Integrated Relativistic Velocity and Acceleration Composition

The Fully Integrated Principles of Velocity and Acceleration Composition

## Abstract

Presented in this work are the four fundamental cases involving the integration of relativistic velocity and acceleration composition principles made obvious by the two preceding millennium relativity works on velocity composition and acceleration composition.  Through systematic analysis it is clearly demonstrated that the fundamental nature of unidirectional motion and acceleration is accurately definable in a newly introduced relativistic coordinate system.  It is further shown that velocity composition is fundamentally a Y-axis consideration of this new coordinate system and acceleration composition is fundamentally an X-axis consideration.  Taken together in their integrated form, the principles presented here provide the entire mathematical basis for all fundamental relativistic compositional behavior involving uniform motion and acceleration.

Integrated Relativistic Velocity and Acceleration Composition

The Fully Integrated Principles of Velocity and Acceleration Composition

1. Introduction

In the two previous works on velocity composition1 and acceleration composition2 the correct formulas for relativistic velocity composition, and relativistic acceleration composition were introduced.  In deriving these formulas, it was shown that although speeds, whether associated with uniform motion, or acceleration are not directly additive, the corresponding travel distances are.  It logically follows from this that the mathematical expressions for distances associated with uniform motion may be interchanged with the mathematical expressions for distances associated with acceleration to derive composition formulas that integrate uniform motion speeds with the instantaneous speeds of acceleration.  Less obvious with regard to the preceding works is the finding that velocity composition is primarily a function of the Y-axis whereas acceleration composition is primarily a function of the X-axis.  Our purpose here is to finish the job started in the earlier works and expand the relativistic composition process into the area involving integration of uniform motion speeds with the instantaneous speeds of acceleration.

2. The Four Fundamental Cases Involving Acceleration and Uniform Motion

Even if we limit our study to only the relationships involving two objects where each is either accelerating or moving at a uniform rate of speed in the same direction relative to a stationary frame (SF) of reference, there are four possibilities that must be covered.  If R is the number of relational conditions (in this case 2, i.e., acceleration and uniform motion), N is the number of objects, and P is the number of possible combinations, this relationship can be expressed mathematically by the formula, To make the study more complete, however, a third object is needed, and even a fourth.  That is, one that will travel the resulting distance between the other two objects during the same time interval, and another that will transition between the reference frames of the other two objects during that same time interval.  Using the above formula, this would appear to give us as many as 16 different cases to examine.  As it turns out, however, there are still only four fundamental cases to evaluate, but each has two primary sets of conditions to consider, those involving the Y-axis, and those involving the X-axis.  Our approach will be to cover the four cases in two separate treatments, first with regard to the Y-axis and its relationship to velocity composition, and lastly with regard to the X-axis and its relationship to acceleration composition.

3. Defining the Relativistic Coordinate System

Whereas the spherical reference frames used in the millennium theory of relativity3 have proven themselves beyond question by uncovering the fact that Einstein’s special relativity velocity composition formula is invalid1 and then going on to show what the correct formula is1, such reference frames are impractical for use with acceleration.  The reason is simple.  During acceleration the velocities are constantly changing.  Thus, no single representation using spherical frames can effectively portray the most important aspects of acceleration.  That is, the change in speed over time, or distance.  Moreover, even the rectangular coordinate systems normally associated with special relativity are inadequate for such purposes, because, they either ignore the Y-axis, as in the system used to develop the velocity composition formula4, or they fail to show the relativistic relationships between the stationary frame and moving frame of reference.  With the progress made during the earlier relativistic motion perspective research5, and more recently, the acceleration composition research2, it is now possible to formalize the approach first introduced in the latter of the two respective research papers.  Such formalization is made possible by a retrospectively obvious, yet somewhat surprising relationship between the earlier derived millennium principles for velocity composition and the subsequent principles for acceleration composition.  Implicit during the integration of these principles is the fact that for any instant in time, the velocity composition formula correctly reports the relative speed between two objects moving in the same direction regardless of the fact that one or both are engaged in acceleration.  This revelation used in conjunction with other previously explored relationships between two accelerating objects leads to a formally definable relativistic coordinate system that correlates the coordinates of the moving frame with those of the stationary frame.

To understand the role of velocity composition in regard to acceleration we need only to evaluate the conditions that apply to two objects, synonymous here with points of mass, accelerating at different rates in the same direction away from a stationary frame of reference.  Quite obviously both objects can discontinue their acceleration at any instant in time.  If that is the case, and if their speeds are referenced to the same stationary frame, then the uniform speeds attained by these objects relative to the stationary frame, and also the speed of the faster object relative to the slower object are in agreement with the relationships defined by the millennium relativity velocity composition formula.  When used in conjunction with other millennium formulas, if the rates of acceleration and the SF time interval during which the accelerations occur are known, then we not only have the means to determine the instantaneous speeds achieved by these objects during any instant in time, but also the distance traveled and the instantaneous speed of the faster object relative to the slower object.  This is true, even if the objects continue to accelerate.  That is, objects do not actually have to stop accelerating for us to evaluate the conditions that would exist if they did stop accelerating at any given instant.  Thus, whereas the previously derived acceleration formulas6 define the time, distance, speed, and acceleration rate relationships for each of the two objects separately for any instant in time, the velocity composition formula defines the speed relationships between the two objects for that same instant.  These integrated relationships will become clearer as we proceed with the analysis.

Referring now to Figure 1, the two curves designated ac1 and ac represent the accelerations of two objects that were initially at rest at the point of origin of a stationary frame of reference.  These curves show the change in position along the SF X and Y-axes for both objects throughout the SF time interval during which the accelerations occurred.  If the X-axis represents SF distance, then Da1 is the SF distance traveled by the object with the lower acceleration rate, ac1, and Da is the SF distance traveled by the object with the higher acceleration rate, ac during the interval.  Likewise, if the Y-axis represents speed achieved relative to the stationary frame during the interval, then v1 is the SF speed reached by the object with the lower acceleration rate, and v is the SF speed reached by the object with the higher acceleration rate.

Given the proposition that both accelerations can be simultaneously discontinued at any instant in SF time, it can be claimed that for any instant in SF time, the object with the lower rate of acceleration resides at the point of origin of a uniform motion frame relative to which the object with the higher acceleration rate has potential uniform motion.  In plain words, if both accelerations discontinue at the same instant, then both objects will move in the same direction at different uniform rates of speed.  Since the speeds will be uniform, either object can be designated the object at rest, relative to which the other object has uniform motion.  In the illustrated case, the object with the lower rate of acceleration, and thus the lower rate of instantaneous speed, (another way of saying potential uniform motion speed) designated v1, is the object chosen as being at rest in a reference frame that will have a uniform rate of speed, v1 relative to the SF.  The faster accelerating object with the higher instantaneous speed, designated v, relative to the SF, then has some other, yet to be determined, instantaneous speed relative to the rest object and the uniform motion frame it defines. As indicated in the illustration of Figure 1, the difference in magnitude between v and v1 can be exploited to correlate the Y coordinates of the SF with the Y` coordinates of the uniform motion frame (UF) v1.  In a similar fashion, the difference in distance between Da and Da1 can be exploited to correlate the X coordinates of the SF with the X` coordinates of the UF, v1.  In the former case, this is accomplished directly using the millennium relativity instantaneous speeds formulas in conjunction with the millennium relativity velocity composition formula, while in the latter case it is accomplished directly using the millennium relativity relativistic acceleration distance formulas.  Note:  Although specific use of X and Y coordinates may serve some useful purposes in some forms of analyses, this method of working directly with such coordinates in regard to relativistic analyses is considered both primitive, and redundant by the author, and tends to only add an unnecessary layer of complexity to relativistic problems that by their very nature are complicated enough.  Thus, in the ensuing analyses, the problems will be formulated in terms of speeds, times, and distances directly without any unnecessary references to such X and Y coordinates.

4. The Four Velocity Composition Cases

As stated earlier, the four fundamental cases involving acceleration and uniform motion can be grouped together and evaluated first from the perspective of velocity composition, and later from the perspective of acceleration composition.  In each of the four fundamental cases making up this group there are two objects that are either both accelerating relative to the stationary frame, or either the faster, or slower of the two is accelerating while the other is moving at a uniform rate of speed, or both are moving at a uniform rate of speed.  During the initial treatment, we are interested only in the uniform motion relationship (velocity composition) between the two objects, while in the later treatment our focus will shift to the acceleration relationship (acceleration composition) between the two objects.

Case 1 – Part – 1 – Two Different Acceleration Rates

Referring to Figure 2, we will now take a closer look at the velocity relationships of the two accelerating objects previously shown in Figure 1. Perhaps the most important thing to remember when viewing such illustrations is that they provide a history of how the objects arrived at their present states.  Such illustrations do not show the future course of these objects.  Thus, the distances traveled and speeds reached all have to do with the past.  If both objects were to discontinue accelerating at their present locations in the illustration, both would continue in a straight-line path to the right.  It is this new, yet to occur, behavior that is predicted by the velocity composition formula.  This behavior is very much like that involving centrifugal force.  If the restraining force is removed, we know the object will continue in a tangential straight-line path from the axis of rotation.  Similar is true here.  If at any instant in time, the energy needed to sustain, or alter the accelerations in any manner, is absent, the objects will no longer follow the prescribed acceleration curves, but instead continue at the most recent instantaneous speeds achieved.  Thus, such instantaneous speeds may, as stated earlier, be alternately viewed as uniform motion speeds, and the millennium relativity velocity composition formula can be used to show their relationships to each other for any instant in time.  To do this, however, we must first determine the instantaneous speeds, v and v1 for the two objects under investigation.

Given that the constant rates of acceleration, ac and ac1 are known along with the SF time interval during which speeds v and v1 were attained, we can determine these instantaneous speeds by first reconfiguring the millennium relativity acceleration formulas7, and (1) Constant Acceleration Formulas

to show speeds instead of acceleration rates.  This gives us, and (2) Instantaneous Speeds from Acceleration

where tv and tv1 are the corresponding time intervals in the uniform motion frames presently transitioned to by the two accelerating objects.  (The uniform motion frames where each of the two objects are for that instant at rest.  This does not mean that the two objects are at rest relative to the SF, but only that their speeds match the speeds of the two uniform motion frames just transitioned to.)

Next, we use the millennium relativity time transformation formulas3, and (3) Time Transformation Formulas

to correlate the time intervals, tv and tv1 in UF’s v and v1 respectively, with time interval T in the SF frame.  By way of substitution into the earlier given speed formulas, (2), we obtain, and which when solved for v and v1 respectively, gives8, and (4) Instantaneous Speed Formulas

for instantaneous speeds, v and v1.  If we now take the millennium relativity formula for velocity composition1, (5) Velocity Composition Formula

and solve it for V2, we obtain2, (6) Velocity Composition Formula for Vu2

where V2 has been relabeled, Vu2.  This distinction is used to show that Vu2 is a uniform motion speed that correlates with instantaneous speed v and thus distinguishes Vu2 from V2 that would be used to designate an instantaneous speed, and also from U2, to be used later, that correlates with a uniform motion speed, u.  Although this may appear to be an unnecessary distinction, previous experience has convinced the author to be very specific as to what kind of value a given variable represents.

Now that the necessary formulas are in place, we can not only determine the speed of the faster object relative to the slower object, but can also use that information to correlate the Y-axis of the SF with the Y`-axis of the UF, v1.  (I.e. scale the Y`-axis to agree with speed Vu2 as shown in Figure 2.)  For comparison purposes, the values used are the same as those used in the examples of the millennium relativity acceleration composition work.

Given the values, c = 299792458 m/s, g = 9.80665 m/s, ac1 = 1g, ac = 2.309401076758503g, and T = 17649784.21044083 s, the instantaneous speed formulas (4) give,

v = .8c    and    v1 = .5c.  (Throughout paper, all speeds are converted to terms of c.)

When these values are used with the velocity composition formula, (6) we obtain,

Vu2 = .34641c

for the speed of the faster object relative to the slower object as would be observed in UF v1.  Knowing the UF v1 value for speed Vu2, that corresponds to the SF value for v, then permits the proper scaling of the Y`-axis as shown in Figure 2.  (If necessary, to show the entire range of Y` values from 0 to c, the UF v1 coordinate system could be extended upward beyond the range of the SF coordinate system.)  Of interest here is the fact that the corresponding SF value vu2, for UF v1 value Vu2, given by the formula2, (7) Equivalent Speed Relative to SF

used in conjunction with time transformation formula, (3) for tv1, has the value, .3c which is exactly what is expected.  (I.e. equal to the difference between v and v1)  This concludes the Part 1 analysis involving two objects with different constant rates of acceleration.  In the next example we will replace the object with the higher rate of acceleration with an object moving at a uniform rate of speed that equals the instantaneous speed achieved by the former object in the previous example.

Case 2 – Part 1 – Low Instantaneous Speed vs. High Uniform Speed

The only difference then, between the example illustrated in Figure 3 and the previous example, is this substitution of a uniform motion object for one that was accelerating at the rate ac in the previous example.  For the purpose of direct comparison with the previous example, speed u of the faster object in this example is .8c, which as previously stated, is the same as the instantaneous speed achieved by the faster object during SF time interval T in the previous example.  With some thought, it should become apparent that since the uniform motion speed does not change, the results for this example will equal those of the last example only when the same SF time interval T is used.  That is, a shorter or longer time interval will produce different results than would be obtained previously.  Thus, even though the same velocity composition formulas can be used, the designators will be changed to show this difference and therefore reduce the potential for confusion between the different examples. Modifying the velocity composition formula, (5) to agree with the present example gives, (8) Velocity Composition Formula Modified

where u is the SF uniform motion speed of the faster object, and U2 is the corresponding speed in UF v1.  Solving for U2 then gives, (9) Velocity Composition Formula for U2

for speed U2 of the faster object relative to the accelerating object for any SF time interval in which the accelerating object does not overtake the faster object.

Given the same values as before with the exception that the variable ac has been replaced by the variable u, we have, c = 299792458 m/s, g = 9.80665 m/s, ac1 = 1g, u = .8c, and T = 17649784.21044083 s.  Since in this case u is already given, we need only to find the value for instantaneous speed v1 using the appropriate formula (4) to arrive at,

u = .8c    and    v1 = .5c.

As would be expected, when these values are used with the modified velocity composition formula (9) we obtain,

U2 = .34641c

which is the same value for the speed of the faster object relative to the slower object previously obtained.  And as before, knowing the UF v1 value for U2, that corresponds to the SF value for u, permits the proper scaling of the Y`-axis as shown in Figure 3.  (Also as before, the UF v1 coordinate system could be extended upward if necessary.)  And finally, the SF value for u2, corresponding to the UF v1 value for U2, as given by the formula, (10) Uniform Motion Speed Relative to SF

has the value, .3c which in this case is equal to the difference between speeds u and v1.  This concludes the Part 1 analysis involving the relationships between a lower speed accelerating object and the higher speed uniform motion object it is compared with.  Of particular interest in this example is the fact that the change in distance along the X-axis for the faster moving object has no bearing on the result.  This serves as a good reminder that velocity composition involves current conditions and not those of the past.  Stated perhaps an even a better way, velocity composition involves changes along the Y-axis, and not the X-axis.

Case 3 – Part 1 – Low Uniform Speed vs. High Instantaneous Speed

In the next example we will reverse the roles of the two objects just covered.  This time it is the lower speed object that travels at a uniform rate of speed while the higher speed object accelerates at a constant rate.  In referring to Figure 4, it should be noted that it is now the uniform motion object moving at speed u1 that defines the origin of the uniform motion frame.  Accordingly, this frame is now designated, UF u1.  Modifying velocity composition formula (5) to appropriately show the discussed changes, give us, (11) Velocity Composition Formula Modified

that when solved for Vu2, gives, (12) Velocity Composition Formula for Vu2

where Vu2 is the instantaneous speed of the faster object relative to the slower object moving at speed u1. Again using the same values as in the original example, this time with the exception that speed u1 is used in place of acceleration rate ac1, we have, c = 299792458 m/s, g = 9.80665 m/s, u1 = .5c, ac = 2.309401076758503g, and T = 17649784.21044083 s.  Then, using the instantaneous speed formula, (4) for speed v, and the given value for u1, we have,

v = .8c    and    u1 = .5c.

When these values are used with the velocity composition formula, (12) we obtain,

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