would have been if the attained speeds had remained equal. Sine the uniform motion objet traveling at speed relative to the stationary frame, is traveling at a slower speed than the seond objet is relative to the stationary frame, the uniform motion objet enounters less severe time variane than the seond objet and therefore the uniform motion distane traveled relative to the stationary frame will be less affeted. hus, in this ase the ombined distanes traveled by the two aelerating objets will be less than the distane traveled by the uniform motion objet and a gap will appear between the two distanes suh as that shown in Figure 5 between distanes a and v. (It bears reminding here that time variane is a nonlinear funtion. It will always be muh more severe for the faster moving objet than for the slower moving objet.) nder the onditions stated, the gap will ontinue to inrease as the speeds inrease toward light-speed. (Although, due to another effet to be overed later, as is approahed the gap beomes less and less signifiant ompared to the distanes traveled.) It follows from the above analysis, that if the aeleration rate of the seond objet (in the ontext originally given) is greater than the aeleration rate of the first objet, a reverse gap will be evident at the start of the experiment as shown in Figure 6. In this ase, at any point in time during the beginning of the experiment, the ombined aeleration distanes traveled by the two aelerating objets will be greater than the distane traveled by the uniform motion objet. hen, as the experiment ontinues and the instantaneous speeds inrease, the effets of time variane will ause the gap to redue until finally, when the two instantaneous aeleration speeds are equal, the gap will disappear. It is always at the point where the two instantaneous speeds are equal, that the total distane from aeleration will equal the distane from uniform motion. As the experiment ontinues past this point, the gap will our in the opposite manner and as stated earlier, the ombined aeleration distanes of the two aelerating objets will be shorter than the distane of the uniform motion objet. Again, the gap will ontinue to inrease as speed is approahed. Relevant to what was just disussed, it will be shown during the omparisons of the two veloity omposition formulas, that Einstein s formula atually gives the instantaneous speed for aeleration and not the uniform motion speed as stated in speial relativity. Even then, however, Einstein s formula gives orret results only under onditions involving three speial ases, one of them being when the two instantaneous speeds from aeleration are equal. (When there is no gap between distanes a and v.) he other two speial ases will be disussed at the end of the analysis when the data resulting from the analysis is evaluated. 8
0. he Analysis he most revealing way to ondut the analysis is to first selet aeleration rates A and A and then generate data ables based on a uniform series of inreasing time intervals. Even more revealing will be to have the gap between a and v appear in one diretion at the beginning of the series when stationary frame time interval is at its first value, disappear in the middle of the series when the two distanes are equal, and then advane in the opposite diretion out to the end of the series when the time interval is at its highest value. Sine the hosen aeleration rates, and time interval determine the speeds that will be reahed and also the point at whih speed will equal speed, we an start by seleting the aeleration rate for A and then deiding what ahieved speed, do we wish to math. Having deided that, we an then determine the aeleration rate needed for A to bring this about. In the example given in this analysis, an aeleration rate of g is hosen for A, and a speed of.5 is hosen for. hen, by taking equation () and solving it for time interval we obtain, (48) Formula for SF ime Interval Based on alues of A and A where is the SF time interval upon whih the seleted value for will be ahieved at the hosen rate of aeleration, A. Now, sine by definition, a (8) Relationship of a to a and a a a we an by way of substitution, using the right sides of equations, (5, 4, and 8) respetively, arrive at, A t v ( A ) and sine must equal, by substituting in plae of obtain, A t v ( A ) whereupon, with one final substitution using the right side of equation (7) in plae of t v, arrive at, A ( A ). Solving this final equation for A we obtain, 9
A (49) Formula for Finding A for the ase, ( ) whih is the formula for determining the value needed for A in order for to equal when time interval is reahed. Now let us go through the steps neessary to generate the data for the ables. Given the following values, 9979458 m/s g 9.80665 m A g let us determine the SF time interval needed for to equal.5. sing formula (48) we obtain,.5 i 7649784. 044083 s 9.80665 (.5) where i is the SF time interval where.5. Next, using formula (49) we get,.5 A.309400767 g (.5) 58503 i for the value of A in terms of g. his is the value that A must have for when time interval i is reahed. Now let us divide time interval i by 0 to get the inremental time intervals that will be used for the ables. In this ase, let us simply say, z i.7650 0 where z. i, or one tenth of the interval needed for to equal. And finally, let us define a variable alled, Int, that we an assign the values of to 0 in suessive order. Int represents the short intervals we will step through during the analysis. hus with Int so defined, the formula for the SF time interval beomes, 6 6 z Int.7650.7650 6 s where the value of will inrease in one tenth inrements of the value needed for to equal. hus, when Int, a will be greater that v beause we will have only one tenth of the value needed, and when Int, we will have two tenths, and so on. At Int 0, will equal, and subsequently, a will equal v and the gap will disappear. hen as we ontinue to inrease the value of Int toward 0, a will beome less than v and the gap will reappear in the opposite manner and ontinue to inrease out to the end of the series. We are now ready to begin generating the data that will be used in the ables. Starting again with, g, and A given, and A determined for at Int 0, and now determined, for Int, we an begin alulating the remaining data as follows: 0
Given: 9979458 m/s g 9.80665 m/s A g or, 9.80665 m/s Just etermined: A.30940 g or,.64749 m/s 6 Int.765 0 s Results: A () giving, 0.05764 ( A ) A () giving, 0.36 ( A ) (39) giving, 0.0884 (37) giving, 0.06637.76040 s (7) t v 6 t v u (47) u (43) giving, 0.03759 (45) giving, 0.07508 A A A.80983 m s (0) giving,. 3064 g t g / v A A.77367 m s () giving, r. 3055 g g r /
a (3) giving, a 0. 0037 LY LY a 006 a (4) giving, 0. LY LY a t a 00 v (8) giving, 0. LY LY v () giving, v 0. 00739 LY LY v (3) giving, v 0. 003 LY LY (6) giving, v 0. 0047 LY LY v v v u 07465 v u (7) giving, 0. tv. Struturing the eloity Composition Formulas for Comparison Before the results of the generated data an be ompared with the preditions of the two different veloity omposition formulas, millennium, and speial relativity, the formulas have to be properly strutured for the variables that are used. In their basi form, the two different formulas an be shown as, R (50) Basi Millennium Formula for eloity Composition and E (5) Basi Speial Relativity Formula for eloity Composition were R and E are the omposite values for uniform motion speeds, and. he R, and E subsripts will be used to distinguish any variable in the millennium formula from the same variable in the speial relativity formula where diret omparisons of the variables will be made. Although both formulas are orretly shown for general purposes, the variable is inonsistent with the variable, u used to show the uniform motion speed relative to F in the present analysis. In other words, if both of these formulas are intended to show omposition
of uniform motion speeds, as stated in both parent theories, all of the variables used must express uniform motion values. In the given analysis, and are used to express both uniform motion speeds and instantaneous speeds related to aeleration. hus, no distintion is neessary in using these two variables in either role. his is not true of variable. Although this variable does represent both kinds values in the present paper, it was shown that its uniform motion value annot be used in onjuntion with veloity omposition of uniform motion speeds. hat is, it was shown that u, and not represents the differene between uniform motion speeds,, and. Aordingly, u must be substituted for in both of the above formulas for them to orretly perform their stated purposed. In plain words, in the present analysis was used only in relation to instantaneous speed related to aeleration and never in relation to uniform motion speeds that orrespond to those instantaneous speeds. he orretly stated formulas for veloity omposition are then, Ru u (5) Millennium Formula for eloity Composition and u Eu (53) Speial Relativity Formula for eloity Composition u where the u subsript was added to the R, and E variables in both formulas to distinguish them from variables that will be used in formulas involving aeleration omposition. In regard to the instantaneous speeds assoiated with onstant aeleration, the millennium formula is, tv Ri (35) Aeleration Composition (Speed) tv whereas, albeit unintended, the speial relativity formula would be, Ei (54) Speial Relativity Formula for Instantaneous eloity Composition where the variables Ri and Ei are the millennium and speial relativity variables for the omposition of the instantaneous speeds resulting from aeleration rates, A, and A. Sine,, and are also uniform motion speeds, we an struture the millennium veloity omposition formula to work with these speeds too. In this ase, we have, 3
R (55) Millennium Formula for eloity Composition where R is the omposite speed of, and. It should be noted that the transformation fator must still referene F, beause, it is that frame that the value of is referened to. his gives the millennium formula a deisive advantage over the speial relativity formula, E (56) Speial Relativity Formula for eloity Composition sine there is no way to make suh a distintion. Another limitation of the speial relativity formula avails itself if we try to struture the formulas to return the value for u. his is aomplished by taking the basi formulas (50, and 5) and solving for. For the millennium formula, this gives us, u R ( ) (57) Millennium eloity Composition Formula for u while for the speial relativity formula we get, E ( ) (58) Speial Relativity eloity Composition Formula for where again there are no provisions for making the speial relativity formula return the orret value, u. hus, the most we an do here is to show that it not only returns the wrong value, instead of u, but does so orretly only for the speial ases where. It may also prove instrutional to inlude at least one millennium aeleration formula for finding the rate of aeleration A in order to ompare it with the value that was atually used to generate the data. A A tv ( ) ( ) A A tv A R (30) Aeleration Composition (Rate) A A tv ( ) ( ) A A tv he only hange made to formula (30) was the addition of the R subsript to distinguish the results from the value, A that it will be ompared to. We are now ready to ompare the results of all of these omposition formulas to the data that were generated.. he ata he data from the analysis were organized into six separate ables, of whih ables through 5 allow diret omparison of the millennium omposition formulas to those of speial 4
relativity. o the author s knowledge there is no speial relativity equivalent to the millennium theory aeleration rate formula, (30) overed in able. In addition to validating the millennium formulas and showing the true meaning of the speial relativity formulas, the data in these ables reveal the remaining two speial ases where the speial relativity formulas give the orret results for the instantaneous speeds related to aeleration. o inrease the thoroughness of the analysis, additional time intervals, in exess of the 0 originally disussed, have been inluded in the ables. Referring now to able, it an be seen that the first Int value is extremely small and leads to a SF time interval that is only a small fration of a seond that in turn leads to a very low omposite speed. By way of ontrast, the final four Int values dramatially inrease the time interval to raise speed to a high fration of the speed of light. he purpose for adding Int was beause it helps identify a point where the value of A hanges diretion. In viewing able, it an be learly seen that the millennium formula (30) for aeleration rate A R gives results that are idential to those arrived at for A using the data generation proedure overed in Setion 0. Of partiular interest in able is the fat that the relative aeleration rate A r equals the onstant aeleration rate, A when Int 0. his is exatly the result that is expeted if at this instant in time. his is beause with regard to inertial frame, it is the relative rate of aeleration, A r, and not the onstant rate of aeleration, A that has the same relationship to the stationary frame as does aeleration rate A regarding inertial frame,. he final two points of interest regarding able onern the range of values shown for A. As an be seen, the value of A dereases as speed inreases toward, and then after Int 0, the value of A hanges diretion and inreases as speed ontinues to approah the speed of light. his hange in diretion is also refleted in the value of A r as seen at Int 00. his appears to be the harateristi behavior of aeleration rate A. he final point of interest is this: If aeleration rate A is hanging, then it annot be a onstant rate of aeleration in the same regard as aeleration rates A, and A. It is rather the onstant rate of aeleration required to travel distane, a, during the same interval that distanes a and a are traveled. In other words, for any given interval, a different rate of onstant aeleration is needed to travel the differene in distanes between a, and a. able - Millennium Relativity Aeleration Composition Int s % of A m/s A R m/s A m/s A m/s A r m/s 0-8 0.0765.3 0-7.64749.64749 9.80665.84084.84084.76498 0 6 3.6.64749.64749 9.80665.80983.77367 3.5996 0 6 5.766.64749.64749 9.80665.7049.5898 3 5.9494 0 6 37.39.64749.64749 9.80665.5889.964 4 7.0599 0 6 47.059.64749.64749 9.80665.4080.9365 5 8.8489 0 6 55.47.64749.64749 9.80665.880.54868 6.05899 0 7 6.47.64749.64749 9.80665.000.5566 7.3548 0 7 68.3.64749.64749 9.80665.8490 0.77604 8.498 0 7 7.954.64749.64749 9.80665.64097 0.438 9.58848 0 7 76.8.64749.64749 9.80665.47303 0.09758 0.76498 0 7 80.0.64749.64749 9.80665.3374 9.80665.9448 0 7 8.63.64749.64749 9.80665.943 9.5485.797 0 7 84.8.64749.64749 9.80665.0840 9.30 3.9447 0 7 86.69.64749.64749 9.80665 0.998 9.09 4.47097 0 7 88.48.64749.64749 9.80665 0.9903 8.94554 5.64747 0 7 89.443.64749.64749 9.80665 0.866 8.7936 5
6.8397 0 7 90.546.64749.64749 9.80665 0.8795 8.66048 7 3.00046 0 7 9.49.64749.64749 9.80665 0.78753 8.54507 8 3.7696 0 7 9.308.64749.64749 9.80665 0.76854 8.4447 9 3.35346 0 7 93.06.64749.64749 9.80665 0.75959 8.3575 0 3.5996 0 7 93.633.64749.64749 9.80665 0.7594 8.87 3.70645 0 7 94.74.64749.64749 9.80665 0.76687 8.589 00.76498 0 8 99.7.64749.64749 9.80665 3.4778 8.0069 0 3.76498 0 9 99.997.64749.64749 9.80665 6.00587 8.3686 0 4.76498 0 0 99.99997.64749.64749 9.80665 6.3497 8.40986 0 5.76498 0 99.9999997.64749.64749 9.80665 6.3854 8.4468 In able, the results of millennium formula (5) indiated by the variable, Ru are ompared to the results of speial relativity formula (53) indiated by the variable, Eu. Here we see the first indiation of the seond speial ase in whih the speial relativity formula gives the orret result. As an be seen, at very low speeds where time variane is not yet a fator (Int, 0-8 ) the orretly strutured speial relativity formula for veloity omposition gives the same result as the millennium formula for veloity omposition and thus in like manner, a result that is in agreement with the value generated for speed. Whereas the millennium formula ontinues to return the orret value for throughout the entire range of values given in the able, however, the speial relativity formula varies signifiantly until speed is approahed. his being the first indiation of the third and final speial ase referred to earlier in the paper. In this final ase, where has reahed a value of 99.9999997 perent of speed, the effets of time variane are at an extreme, and as shown in the relativisti motion perspetive paper, the distane traveled by way of onstant aeleration approahes the distane traveled via uniform motion. able Millennium Relativity eloity Composition Int % of u Ru Eu 0-8.3 0-7 5.77350 0-0 7.55983 0-0.33333 0-9.33333 0-9.33333 0-9 3.6 0.05764 0.07465 0.36 036 0.37 5.766 0.47 0.439 0.5766 0.5766 0.544 3 37.39 0.7066 0.037 0.3739 0.3739 0.368 4 47.059 0.50 0.503 0.47059 0.47059 0.4545 5 55.47 0.7735 0.8868 0.5547 0.5547 0.5407 6 6.47 0.3733 0.347 0.647 0.647 0.5807 7 68.3 0.3747 0.3379 0.683 0.683 0.6837 8 7.954 0.493 0.347 0.7954 0.7954 0.66565 9 76.8 0.4608 0.3463 0.768 0.768 0.696 0 80.0 0.5 0.3464 0.8 0.8 0.745 8.63 0.536 0.34368 0.863 0.863 0.749 84.8 0.56949 0.3388 0.848 0.848 0.764 3 86.69 0.6008 0.3347 0.8669 0.8669 0.77757 4 88.48 0.686 0.353 0.8848 0.8848 0.799 5 89.443 0.65465 0.379 0.89443 0.89443 0.80474 6 90.546 0.67855 0.3089 0.90546 0.90546 0.8634 7 9.49 0.70047 0.30048 0.949 0.949 0.869 8 9.308 0.7058 0.905 0.9308 0.9308 0.83657 9 93.06 0.7390 0.837 0.9306 0.9306 0.84546 0 93.633 0.75593 0.7557 0.93633 0.93633 0.85367 94.74 0.7745 0.6763 0.9474 0.9474 0.866 6