Relativistic Constant Acceleration Distance Factor. Copyright 2010 Joseph A. Rybczyk

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1 Relativistic Constant Acceleration Distance Factor Copyright 2010 Joseph A. Rybczyk Abstract A formalized treatment of a previously discovered principle involving relativistic constant acceleration distances is presented for the first time. The principle was first discovered in an earlier work by this investigator in 2001 and expounded upon in succeeding works but never singled out for separate treatment as will be given here. 1. Introduction When the principles of relativistic physics are properly understood it is apparent that the fundamental relationships between Newtonian uniform motion and constant acceleration including the distances traveled do not hold for relativistic speeds. In the case of distances, which is our focus in this present work, an earlier defined relationship 1 between relativistic constant acceleration and uniform motion was discovered that showed a divergence from what is observed regarding Newtonian speeds. Our purpose here is to formalize that relationship and reduce it to a new fundamental principle of relativistic physics to be referred to as the relativistic constant acceleration distance factor. 2. The Newtonian Constant Acceleration Distance Factor Given the Newtonian formulas for uniform motion and constant acceleration distances we have 1 and where 3 is the rate of constant acceleration used in equation (2). In the case of equation (1) v is the uniform motion speed used during time interval T to travel uniform motion distance D u. In the case of equation (2) a is the constant rate of acceleration used during time interval T to travel constant acceleration distance D a. By substituting the right side of equation (3) in place of a in equation (2) we then derive where v is the speed reached at the end of time interval T during which distance D a is traveled. In viewing equations (1) and (4) it is apparent by way of substitution that 1

2 1 2 5 gives the relationship of uniform motion distance D u to constant acceleration distance D a. With that being the case, we can then refer to the fraction as the Newtonian constant acceleration distance factor. This then provides a convenient way to correlate the distances traveled at uniform motion speeds to the distances traveled during constant acceleration when transitioning between inertial frames. In other words if we want to know what distance we will travel relative to the distance traveled by a uniform motion object during the time we accelerate at a constant rate of acceleration to reach that same speed from a state of rest, we can simply apply the distance factor given in expression (6) to the uniform motion distance as shown in equations (4) and (5). The distance factor then isolates and defines the relationship between uniform motion distances and constant acceleration distances and helps us to understand the difference in behavior involved in the two different phenomena. Now, let us see how this understanding can benefit us when it is applied to the far more difficult to understand phenomena involving the total relationship between uniform motion and constant acceleration that includes both Newtonian and relativistic speeds from 0 to c. 3. The Relativistic Constant Acceleration Distance Factor Whereas equation (1) for uniform motion distances traveled at Newtonian speeds is equally valid at relativistic speeds, i.e., speeds from 0 to c, the same in not true for equations (2) through (5). As shown in earlier works 2 by this investigator, the relativistic form of distance equation (2) for relativistic constant acceleration is given by 7 Relativistic Constant Acceleration Distance where acceleration rate a r is given by 8 and is the rate of acceleration relative to the stationary frame for an object under constant acceleration as determined in the frame of the accelerating object. The constant rate of acceleration is given by 9 where a c is the constant rate of acceleration experience in the acceleration frame and t is the time interval in the transitioned to uniform motion frame v that corresponds to stationary frame time 2

3 interval T during which the acceleration takes place. The relationship of uniform motion frame time interval t to stationary frame time interval T is then given by 10 that is the millennium relativity 3 (MR) version of the Lorentz time transformation formula 4. If we now take equations (8) and (9) and express them in terms of speed v, we obtain 11 by which the left sides of the resultant equations can be place equal to each other to derive 12 for the relationship between the two. We can then substitute the right side of equation (10) in place of interval t of equation (12) to derive, upon simplification 13 giving the transformation relationship between the constant rate of acceleration a c in the frame of the accelerating object and its value a r relative to the stationary frame to which the acceleration is referenced. Next, we take the right side of equation (8) and substitute it in place of acceleration rate a r in equation (7) to derive 14 Relativistic Constant Acceleration Distance that upon simplification gives 15 Relativistic Constant Acceleration Distance for the stationary frame constant acceleration distance resulting from constant acceleration during stationary frame time interval T. By re-expressing equation (15) in the form 16 Relativistic Constant Acceleration Distance or by way of substitution with the left side of uniform motion distance equation (1) in place of the term vt in equation (16) in the form 3

4 17 Relativistic Constant Acceleration Distance we have derived the relativistic version of the Newtonian equations (4) and (5) derived at the beginning. These last two equations show clearly the distance relationship between uniform motion and constant acceleration for all values of v from 0 to c. The resultant expression 18 for the relativistic constant acceleration distance factor can now be analyzed to develop an improved understanding of what happens during constant acceleration to relativistic speeds. 4. Analysis of the Relativistic Constant Acceleration Distance Factor Before analyzing the relativistic constant acceleration distance factor it is necessary to clarify the meanings of all of the variables involved in deriving the factor. This is necessary because those variables must be used to determine the value of v used in the distance equations (15), (16), and (17). Due to the asymptotic nature of accelerating to relativistic speeds, the rate of acceleration a r measured in the stationary frame, to which the constant rate of acceleration a c in the frame of the accelerating object is referenced, will drop toward zero as speed v of the accelerating object approaches light speed c relative to the stationary frame. That is, the constant rate of a c is the rate that applies to the frame of the accelerating object and is not affected by the acceleration. The relative rate of acceleration a r is the corresponding rate that is measured in the stationary frame, and this rate diminishes inversely with the rise in speed v relative to the stationary frame. As for speed v, although it has the same value for acceleration as it does for uniform motion there is a difference in its meaning in each of those cases. With regard to acceleration, it is the speed reached at any instant in time. Thus, with respect to acceleration from a state of rest in the stationary frame, it begins with a value of 0 and increases toward speed c as the acceleration continues during stationary frame time interval T. In the uniform motion frames that are transitioned to at each instant during the acceleration, speed v is the non-changing uniform motion speed that takes place during the entire same time interval T. For example, assume that an object is accelerated at a constant rate of acceleration a c, from a state of rest in the stationary frame to a speed v, during some stationary frame time interval T. From the beginning of the period of acceleration, an object in the inertial frame visited at the end of time T, has traveled a distance vt relative to the stationary frame that the acceleration is referenced to. Thus, v was always the speed traveled at in the visited frame relative to the stationary frame, but it is only the most recent speed achieved by the accelerating object, relative to the stationary frame when the two speeds are equal. Now let us analyze the behavior defined by the relativistic constant acceleration distance factor. If we set speed v in the distance factor to v = 0 we obtain 0 19 giving 4

5 T Relativistic Constant Acceleration Distance for a constant acceleration distance of 0 as is expected. If we next set speed v in the distance factor to a speed that is just on the border of a relativistic speed, say at v = 0.01c, then the distance factor will reduce to approximately giving Relativistic Constant Acceleration Distance for constant acceleration distance D a relative to uniform motion distance D u that occurs simultaneously during the time interval in which the acceleration takes place. This, of course is a Newtonian result as it should be for the speed used. For the final example, let us set speed v to v = c. This then gives 0 23 for the distance factor which in turn gives 24 Relativistic Constant Acceleration Distance or an equality of distances traveled by both constant acceleration and uniform motion. As can be seen then, as the speed from constant acceleration approaches light speed c, the distance traveled by an object under constant acceleration approaches the distance traveled at a uniform motion rate of speed. From the analysis just completed it is apparent that an object under constant acceleration will travel distances that follow the principles of Newtonian constant acceleration at Newtonian speeds and in an asymptotic manner change to Newtonian uniform motion distances as the speed of light is approached. This behavior was originally discovered during an earlier research effort by the investigator involving kinetic energy Dual Definitions for Uniform Motion and Constant Acceleration Distances If an object traveling at a uniform motion speed v relative to the stationary frame travels distance vt during stationary frame time interval T, and if the corresponding time t in the frame of that moving object is given by time transformation equation (10), then that same stationary frame distance can be expressed as Vt using values of V and t that are valid in the moving frame of the uniform motion object. Since time interval t is smaller than time interval T and since it is the same stationary frame distance vt that is being defined, it follows that V must be a faster speed than v. More specifically, if we redefine distance D u given by equation (1) as 25 5

6 we can then place the right side of equation (25) equal to the right side of equation (1) and solve for speed V to obtain 26 where speed V is the moving frame uniform motion speed that corresponds to stationary frame speed v. Substituting the right side of time transformation equation (10) for t in equation (26) then gives 27 that simplifies to 28 for the final version of the formula that relates moving frame uniform motion speed V to stationary frame uniform motion speed v. It should be noted that the fraction to the right of speed v in equation (28) is simply the MR form of the special relativity 6 Lorentz gamma transformation factor. As might be expected, a similar, although a bit more complicated, distance conversion process is also available for constant acceleration distances. If equation (15) gives the constant acceleration distance for speed v relative to the stationary frame during stationary frame time interval T, and if V and t are the respective corresponding variables for speed and time in the moving inertial frame, it stands to reason that 29 Relativistic Constant Acceleration Distance gives the same constant acceleration distance as equation (15) but in terms of moving frame speed V and time interval t. If we now solve equation (29) for speed V, we obtain

7 or in rearranged form 2 41 for the constant acceleration speed V in moving frame terms. By substituting the right side of equation (15) for D a in equation (41), we obtain 2 42 for the final version of the formula for moving frame speed V that corresponds to stationary frame speed v. By substituting the right side of time transformation equation (10) in place of t in equation (42) and simplifying, the remaining time variable T will cancel out of the equation. The resulting equation is more complex then equation (42) and adds nothing to the intended purpose of this paper, so it has been left out of the analysis. Important: In consideration of that fact that the same constant acceleration distance can be expressed in terms of the lower stationary frame speed v or its corresponding higher moving frame speed V, it must be understood that the two different values must not be intermixed within the expression that contains the distance factor. For example, in referring to the bracketed expressions in equation (42), it must be understood that the variable v in the denominator must be the same as the variable v in the numerator. Stated another way, the variable v used in the distance factor of equation (16) must be the same as the variable v acted upon at the right side of the equation. 6. The Special Relativity Version of the Constant Acceleration Distance Factor The constant acceleration distance factor formalized in the form of expression (18) can be expressed in the more familiar form typical of Einstein s special theory of relativity 6. By dividing the top and bottom of expression (18) by c, we obtain 7

8 where the similarity to the Lorentz gamma factor is unmistakable. 7. Conclusion The relativistic constant acceleration distance factor formalized in the form of expression (18) in the present paper is an important new factor in relativistic physics. When use in conjunction with the millennium relativity treatment of constant acceleration it helps clarify the fundamental nature of what takes place during the transition process between different inertial frames. These insights then allow a better understanding of the relativistic acceleration addition process and its comparison to the relativistic velocity addition process involving uniform motion. In that regard, it is the author s contention that Einstein s velocity addition formula is actually an acceleration addition formula as shown in the author s previous works. REFERENCES 1 Joseph A. Rybczyk, Time and Energy, (2001); Time and Energy, Inertia and Gravity, (2002); Relativistic Motion Perspective, (2003); Millennium Relativity Acceleration Composition, (2003); Integrated Relativistic Velocity and Acceleration Composition, (2004) 2 Joseph A. Rybczyk, Time and Energy, (2001); Relativistic Motion Perspective, (2003) 3 Joseph A. Rybczyk, A series of 30 main papers beginning with the Millennium Theory of Relativity in 2001 in addition to 35 supplemental papers including this present paper in All are available from the Millennium Relativity web site at 4 H. A. Lorentz, Dutch physicist, discovered a new way to transform distance and time measurements between two moving observers so the Maxwell s equations would give the same results for both, (1895); The use of the formula was expanded upon by Einstein in his special theory of relativity. 5 Joseph A. Rybczyk, Time and Energy, (2001) 6 Albert Einstein, On the Electrodynamics of Moving Bodies, now known as the Special Theory of Relativity, Annalen der Physik, (1905) 8

9 Relativistic Constant Acceleration Distance Factor Copyright 2010 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 9