The Speed Limit Theory of Applied Force
The theory to be presented in this work will show that contrary to current scientific beliefs, from one perspective, the classical principles of physics involving Newton's laws of motion are not correct even at speeds that are not a significant fraction of the speed of light. In the same context it is also shown that the special theory of relativity is not correct at any speed including those approaching the speed of light. That is, neither theory is actually entirely correct at any speed from 0 to light speed c. In support of this finding it is shown that a fundamental principle of physics regarding an inherent limit to the speed at which force can be applied, has never been addressed. In fact, in the author's opinion, it was the very absence of this principle in the classical principles of physics that led to the falsely assumed need for the mass increase principle put forth by Einstein in the special theory of relativity. When all of these considerations are properly addressed as accomplished in this work, a new theory of physics arises that replaces all existing theories that do not include the speed limit factor presented here.
When properly understood, it is apparent that Newton's laws of motion1 only account for the effect that force has on mass and not the actual amount of force employed. As a direct consequence the laws then fail to take into consideration an inherent speed limit associated with all scientific methods of applying force, including mechanical, chemical, electrical, electro-mechanical, electromagnetic, etc. It was because of this omission, which is evidenced in particle physics where the speed of light cannot be exceeded, that ultimately resulted in the mass increase principle of Einstein's special theory of relativity2. By applying the gamma factor3 to the, so called, rest mass of the object being acted upon by the force, Einstein claimed that such mass increases toward infinity as its speed approaches the speed of light, and therefore it can never actually reach light speed. Although Einstein was on the right path in attempting to correct Newton's laws of motion, his solution was not correct and only added to the confusion involving the application of force. The real solution is to replace the mass increase factor with a speed limit factor as will be shown next.
2. The Speed Limit Factor in Applied Force
It is abundantly clear when applying force in normal activities, such as pushing an automobile, for example, that we cannot push it faster than the maximum speed of those doing the pushing4. To account for this missing factor in Newton's laws of motion, we simply modify the classical formula for the second law of motion
with the new speed limit factor
to obtain the applied force version of the formula for the second law of motion
as just given5. In this correct formula for the applied force version of the second law of motion, F is the applied force, m0 is the unaffected rest mass, a is the rate of acceleration, v is the resulting speed, and vL is the speed limit of the applied force. Note: m0 is used for the mass only to avoid confusion with the m that is used for mass increase in Einstein's special theory of relativity. Mass m0 in the above formula is not affected by the increase of speed v of its motion resulting from the applied force. In that regard it is important to understand that Einstein's special theory of relativity implies, via the principle of mass increase, the relationship given as
which is not truly correct, even at speeds approaching the speed of light c, because it attributes the increase in force to an increase in the mass of the accelerating object. Ignoring that false assumption for the moment, it subsequently follows from his theory that equation (4) can then be rewritten in the simplified form of
where mass m is defined by
to show the effect of motion on the rest mass m0 of the accelerating object.
3. Definition of Speed Limit Term, vL
Getting back to the speed limit principle that this paper is really about, it should be understood that the speed limit term, vL in equation (3) can have any assigned value from 0 to light speed c, dependent only on the type and nature of the applied force. It is a speed just beyond that which is achievable by the type and nature of the force being applied. For example, in the case of a rocket engine, it could be a speed just beyond that which the rocket would achieve if it had no payload to propel, just its own mass and that of the propellent. Thus, it wouldn't matter how many of such engines were used simultaneously in parallel to propel a payload. The more that are used to propel a certain payload mass, the closer the mass would come to achieving the determined speed limit. But even an infinite quantity of such engines used simultaneously in parallel, would never reach the speed limit. If the speed limit were reached, it would simply mean that the wrong value was used for vL and the limit should be raised accordingly. At the upper end of the scale we have vL = c for particle accelerators for example. This is the limit used in the special relativity form of the equation, (4), but it is only valid for use where such speeds of v = c are closely approached, such as in the acceleration of subatomic particles. At the lower end of the scale, involving humans pushing an automobile, for example, it could be the speed of the fastest running human on record, i.e., 44.72 km/h (27.8 mph). Thus, such humans burdened by the mass of the automobile being propelled, would never reach that speed. I.e., the more humans that are used in assistance the closer they would approach such limit, but no matter how many humans are added, the limit would never be reached.
From what was just discussed, it should be clear that equation (3) gives the same kind of results for used at the lower end of the scale, where vL approaches 0, as it does at the upper end of the scale where vL approaches c. Thus, the implied special relativity form of the formula, equation (4) where vL = c, is only valid for speeds approaching c, and Newton's form of the formula, equation (1) is not truly valid at any speed, that is, with regard to applied force. I.e., Newton's formula gives a linear increase in speed as the force is increased and not an asymptotical increase in speed that levels off as the speed limit is reached, such as in the special relativity form of the equation. The special relativity form of the equation, on the other hand, gives results that are virtually identical to Newton's at speeds that do not approach c. In fact, it is for this reason that Einstein's mass increase principle is thought to be valid for all speeds from 0 to c. I.e., it transitions from Newtonian behavior at low speeds to relativistic behavior as the speed increases toward light speed c. And, for this reason it is believed to give the correct results for all speeds when it actually doesn't, as already discussed.
4. Applied Force and Experienced Force
One of the biggest problems concerning our understanding of inertial force involves the fact that there are two different aspects to consider. There is the applied force and the resultant experienced force. Using the automobile example, the applied force is the total force of all the humans involved in pushing it. If there were a person inside the accelerating automobile, they would only experience the force related to the actual acceleration that occurs. More specifically, when these two different aspects of inertial force are properly understood, it becomes apparent that Newton's second law formula, equation (1), is valid for determining the experienced force. I.e., when used to determine the force from acceleration that is felt by the occupant of the automobile, it will give the correct result for acceleration rate a at all speeds from 0 to c. So, in that regard Newton's second law of motion is actually correct. However, the applied force that causes the acceleration is another thing entirely. To actually measure it, we would have to apply an opposing force to inhibit the motion of the automobile or even bring the motion of the automobile to a complete stop. For example, if the automobile was pushed into an impenetrable, unmovable, stone wall, the damage done to the vehicle would give an indication of how much force was being applied. The greater the number of people pushing the automobile during the collision, the greater the applied force and therefore the greater the subsequent damage to the automobile. This would be the case even if the speed limit was approached and the acceleration was approaching zero. The occupant of the automobile would experience almost no force from the acceleration but could be severely injured from the applied force when the automobile strikes the wall. Note, the damage done would be greater than that resulting from just the speed v of the automobile at the moment of impact as discussed next.
5. Breakthrough Involving Mass Increase
Upon evaluating the automobile pushing example it becomes clear that if the individuals doing the pushing are still pushing when the automobile encounters a resistance to its motion, then their mass is included in determining the force given by applied force formula (3). I.e., they are moving along with the automobile at the same speed and in direct contact with it, so their mass is included along with that of the automobile and its passenger in countering the resistance to its motion. In that regard, (discounting the speed limit c) the implied special relativity formula (4) is correct, but for the wrong reason. I.e., it is the total mass encountering the resistance that has increased and not the mass of the object being pushed. In short, real mass increase involves the total mass being measured and not just the mass of the object being accelerated. This means that the energy used in the acceleration process in particle accelerators has mass which in turn is in agreement with Einstein's
mass-energy equivalence formula6. And in this regard the author has never been in disagreement with Einstein. I.e., the greater the amount of energy used in the acceleration process, the greater the amount of mass. This is in agreement with the dual contact applied force principle discussed next.
6. Dual Contact and Single Contact Applied Force
A second contributing factor to the confusion involving applied force is the fact that there are actually two different types of applied force. There is a dual contact (DC) type and a single contact (SC) type. The automobile example discussed in Sections 2 through 5 involves the dual contact type. In this usage the force is applied between two bodies. In the quoted example the one body is the Earth and the other is the automobile. I.e. the pushers are exerting a force between the Earth and the automobile. The single contact type of force was briefly mentioned at the beginning of Section 3. It was in relation to the rocket. In that case the force from the expanding gases in the rocket engine contacts only the rocket itself. (Discounting the atmosphere, of course.) Note. in outer space there is no atmosphere to contact, yet the rocket still works. The main point being that a rocket engine is designed to work through the reaction to the expulsion of the expanding gases from the combustion chamber. Another example of single contact force is the solar sail. It is propelled forward from the force of the light that is reflected from it. Low speed examples of this kind of force are too numerous to discuss in detail. They involve the throwing of a baseball by someone who first jumps into the air, or the firing of a projectile from a gun that was tossed into the air, for example, or a series of objects tossed from any kind of vehicle that is free to move in the opposite direction, etc. The only contact involving such force is that between the propelled object and the mass that it expelled. Note: In the vehicle example it is assumed that there is an insignificant amount of resistance in moving along the Earth's surface due to the low friction wheels and wheel bearings of the vehicle.
7. Single Contact Experienced Force
Unlike the case with applied force, there is only a single contact type of experienced force. Such, for example, was the case involving the occupant of the automobile discussed in Section 4. In that example it was the force exerted between the automobile and the occupant. However, whereas the resultant acceleration from dual contact applied force is the same for both, the automobile and the occupant, the experienced force is directly proportional to the mass of the object involved. It then follows, as previously discussed in Section 4, that it is Newton's formula, equation (1), that correctly defines this form of inertial force. Thus, Newton's second law involves single contact experienced inertial force. But, it does not involve dual contact nor single contact applied force. In all cases involving applied force, there is a speed limit involved in the application. Such speed limit, however, would appear to have a different effect regarding single contact applied force from that regarding dual contact applied force as discussed next.
8. The Different Speed Limits of SC Applied and Experienced Force
Perhaps one of the most confusing things involving what the author is trying to show, is the fact that single contact applied forces are different from single contact experienced forces. In the case of single contact experienced force, Newton's formula (1) is valid. But, in the case of single contact applied force, the author's applied force formula (3) is valid. The confusion arises from the fact that both types of forces are in effect simultaneously. The single contact experienced force is that which is experienced by the mass being accelerated, the rocket for example. The single contact applied force is that involving the mass being expelled by the rocket engine to propel the rocket forward. Whereas Newton's formula shows that there is no limit to the speed that the rocket can reach, the applied force formula shows that there is a limit to the speed at which the propellent is forced out of the engine. Thus, no matter how large the engine or how many engines are used, the ejected gases can never exceed the speed limit, vL that is valid for the propellent and type of engine being used. In short, there is a speed limit to the rate of acceleration of the expelled gases at the moment of expulsion but there is no speed limit to the rate of acceleration of the rocket. This is because each succeeding instant of propellant expulsion occurs at an increased rocket speed resulting from the previous instant of propellant expulsion. Theoretically, the rocket could eventually exceed the speed of light, assuming it could continue to replenish its fuel supply along the way.
9. The Same Speed Limits at Both Contact Points of DC Applied Force
According to the case of single contact applied and experienced forces just discussed, there is no speed limit to the experienced force involving single contact forces. Thus, Newton's formula (1) would be valid. That is, in relation to the automobile example, the force experienced by the automobile would be given by
where Fr is the force experienced by the recipient, i.e., automobile, m0r is the recipient's mass, and ar is the resultant rate of acceleration.
However, unlike the case with the single contact applied and experienced forces just discussed in Section 8, the same speed limit is in effect for both points of contact involving the applied dual contact forces. Subsequently, the resulting speed limit cannot be reached, let alone exceeded by the object to which the force is applied. Using the automobile example, it should be understood that the opposing mass that is being pushed upon to accelerate the automobile never changes. It is always the same Earth via its surface, and thus the same Earth mass, that is used from the beginning to the end of the acceleration process. With that understood, it would appear then, that the same applied force speed limit that applies to the automobile, given here as
where Fp is the force of the producer, m0t is the total mass that includes the mass of all of the pushers plus the mass of the automobile, including its occupant, and ap is the rate of acceleration, also applies to the Earth. This then gives
where Fo is the opposing force of Earth's mass, m0E is Earth's mass, and aE is Earth's rate of acceleration. Consistent with what might be expected, when these two forces are equated to each other, the speed limit factors cancel out and we are left with
for Earth's resultant acceleration. Given the small value expected for the producer's rate of acceleration ap, in addition to the vast mass of the Earth m0E in comparison to the small total combined mass m0t of the force producers and automobile, including its occupant, the value for Earth's rate of acceleration would be virtually zero. I.e., it would be unmeasurable. Note: The author is fully aware that the Earth, as used in this example, is not actually being pushed away by the force producers. It is actually being rotated about its center of mass were such motion measurable. Nonetheless, the author was of the opinion that the automobile example would be satisfactory for the purpose of enabling the reader to grasp the points the author is trying to convey.
The author readily acknowledges that the principles covered in this paper can be very confusing. In fact, he himself has struggled over a period of many months in trying to present them in the simplest and easiest to understand manner possible. The two main points to take away from this paper are that Einstein's mass increase principle of special relativity is not valid and that the application of inertial force contains a speed limit that Newton also missed in deriving his laws of motion. When these two factors are addressed as accomplished in this work, the principles of, applied and experienced, dual and single contact, inertial forces emerge. It is then up to all of us who have a devoted interest in the science of physics and mathematics to perfect, refine, optimize and formalize such principles to the best of our abilities. In summation, mass does not increase and there is a speed limit inherent in the application of inertial force that extends from zero velocity to the speed of light c.
1 Isaac Newton, The Principia, (1687) as presented in Physics for Scientists and Engineers, second addition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)
2 Albert Einstein, Special Theory of Relativity, (1905), originally published under the title, On the Electrodynamics of Moving Bodies, in the journal, Annalen der Physik.
4 Joseph A. Rybczyk, Breaking the Light Speed Barrier, (2007) Section 7 Cause and Effect Laws of Motion (2014) Section 3. Both papers available from Millennium Relativity website at www.mrelativity.net
5 Joseph A. Rybczyk, Relationship of E = mc2 to F = ma and Gravity (2016) Section 7. Available from Millennium Relativity website at www.mrelativity.net
6 The energy equation E = mc2 first appeared in a paper by Max Planck, "The Principle of Relativity and the Fundamental Equations of Mechanics," Verhandlungen der Deutschen Physikalischen Gesellschaft 4 (1906):136141. With regard to Einstein, the energy equation is first mentioned in the paper, Does the inertia of a body depend upon its energy-content? (1905), It is unclear when the equation was first given in the form E = mc2, Einstein's first use of the equation was in, "On the Inertia of Energy Required by the Relativity Principle," Annalen der Physik 23 (1907):371-384.