Corrections to Special Relativity
By:- H. E. Retic
Published by-
The H. E. Retic Co.
West Caldwell, NJ
1:- Special Relativity Made Valid for Accelerated Reference Frames
2:- Eliminating the Discrepancy Between Relativistic and Classical Inertial Mass
3:- Correcting the Error in the Lorentz Transformation for Transverse Force
4:- Resolving the Right Angle Lever Paradox
5:- The Source of Inertial Mass
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Section 0 - Summary:-
This paper derives the Lorentz Transformations for Force parallel to and transverse to the relative velocity vector between velocity reference frames. In the process it demonstrates that both the accepted Lorentz Transformation for Force in the transverse direction and the interpretation of the Right Angle Lever Thought Experiment, as described in many texts, are incorrect. A final result is the determination of the manner in which the kinetic energy associated with a moving object (particle) is stored. It is also shown that the reason that it is considered that Special Relativity is not valid for accelerated reference frames is that it defines mass in different terms than those commonly used in mechanics. The difficulty with acceleration in Special Relativity vanishes when the conventionally employed definition of mass is used and it is recognized that inertial mass is the incremental impulse required to produce an incremental change in velocity rather than the integral of that incremental impulse between the velocity limits of zero and 'V' as is the current practice.
Sections
Section Titles 0 Summary 1 Introduction 2 Groundwork of Discussion 3 A Comparison of the Velocity Difference Between Velocity
Reference Frames B and C as Observed in Reference Frame B
and as Observed in Reference Frame A 4 Determination of the Lorentz Transformation for Incremental Mass and
for Force Between Reference Frames Having Relative Velocity References 5 The Balance of Moments Applied to a Right Angled Lever in Velocity
Reference Frame B Moving with Velocity 'V' with Respect to Velocity
Reference Frame A as Observed in Reverence Frames A and B 6 The Conventional Lorentz Transformation for Transverse Force as
Related to the Right Angle Lever Thought Experiment 7 The Lorentz Transformation for Parallel and Transverse Force as
Related to a Compressed Spring Thought Experiment
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Section 1 - Introduction:-
1.1- The purpose of this paper is to develop a system of force, length, and time transformations between reference frames having a relative velocity equivalent to the mass, length, and time system of Lorentz Transformations currently in use. The reason for revising the system of Lorentz Transformations is that, unlike force, mass is not directly observable. The mass of an object (particle) is a property which can only be determined by a measurement which involves force, length, and/or time.
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Section 2 - Groundwork of Discussion:-
2.1 - At first glance, the conversion to a force, length, and time system of transformations would appear to have already been accomplished. Minkowski has already provided the Lorentz Transformations for Force in directions parallel to and transverse to the relative velocity vector between reference frames. Unfortunately, an error has been made in the development of the Lorentz Transformation for Transverse Force. This error was revealed by the classical Right Angle Lever Thought Experiment (described later) but, instead of this thought experiment resulting in the correction of the error, the error was rationalized by an explanation which ignored basic mechanical principles. Investigation suggests that the error in the determination of the Lorentz Transformation for Transverse Force resulted from the fact that it was derived by foolishly using Maxwell's Equations in a manner for which they were not suitable.
2.2- The procedure employed by this paper in developing the correct Lorentz Transformation for Force in directions parallel to and transverse to the relative velocity vector between reference frames is, for the parallel and transverse directions:
- To determine the incremental energy required in the 'stationary' and the 'moving' reference frame to produce an incremental change in velocity.
- To determine the force required in the 'stationary' and the 'moving' reference frame required to impart that energy.
- To determine the Lorentz Transformation for Force by comparing the force in the 'stationary' and the 'moving' reference frame.
2.3- The first step in the procedure consists of determining the Lorentz Transformations for Parallel and Transverse Velocities between the 'stationary' and the 'moving' reference frames. This is accomplished by employing the equation for the addition of velocities provided by Special Relativity to provide the relationship between a small velocity as observed in the 'moving' reference frame and that same small velocity as observed in the 'stationary' reference frame. It is then shown that the Lorentz Transformation for Incremental Velocity can be approximated by the by the Lorentz Transformation for Length divided by the Lorentz Transformation for Time with negligible error.
2.4- The Lorentz Transformation for Parallel and for Transverse Forces are then determined by determining the incremental impulse required to produce an incremental velocity change in the 'stationary' and 'moving' reference frames to determine the Incremental Mass (mass in the classical mechanical sense) of the object (particle) for both of these reference frames and the Lorentz Transformations for that 'Incremental Mass' between those reference frames. The Lorentz Transformations for Force are determined by dividing the product of the Lorentz Transformations for 'Incremental Mass' and for incremental velocity by the Lorentz Transformation for Time.
2.5- The results obtained agree with the Lorentz Transformation for Parallel force as derived by Minkowski. The results are reciprocal to the conventionally accepted Lorentz Transformations for Transverse Force, however, they are in agreement with a rigorous treatment of the classic Right Angle Lever Thought Experiment. In order to deal with the discrepancy, the textbook treatment of the Right Angle Lever Thought Experiment is discussed. It is shown where the conventional textbook treatment of that thought experiment is deficient.
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Section 3 -A Comparison of the Velocity Difference Between Velocity Reference Frames B and C as Observed in Reference Frame B and as Observed in Reference Frame A.
3.1- Consider three collinear velocity reference frames designated as A, B, and C. As observed in reference frame B, reference frame C has a velocity of +(dv) and reference frame A has a velocity of -vA. As observed in reference frame A, reference frame B has a velocity of +VB and reference frame C has a velocity of +VC. What is to be determined is the velocity, (dV), of reference frame C with respect to reference frame B as observed in reference frame A:
Eq. 3.1.1 (dV) = VC-VB
3.2- Special Relativity provides the velocity of reference frame C with respect to reference frame A in terms of the velocity of reference A and C with respect to reference frame B. In terms of the previous paragraph, this expression is:
Eq. 3.2.1 VC = [-vA+(dv)]/[1 -vA*(dv)/C2]
But, since the relative velocity between reference frames A and B must be equal and opposite in their respective reference frames if a preferred reference frame is not to be directly observable:
Eq. 3.2.2 vA = -VB
Then:
Eq. 3.2.3 VC = [VB+(dv)]/[1 + VB*(dv)/C2]
Combining Equations 3.1.1 and 3.2.3 provides:
Eq. 3.2.4 (dV) = (dv)*[1-VB2/C2]/[1 +VB*(dv)/C2]
And, if (dv) and V are sufficiently small compared to the velocity of light, one may write without significant error:
Eq. 3.2.5 (dv)= (dV)/[1-VB2/C2]
Where:
- '(dv)' is the velocity of reference frame C with respect to reference frame B as observed in reference frame B.
- '(dV)' is the velocity of reference frame C with respect to reference frame B as observed in reference frame A.
- The upper case letter 'V' refers to observations of velocity made with the units of measurement of reference frame A.
- The lower case letter 'v' refers to observations of velocity made with the units of measurement of reference frame B.
3.3 - The conventional Lorentz Transformations for Length and Time are given by:
Eq. 3.3.1 lp = Lp/[1-V2/C2]0.5
Eq. 3.3.2 lt = Lt
Eq. 3.3.3 t = T*[1-V2/C2]0.5
Where:
- The symbol 'l' refers to a length in reference frame B as observed in reference frame B.
- The symbol 'L' refers to the same length in reference frame B as observed in reference frame A.
- The symbol 't' refers to a time duration in reference frame B as observed in reference frame B.
- The symbol 'T' refers to that same duration of time in reference frame B as observed in reference frame A.
- The subscript 'p' refers to directions parallel to the velocity vector between reference frames A and B.
- The subscript 't' refers to directions transverse to the velocity vector between reference frames A and B.
3.4 - It will be noted that Equation 3.2.5 is the same as that which would be obtained by dividing the Lorentz Transformation for Parallel Length by the Lorentz Transformation for Time to obtain the Lorentz Transformation for Velocity (length/time). It follows, therefore that Equation 3.2.5 may be restated as:
Eq. 3.4.1 vp = Vp/[1-VB2/C2]
Where:
- 'vp' refers to a small velocity within reference frame B as observed in reference frame B in a direction parallel to the relative velocity between reference frames A and B.
- 'Vp' refers to that same velocity as observed in reference frame A.
The Lorentz Transformation between reference frames A and B for incremental velocities in reference frame B which are transverse to the velocity vector between reference frames A and B can be obtained by dividing the Lorentz Transformation for Transverse Length, Equation 3.3.2, by the Lorentz Transformation for time, Equation 3.3.3 to obtain:
Eq. 3.4.2 vt = Vt/[1-VB2/C2]0.5
The error in the determination of the Lorentz Transformation for Transverse Velocity given by Equation 3.4.2 is negligible compared to the already negligible error in the determination of Equation 3.4.1 since the simultaneity correction represented by the denominator of Equation 3.2.4 does not apply.
3.5- Since acceleration is defined as the rate of change of velocity with respect to time, the parallel and transverse transformations for acceleration may be obtained by combining Equations 3.3.2, 3.4.1, and 3.4.2. Providing the velocities represented by Equations 3.4.1 and 3.4.2 represent an incremental change in velocity within reference frame B which is changing uniformly with respect to time, accelerations observed in each reference frame are given by:
Eq. 3.5.1 Ap = (dVp)/(dT)
Eq. 3.5.2 ap = (dvp)/(dt)
Eq. 3.5.3 At = (dVt)/(dT)
Eq. 3.5.4 at = (dvt)/(dt)
Where:
- 'A' is an acceleration in reference frame B as observed in reference frame A.
- 'a' is the same acceleration in reference frame B as observed in reference frame B.
The Lorentz Transformations for Acceleration then become:
Eq. 3.5.5 ap = Ap/[1-VB2/C2]1.5
Eq. 3.5.6 at = At/[1-VB2/C2]
It should be noted that, unlike the equations for the velocity transformations, the acceleration transformations of Equations 3.5.5 and 3.5.6 are not limited to incremental values. Large accelerations for infinitesimal periods of time produce the infinitesimal changes in velocity required to satisfy the approximation represented by Equation 3.3.4. It should also be noted that Equations 3.5.5 and 3.5.6 are compatible with the classical meaning of acceleration as the second derivative of position with respect to time.
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Section 4 - Determination of the Lorentz Transformation for Incremental Mass and for Force Between Reference Frames Having Relative Velocity
4.1 - In this Section, the Lorentz Transformations for Parallel and for Transverse Force will be determined to allow the description of phenomena defined by Special Relativity to be treated in terms of a force (F), length (L), and time (T) system of units rather than the conventional mass (M), length (L), and time (T) system. The reason for this change is the fact that, unlike force, length, and time, mass cannot be observed directly but must be inferred from measurements which involve force, length, and time. The mass of an object (particle) can only be measured in terms of:
- Inertial properties which involve the incremental impulse (force-time product) required to produce an incremental velocity (length/time ratio) change. (An incremental velocity change occurs when photons undergo grazing angle reflection.)
- Gravitational properties which involve the force-length2 product which exists between it and another mass having a known magnitude.
- The energy (force-length product) released when an object (particle) is annihilated. The released energy retains the inertial properties of the original object (particle).
4.2 - This discussion will be based upon the classical definition of inertial mass. This mass, Mi, designated as the incremental mass of an object (particle), is defined as the infinitesimal impulse required to produce an infinitesimal velocity change. The Lorentz Transformation for Incremental Mass in the parallel and transverse directions and the Lorentz Transformations for Parallel and Transverse force is derived below.
4.3 - Consider again two reference frames, A and B, having a relative velocity V and with a material object (particle) in reference frame B. As observed in reference frame B, the object has a mass of 'm'. If that object is brought to rest in reference frame A it will impart a momentum to reference frame A which is equal to M*V where 'M' is the mass of the object as observed in reference frame A as provided by the conventional Lorentz Transformation for mass:
Eq. 4.3.1 M = m/[1-V2/C2]0.5
4.4 - Let us now consider that the velocity of the object in reference frame B is changed incrementally in a direction parallel to the velocity vector between reference frames A and B. What is to be determined first is the incremental mass of the object as observed in reference frame A, as revealed by the incremental impulse required to produce the incremental velocity change. The first step is to differentiate Equation 4.3.1 with respect to V to provide:
Eq. 4.4.1 (dMp) = m*V*(dVp)/(C2*[1-V2/C2]/C2]1.5)
The relationship between energy and mass is:
Eq. 4.4.2 E = M*C2
Differentiating Equation 4.4.2 provides:
Eq. 4.4.3 (dEp) = C2*(dMp)
Then, combining Equations 4.4.1 and 4.4.3:
Eq. 4.4.4 (dEp) =m*V*(dVp)/[1-V2/C2]1.5
But (dEp) is the energy, as observed in reference frame A, which was added to (or removed from) the object in order to produce the incremental change in velocity. This energy is given by:
Eq. 4.4.5 (dEp) = Fp(dLp)
Where:
- '(dLp)' is the distance the object travels in reference frame A during the incremental velocity change which occurs in the incremental time (dT)
- 'Fp' is the force applied to the object to change its velocity, all as observed in reference frame A. Equation 4.4.5 may then be written as:
Eq. 4.4.6 (dEp) = Fp*V*(dT)
Combining Equations 4.4.4 and 4.4.6 provides:
Eq. 4.4.7 Fp = m*(dVp)/[(dT)*(1-V2/C2)1.5]
The incremental mass, Mip, of the object in reference frame B as observed in reference frame A is given by:
Eq. 4.4.8 Mip = Fp*(dT)/(dVp)
Therefore:
Eq. 4.4.9 Mip = m/[1-V2/C2]1.5
4.5 - Since the Lorentz Transformation for the derivative of a quantity is identical to the Lorentz Transformation for the quantity itself, combining Equation 4.4.7 with Equations 3.3.2 and 3.4.1 provides:
Eq. 4.5.1 Fp = m*(dvp)/(dt)
But, since the terms in lower case letters refer to observations of quantities in reference frame B made within reference frame B, the conventional laws of mechanics apply:
Eq. 4.5.2 fp = m*(dvp)/(dt)
Then the Lorentz Transformation for Force in a direction parallel to the relative velocity vector becomes:
Eq. 4.5.3 fp = Fp
It will be noted that Equation 4.5.3 is identical to the transformation provided by Minkowski.
4.6 - The determination of the Lorentz Transformation for a force in a direction perpendicular to the relative velocity vector between reference frames A and B for the incremental mass is accomplished in a similar manner, Consider an object in reference frame B which is subjected to an incremental impulse in a direction perpendicular to the relative velocity vector between reference frames A and B. As a result of this impulse, the object acquires an incremental velocity transverse to the velocity vector between A and B of (dvt), as observed in reference frame B, and of (dVt) as observed in reference frame A. Equation 3.2.5 provides:
Eq. 4.6.1 (dvt) = (dVt)/[1-V2/C2]0.5
4.7 - Prior to the incremental acceleration, the mass of the object in reference frame B as observed in reference frames A and B is given by Equation 4.3.1. Following the incremental transverse acceleration, the relative velocity, V1, between the object and reference frame A has increased and is equal to:
Eq. 4.7.1 V1 = [V2 +(dVt)2]0.5
The mass of the object, as observed in reference frame A is now equal to:
Eq. 4.7.2 M1 = m/[1-v2/C2-(dVt)2/C2]0.5
The incremental change in mass of the object as observed in reference frame A is given by:
Eq. 4.7.3 (dMT) = M1-M
Or, since (dVt) represents an infinitesimal velocity change, we may write without significant error:
Eq. 4.7.4 (dMt) = m*C*(dVt)2/(2*[C2-V2]1.5)
Differentiating Equation 4.4.2 for the transverse direction provides:
Eq. 4.7.5 (dEt) = C2*(dMt)
Then:
Eq. 4.7.6 (dEt) = m*(dVt)2/(2*[1-V2/C2])1.5
Assuming for simplicity that the force, F, applied to the object to produce the incremental velocity change, as observed in reference frame A, was unchanged during the velocity change, the energy supplied, as observed in reference frame A, is given by:
Eq. 4.7.7 (dEt) = F*(dVt)*(dT)/2
Where (dT) is the time which the force was applied. Then:
Eq. 4.7.8 Ft*(dT) = m*(dVt)/([1-V2/C2]1.5)
Or, since in terms of reference frame A:
Eq. 4.7.9 Ft*(dT) = Mit*(dVt)
Then:
Eq. 4.7.10 Mit = m/([1-V2/C2}1.5)
Combining Equation 4.4.9 with Equation 4.7.10 provides:
Eq. 4.7.11 Mit = Mip
Equation 4.7.11 shows that, as is the case with relativistic mass, the instantaneous mass of an object is not a vector quantity. Therefore, we may write, for the instantaneous mass in any direction, Mi:
Eq. 4.7.12 Mi = m/([1-V2/C2]1.5)
4.8 - Equation 4.7.10 allows the transformation for force in the transverse direction to be determined since, in reference frame B:
Eq. 4.8.1 f*(dt) = m*(dv)
And in reference frame A:
Eq. 4.8.2 F*(dT) = Mi*(dV)
Combining the Lorentz Transformation for Transverse Length (3.3.2) and for Time (3.3.3) with Equations 4.7.10, 4.8.1 and 4.8.2 provides:
Eq. 4.8.3 ft = Ft*([1-V2/C2]0.5)
4.9 - It will be noted that Equation 4.8.3 is the reciprocal of the conventionally accepted Lorentz Transformation for Transverse Force. The next Section will discuss the significance of the reciprocal relationship between Equation 4.8.3 and the conventionally accepted Lorentz Transformation for Transverse Force in terms of the Moving Right Angle Lever Thought Experiment. Applied in a straight forward manner, that thought experiment will now yield results with Equations 4.5.3 and 4.8.3. The arguments which have been employed to reconcile the thought experiment with the conventional (and erroneous) Lorentz Transformation for Transverse Force will be discussed in a later Section.
4.10 - It will also be noted that, by incorporating the concept of instantaneous mass, Mi, (mass in the classical sense), Special Relativity becomes valid for accelerated reference frames as well as for reference frames having a constant velocity. This result obtains because the Lorentz Transformation for Instantaneous Mass is identical to the Lorentz Transformation for Momentum (M*V) which is a component of the four vector mass transformation currently in use. The conventional Lorentz Transformation for Mass may be derived from the Lorentz Transformation for Incremental Mass by integrating the incremental impulse required to produce the relative velocity of the incremental mass with respect to the observer's reference frame and dividing that integral by the velocity of the object (particle).
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Section 5 - The Balance of Moments Applied to a Right Angled Lever in Velocity Reference Frame B Moving with Velocity V with Respect to Velocity Reference Frame A as Observed in Reverence Frames A and B.
5.1 - Consider a right angled level, as shown in Figure 5.1.1, located in reference frame B which is moving with velocity V with respect to reference frame A. One arm of the lever is parallel to the relative velocity between A and B and the other arm is transverse to that velocity. For simplicity, both lever arms are considered to be of equal length as observed in reference frame B. The definitions provided in Sections 3 and 4 are retained.
5.2 - A force, Fp, parallel to the relative velocity is applied to the transverse arm and generates a force, Fhp, equal in magnitude and opposite in direction to Fp at the pivot pin resulting in a couple which causes a torque to be applied to the lever. Another force, Ft, transverse to the relative velocity is applied to the parallel arm of the lever and generates a force, Fht, equal in magnitude and opposite in direction to Ft at the pivot pin resulting in a couple which causes a torque to be applied to the lever opposing the torque generated by Fp. The lever is observed not to rotate in either reference frame.
5.3 - As observed in reference frame A, energy is added to the transverse arm of the lever by force Fp at a rate given by:
Eq. 5.3.1 (dEp) = Fp*V*(dT)
and the energy added to the transverse arm of the lever by the force at the hinge pin at a rate given by:
Eq. 5.3.2 (dEhp) = Fhp*V*(dT)
The total rate that energy is added to the transverse arm of the lever is given by:
Eq. 5.3.3/TD>(dE) = (dEp)+(dEhp)
But, since Fp and Fhp are equal in magnitude and opposite in direction:
Eq. 5.3.4 (dE)/(dT) = 0
5.4 - From the preceding, in terms of reference frame A, energy enters the lever at the end of the transverse arm, flows laterally through the transverse arm, and exits the lever through the hinge pin. None of this energy remains in the lever. Since the transverse velocity of the lever is zero in both reference frames (A and B), no energy enters or leaves the parallel arm in either reference frame.
5.5 - The angular momentum of the lever is the product of its moment of inertia and its angular velocity, Since the lever is observed not to rotate in either reference frame A or B, its angular velocity remains at zero in both reference frames as does the rate of change of that angular velocity. The zero rate of change of angular velocity means that the rate of change of angular momentum is also zero in both reference frames. Since the rate of change of angular momentum is zero in both reference frames, the net torque applied to the lever in both reference frames must also be zero. We may write, therefore:
Eq. 5.5.1 Ft*Lp = Fp*Lt
Eq. 5.5.2 ft*lp = fp*lt
Combining Equations 3.3.1, 3.3.1, 4.5.3, 5.5.1, and 5.5.2 provides:
Eq. 5.5.3 ft = Ft*([1-V2/C2]0.5)
5.6 - It will be noted that Equations 4.8.3 and 5.5.3 are identical. This identity shows that a rigorous application of the Right Angle Lever Thought Experiment validates the Lorentz Transformation for Transverse Force provided by Equation 4.8.3. It will also be noted that Equation 4.8.3 is the reciprocal of the conventionally accepted Lorentz Transformation for Transverse Force. This difference is discussed in the next Section.
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Section 6 - The Conventional Lorentz Transformation for Transverse Force as Related to the Right Angle Lever Thought Experiment
6.1 - Equation 4.8.3, which provides the Lorentz Transformation for Transverse Force, is the reciprocal of the conventionally accepted equation for that transformation. As was shown in the previous Section, Equation 4.8.3 is consistent with the results of a rigorous analysis of the Right Angle Lever Thought Experiment. It would appear, therefore, that an error was made in the derivation of the conventional Lorentz Transformation for Transverse Force, and rather than using the paradox introduced into the Right Angle Lever Thought Experiment by this error as an indication of its existence and responding by correcting the error, a rather convoluted and faulty reasoning process was employed to explain away the Right Angle Lever Paradox.
6.2 - Considerable effort was expended in the resolution of this conflict in the years following the publication of the Special Theory of Relativity. This result of this effort produced some rather surprising arguments. These arguments, which occur in more than one recognized text, are typified by the description of the Right Angle Lever Thought Experiment provided in one of the accepted texts on the subject which shall be unnamed (see note at the end of this Section). Summarized and converted to the terminology of this discussion, these statements are:
Eq. 6.2.1 ft = Ft/([1-V2/C2]0.5)
The transformation of Equation 6.2.1 is the inverse of Equation 4.8.29, and, as observed in reference frame A, the external forces applied to the right angle lever produce a torque, T1 as given by:
Eq. 6.2.2 T1 = Fp*Lt*V2/C2
The force, Fp, is doing work on the lever at the rate:
Eq. 6.2.3 (dE)/(dT) = Fp*V
The angular momentum, J, of the lever is increasing at the rate:
Eq. 6.2.4 (dJ)/(dT) = -Fp*Lt*V2/C2
The increase in angular momentum produces a counter torque, T2, of:
Eq. 6.2.5 T2 = -Fp*Lt*V2/C2
The net torque, TN, applied to the lever is given by:
Eq. 6.2.6 TN = T1+T2
Or:
Eq. 6.2.7 TN = 0
And, as a result, the lever does not rotate.
6.3 - In analyzing the Right Angle Lever Thought Experiment, as described in conventional texts, one must conclude that:
- No energy is added to the lever as observed in reference frame A as a result of the application of the application of the force FP. The energy added by FP flows along the transverse arm of the lever and exits the lever as a result of the equal and opposite force, FHP, existing at the hinge pin (Eq. 5.2).
- The effects of the force, FHP, have been ignored in textbooks on Special Relativity.
- Since the lever is observed not to rotate in either reference frame its angular momentum is constant at a value of zero since its angular momentum is the product of its moment of inertia and its angular velocity. The rate of change of its angular momentum is therefore also equal to zero.
- Any net torque applied to the Right Angle Lever by the forces Fp and Ft cannot be compensated by a change in angular momentum of the right angle lever. The torque resulting from a change in angular momentum is equal to zero.
6.4 - The Right angle Lever Thought Experiment requires that the torques applied to the lever by Fp and Ft must be equal and opposite unless:
- As observed in reference frame A, the flow of energy along the transverse arm of the lever produces a torque couple acting on it which is independent of the couple represented by Fp and Fhp.
- A parallel force exerted on the transverse arm by the flow of energy would not satisfy the requirement. It would result in a readily observable inequality between Fp and Fhp.
- For such a couple to exist, there must be an entity for the couple to react against. No such entity has been proposed.
6.5 - Since the torques applied to the lever by Fp and Ft must be equal and opposite, it follows that Equation 4.8.3 is the correct expression for the Lorentz Transformation for Transverse force and the presently accepted transformation (Equation 6.2.1) is incorrect. (The author does not know the nature of the error in the derivation of Equation 6.2.1 since he has not examined the derivation, nor is he interested. The fact that the error(s) produce an invalid result is sufficient to show that at least one error exists. It is unfortunate that theoreticians have not spent as much time searching for the error(s) as they did in explaining away the Right Angle Lever Paradox.)
Note:- The text referred to is not named because the error involved is of a type which would not be excused if made by a student of freshman physics. For the error to be included in texts written by individuals possessing PhD's in Physics which are used for teaching is appalling. It is not, however, the author's wish to embarrass individuals by naming them.
6.6 - (Update to Corrections to Special Relativity [2005]). The author has received an E-mail response which asserted that he had re-derived the Lorentz Transformation for Transverse Force using Maxwell's Electromagnetic Equations and verified that the conventionally accepted expression was correct and that the expression provided in this article is incorrect. Unfortunately for such a conclusion, it must be pointed out that his assertion is false! Since Maxwell's Equations involve the velocity of light, in order to apply them for this purpose, it is first necessary to correct them for the change in the ABSOLUTE velocity of light caused by a change of a velocity reference frame.
6.7 - (Update to Corrections to Special Relativity [2005])When Special Relativity was published, it was asserted that the Special Theory of Relativity showed that the velocity of light was the same in all reference frames. This statement is quite true as long as one considers effects which occur within a single reference frame. Special Relativity does not show that the velocity is the same, in absolute terms, between reference frames.
6.8 - (Update to Corrections to Special Relativity [2005]). In order to determine whether the velocity of light, which is observed to be the same not only within a refernce frame but between all reference frames, one must first determine the Lorentz Transformation for Velocity. Since velocity is equal to length/time, it followsd that the Lorentz Transformation for Velocity must be equal to the Lorentz Transformation for time (1/(1-V^2/C^2)^0.5 divided by the Lorentz Transformations for Time (1-V^2/C^2)^0.5. It follows, therefore, that the Lorentz Transformation for Velocity is equal to 1/(1-V^2/C^2)! It is not equal to unity. This transformation tells us that if, as we observe, the velocity of light has its nominal velocity of C WITHIN any given reference frame, it cannot retain that property BETWEEN reference frames which have a relative velocity. Einstein's UNPROVEN assertion that the velocity of light is constant is misleading. The velocity of light is "a constant" when measured within a reference frame. IT IS NOT CONSTANT when observed between reference frames which differ in velocity and/or elevations. The alleged refutation of the Lorentz Transformation for Transverse Force does not stand, The Transformation provided in this article remains valid.
:
Section 7 - The Lorentz Transformation for Parallel and Transverse Force as Related to a Compressed Spring Thought Experiment.
7.1 - The revision of the Lorentz Transformation for Transverse Force provided by Equation 4.8.29 satisfies two requirements:
- It is consistent with a rigorous analysis of the Right Angle Lever Thought Experiment.
- It is consistent with the effects which must be observed when an object (particle) having a velocity with respect to the observer acquires an additional incremental velocity which is perpendicular to the original velocity. In the observer's reference frame:
- The magnitude of the relative velocity will increase incrementally to produce an incremental increase in the relativistic mass of the object (particle).
- The energy (E=M*C2) represented by the increase in relativistic mass equals the incremental work done in the observer's reference frame in producing the incremental transverse velocity change.
On casual evaluation, the revision of the Lorentz Transformation for Transverse Force would seem to lead to a different difficulty, as shown by the Compressed Spring Thought Experiment illustrated in Figure 7.1.1.
7.2 - Consider an object (particle) in which energy is stored anisotropically. Such an object might be a compressed spring as illustrated in Figure 7.1.1. In this thought Experiment, two ideal and massless springs are compressed and tied to store energy. One spring is oriented so as to store its energy in a direction parallel to the velocity vector between reference frame A and B. The other spring is orientated so as to store its energy in a direction transverse to that velocity vector. For simplicity, it will be assumed that the energy storage and compressed distance of both springs are identical, as observed in reference frame B.
7.3 - As observed in reference frame B, the energy stored in the transverse spring is:
Eq. 7.3.1 et = dt*ft/2
:
And the energy, ep, stored in the parallel spring is:
Eq. 7.3.2 ep = dp*fp/2
As observed in reference frame A, the energy, Et, stored in the transverse spring is:
Eq. 7.3.3 Et = Dt*Ft/2
And the energy, Ep, stored in the parallel spring is:
Eq. 7.3.4 Ep = Dp*Fp/2
Where:
- 'D' is the deflection of a spring as observed in reference frame A.
- 'd' is the deflection of the same spring as observed in reference frame B.
- 'E' is the energy stored in a spring as observed in reference frame A.
- 'e' is the energy stored in the same spring as observed in reference frame B.
- All other symbols and subscripts are as previously defined. Since the symbols 'D' and 'd' refer to lengths and the symbols 'F' and 'f' refer to forces, the Lorentz Transformations for Parallel and Transverse Lengths and Forces apply.
Since it has been arbitrarily defined that, as observed in reference frame B, the stored energy and the compression distance of the parallel and transverse s prings are equal, we may write:
Eq. 7.3.5 et = ep
Eq. 7.3.6 ft = fp
Eq. 7.3.7 dt = dp
Combining Equations 3.3.1, 3.3.2, 4.5.3, 4.8.3, 7.3.1, 7.3.2, 7.3.3, 7.3.4, 7.3.5, 7.3.6 and 7.3.7 provides:
Eq. 7.3.8 Ep = Et*[1-V2/C2]
Eq. 7.3.9 ep = Ep/([1-V2/C2]0.5)
Eq. 7.3.10 et = Et*([1-V2/C2]0.5)
7.4 - In combination, Equations 7.3.5 and 7.3.8 appear to pose a dilemma. Special Relativity was derived based upon the principle that all velocity reference frames within the limits of +/-C are equally valid as a basis for all physical measurements (Principle of Relativity). Equation 7.8 shows that, as observed in reference frame A, the compressed spring that is parallel to the relative velocity vector has stored less energy than did the compressed spring which is transverse to the relative velocity vector. Equation 7.3.5 shows that, as observed in reference frame B, the energies stored in the two springs are equal.
7.5 - As observed in reference frame A, the transverse spring loses stored energy as it is rotated to the parallel direction, As observed in reference frame B it does not lose stored energy. One would expect that a loss of energy, as observed in reference frame A, stored in the spring as a result of the change in its orientation would result in a torque which would tend to rotate the spring towards the parallel orientation. The existence of such a torque in reference frame A and its non-existence in reference frame B would compromise the Principle of Relativity by allowing a 'preferred' velocity reference frame to be established.
7.6 - The effect suggested by the previous paragraph should not occur. In order for an object (particle) to acquire a velocity with respect to reference frame A, kinetic energy must be added to the object (particle) and the mass of that object (particle) is increased by the amount of the kinetic energy added in accordance with E=M*C2. When the object (particle) is eventually brought to rest with respect to reference frame A, it gives up that kinetic energy and reverts to its original rest mass, One would conclude, therefore, that the energy is stored in the moving object (particle) and/or in the space in the immediate vicinity of the moving object (particle). One is then faced with the question as to where and how that kinetic energy is stored. To investigate that question, let us consider that the rest mass of an object (particle) results form the storage of energy within the object (particle) in a manner equivalent to the storage of energy in the parallel and transverse springs of Figure 7.1.1.
7.7 - Let us first consider the storage of kinetic energy in the spring which is transverse to the velocity vector, as revealed by Equation 7.3.10. Dividing this Equation by C2 provides:
Eq.7.7.1 mt = Mt*([1-V2/C2])
Since Equation 7.7.1 is the conventional Lorentz Transformation for Mass, it follows that, as observed in terms of reference frame A, the kinetic energy of energy stored along a transverse axis of the object (particle) in reference frame B is stored as an increase of the stored energy along that axis. This increase in stored energy results from an increase in stiffness of the 'spring' which stores the 'rest mass' energy. (The deflection of the 'spring' is the same in both reference frames while the force, in reference frame A, producing the deflection has increased. This is equivalent to an increase in the 'stiffness' of the 'spring'.)
7.8 - The storage of the kinetic energy of the compressional energy stored in the parallel spring is not so readily explained. Dividing Equation 7.3.9 by C2 provides:
Eq.7.8.1 mp = Mp*([1-V2/C2])
Since this Equation is the reciprocal of the Lorentz Transformation for Mass, it indicates that the relativistic mass of the compressional energy of the parallel spring decreases as observed in reference frame A as the velocity of reference frame B with respect to A is increased. The decrease in the relativistic mass of the compressed energy in the parallel spring results from a decrease in the deflection of the spring, as observed in reference frame A, while the force applied to the spring remains unchanged. Unlike the transverse spring, the parallel spring does not store the kinetic energy associated with its compressional energy within itself. In terms of reference frame A, its kinetic energy and part of its rest mass energy must be stored in the space Figure 7.8.1 around the spring since that energy is transported with the spring but is not stored in the spring itself. (It should be noted that such a conclusion would also have been erroneously attributed to the transverse spring under the conventional Lorentz Transformation for Transverse Force, Equation 6.2.1)
7.10 - Since Equation 5.11 shows that, as observed in reference frame A, the storage of both rest mass and kinetic energy in the transverse spring , Et, accounts for the relativistic mass of the energy stored in the transverse spring, it follows that:
Eq. 7.10.1 Ert = Et
Where:
- 'Ert' is the energy equivalent of the transversely stored relativistic mass of the object (particle), as observed in reference frame
A.
Similarly, the total energy stored in the parallel spring, Ep is given by:
Eq. 7.10.2
Ep = Erp*[1-V2/C2]
In order for this to occur, the parallel spring must shed energy from within itself and carry that energy along with it in a disk shaped region of the space around it, with the disk oriented perpendicular to the velocity vector. The energy contained in that disk, as observed in reference frame A, must equal:
Eq. 7.10.3 Es = Er*(V2/C2)/(1-V2/C2)0.5
Where:
- 'Es' is the energy, as observed in reference frame A, stored in the space in the vicinity of the object (particle) as a result of its velocity with respect to reference frame A.
:
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2:- Eliminating the Discrepancy Between Relativistic and Classical Inertial Mass
3:- Correcting the Error in the Lorentz Transformation for Transverse Force
4:- Resolving the Right Angle Lever Paradox
5:- The Source of Inertial Mass
Sections | Section Titles |
|---|---|
| 0 | Summary |
| 1 | Introduction |
| 2 | Groundwork of Discussion |
| 3 | A Comparison of the Velocity Difference Between Velocity Reference Frames B and C as Observed in Reference Frame B and as Observed in Reference Frame A |
| 4 | Determination of the Lorentz Transformation for Incremental Mass and for Force Between Reference Frames Having Relative Velocity References |
| 5 | The Balance of Moments Applied to a Right Angled Lever in Velocity Reference Frame B Moving with Velocity 'V' with Respect to Velocity Reference Frame A as Observed in Reverence Frames A and B |
| 6 | The Conventional Lorentz Transformation for Transverse Force as Related to the Right Angle Lever Thought Experiment |
| 7 | The Lorentz Transformation for Parallel and Transverse Force as Related to a Compressed Spring Thought Experiment |
- To determine the incremental energy required in the 'stationary' and the 'moving' reference frame to produce an incremental change in velocity.
- To determine the force required in the 'stationary' and the 'moving' reference frame required to impart that energy.
- To determine the Lorentz Transformation for Force by comparing the force in the 'stationary' and the 'moving' reference frame.
Eq. 3.1.1 (dV) = VC-VB
Eq. 3.2.1 VC = [-vA+(dv)]/[1 -vA*(dv)/C2]
But, since the relative velocity between reference frames A and B must be equal and opposite in their respective reference frames if a preferred reference frame is not to be directly observable:
Eq. 3.2.2 vA = -VB
Then:
Eq. 3.2.3 VC = [VB+(dv)]/[1 + VB*(dv)/C2]
Combining Equations 3.1.1 and 3.2.3 provides:
Eq. 3.2.4 (dV) = (dv)*[1-VB2/C2]/[1 +VB*(dv)/C2]
And, if (dv) and V are sufficiently small compared to the velocity of light, one may write without significant error:
Eq. 3.2.5 (dv)= (dV)/[1-VB2/C2]
Where:
- '(dv)' is the velocity of reference frame C with respect to reference frame B as observed in reference frame B.
- '(dV)' is the velocity of reference frame C with respect to reference frame B as observed in reference frame A.
- The upper case letter 'V' refers to observations of velocity made with the units of measurement of reference frame A.
- The lower case letter 'v' refers to observations of velocity made with the units of measurement of reference frame B.
Eq. 3.3.1 lp = Lp/[1-V2/C2]0.5
Eq. 3.3.2 lt = Lt
Eq. 3.3.3 t = T*[1-V2/C2]0.5
Where:
- The symbol 'l' refers to a length in reference frame B as observed in reference frame B.
- The symbol 'L' refers to the same length in reference frame B as observed in reference frame A.
- The symbol 't' refers to a time duration in reference frame B as observed in reference frame B.
- The symbol 'T' refers to that same duration of time in reference frame B as observed in reference frame A.
- The subscript 'p' refers to directions parallel to the velocity vector between reference frames A and B.
- The subscript 't' refers to directions transverse to the velocity vector between reference frames A and B.
Eq. 3.4.1 vp = Vp/[1-VB2/C2]
Where:
- 'vp' refers to a small velocity within reference frame B as observed in reference frame B in a direction parallel to the relative velocity between reference frames A and B.
- 'Vp' refers to that same velocity as observed in reference frame A.
Eq. 3.4.2 vt = Vt/[1-VB2/C2]0.5
The error in the determination of the Lorentz Transformation for Transverse Velocity given by Equation 3.4.2 is negligible compared to the already negligible error in the determination of Equation 3.4.1 since the simultaneity correction represented by the denominator of Equation 3.2.4 does not apply.
Eq. 3.5.1 Ap = (dVp)/(dT)
Eq. 3.5.2 ap = (dvp)/(dt)
Eq. 3.5.3 At = (dVt)/(dT)
Eq. 3.5.4 at = (dvt)/(dt)
Where:
- 'A' is an acceleration in reference frame B as observed in reference frame A.
- 'a' is the same acceleration in reference frame B as observed in reference frame B.
Eq. 3.5.5 ap = Ap/[1-VB2/C2]1.5
Eq. 3.5.6 at = At/[1-VB2/C2]
It should be noted that, unlike the equations for the velocity transformations, the acceleration transformations of Equations 3.5.5 and 3.5.6 are not limited to incremental values. Large accelerations for infinitesimal periods of time produce the infinitesimal changes in velocity required to satisfy the approximation represented by Equation 3.3.4. It should also be noted that Equations 3.5.5 and 3.5.6 are compatible with the classical meaning of acceleration as the second derivative of position with respect to time.
- Inertial properties which involve the incremental impulse (force-time product) required to produce an incremental velocity (length/time ratio) change. (An incremental velocity change occurs when photons undergo grazing angle reflection.)
- Gravitational properties which involve the force-length2 product which exists between it and another mass having a known magnitude.
- The energy (force-length product) released when an object (particle) is annihilated. The released energy retains the inertial properties of the original object (particle).
Eq. 4.3.1 M = m/[1-V2/C2]0.5
Eq. 4.4.1 (dMp) = m*V*(dVp)/(C2*[1-V2/C2]/C2]1.5)
The relationship between energy and mass is:
Eq. 4.4.2 E = M*C2
Differentiating Equation 4.4.2 provides:
Eq. 4.4.3 (dEp) = C2*(dMp)
Then, combining Equations 4.4.1 and 4.4.3:
Eq. 4.4.4 (dEp) =m*V*(dVp)/[1-V2/C2]1.5
But (dEp) is the energy, as observed in reference frame A, which was added to (or removed from) the object in order to produce the incremental change in velocity. This energy is given by:
Eq. 4.4.5 (dEp) = Fp(dLp)
Where:
- '(dLp)' is the distance the object travels in reference frame A during the incremental velocity change which occurs in the incremental time (dT)
- 'Fp' is the force applied to the object to change its velocity, all as observed in reference frame A. Equation 4.4.5 may then be written as:
Combining Equations 4.4.4 and 4.4.6 provides:
Eq. 4.4.7 Fp = m*(dVp)/[(dT)*(1-V2/C2)1.5]
The incremental mass, Mip, of the object in reference frame B as observed in reference frame A is given by:
Eq. 4.4.8 Mip = Fp*(dT)/(dVp)
Therefore:
Eq. 4.4.9 Mip = m/[1-V2/C2]1.5
Eq. 4.5.1 Fp = m*(dvp)/(dt)
But, since the terms in lower case letters refer to observations of quantities in reference frame B made within reference frame B, the conventional laws of mechanics apply:
Eq. 4.5.2 fp = m*(dvp)/(dt)
Then the Lorentz Transformation for Force in a direction parallel to the relative velocity vector becomes:
Eq. 4.5.3 fp = Fp
It will be noted that Equation 4.5.3 is identical to the transformation provided by Minkowski.
Eq. 4.6.1 (dvt) = (dVt)/[1-V2/C2]0.5
Eq. 4.7.1 V1 = [V2 +(dVt)2]0.5
The mass of the object, as observed in reference frame A is now equal to:
Eq. 4.7.2 M1 = m/[1-v2/C2-(dVt)2/C2]0.5
The incremental change in mass of the object as observed in reference frame A is given by:
Eq. 4.7.3 (dMT) = M1-M
Or, since (dVt) represents an infinitesimal velocity change, we may write without significant error:
Eq. 4.7.4 (dMt) = m*C*(dVt)2/(2*[C2-V2]1.5)
Differentiating Equation 4.4.2 for the transverse direction provides:
Eq. 4.7.5 (dEt) = C2*(dMt)
Then:
Eq. 4.7.6 (dEt) = m*(dVt)2/(2*[1-V2/C2])1.5
Assuming for simplicity that the force, F, applied to the object to produce the incremental velocity change, as observed in reference frame A, was unchanged during the velocity change, the energy supplied, as observed in reference frame A, is given by:
Eq. 4.7.7 (dEt) = F*(dVt)*(dT)/2
Where (dT) is the time which the force was applied. Then:
Eq. 4.7.8 Ft*(dT) = m*(dVt)/([1-V2/C2]1.5)
Or, since in terms of reference frame A:
Eq. 4.7.9 Ft*(dT) = Mit*(dVt)
Then:
Eq. 4.7.10 Mit = m/([1-V2/C2}1.5)
Combining Equation 4.4.9 with Equation 4.7.10 provides:
Eq. 4.7.11 Mit = Mip
Equation 4.7.11 shows that, as is the case with relativistic mass, the instantaneous mass of an object is not a vector quantity. Therefore, we may write, for the instantaneous mass in any direction, Mi:
Eq. 4.7.12 Mi = m/([1-V2/C2]1.5)
Eq. 4.8.1 f*(dt) = m*(dv)
And in reference frame A:
Eq. 4.8.2 F*(dT) = Mi*(dV)
Combining the Lorentz Transformation for Transverse Length (3.3.2) and for Time (3.3.3) with Equations 4.7.10, 4.8.1 and 4.8.2 provides:
Eq. 4.8.3 ft = Ft*([1-V2/C2]0.5)
Eq. 5.3.1 (dEp) = Fp*V*(dT)
and the energy added to the transverse arm of the lever by the force at the hinge pin at a rate given by:
Eq. 5.3.2 (dEhp) = Fhp*V*(dT)
The total rate that energy is added to the transverse arm of the lever is given by:
Eq. 5.3.3/TD>
But, since Fp and Fhp are equal in magnitude and opposite in direction:
Eq. 5.3.4 (dE)/(dT) = 0
5.4 - From the preceding, in terms of reference frame A, energy enters the lever at the end of the transverse arm, flows laterally through the transverse arm, and exits the lever through the hinge pin. None of this energy remains in the lever. Since the transverse velocity of the lever is zero in both reference frames (A and B), no energy enters or leaves the parallel arm in either reference frame.
5.5 - The angular momentum of the lever is the product of its moment of inertia and its angular velocity, Since the lever is observed not to rotate in either reference frame A or B, its angular velocity remains at zero in both reference frames as does the rate of change of that angular velocity. The zero rate of change of angular velocity means that the rate of change of angular momentum is also zero in both reference frames. Since the rate of change of angular momentum is zero in both reference frames, the net torque applied to the lever in both reference frames must also be zero. We may write, therefore:
Eq. 5.5.1 Ft*Lp = Fp*Lt
Eq. 5.5.2 ft*lp = fp*lt
Combining Equations 3.3.1, 3.3.1, 4.5.3, 5.5.1, and 5.5.2 provides:
Eq. 5.5.3 ft = Ft*([1-V2/C2]0.5)
5.6 - It will be noted that Equations 4.8.3 and 5.5.3 are identical. This identity shows that a rigorous application of the Right Angle Lever Thought Experiment validates the Lorentz Transformation for Transverse Force provided by Equation 4.8.3. It will also be noted that Equation 4.8.3 is the reciprocal of the conventionally accepted Lorentz Transformation for Transverse Force. This difference is discussed in the next Section.
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Section 6 - The Conventional Lorentz Transformation for Transverse Force as Related to the Right Angle Lever Thought Experiment
6.1 - Equation 4.8.3, which provides the Lorentz Transformation for Transverse Force, is the reciprocal of the conventionally accepted equation for that transformation. As was shown in the previous Section, Equation 4.8.3 is consistent with the results of a rigorous analysis of the Right Angle Lever Thought Experiment. It would appear, therefore, that an error was made in the derivation of the conventional Lorentz Transformation for Transverse Force, and rather than using the paradox introduced into the Right Angle Lever Thought Experiment by this error as an indication of its existence and responding by correcting the error, a rather convoluted and faulty reasoning process was employed to explain away the Right Angle Lever Paradox.
6.2 - Considerable effort was expended in the resolution of this conflict in the years following the publication of the Special Theory of Relativity. This result of this effort produced some rather surprising arguments. These arguments, which occur in more than one recognized text, are typified by the description of the Right Angle Lever Thought Experiment provided in one of the accepted texts on the subject which shall be unnamed (see note at the end of this Section). Summarized and converted to the terminology of this discussion, these statements are:
Eq. 6.2.1 ft = Ft/([1-V2/C2]0.5)
The transformation of Equation 6.2.1 is the inverse of Equation 4.8.29, and, as observed in reference frame A, the external forces applied to the right angle lever produce a torque, T1 as given by:
Eq. 6.2.2 T1 = Fp*Lt*V2/C2
The force, Fp, is doing work on the lever at the rate:
Eq. 6.2.3 (dE)/(dT) = Fp*V
The angular momentum, J, of the lever is increasing at the rate:
Eq. 6.2.4 (dJ)/(dT) = -Fp*Lt*V2/C2
The increase in angular momentum produces a counter torque, T2, of:
Eq. 6.2.5 T2 = -Fp*Lt*V2/C2
The net torque, TN, applied to the lever is given by:
Eq. 6.2.6 TN = T1+T2
Or:
Eq. 6.2.7 TN = 0
And, as a result, the lever does not rotate.
6.3 - In analyzing the Right Angle Lever Thought Experiment, as described in conventional texts, one must conclude that:
- No energy is added to the lever as observed in reference frame A as a result of the application of the application of the force FP. The energy added by FP flows along the transverse arm of the lever and exits the lever as a result of the equal and opposite force, FHP, existing at the hinge pin (Eq. 5.2).
- The effects of the force, FHP, have been ignored in textbooks on Special Relativity.
- Since the lever is observed not to rotate in either reference frame its angular momentum is constant at a value of zero since its angular momentum is the product of its moment of inertia and its angular velocity. The rate of change of its angular momentum is therefore also equal to zero.
- Any net torque applied to the Right Angle Lever by the forces Fp and Ft cannot be compensated by a change in angular momentum of the right angle lever. The torque resulting from a change in angular momentum is equal to zero.
- The effects of the force, FHP, have been ignored in textbooks on Special Relativity.
- Any net torque applied to the Right Angle Lever by the forces Fp and Ft cannot be compensated by a change in angular momentum of the right angle lever. The torque resulting from a change in angular momentum is equal to zero.
6.4 - The Right angle Lever Thought Experiment requires that the torques applied to the lever by Fp and Ft must be equal and opposite unless:
- As observed in reference frame A, the flow of energy along the transverse arm of the lever produces a torque couple acting on it which is independent of the couple represented by Fp and Fhp.
- A parallel force exerted on the transverse arm by the flow of energy would not satisfy the requirement. It would result in a readily observable inequality between Fp and Fhp.
- For such a couple to exist, there must be an entity for the couple to react against. No such entity has been proposed.
- A parallel force exerted on the transverse arm by the flow of energy would not satisfy the requirement. It would result in a readily observable inequality between Fp and Fhp.
- For such a couple to exist, there must be an entity for the couple to react against. No such entity has been proposed.
6.5 - Since the torques applied to the lever by Fp and Ft must be equal and opposite, it follows that Equation 4.8.3 is the correct expression for the Lorentz Transformation for Transverse force and the presently accepted transformation (Equation 6.2.1) is incorrect. (The author does not know the nature of the error in the derivation of Equation 6.2.1 since he has not examined the derivation, nor is he interested. The fact that the error(s) produce an invalid result is sufficient to show that at least one error exists. It is unfortunate that theoreticians have not spent as much time searching for the error(s) as they did in explaining away the Right Angle Lever Paradox.)
Note:- The text referred to is not named because the error involved is of a type which would not be excused if made by a student of freshman physics. For the error to be included in texts written by individuals possessing PhD's in Physics which are used for teaching is appalling. It is not, however, the author's wish to embarrass individuals by naming them.
6.6 - (Update to Corrections to Special Relativity [2005]). The author has received an E-mail response which asserted that he had re-derived the Lorentz Transformation for Transverse Force using Maxwell's Electromagnetic Equations and verified that the conventionally accepted expression was correct and that the expression provided in this article is incorrect. Unfortunately for such a conclusion, it must be pointed out that his assertion is false! Since Maxwell's Equations involve the velocity of light, in order to apply them for this purpose, it is first necessary to correct them for the change in the ABSOLUTE velocity of light caused by a change of a velocity reference frame.
6.7 - (Update to Corrections to Special Relativity [2005])When Special Relativity was published, it was asserted that the Special Theory of Relativity showed that the velocity of light was the same in all reference frames. This statement is quite true as long as one considers effects which occur within a single reference frame. Special Relativity does not show that the velocity is the same, in absolute terms, between reference frames.
6.8 - (Update to Corrections to Special Relativity [2005]). In order to determine whether the velocity of light, which is observed to be the same not only within a refernce frame but between all reference frames, one must first determine the Lorentz Transformation for Velocity. Since velocity is equal to length/time, it followsd that the Lorentz Transformation for Velocity must be equal to the Lorentz Transformation for time (1/(1-V^2/C^2)^0.5 divided by the Lorentz Transformations for Time (1-V^2/C^2)^0.5. It follows, therefore, that the Lorentz Transformation for Velocity is equal to 1/(1-V^2/C^2)! It is not equal to unity. This transformation tells us that if, as we observe, the velocity of light has its nominal velocity of C WITHIN any given reference frame, it cannot retain that property BETWEEN reference frames which have a relative velocity. Einstein's UNPROVEN assertion that the velocity of light is constant is misleading. The velocity of light is "a constant" when measured within a reference frame. IT IS NOT CONSTANT when observed between reference frames which differ in velocity and/or elevations. The alleged refutation of the Lorentz Transformation for Transverse Force does not stand, The Transformation provided in this article remains valid.
:
Section 7 - The Lorentz Transformation for Parallel and Transverse Force as Related to a Compressed Spring Thought Experiment.
7.1 - The revision of the Lorentz Transformation for Transverse Force provided by Equation 4.8.29 satisfies two requirements:
- It is consistent with a rigorous analysis of the Right Angle Lever Thought Experiment.
- It is consistent with the effects which must be observed when an object (particle) having a velocity with respect to the observer acquires an additional incremental velocity which is perpendicular to the original velocity. In the observer's reference frame:
- The magnitude of the relative velocity will increase incrementally to produce an incremental increase in the relativistic mass of the object (particle).
- The energy (E=M*C2) represented by the increase in relativistic mass equals the incremental work done in the observer's reference frame in producing the incremental transverse velocity change.
On casual evaluation, the revision of the Lorentz Transformation for Transverse Force would seem to lead to a different difficulty, as shown by the Compressed Spring Thought Experiment illustrated in Figure 7.1.1.
7.2 - Consider an object (particle) in which energy is stored anisotropically. Such an object might be a compressed spring as illustrated in Figure 7.1.1. In this thought Experiment, two ideal and massless springs are compressed and tied to store energy. One spring is oriented so as to store its energy in a direction parallel to the velocity vector between reference frame A and B. The other spring is orientated so as to store its energy in a direction transverse to that velocity vector. For simplicity, it will be assumed that the energy storage and compressed distance of both springs are identical, as observed in reference frame B.
7.3 - As observed in reference frame B, the energy stored in the transverse spring is:
Eq. 7.3.1 et = dt*ft/2
:
And the energy, ep, stored in the parallel spring is:
Eq. 7.3.2 ep = dp*fp/2
As observed in reference frame A, the energy, Et, stored in the transverse spring is:
Eq. 7.3.3 Et = Dt*Ft/2
And the energy, Ep, stored in the parallel spring is:
Eq. 7.3.4 Ep = Dp*Fp/2
Where:
- 'D' is the deflection of a spring as observed in reference frame A.
- 'd' is the deflection of the same spring as observed in reference frame B.
- 'E' is the energy stored in a spring as observed in reference frame A.
- 'e' is the energy stored in the same spring as observed in reference frame B.
- All other symbols and subscripts are as previously defined. Since the symbols 'D' and 'd' refer to lengths and the symbols 'F' and 'f' refer to forces, the Lorentz Transformations for Parallel and Transverse Lengths and Forces apply.
Since it has been arbitrarily defined that, as observed in reference frame B, the stored energy and the compression distance of the parallel and transverse s prings are equal, we may write:
Eq. 7.3.5 et = ep
Eq. 7.3.6 ft = fp
Eq. 7.3.7 dt = dp
Combining Equations 3.3.1, 3.3.2, 4.5.3, 4.8.3, 7.3.1, 7.3.2, 7.3.3, 7.3.4, 7.3.5, 7.3.6 and 7.3.7 provides:
Eq. 7.3.8 Ep = Et*[1-V2/C2]
Eq. 7.3.9 ep = Ep/([1-V2/C2]0.5)
Eq. 7.3.10 et = Et*([1-V2/C2]0.5)
7.4 - In combination, Equations 7.3.5 and 7.3.8 appear to pose a dilemma. Special Relativity was derived based upon the principle that all velocity reference frames within the limits of +/-C are equally valid as a basis for all physical measurements (Principle of Relativity). Equation 7.8 shows that, as observed in reference frame A, the compressed spring that is parallel to the relative velocity vector has stored less energy than did the compressed spring which is transverse to the relative velocity vector. Equation 7.3.5 shows that, as observed in reference frame B, the energies stored in the two springs are equal.
7.5 - As observed in reference frame A, the transverse spring loses stored energy as it is rotated to the parallel direction, As observed in reference frame B it does not lose stored energy. One would expect that a loss of energy, as observed in reference frame A, stored in the spring as a result of the change in its orientation would result in a torque which would tend to rotate the spring towards the parallel orientation. The existence of such a torque in reference frame A and its non-existence in reference frame B would compromise the Principle of Relativity by allowing a 'preferred' velocity reference frame to be established.
7.6 - The effect suggested by the previous paragraph should not occur. In order for an object (particle) to acquire a velocity with respect to reference frame A, kinetic energy must be added to the object (particle) and the mass of that object (particle) is increased by the amount of the kinetic energy added in accordance with E=M*C2. When the object (particle) is eventually brought to rest with respect to reference frame A, it gives up that kinetic energy and reverts to its original rest mass, One would conclude, therefore, that the energy is stored in the moving object (particle) and/or in the space in the immediate vicinity of the moving object (particle). One is then faced with the question as to where and how that kinetic energy is stored. To investigate that question, let us consider that the rest mass of an object (particle) results form the storage of energy within the object (particle) in a manner equivalent to the storage of energy in the parallel and transverse springs of Figure 7.1.1.
7.7 - Let us first consider the storage of kinetic energy in the spring which is transverse to the velocity vector, as revealed by Equation 7.3.10. Dividing this Equation by C2 provides:
Eq.7.7.1 mt = Mt*([1-V2/C2])
Since Equation 7.7.1 is the conventional Lorentz Transformation for Mass, it follows that, as observed in terms of reference frame A, the kinetic energy of energy stored along a transverse axis of the object (particle) in reference frame B is stored as an increase of the stored energy along that axis. This increase in stored energy results from an increase in stiffness of the 'spring' which stores the 'rest mass' energy. (The deflection of the 'spring' is the same in both reference frames while the force, in reference frame A, producing the deflection has increased. This is equivalent to an increase in the 'stiffness' of the 'spring'.)
7.8 - The storage of the kinetic energy of the compressional energy stored in the parallel spring is not so readily explained. Dividing Equation 7.3.9 by C2 provides:
Eq.7.8.1 mp = Mp*([1-V2/C2])
Since this Equation is the reciprocal of the Lorentz Transformation for Mass, it indicates that the relativistic mass of the compressional energy of the parallel spring decreases as observed in reference frame A as the velocity of reference frame B with respect to A is increased. The decrease in the relativistic mass of the compressed energy in the parallel spring results from a decrease in the deflection of the spring, as observed in reference frame A, while the force applied to the spring remains unchanged. Unlike the transverse spring, the parallel spring does not store the kinetic energy associated with its compressional energy within itself. In terms of reference frame A, its kinetic energy and part of its rest mass energy must be stored in the space Figure 7.8.1 around the spring since that energy is transported with the spring but is not stored in the spring itself. (It should be noted that such a conclusion would also have been erroneously attributed to the transverse spring under the conventional Lorentz Transformation for Transverse Force, Equation 6.2.1)
7.10 - Since Equation 5.11 shows that, as observed in reference frame A, the storage of both rest mass and kinetic energy in the transverse spring , Et, accounts for the relativistic mass of the energy stored in the transverse spring, it follows that:
Eq. 7.10.1 Ert = Et
Where:
- 'Ert' is the energy equivalent of the transversely stored relativistic mass of the object (particle), as observed in reference frame
A.
Similarly, the total energy stored in the parallel spring, Ep is given by:
Eq. 7.10.2
Ep = Erp*[1-V2/C2]
In order for this to occur, the parallel spring must shed energy from within itself and carry that energy along with it in a disk shaped region of the space around it, with the disk oriented perpendicular to the velocity vector. The energy contained in that disk, as observed in reference frame A, must equal:
Eq. 7.10.3 Es = Er*(V2/C2)/(1-V2/C2)0.5
Where:
- 'Es' is the energy, as observed in reference frame A, stored in the space in the vicinity of the object (particle) as a result of its velocity with respect to reference frame A.
: