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Derive the mass to velocity relation William J. Harrison [email protected] Based on my Mathcad effort last modified: Tuesday, January 04, 2000 7:31:14 AM Last update/modification: Dec-18-2003 The purpose of this paper is to assess the value of ‘known’ theories and relations. Each person must decide for himself what he or she will believe and incorporate into their ‘body of knowledge. It is hoped that this paper will help. No new theories will be introduced. Einstein's relativity includes some simple relations that any high school student can manipulate to determine how Einstein expects things to look and happen when speeds are so great that the usual relations fail. We will derive or disprove them here -- 'scientifically'. The derivations will be based not entirely on scientific knowledge but partly on assumptions that we will keep track of very resolutely (the way theorists do). Note: a list of 34 assumptions of SRT can be found at: http://www.dipmat.unipg.it/~bartocci/quest.htm. To find the list, perform a 'find on this page' of: "Thirty-four" (no quotes.) Here, we will begin. We will assume a linear relationship. Assumption 1: E = m*c2 That’s all there is to it. Making assumptions is so easy that it can get you into a lot of trouble. See: E= mc2 is Not Einstein's Discovery at Dr. Herrmann’s site for a good discussion of the assumptions’ foundations. Using considerable foreknowledge of its effects let me introduce some consequences of assumption 1 coupled with questions that might entail further assuming. Long before Einstein's time some mathematical definitions of terms was worked out such as: Work=Force*Distance. That is a definition, not an assumption. Work was found to be the same thing as energy. So now the relation is Energy=Force*Distance. Other relations were defined and derived from definitions. We will use them, and will look at their application in an effort to assure that we use them correctly. (It is not realistic to expect to overlook nothing. Also many people have allowed themselves to come to despise the 'trivial' matters. Here, I try to deal with the trivial effects mostly for the high school students who are attempting to grow, but also in preparation for challenges to my assertions. I also try to do that in my personal analyses and assessments.) When we push an object, we accelerate it and say that it has kinetic energy. When we lift an object we say that it has potential energy. Now if E=m*c2 is true, will the energy that we say it possesses as a result of these processes be reflected in its mass? Since energy can take different forms, it may be that kinetic and potential energies are forms like light which possess no mass. We will assume that kinetic and potential energy is contained in the mass of the object which possesses it. Assumption 2: Kinetic and/or potential energy increases the mass of an object according to assumption 1. If either of these assumptions conflict with the scientific communities' working definitions, we are going to have a very hard time with the scientific community. Let's check. Impulse and momentum have been found to be exactly equal to each other in magnitude and units. Of these definitions, only the third employs the value of mass and would be affected by the brand new assumption that mass is not constant. If the third must be changed, it is no small matter. The momentum must still be equal to the unchanged impulse relation. We need to examine the possibility that our assumptions will result in conclusions that are in conflict with the body of scientific knowledge. Starting our conceptual analysis with the fact that if we impart to a moderate body a large momentum it must have a large velocity and kinetic energy. The body must therefore have greater mass. Here, assumption 2 enters our analysis: if it is true and we imparted a definite impulse by applying a constant force for some specific time, therefore the product of m and v must still be the same as that impulse even though the mass has changed. Therefore we must now expect that a body possessing an increased momentum will have both a higher mass and a lower velocity than we had previously expected but their product will be just the same as before. But we now have that momentum is a function of two variables in its definition: . If assumption 1 is correct, even very large velocities must not appreciably or measurably increase the mass of a 1 kg object. We must verify that our assumptions’ effects are beneath the notice of the early scientific community and thus not in measurable conflict with all their careful measurements. It might be correct and no one had yet found physical evidence of it by measurement, but they eventually would. The definition still correctly defines momentum, but with two variables. Note: If you are going to theorize, you will be faced with just such problems. We can now use assumption 1, assumption 2, and the three definitions above with calculus to derive a relationship that will restore the definition of momentum to a function of one variable, v instead of two: m and v. In this derivation, force is constant. To evaluate the constant we see that when the velocity is zero the constant is related to only one of our variables, m. Assumption 2 tells us that the mass is constant when the object is stationary because velocity is what alters it. The constant value it has is, of course, the rest mass m0. Evaluating at v=0and m=m0, we can solve for the constant. Conclusion 1: m = m0/(1-v2/c2)1/2 Notice 4 things about this conclusion: 1. It implies that c is a limiting velocity above which mass is imaginary. 2. It implies an eventual infinite mass at the limiting velocity. 3. It results in a new form for the definition of momentum. Conclusion 2: New formula for momentum -- However, we have to verify that physicists might accept that they never noticed the discrepancy between the theoretical momentum and the results of their careful measurements: Note that the definition of momentum is changed so slightly that in a real test mass at a velocity of mach 6, the increase of mass, and therefore momentum, is very tiny: Mmach 6*v=M0*( 1+2.394*10-11 )*v. It is obvious that no one would find the resultant difference in the 11th significant figure of the momentum by measurement. Now we will begin to use the standard symbol, γ (Gamma), to consistently represent 1/(1-v2/c2)1/2 in our equations much as m represents mass in our equations. Note here that particle physicists "always" represent mo using the variable, m. Then they use gamma*m to represent the mass of a moving particle. I have seen m used where γm was correct and the results of the computation revealed that the correct value was intended and understood by the author. Reading critically (required of all people) does not mean being a difficult and demanding critic. At this point, we know that if: 1. Our definitions apply to real physical quantities as intended... 2. Our two assumptions are correct... We have derived a mathematical relationship for the mass of moving bodies as a function of their velocity. ACKNOWLEDGEMENT I learned of this approach to the derivation from: Richard Feynman’s Six Not-So-Easy Pieces p.69, Addison-Wesley, 1997 Discussion of Consequences: The range of the consequences of those two assumptions and three definitions is astounding. We will get a decent sample of those consequences here. (1) Is acceleration disallowed? (2) Is this ‘longitudinal’ mass? (3) The ‘old’ formula, F=m*a, is no longer valid. (4) Is it ‘relative’ mass? (5) Infinite mass at v=c? (6) How is potential energy accommodated? (7) Is the mass real? (8) Absolutes? (1) Is acceleration disallowed? Acceleration is readily accommodated in these relations. An accelerating body always has an instantaneous speed and therefore an instantaneous mass. That truth is incorporated in the method of calculus. When we took derivatives with respect to time, we made it mandatory that the derived relation always mathematically relates instantaneous values to each other. The calculus is not repealed. But the correct acceleration relations must be used. The validity of a variable force is directly related to the aspect of validity under acceleration via a force that is not constant. Here it is relevant that we definitely did include time as a relevant factor in the derivation. Then, the time factor dropped out of the relations during correct mathematical manipulation. This tells us that time is not relevant to the final result. Thus, it does not matter how long it took to establish the velocity it has at some instant. This indicates that the result is the same without regard to the way the velocity was acquired. Variable force is thus accommodated too, but we haven’t yet established here how to do it. (2) Is this ‘longitudinal’ mass? This brings us to the fact that math and mathematical models cannot prove physics. They are used to prove what our theories predict and they do prove what our theories predict. But they may not predict what really happens! So, theory must be tested by experiment. The three definitions in our starting point are all vector equations. The first one says that the scalar value of energy is equal to the dot product of the vectors force and distance. It is therefore required that the change of the velocity and momentum vectors must have the same direction as the force vector when we apply these relations to one mass with one (possibly resultant) force on it. Changing that vector relation to the required scalar one mandates the inclusion of the factor cosine (φ). This factor is 1 (and is properly considered evaluated and included) only when the force is in fact in the direction of the distance. Then the scalar magnitudes of impulse equals momentum in any (one) physical situation on the condition that the force and velocity vectors have the same direction. That is in fact what the definitions demand. That requirement must be met before the vector relations can properly be converted to scalar ones. Therefore, the relation refers to longitudinal mass. Since the vector definitions also indicate that it is impossible to achieve any velocity you might need to use in your calculations by applying forces that are transverse to that velocity, we must conclude that the ‘transverse’ mass is not addressed. It will require use of the definitions of torque and angular momentum to address that. By experiment, transverse mass has been found to agree with the above relation to the extent that it is measured to be within 10% of the predicted longitudinal mass. Here, definition 3, ( ), requires that mass is a scalar. But relativists describe it as a two-valued scalar with orthogonal values. The reason for this is that force is no longer equal to mass times acceleration, or even mass times gamma times acceleration. (3) The ‘old’ formula, F=m*a, is no longer valid. The derivation of was the result of using mathematics to solve for the ‘m’ in , (which was arrived at by the equality of two definitions), so that we know how the mass varies with time. Having done so, we can take the derivative of the new relation, acceleration relation. , as our starting point to find the instantaneous force- Here, note that F and v are the only vectors in the relation and must be aligned parallel to each other or the relation is false. We need a different starting point for centripetally directed components of a force not parallel to the direction of movement. A decelerating force is mathematically allowed because the minus sign can legitimately be moved out of the vector as a scalar multiplier and the direction vectors then match term for term. The force and acceleration directions are identical and both are in the direction of any already existing velocity. Now we have that during constant velocity (inertial frame) movement the longitudinal mass is mo*γ, and during longitudinal acceleration its inertia is mo*γ3. If we are going to accept both statements as simultaneously true, we are obligated to qualify or interpret the statements. Let’s say, during acceleration, its instantaneous mass is mo*γ and its resistance to acceleration is mo*γ3. Then we have that the increase (γ2) can be considered a reactive resistance to acceleration much like inductors have a reactive resistance to changing currents which disappears when the current stops changing and the steady-current resistance remains. Standard relativists (as I understand) do not like the analogy. It may or may not be misleading. It may set one (myself, for example) up to make unfounded assumptions without realizing it. If ones assumptions become unconscious, then when problems develop, one has difficulty tracking down the cause. We must always deal with the tendency to unconscious assumptions and get used to it. I am used to it; so I am not going to let it bother me. I have made mistakes before and I will again. The opportunity to publish to interested parties on the web will surely assist me toward eventually getting it right. We now must recognize that the added factor will result in less acceleration while the definition; , must remain true. Also the kinetic energy relation and so on must remain true. I am not going to track the consequences of the consequences of my derivation here. It does remain to be done: it is relevant to ultimately deciding to accept or reject the assumptions and all these consequences. (4) Is it ‘relative’ mass? Who has increased mass? Standard relativists desire to have a system without absolute or preferred frames. (They readily propound the beauty of the systems incorporation of the idea that any observer may consider himself at rest or not at rest.) As one goes around the universe assuming, as convenience dictates, that this or that object is at rest, then the masses of all objects change in a fashion that would give different results for collision calculations. Unfortunately, relativistic perceptions are so interwoven with this (so far) that I must put off resolving it until the relevant relations are developed and the limits of their usefulness is discussed. (5) Infinite mass at v=c? It is mathematically clear that the derived relation predicts that as the velocity approaches c, the mass approaches an infinite value. However, this is in conflict with the body of scientific knowledge. Cerenkov radiation is emitted by a particle that is traveling faster than local c in a medium. The prime example is helium nuclei in water. Shedding energy via that radiation, the particle slows to a velocity less than c in its medium. At no time can it be considered to have infinite mass. Also, at no time can it be considered to be so massive that it is inside its own Schwartzchild radius – i.e. it is not even momentarily a black hole. It’s clear that such a black hole would have to ‘remember’ more than mass, charge, and spin in order to return to us another alpha particle with all its constituent parts assembled for us. Here, the velocity which is greater than c in water is never greater than c in a vacuum. Relativists use that to assure us that the mass should not be infinite during transit according to theory. That still leaves us with the fact that the velocity it does have, which is less than c, is above some sustainable maximum. That establishes the empirical existence of a maximum velocity, (different values in different materials), less than c, and casts doubt on the possibility that the mass of a particle can approach an infinite value. Another such situation is encountered in particle accelerators with protons. According to data presented by Dr. P. Gibson, if I interpret it correctly, the maximum velocity of a proton in the local vacuum of the accelerator is about 0.9998285*c. The data is available to you at: http://wbabin.net. Look on that page for the visitor's comments box. Select Dr. Paul Gibson and click the "GO" button to read the data in his comments to Mr. Babin. I had hoped that the super conducting super collider planned for Texas might be powerful enough to find a limiting velocity for the electron if one exists. A table of such values for all isotopes of hydrogen and helium (and more) is desirable. That will not change the fact that the mass-velocity relation derived here and used throughout relativity is going to stay in conflict with the body of scientific knowledge. In this, it is much like Newton’s derived formula for gravitational attraction. Newton told us that it could be depended upon only when the distance between the masses is great enough that the increments of mass may be considered concentrated at a point. Mathematically, that can be said to occur when the vector sum of the incremental forces is equal to the algebraic sum of those same forces within the accuracy of your chosen number of significant figures. (His formula does work closer than that - presumably because large gravitating masses have much to most of their mass concentrated in their cores.) Technically, Newton was never proven wrong. Instead, the limits of applicability of his relation have been more accurately defined. Theoretical considerations did not lead Einstein to expect any discrepancy (to the best of my knowledge). Therefore, Einstein’s relation as we derived it here has been proven wrong in the laboratory (unlike Newton’s). We will stick with this relation, though, and remain aware that it provably has limits of applicability at the high end of velocity. Note that no one knows for sure that a fixed terminal velocity for any particle exists. It may be that an energy region is entered wherein, as in gravitational acceleration; the rest mass begins to reduce. Consider that the particle may be on the verge of becoming pure electromagnetic and neutrino energy. In order that such a (proposed or fantasized) transition may proceed in a gradual fashion, the velocity must increase somewhat during the process. In this region, the particle emits Cerenkov radiation in what might be another kind of ‘reactive resistance’. (Rough analogies can be helpful or harmful.) (6) How is potential energy accommodated? Here I narrow the scope of discussion to gravitational potential in order to manage the size of the discussion. Gravitational potential energy requires a different point of view. The consequences of including it in assumption 2 are very involved. Standard relativity includes it and does have the problems described in this discussion. Since any object accelerating due exclusively to gravitational force has a constant total energy and a varying speed, the relation: has to be interpreted to mean that the rest mass is varying under gravitational acceleration. Thus, as an object falls, its rest mass is converted to kinetic energy; kinetic energy is part of the total mass, and total mass is constant. So, the expression for kinetic energy has to be proportional to the difference between the fixed total mass and the variable rest mass. That is: algebraic sum of the component energies.) . (This makes use of the fact that total energy is the Since the variable rest mass is usually not known, a more useful form is: . This is the same kinetic energy found in conclusion 3 but it is expressed here as a function of the gravitationally constant total energy. Note that the particle physicist’s use of m to always represent rest mass is inconvenient here, but they are well aware of it and take proper care. These relations tell us that a fixed gravitational potential at every distance from the center of a gravitating body yields a corresponding fixed rest mass for a body brought from infinity to that distance. It means also, that when a mass falls to a gravitating surface and comes to rest, we now have a mathematical expression that allows us to determine the new rest mass of that object. This, in turn, tells us that the method and time taken to move an object from one distance to another will not influence the value of the rest mass. Therefore, it is shown that conclusion 3 does not conflict with subsequent conclusions here or with the body of science in regard to potential energy. Applying that relation to an object which has fallen (theoretically) from an infinite distance to the surface of the Sun we find that it has lost about 2 parts per million of its rest mass at infinity. This is a standard result that can be found in many books and papers. I repeat it to show that standard relativists do apply these relations in that way. However, this is where standard relativity develops ‘problems’ as mentioned earlier. The limit of kinetic energy as velocity approaches c is: That is; all the mass is converted to kinetic energy. The Schwartzchild radius formula indicates that ½ the mass is converted to energy when v=c. That is in definite disagreement with predictions; requiring that the cause of disagreement be tracked down and assessed. The proper way to do the tracking down is to study the original publication of the result authored by Mr. Schwartzchild. They charge money for those things. So, I was lucky to find the information in: The international edition of “Sears and Zemansky’s University Physics With Modern Physics”, tenth edition, by Young & Freedman. It is published by Addison Wesley Longman Inc. My copy is “…authorized for sale only in the Philippines.” The conclusions I reach should not be taken as reflecting badly on either the authors or the publisher on the grounds that working within the system is compulsory. They did not design the system. Having acknowledged that I did not do proper research, I continue: On page 380, we read that using the relation for escape velocity and setting the escape velocity to c arrives at the Schwartzchild radius. Then on pages 365-366 we find that the relation for escape velocity is found by setting kinetic energy equal to the energy required to escape. But the relation chosen to represent the kinetic energy there is: . We recognize that as an approximate formula that yields useful results only when the velocity is an insignificant fraction of c. Schwartzchild has every intention of using it all the way to c. That use of the pre-relativistic expression for kinetic energy invalidates Schwartzchild’s result. Ancillary conclusion (a): The Schwartzchild radius is wrong. If one accepts the two assumptions and the three definitions we started with as valid statements and can find no fault with the derivation of the mass-velocity relation, then one must conclude that the Schwartzchild radius is indeed wrong. Further, to accept the Schwartzchild radius is to conclude that the mass-velocity relation does not apply to velocities induced by gravitational effect. And then one has at least two ‘flavors’ of velocity -- that must be handled with differentl mathematical relations between mass and velocity. Relativists do differentiate between linear accelerations that change space velocity and accelerations in circular paths that do not. But I have never heard of anyone stating that a gravitationally induced velocity is different from a force-induced velocity. The logical acceptability and utility of using gravitationally induced velocities in the usual relativistic mass-velocity relation no longer ‘proves’ that it is correct. This is because standard relativity has become such an ingrained mess of known self-contradiction that now we can seriously entertain the idea that there might be different kinds of velocity. Again, if you accept the Schwartzchild radius, there are different kinds of velocity – in your physics – that you should conscientiously use without the usual mass-velocity relation. Standard relativists often use mutually contradictory statements as though they were simultaneously true. (7) Is the mass real? Relativity has problems such that some scientists have concluded that the mass increase is not real. According to the derived relation, if you travel in a space ship close enough to the speed of light, you would be in danger of being crushed by your own gravity! If this is true, then surely you would feel the effect long before it became dangerous. Then, the standard relativistic statement that there is no way to measure your velocity inside a closed room would be proven wrong. That assumption is so fundamental to relativity that disproving it would be disastrous to standard relativity (which in its turn is disastrous to standard logic). All the usual systems of viewing and measuring the increased mass of high velocity particles in particle accelerators tell us only that they do indeed behave kinetically as though they had increased mass. That behavior does not necessarily imply that they will gravitationally behave as though the mass were increased by attracting other masses more strongly. In this derivation, we assumed increased mass but as gravitational relations had nothing to do with the derivation we have not shown that fast particles attract other masses with greater force. The kinetic mass is also called the inertial mass (the kinetics of mass is inertia – and, apparently, reactance to acceleration). The experimentalists have repeatedly measured our solar system with great accuracy. They have determined that inertial mass and gravitational mass are precisely equal all the way to the limits of measurement accuracy. I cannot quote those limits, however. It may be that they have confirmed this to the accuracy of Newtonian theory. Since relativistic velocities cannot be achieved with lab-quality scientific equipment, there is no empirical evidence that the mass is real or not real. Speculation on the matter may lead to tests that can be performed with available equipment. Others might already know of such tests. (8) Absolutes? This derivation does not provide any information regarding absolutes as far as I can tell. Again, the results of collision calculations are expected to be revealing in this regard. Email W. J. Harrison Please include 'relativity' in the subject line. .