i.e. `m.Uy(v) = m.Uy(0)` but `Uy(v) = Uy(0).L(v)`

Hence to fix the contradiction, we follow the precedent for time and suppose
`m` depends on `v`. That is let:

` m(v) = m(0)/L(v) *EQ12`

`For ``v < c` the mass of an object is greater as measured by
a stationary observer than the measure made by an observer travelling with the
object, or as measured afterwards when the object is stationary relative to an
observer. Note that as speed builds towards `c`, the objects mass
increases without bound, hence no object with non zero mass can be given
sufficient energy to reach the speed of light.

Using *EQ12*

`
`substituting for` L(v)`

Which is modified to

`m(v) ^{2}c^{2} = m(0)^{2}c^{2} +
m(v)^{2}v^{2}`

where `m(v).v` is the momentum, commonly called `P`

` m(v) ^{2}c^{2} = m(0)^{2}c^{2} +
p^{2} *EQ13`

Expanding `1/L(v)` by Taylor series, ` EQ12` becomes

` m(v) = m(0) [1 + v ^{2}/2c^{2} +
3v^{4}/8c^{4} + ...]`

multiplying by `c ^{2}` and expanding

`m(v)c ^{2} = m(0)c^{2} + m(0)v^{2}/2 +
3m(0)v^{4}/8c^{4} + ... *EQ14`

The units of both sides of ` EQ14` are those of energy and the
familiar Newtonian Kinetic energy term appears second on the right hand side of
the equation. The terms involving

` E = m(v)c ^{2} *EQ15`

To suppose that the intrinsic energy of a mass is `m(0)c ^{2}`
requires that an object of zero rest mass and velocity of less than

If the angle `A` is small then the resultant velocity is small and so
the relativistic mass increase is negligible, the mass being close to
`2m(0)`. However this is not the case if it is accepted that mass is
conserved in the collision. The resultant mass is `M=2m(v)`, if
`v` was large then this is noticeably greater than `2m(0)`, the
combined rest mass. Hence the mass of the resultant object depends on the
speed or kinetic energy of its precursors. Moreover, should a particle of mass
`M` fly apart into two fast parts, as in nuclear fission, each will
dissipate its kinetic energy, hence velocity, hence relativistic mass as it
slows through interactions with neighbouring particles. The energy released
`(KE)` is a function of the mass deficit of the initial particle and its
now stationary products. Taking a clue from ` EQ15` this function
is simply the multiplicative constant

Change of `
`

It would seem that energy can be stored as mass and so this is evidence to
support the claim that the intrinsic energy of a mass could be
`m(0)c ^{2}`. There are other implications of mass depending on
creation velocity. The insides of a particle consist of the intrinsic masses
of Protons etc. minus energy used to stick them together plus energy left over
from the velocity of the creation of the particle, so what does a masson look
like? Or is it that the insides of particles may emit protons etc. but that
they do not actually exist inside the particle. As Richard Feynman put it, one
can emit the word cat without having it ready made inside you, there is no
finite supply of this word, only the energy used to make it. So it is for the
contents of particles, the emission products do not necessarily exist
internally.

A more useful and thought provoking version of ` EQ15` is obtained
by squaring and dividing by

`
`

then using **EQ13*

`
`

or

`
*EQ16`

`In the preceding development, momentum was defined as a non zero mass
multiplied by velocity. Although this is valid it does not exclude other forms
of momentum. Consider an object with zero rest mass but with velocity
``c`, calculating `m(c)` we have an ambivalent situation, both
`L(c)` and `m(0)` are zero. Is their quotient zero, finite or
infinite? If the answer is finite then there may be objects which have
momentum and energy but only at the speed of light. Photons seem to fit this
description. For example consider the momentum transferred from one particle
to another by light, the source particle recoils on emission thus losing
momentum, there is a slight travel time for the photon between particles, then
the target particle gains momentum when hit by the photon. So if momentum is
always conserved the photon must carry it. Hence let ` *EQ16`
describe these as well.

Go back

This document was created on 23 August 1995

last modified on 23 August 1995

and is written by and copyright to btaylor@taylormade.com.au