Download Derive the mass to velocity relation

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.
.