# XYLENE POWER LTD.

## BASIC PHYSICAL CONCEPTS - PART A

#### By Charles Rhodes, P. Eng., Ph.D.

RELATIVITY:
Modern science rests on belief in the existence of a set of physical laws, which are independent of position and time, that govern the evolution of the universe. These physical laws account for the behavior and interaction of all observeable objects. From a religious perspective, these physical laws are an expression of the will of God. The physical laws must be consistent with the existence of the universe and the evolution of life as we know it.

We assume the existence of three real orthogonal spacial dimensions with orthonormal unit vectors x, y, z so that the position of any particle in absolute space can be designated by xix, yiy, ziz.

By definition of the orthonormal unit vectors:
x.x = 1
y.y = 1
z.z = 1
x.y = y.x = 0
y.z = z.y = 0
z.x = x.z = 0

However, we have no means of measurement of absolute position. All of our position measurements are relative to an arbitrary reference position xox, yoy, zoz in the frame of reference of an inertial (non-accelerating) observer. Hence all measurements of position of particle i take the component form:
(xi - xo)x, (yi - yo)y, (zi - zo)z.

It is convenient to represent this position with vector notation as Xi.

VECTOR POSITION:
Thus Xi denotes a relative position vector of paarticle i with orthogonal x, y, z unit vectors and (xi - xo)x, (yi - yo)y and (zi - zo)z components. Hence:
Xi = (xi - xo)x + (yi - yo)y + (zi - zo)Z
and if:
xi = xo, yi = yo , zi = zo
then:
Xi = 0

TIME:
There is no such thing as a measurement of absolute time. Att time measurements are relative. An observer can only measure time periods of the form (t - to) where t is the present time for that observer and to is the time of an an arbitrarly chosen prior event in that observers frame of reference.

Elapsed time (t - to) is measured by the observer counting ticks of a clock which is stationary with respect to his reference location:
xox + yoy + zoz.

Each clock tick might be the time period for one rotation of the Earth with respect to distant stars or might be the time period of an absorption/emission photon oscillation corresponding to a well defined electron energy transition within an atomic isotope such as Caesium-133. Since to = constant:
dto = 0
giving:
d(t - to) = dt

Time is complex because special relativity shows that clocks that are moving with respect to each other indicate different elapsed times. Similarly general relativity shows that clock indications are affected by gravitational fields.

VELOCITY:
An inertial observer has no means of determining:
(dxo / dt), (dyo / dt) or (dzo / dt).

All his velocity measurements are relative and are of the form:
Vi
= [d(xi-xo)/ dt]x + [d(yi - yo) / dt]y + [d(zi - zo) / dt]z
= dXi / dt

MOTION:
The relative velocity vector Vi of particle i located at relative position:
Xi
is given by:
Vi = d(Xi) / dt

The relative velocity vector:
Vi
may have a direction and magnitude that are arbitrary with respect to the direction and magnitude of the relative position vector:
(Xi)

The relative velocity Vi of particle i at relative position (Xi) is:
Vi = d(Xi) / dt.
= [d(Xi - Xc) + dXc] / dt
Note that both of these velocity vectors are measured at relative position:
(Xi).

For now Xc and dXc / dt are arbitrary but later they will be chosen to be the position and velocity of the center of momentum.

INERTIAL OBSERVER:
Mathematical description of the observed universe becomes very complicated if the observer is subject to acceleration. To simplify the mathematics it is assumed herein that the observer's absolute velocity (dxo / dt)x + (dyo / dt)y + (dzo / dt)z
= constant
so the observer's acceleration i = 0

Such an observer is known as an inertial observer.

Then for an inertial observer:
The relative position of particle i is:
Xi
and the relative velocity of point i is:
Vi = dXi / dt

PHYSICAL LAWS:
Since measurements of position and time are purely relative, the physical laws of the universe should be independent of the choices of xo, dXo / dt, yo, dyo / dt, zo, dzo / dt and to.

The physical laws are expressed as mathematical equations that are functions of relative position Xi, relative velocity Vi and relative time (t - to) or dt.

In macroscopic behavior of physical objects most systems are governed by mathematical equations that have unique physical solutions. However, there are exceptions.

A well known macroscopic system that has many stable states are the bits of computer memory. The actual status of the computer memory bits is dependeent on their previus history.

At the microscopic level the mathematical equations which describe atoms have multiple discrete real solutions known as energy states. A particular particle reality can adopt any one of those discrete states. The existence of these multiple real solutions introduces a degree of randomness that makes evolution somewhat non-deterministic and gives life forms a limited degree of free will.

The survival of life forms in adverse circumstances depends on how wisely they exercise their limited free will to adapt to changes in their environment.

PARTICLES:
Energy is the structural material of the universe. Everything that exists contains some energy. A condition of:
Ei = 0
implies non-existence.

In essence the universe at relative time (t - to) consists of a spacial distribution of particles each with:
a characteristic designator
i,
a relative position:
Xi,
an energy:
Ei
and an energy flux at Xi of:
Ei Vi = Ei dXi / dt.

TOTAL ENERGY:
Any real object involves a spacial distribution of N particles, each with energy Ei. The object's total energy Et is given by:
Et = Sum from i = 1 to i = N of:
Ei

LAW OF CONSERVATION OF ENERGY:
The law of conservation of energy is one of the most fundamental physical laws. It states that energy can neither be created nor destroyed but can be changed in form. From the perspective of an inertial observer, for every isolated particle:
dEi / dt = 0

For an isolated collection of N interacting particles the total energy Et is givenby:
Et = Sum from i = 1 to i = N of Ei

Then:
dEt / dt = Sum from i = 1 to i = N of:
dEi / dt
= 0
so for any isolated collection of interacting particles the total energy Et is constant.
Hence:
dEt / dt = 0

LAW OF CONSERVATION OF ENERGY FLUX (LINEAR MOMENTUM):
Recall that particle i at pointXi exhibits energy Ei and energy flux:
Ei Vi = Ei dXi /dt

The vector:
Pi = Ei Vi / C^2
where C = the speed of light
is known as the linear momentum of mass Ei at point Xi.

Energy flux (momentum) is a conserved quantity.
For a single isolated particle since:
Ei = constant
then;
Vi = constant.

Consider the interaction of two particles i and j at point Xi. Let "a" denote the state just before the interaction at which point:
Xia = Xja
and the total energy Et is:
Et = Eia + Eja
and the total energy flux is:
Eia Via + Eja Vja

During the interaction there is an exchange of energy. Immediately after the interaction the total energy is:
Et = Eib + Ejb
and the total energy flux is:
Eib Vib + Ejb Vjb

The Law of Conservation of Energy requires that:
Eia + Eja = Eib + Ejb
while the law of conservation of energy flux (momentum) requires that at Xi, (t- to):
Eia Via + Eja Vja = Eib Vib + Ejb Vjb

Recall that:
Pi = Ei Vi / C^2
where:
C = speed of light
and
Pi = Linear Momentum of particle i.

Then:
Pia + Pja = Pib + Pjb

Hence the Law of Conservation of Energy Flux is also known as the Law of Conservation of Linear Momentum.

FUNDAMENTAL LAW OF MECHANICS:
At all times for particle i its energy and momentum are related by the differential equation:
Ei dEi = C^2 Pi.dPi
or
Ei^2 - Eio^2 = C^2 (Pi^2 - Pio^2)
where Eio is the particle energy when:
Pio = 0
corresponding to Vi = 0

During an interaction between particles for each particle i there is a change in both Ei and Pi. Recall that:
Ei dEi = C^2 Pi.dPi
or
dEi = C^2 (Pi dPi / Ei)
= C^2 (Vi dPi) / C^2
= Vi dPi
= [dXi / dt] dPi
= [dPi / dt] [dXi]
= Fi.dXi
which gives a change in energy of particle i as being caused by a force Fi acting over a distance dXi.

Note that:
Fi = [dPi / dt] = dEi / dXi
A force Fi causes a change in energy dEi with respect to change in position dXi which is equal to the change in linear momentum dPi with respect to a change in time dt.

This equation can be written as:
dEi dt = dPi.dXi
which equation is of fundamental importance in quantum mechanics.

Recall that:
Pi = Ei Vi / C^2
and
dXi / dt = Vi
giving:
dEi = d[Ei Vi].Vi / C^2
= [Ei Vi.dVi + Vi^2 dEi] / C^2
or
dEi [1 - (Vi^2 / C^2)] = (Ei / C^2) Vi.dVi
or
dEi / Ei = Vi.dVi / (C^2 - Vi^2)
or
Ln(Eib / Eia) = Integral from Via to Vib of
Vi.dVi /(C^2 - Vi^2)

Make substitution:
Try Ui = C^2 - Vi^2
giving:
dUi = - 2 Vi.dVi
and
Vi.dVi / (C^2 - Vi^2) = (- 1 / 2) dUi / Ui
and
Uia = C^2 - Via^2
and
Uib = C^2 - Vib^2

Thus:
Ln(Eib / Eia) = (-1 / 2) Ln[Uib / Uia]
= (-1 / 2) Ln[(C^2 - Vib^2) / (C^2 - Via^2)]
or
Ln(Eib / Eia) = Ln[(C^2 - Via^2) / (C^2 - Vib^2)]^(1 / 2)
or
Eib / Eia = [(C^2 - Via^2) / (C^2 - Vib^2)]^(1 / 2)
or for the special case of Eia = Eio corresponding to Via^2 = 0:
Eib = Eio / [1 - (Vib / C)^2]^1 / 2

Note that Eib becomes very large as Vib^2 approaches C^2.

SIMPLIFICATION:
The energy Ei of a particle can be expressed as:
Ei = Eio + Ek
where:
Eio = particle energy at rest
and
Eik = particle kinetic energy
= particle energy component due to particle motion

Hence:
Ei^2 = Eio^2 + 2 Eio Eik + Eik^2

Similarly:
Pi = Ei Vi / C^2
or
Pi^2 = Ei^2 Vi^2 / C^4
= [Eio^2 + 2 Eio Eik + Eik^2] Vi^2 / C^4

Recall that:
Ei^2 - Eio^2 = C^2 Pi^2
or
Eio^2 + 2 Eio Eik + Eik^2 - Eio^2
= C^2 [Eio^2 + 2 Eio Eik + Eik^2] Vi^2 / C^4
or
2 Eio Eik + Eik^2 = [Eio^2 + 2 Eio Eik + Eik^2] Vi^2 / C^2

There are a large class of physical problems for which:
Eik << Eio
For these problems:
2 Eio Eik ~ Eio^2 V^2 / C^2
or
Eik ~ Eio V^2 / 2 C^2
= Mio V^2 / 2
where:
Mio = Eio / C^2.

Similarly:
Pi = Ei Vi / C^2
= (Eio + Ek) Vi / C^2
~ Eio Vi / C^2
= Mio Vi

Thus, if:
Eik << Eio
Then Eik and Pi simplify to the Newtonian expressions for particle kinetic energy and particle momentum.

GENERAL APPLICATION OF LAWS OF CONSERVATON OF ENERGY AND CONCERVATION OF ENERGY FLUX (MOMENTUM):
The rate of change of energy contained within an isolated closed surface is always equal to the net flow of energy through that closed surface.

JOINING OF PARTICLES:
If at time (t - to) two particles with energies Ei and Ej are at the same position in space then:
(Xi - Xo) = (Xj - Xo)
and the energy packets can join additively to form a new energy packet Ek given by:
Ek = Ei + Ej

As before:
dEk / dt = (dEi / dt) + (dEj / dt)
= 0 + 0
= 0

SPLITTING OF PARTICLES:
At time (t - to) and position (Xk - Xo) a particle with energy Ek can split into two particles with energies Ei and Ej where:
Ei + Ej = Ek
and at the instant of interaction:
(Xk)
= (Xi)
= (Xj)

As before:
dEi / dt = 0
and
dEj / dt = 0

FIX FROM HERE

RELATIVE CENTER OF ENERGY:
For any isolated cluster of N energy packets the position of the center of energy is:
(Xce)
where Xce is defined by:
Sum over all i = 1 to i = N of:
{Ei ((Xi - Xce))}
= 0
or
Sum over all i = 1 to i = N of:
Ei [Xi]
= Sum over all i = 1 to i = N of:
Ei Xce = Et Xce
Thus the total energy of an assembly of N particles can be thought of as being concentrated at the assembly relative center of energy defined by:
Xce = Sum over all i= 1 to i = N of:
(Ei / Et)[Xi]

RELATIVE CENTER OF MOMENTUM:

OUT OF ORDER NOTE:
Consider a rigid body rotating at angular rate W. The centrifugal force on energy packet Ei is:
(Ei / C^2) Ri W^2
where:
Ri = radius from the axis of rotation

Note that for a rigid body W is common for all i values.

Hence at the center of energy of a rigid body the centrifugal forces cancel each other. Thus a free rigid body rotates about an axis passing through its center of energy.

VELOCITY OF ENERGY PACKET Ei:
The velocity Vi of energy packet Ei at position (Xi - Xo) is:
Vi = d(Xi) / dt.
= [d(Xi - Xc) + dXc] / dt
Note that both of these velocity vectors are at position:
(Xi).

For now Xc is arbitrary but later it will be chosen to be the position of the center of momentum.

Delivery of energy to a remote location may be done by closed loop circulation of an energy transport fluid in a pipe. Delivery of energy may also be done by a open loop flow of water or a combustible fuel. Such systems actually deliver a change in energy per particle (Eb - Ea) between states a and b but do not measure the absolute energy of the fluid. This change is generally very small compared to the rest mass potential energy of the fluid.

Delivery of energy may also be done by unidirectional flow of radiation such as sunlight, a laser beam, a wave guide or an electric power transmission line. In each of these cases partial reflection of incident radiation at the receiver reduces the energy absorbed.

POWER:
Power = rate of energy transfer between systems:
= dE / dt.

VECTOR MAGNITUDE:
The magnitude |A| of any vector A is defined by:
|A|^2 = Ax^2 + Ay^2 + Az^2
where:BR> Ax = vector component of A parallel to x axis;
Ay = vector component of A parallel to y axis;
Az = vector component of A parallel to z axis;

UNIT VECTOR:
A unit vector parallel to any vector A is:
[A / |A|]

VECTOR COMPONENT:
An arbitrary vector B at any particular position can be expressed as the sum of a component parallel to vector A at the same position and a component normal to vector A at that same position. At this position the component of vector B parallel to vector A is:
(A * B) / |A|
where:
(A * B) = Ax Bx + Ay By + Az Bz

VELOCITY VECTOR COMPONENTS:
Consider the velocity vector component:
d(Xi - Xc) / dt
at relative position:
Xi - Xo

This velocity vector component can be expressed as the sum of two other othogonal velocity vector components. One of these other vector components lies along the radial position vector:
(Xi - Xc)
and the remaining other vector component is normal to the radial position vector:
(Xi - Xc)

On the web page titled: Vector Identities it is shown that for two arbitrary vectors A and B sharing the same position Xi:
|A X B|^2 + (A * B)^2 = |A|^2 |B|^2
or
|A|^2 |B|^2 [cos(Theta)]^2 + |A|^2 |B|^2 [sin(Theta)]^2
= |A|^2 |B|^2
or
[cos(Theta)]^2 + [sin(Theta)]^2 = 1
where:
(Theta) is the angle between vector A and vector B

Thus:
|A X B|^2 + (A * B)^2 = |A|^2 |B|^2
or
{|A X B|^2 / |A|^2} + {(A * B)^2 / |A|^2} = |B|^2

Let:
A = (Xi - Xc)
and
B = d(Xi - Xc) / dt

The velocity vector component at position vector:
Xi - Xo
parallel to relative position vector:
(Xi - Xc)
is:
[(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)|
and by comparison with the above identity the velocity vector component normal to (Xi - Xc) is:
+/- [(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|

TOTAL VELOCITY OF ENERGY PACKET Ei:
Recall that:
Vi = d(Xi - Xc) / dt + d(Xc - Xo) / dt
= [(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)|
+/- [(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|
+ d(Xc - Xo) / dt
= component of d(Xi - Xc)/dt at Xi radial to position:
Xc - Xo
+ component of d(Xi - Xc) / dt at Xi on a path revolving around position:
Xc - Xo
+ velocity of position:
Xc - Xo

MOMENTUM:
Momentum is energy motion. The momentum of energy packet Ei is:
(Ei / C^2) Vi = (Ei / C^2) d(Xi - Xo) / dt

However, each element of momentum occurs at a different point in space. In order to find the total momentum of a cluster of energy elements we need to define a center of momentum Xc where:
Sum from i = 1 to i = N of
(Ei / C^2) d(Xi - Xc) / dt = 0

The momentum of total energy Et is:
Sum from i = 1 to i = N of:
(Ei / C^2) Vi
= Sum from i = 1 to i = N of:
(Ei / C^2) {[(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)|
+/- [(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|
+ d(Xc - Xo) / dt}

= (Et / C^2) d(Xc - Xo) / dt
+ Sum from i = 1 to i = N of:
(Ei / C^2) {[(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)|
+/- [(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|}

= linear momentum
+ angular momentum

CONSERVATION OF LINEAR MOMENTUM:
The law of conservation of linear momentum states that for an isolated system the center of energy velocity is constant, or:
d(Xc - Xo) /dt
= constant
which when energy Et is constant is equivalent to saying that the linear momentum P with respect to Xo is:
P = (Et / C^2) d(Xc - Xo) / dt
= constant.
where Xc is the center of momentum.

This is a fundamental conservation law of physics.

RADIAL MOMENTUM OF A RIGID BODY ROTATING AROUND (Xc - Xo):
Define the center of momentum:
(Xc - Xo)
for a rigid body as the vector position where:
Sum from i = 1 to i = N of:
(Ei / C^2) {[(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)|
= 0

This equation defines Xc which is the center of momentum. A rigid body rotates about its center of momentum.

For every individual energy element Ei revolving around the center of momentum(Xc -Xo)
the motion vector:
d(Xi - Xc) / dt
is normal to the relative position vector:
(Xi - Xc)
so the product:
[(Xi - Xc) * d(Xi - Xc) / dt] = 0

Hence for a rigid body:
= Sum from i = 1 to i = N of:
(Ei / 2 C^2) [(Xi - Xc) * d(Xi - Xc) / dt] / |(Xi - Xc)| = 0

ANGULAR MOMENTUM:
Sum from i = 1 to i = N of:
+/- {(Ei / C^2)[(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|}
= constant

MOMENTUM SPECIAL CASES:
For a single point energy packet:
Et = Ei
and Xc = Xi
so the momentum simplifies to:
Pt = (Et / C^2) d(Xc - Xo) / dt

For a symetrical rigid body like a disk or a sphere rotating about a central axis through the position (Xc - Xo) with no radial motion then for all i values:
(Xi - Xc) * d(Xi - Xc) / dt] = 0
and the angular maomentum is:
L = Sum from i = 1 to i = N of:
+ / - (Ei / C^2)[(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|

If there is no revolution around Xc then:
[(Xi - Xc) X d(Xi - Xc) / dt] = 0
causing the angular momentum to be zero.

CONSERVATION OF ANGULAR MOMENTUM IN NEWTONIAN MECHANICS:
Consider a symetrical body rotating around an axis which passes through Xc:
Recall that:
L = Sum from i = 1 to i = N of:
+ / - (Ei / C^2)[(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|

which is the angular momentum about the center of energy at Xc.

However for the special case of a rigid body rotating about Xc at angular frequency W:
d(Xi - Xc) / dt = [W X (Xi - Xc)]

Hence:
L = Sum from i = 1 to i = N of:
+ / - (Ei / C^2)[(Xi - Xc) X d(Xi - Xc) / dt] / |(Xi - Xc)|
= Sum from i = 1 to i = N of:
+ / - (Ei / C^2)[(Xi - Xc) X [W X (Xi - Xc)]] / |Xi - Xc|

With real large rigid bodies the speeds involved are always much less than the speed of light, so the Newtonian expression for kinetic energy can be used. In Newtonian mechanics the kinetic energy of rotation Eki of Energy Ei rotating about the center axis W passing through center of energy Xc is:
Eki = (Ei / 2 C^2) |Xi - Xc) X W|^2

The kinetic energy of rotation is:
Ekr = Sum from i = 1 to i = N of:
(Ei / 2 C^2) |(Xi - Xc) X W|^2
= Sum from i = 1 to i = N of: (Ii / 2) |W|^2
where Ii = moment of inertia of Ei.

Thus:
Ekr = Sum from i = 1 to i = N of:
(Ii / 2) |W|^2

Note that for a rigid body in free rotation:
I = Sum from i = 1 to i = N of:
Ii
is constant so conservation of rotational kinetic energy keeps |W| constant and hence L constant. Thus apparent conservation of angular momentum is really conservation of rotational kinetic energy Ekr. If while rotating I can be reduced by doing work then both
|W| and Ekr can be increased.

ANGULAR FREQUENCY OF A FIGURE SKATER:
Note that if while in free rotation I is reduced by a figure skater pulling his/her limbs close to his/her vertical axis of rotation conservation of energy forces the angular frequency:
|W|
to increase. Furthermore, since the act of reducing |Xi - Xc| while spinning requires work there is an increase in Ekr which further increases |W|.

This same effect is also important in execution of the rapid kinetic energy accumulating turns used in Fisher Shotokan karate.

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PARTICLES, POTENTIAL ENERGY, KINETIC ENERGY:
It is assumed that the known universe is primarily composed of particles and photons. Each particle i is an entity with a nominal position:
(Xi - Xo),
and a velocity:
d(Xi - Xo) / dT,
and a net electric charge Qi,
and an energy:
Ei = (Eoi + Eki + Ebi)

PARTICLE ENERGY:
Energy tends to concentrate in packets. We call these packets particles.

Each particle has a total energy Et which has components Eo, Ek and Eb.
Et = Eo + Ek + Eb

Eo = the rest energy of a particle. Eo is the energy component with respect to a field free vacuum which is independent of position and velocity. Note that Et = Eo when the particle is at rest and is far from other particles. The rest energy Eo usually consists of a concentrated non-field component and a field component. The non-field component of Eo, which is typically about 98% of Eo, cannot do work unless a particle and its corresponding anti-particle annihilate each other to become a radiant energy photon.

Ek = particle kinetic energy with respect to Eo that is a function of the particle velocity (Vi - Vo). When the particle is at rest:
d(X - Xo) / dT = 0
then:
Ek = 0

If:
(Vi - Vo) = 0
then:
Ek = 0
and
Et = Eo + Eb

Note that rotational kinetic energy about the particle CM does not result in linear motion and hence is part of the potential (rest) energy.

Eb = The binding energy of a particle which is its energy component with respect to Eo that depends on the particle's position with respect to other particles. When the particle is far from other particles Eb = 0. The binding energy Eb arises from overlap of the different particle fields. That field overlap usually causes the volume integral of the field energy density to decrease, making Eb negative.

Total energy Et is the integral over volume of the energy density. The energy density is measured relative to a field free vacuum for which the energy density is usually assumed to be zero. However, note that astronomical observations of other galaxies suggest that on a cosmological scale the interstellar space has a non-zero energy density. The cause of "dark energy" is presently not well understood.

Note that a particle's energy is not highly localized. It is actually distributed through space via its fields. However, an object's energy is sufficiently localized that the energy integral over all space volume has a finite result. The nominal position of a particle is defined by the particle's center of linear momentum.

When there are two or more particles present their extended fields may overlap and causing conversion of potential energy into kinetic energy.

In some cases the interparticle interaction can lead to formation of a new particle or the emission or absorption of a photon. Photon emission often causes free particles to become trapped in a mutual potential energy well.

TOTAL ENERGY:
The total absolute energy E of an isolated system as seen by an inertial observer at Xo, Vo is given by:
E = Sum of all Ei

Each particle has a total energy Ei which includes both potential (rest) energy Eoi and kinetic (motion) energy Eki in the observer's frame of reference.

Thus:
Ei = Eki + Eoi

When there is no particle motion in the observer's frame of reference:
Eki = 0
and
Ei = Eoi
where:
Eoi = rest energy = total energy of the particle as measured by an inertial observer when:
(Pi - Poi) = 0

MUTUAL POTENTIAL ENERGY WELL:
Consider the case of a particle that enters and becomes trapped in a mutual potential energy well. For example a hydrogen molecule that is gravitationally absorbed by the sun.
When the particle first enters the potential energy well:
Ei = Eoi + Eki + Eb
Initially while the particle is far away the particle does not interact with other particles. Hence:
Ebi = 0
and
dEi / dT = 0.

As the particle slides into the potential well Ebi decreases and Ek increases. Then while in the potential energy well some of the particle's kinetic energy is lost via emission of photons. While photons are being emitted:
Ebi decreases.

Hence after photon emission:
(Eb + Ek) < 0
and particle i cannot escape from the potential energy well except by absorbing kinetic energy and momentum from a photon or from another particle in the potential energy well. If adding more particles to the potential energy well sufficiently increases Eoi there will be ongoing energy aggregation in the potential energy well. The consequences of this situation are explored on the web page titled: BASIC PHYSICAL CONCEPTS PART B - ENERGY AGGREGATION. When there are sufficient particles in the potential energy well Eoi no longer increases sufficiently with addition of each particle to maintain system stability. Then the structure fails. This failure mechanism happens with both atomic nuclei and stars. There are structural stability limits that affect the maximum sizes of both atomic nuclei and stars.

The photon emission during the energy aggregation process provides the energy flux that supports all life.

LINEAR MOMENTUM:
Linear Momentum is linear energy motion.
The linear momentum:
(Pi - Poi)
of a particle i is given by:
(Pi - Poi) = (Ei / C^2) (Vi - Vo)
where:
C = speed of light
which speed is constant and is the same for all inertial observers
and
(Vi - Vo)
is the nominal linear velocity of the particle in the frame of reference of an inertial observer.
When:
(Pi - Poi) = 0
then:
(Vi - Vo) = 0
giving:
Poi = (Ei / C^2) Vo

Note that linear momentum is a relative vector quantity. There is no means of determining Poi. Only momentum vector differences of the form (Pi - Poi) can be determined. Note that relative velocity vector:
(Vi - Vo)
is located at relative position:
(Xi - Xo)
and can potentially point in any direction.

CONSERVATION OF LINEAR MOMENTUM:
For any isolated object total linear momentum P is conserved.

CONSERVATION OF MOMENTUM FOR A SINGLE ISOLATED FREE PARTICLE:
For an isolated free particle conservation of energy gives:
dEi / dT = 0
and due to isolation there is no acceleration giving:
d(Vi - Vo) / dT = 0 Hence:
(Pi - Poi) = (Ei / C^2) d(Xi - Xo) / dT
= (Ei / C^2) (Vi - Vo)
= constant.

CENTER OF LINEAR MOMENTUM (CM) FOR A CLUSTER OF PARTICLES:
Consider a cluster of particles not subject to external acceleration. Each particle i in the cluster has a nominal position:
(Xi - Xo) = [(Xi - Xc) + (Xc - Xo)]
and has a nominal velocity:
d(Xi - Xo) / dT = d[(Xi - Xc) + (Xc - Xo)] / dT

The linear momentum of the cluster of particles as seen by an observer at Xo moving at velocity dXo / dT is:
(P - Po)
= Sum over all i of
(Ei / C^2) d(Xi - Xo) / dT = (Ei / C^2) d[(Xi - Xc) + (Xc - Xo)] / dT

The law of conservation of momentum requires that there exists a point (Xc - Xo) moving with velocity:
d(Xc - Xo) / dT
such that:
Sum over all i of:
(Ei / C^2) d(Xi - Xc) / dT = 0 Hence:
(P - Po) = Sum over all i of
(Ei / C^2) d(Xc - Xo) / dT

Total energy E is given by:
E = Sum over all i of Ei.
Hence:
(P - Po) = (E / C^2) d(Xc - Xo) / dT
which gives total particle cluster linear momentum (P - Po) at location (Xc - Xo) in terms of total cluster energy E and cluster CM velocity:
(Vc - Vo) = d(Xc - Xo) / dT.

REVIEW OF PARTICLE PARAMETERS:
Energy is concentrated at or near particles. At relative time (T - To) each particle i has a nominal relative position vector:
((Xi - Xo))
and has a nominal relative velocity vector at (Xi - Xo) given by:
(Vi - Vo) = d(Xi - Xo) / dT
and has a nominal energy Ei measured relative to a field free vacuum and has a momentum:
(Pi - Poi) = (Ei / C^2) d(Xi - Xo) / dT

There is some unavoidable uncertainty in simultaneous meaurements of position and momentum or energy and time because the process of accurately measuing one parameter introduces error into measurement of the other parameter. This issue is known as quantum uncertainty.

CHARGED PARTICLES:
It has been experimentally observed that charged particles such as electrons, protons and atomic nuclei always have a quantized net amount of charge. The net charge on a real particle is always an integral multiple of:
Q = 1.602 X 10^-19 coulombs.
The exact mechanism of quantization of net particle charge is not known. However, protons are believed to be assemblies of subquantum particles known as quarks. Quarks occur in triplets and never exist in isolation. Thus quarks can be viewed as components of real particles. In a proton there are two quarks with charge (2 Q / 3) and one quark with charge (- Q / 3).

FIELDS:
The energy density of a particle i diminishes sufficiently rapidly with increasing distance from its nominal location (Xi - Xo) that the total energy of the particle integrated over all of space is finite. However, the particle's energy density remains non-zero out to infinity. The region of declining energy density around a particle is known as its field. This field contains potential energy.

FIELD ENERGY COMPONENTS:
The field energy per unit volume for a particle at rest has gravitational, electric, and magnetic components. Motion of a particle in the observers frame of reference causes an additional kinetic energy component.

ORTHOGONAL VECTOR COMPONENTS OF FIELD ENERGY:
1) Distributed electric charge at rest causes an electric field;
2) Distributed charge in motion causes a magnetic field;
3) Distributed energy causes a gravitational field;
Changes in these fields are waves that propagate at the speed of light. Fields cause energy relative to field free space.

GRAVITATIONAL FIELDS:
Albert Einstein explained gravity in general relativity via precise tensor equations. However, these equations are very difficult to solve and are usually only required for astrophysical situations involving either extremely precise measurements or extremely large masses. Well known applications of General Relativity include the Global Positioning System, analysis of the orbit of the planet Mercury, analysis of Black Holes and gravitational lensing. For most other purposes the Newtonian gravitational approximation of gravitational force:
F = (G Ma Mb / (R*R)) (R / |R|)
gives satisfactory accuracy where:
F = gravitational force vector between mass Ma and mass Mb;
G = 6.67408 × 10-11 m^3 / (kg s^2) = Newton's gravitational constant;
R = relative position vector between center of mass Ma and center of mass Mb.

FIELDS AND FORCES:
Electric charge causes a spherical radial electric field far from the particle. Electric charge motion causes a magnetic field. Electric fields and magnetic fields contain energy which is positive with respect to a field free vacuum. The presence of positive energy causes a gravitational field. A gravitational field contains energy which is negative with respect to a field free vacuum. The gravitational field energy is generally only significant in situations where there are approximately equal numbers of positive and negative charges so that the electric and magnetic fields cancel. Field issues and the apparent forces arising from field overlap are discussed on the web page titled: FIELD THEORY.

A change in potential energy due to a change in particles field overlap changes the particles binding energy and hence changes the particles kinetic energy and the particles linear momentum. These changes are interpreted as being the result of a force.

KINETIC ENERGY OF A PARTICLE:
The kinetic energy Eki is the portion of energy Ei that is a function of the velocity:
(Vi - Vo)
of the particle in the frame of reference of the inertial observer.

If:
(Vi - Vo) = 0
then:
Eki = 0
and
Ei = Eoi + Ebi

Note that rotational kinetic energy about the particle CM does not result in linear motion and hence is part of the potential (rest) energy.

CHANGE IN ENERGY (ABSORBED WORK):
The fundamental relationship that mathematically defines the change in kinetic energy dEk of an object in terms of changes in position, time and momentum is:
dEk dT = dP*dX
= d(Pi - Poi)*d(Xi - Xoi)
or
dEk = [d(Pi - Poi) / d(T - To)]*d(Xi - Xoi)
If the motion is along only one spacial dimension this equation simplifies to:
(Change in Kinetic Energy) = (Force)*(Distance)

In quantum mechanics (dEk dT) is measurement uncertainty. This issue is important in atomic and nuclear binding theory.

SPECIAL RELATIVITY AND KINETIC ENERGY:
Einstein developed equations relating rest mass, rest energy, total energy, momentum and linear velocity in the reference frame of an inertial (non-accelerating) observer. That body of work is known as special relativity.

Einstein showed that since E = M C^2, where M = mass and C = speed of light, so an object's linear momentum (P - Po) is properly defined by:
(P - Po) = (E / C^2) (V - Vo)
where Po and Vo are invisible to an inertial observer moving with velocity Vo. To this observer Po = 0 and Vo = 0.

Then:
dP = (1 / C^2) [dE V + E d(V - Vo)] Recall that a change in energy is defined by:
dEk = dP*dX / dT
= (1 / C^2) [dE (V - Vo) + E dV]*dX / dT
= (1 / C^2) {[dE (V - Vo)*dX / dT] + [E dV]*dX / dT}
= (1 / C^2) {[dE (V - Vo)^2] + [E (V - Vo) dV]}

Recall that:
E = Eo + Ek + Eb

For circumstances where:
dEb = 0
and
dEo = 0

then:
dE = dEk
giving:
dE = (1 / C^2) {[dE (V - Vo)^2] + [E (V - Vo) dV]}

Thus:
dE [1 - ((V - Vo) / C)^2] = (1 / C^2) E (V - Vo) dV
or
dE / E = (V - Vo) dV / [C^2 - (V - Vo)^2]
and since Vo = constant:
dE / E = (V - Vo) d(V - Vo) / [C^2 - (V - Vo)^2]

Integrating from state a to state b gives:
Ln(Eb / Ea) = (-1 / 2) {Ln|C^2 - (Vb - Vo)^2| - Ln|C^2 - (Va - Vo)^2|
or
Ln[(Ea / Eb)^2] = Ln(|C^2 - (Vb - Vo)^2| / |C^2 - (Va - Vo)^2|)
or
(Ea / Eb)^2 = |C^2 - (Vb - Vo)^2| / |C^2 - (Va - Vo)^2|

Choose state a to be the particle at rest so (Va - Vo) = 0, Ea = Eo. Then:
Ea^2 = Eb^2 |C^2 - (Vb - Vo)^2| / |C^2|
= Eb^2 - Eb^2 (Vb - Vo)^2 / C^2
or
Eb^2 = (Eb^2 (Vb - Vo)^2 / C^2) + Ea^2
= (Eb^2 (Vb - Vo)^2 / C^4)C^2 + Ea^2
= Pb^2 C^2 + Ea^2
or more generally:
E^2 = P^2 C^2 + Eo^2
where the kinetic energy is:
Ek = E - Eo

Note that pair production experiments have shown that Eo is absolute energy with respect to vacuum free space.

Hence, provided that the observer is inertial the linear momentum (P - Po) of an object is related to total object absolute energy E and the object's absolute rest energy Eo via the equation:
E^2 = ((P - Po)*C)^2 + Eo^2
where:
C = speed of light along the direction of momentum propagation, which Einstein assumed to be of the same for all inertial observers. This assumption was justified by the results of the Michaelson-Morley experiment.

KINETIC ENERGY AND VELOCITY:
Application of the Einstein relationship to particle i gives:
Ei^2 = (Pi - Poi)^2 C^2 + Eoi^2
where:
Eoi = potential energy of particle i at rest
and
(Pi - Poi) = (Ei / C^2)(Vi - Vo)
and
Poi = (Ei / C^2) Vo

Due to the vector nature of (Pi - Poi) and (Vi - Voi):
(Pi - Poi) = (Pix - Poix)x + (Piy - Poiy)y + (Piz - Poiz)z
and
(Vi - Vo) = (Vix - Vox)x + (Viy - Voy)y + (Viz - Voz)z

Hence the momentum equation is actually three equations, one for each axis. Total particle energy Ei has at least four orthogonal components consisting of rest energy and momentum along the x, y and z axes. The rest energy involves more orthogonal field related components.

Combination of these equations gives:
Ei^2 = (Ei / C^2)^2 (Vi - Vo)^2 C^2 + Eoi^2
or
Ei^2 {1 - [(Vi - Vo)^2 / C^2]} = Eoi^2
or
Ei = Eoi / {1 - [(Vi - Vo)^2 / C^2]}^0.5
where:
C = speed of light in a vacuum.
Note that in this expression if (Vi - Vo) = 0, then Eki = 0 and Ei = Eoi, as required by the above definition of kinetic energy.

This equation is valid for individual particles but requires generalization for proper application to real objects involving spacially distributed particles.

Note that this expression mathematically permits negative rest energy and negative kinetic energy values.

Rearranging this expression gives kinetic energy Eki as:
Eki = Ei - Eoi
= Ei - Ei{1 - [(Vi - Vo)^2 / C^2]}^0.5
= Ei (1 - {1 - [(Vi - Vo)^2 / C^2]}^0.5)

For the special case of |Vi - Vo| << C:
Eki ~ Ei (1 - {1 - (1 / 2)[(Vi - Vo) / C]^2})
~ (Ei / 2)[(Vi - Vo) / C]^2
which with the substitution:
Mi = Ei / C^2
becomes the Newtonian expression for kinetic energy:
Eki = (Mi / 2) (Vi - Vo)^2

The discovery that:
Ei = Mi C^2
was the key to understanding nuclear energy.

Note that special relativity permits the existence of particles that have negative energy with respect to a field free vacuum. Such particles, which are known as anti-matter, have importance in nuclear physics, especially in beta decay sequences that involve creation of an electron-positron pair immediately followed by neutron formation and positron emission.

RATIO OF KINETIC ENERGY TO REST ENERGY:
The rest energy Eoi of particle i is defined as:
Eoi = Ei - Eki

If |Vi - Vo| << C
then:
Eoi ~ Ei
and the ratio of Eki / Eoi becomes:
(Eki / Eoi) ~ [(Mi / 2) (Vi - Vo)^2] / Mi C^2
= (1 / 2)[(Vi - Vo) / C]^2
indicating that Eki <<< Eoi.

INCREMENT OF ENERGY CONVEYED BY A PHOTON:
Recall that:
Ei^2 = Eoi^2 + C^2 |Pi - Poi|^2

For the special case of a photon which has no rest energy:
Eoi = 0
giving:
Ei = Eki = C |Pi - Poi|

Photons can propagate through a vacuum or can be guided along a transmission path by a transmission line or wave guide. In some circumstances photons with specific energies can be selectively reflected or absorbed.

FIELD FREE VACUUM STATE:
If an isolated system contains nothing but a field free vacuum, then:
E = Sum of all Ei = 0

ANTIMATTER:
Normal matter rest mass contains an amount of energy:
Eo = M C^2
Where M is positive, Eo is positive and the energy level of a field free vacuum is zero. Hence Eo is the energy difference between the particle energy and a field free vacuum reference.

Anti-matter rest mass represents lack of occupancy of an energy state that when occupied (as is normal) contains an amount of energy:
E = - M C^2
where M is positive, E is negative, and the energy level of a field free vacuum is zero.

Thus although the absolute energy of an anti-particle is negative since the presence of an anti-particle represents a lack of state occupancy the energy per unit volume is positive.

When matter and anti-matter annihilate each other the change in energy is 2 E. An occupied +ve energy state and an unoccupied -ve energy state simultaneously disappear. The occupied energy state in transitioning to an unoccupied field free vacuum level releases energy E. Filling in the unoccupied energy state from the field free vacuum level releases another increment of energy E. Thus the energy released is 2 E.

A photon conveys a positive increment in energy. Hence on absorption of a photon a particle increases its energy and on emission of a photon a particle decreases its energy.

When a photon carrying energy (+2E) converts into a particle pair it creates a particle with energy E and an anti-particle with energy E. Thus the photon has delivered an energy increment of (+2E).

This relationship between matter, anti-matter and photons has been demonstrated by numerous nuclear pair production experiments. Similarly when a normal matter particle of energy E and an anti-matter particle of energy E anhiliate each other a photon with energy (2E) is emitted.

Charged anti-matter particles have opposite charge sign to normal matter particles. This when pair production occurs there must be enough energy to physically separate the two particles. This energy becomes rest mass energy.

PAIR PRODUCTION / ANNIHILATION:
Under appropriate circumstances an electromagnetic radiation photon with sufficient energy Ep converts into a particle - antiparticle pair. The particle has rest and kinetic energy:
(+ Ep / 2)
with respect to the vacuum state and the anti-particle has rest and kinetic energy:
(- Ep / 2)
with respect to the vacuum state.

The field free vacuum state is assumed to be the top of a fully occupied density of states.

The effect of particle-anti-particle annihilation is to allow the positive energy particle to fall into the anti-particle's negative energy hole in space emitting a photon of energy Ep.

The total energy Ep of a particle - antiparticle pair is given by:
Ep = (+ Ep / 2) - (- Ep / 2)

Note that a photon provides an increment of energy that separates a normal particle from an anti-particle.

Hence a vacuum can be viewed as a region that contains zero rest energy. If the origin of the universe was in radiation then anti-matter corresponding to the known ordinary matter exists somewhere. The existence of negative energy indicates that an anti-particle moving with positive velocity has negative momentum and negative kinetic energy.

It appears from the mathematical symmetry that electrically neutral anti-matter gravitationally attracts other electrically neutral anti-matter but gravitationally repels neutral ordinary matter. That behavior is consistent with astronomical observations which indicate that there is no accumulation of anti-matter anywhere close by. However, this issue remains to be experimantally proven.

There is an additional complication with anti-matter. When a high energy photon causes production of an electron-positron pair the electron has rest mass energy equal to half of the photon energy. The positron absorbs the other half of the photon energy. During pair production a packet of energy (an electron) moves from the field free vacuum energy level to above the field free vaccuum energy level. Another packet of energy (a positron aka anti-electron) with opposite sign simultaneously moves from the vaccuum energy level to below the vacuum energy level. Thus a photon's electric field causes a ripple in the vacuum energy level. Pair production maintains the same average vacuum energy level.

POWER:
Power is rate of energy transfer (dE / dT) between systems. If the potential energy remains constant power is given by:
(dEk / dT) = [d(Pc - Po) / dT] * [d(Xc - Xo) / dT]
= (external force) * (velocity)

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CHARGE, FIELDS, ENERGY:
The energy distribution is composed of particles (charge quanta, closed current paths, vector fields) and radiation which are all mathematically intertwined. The existence of a charge quantum at a position in space relative to an observer causes a specific local vector electric field distribution. Motion of the charge quantum around a closed path causes a specific local magnetic field vector distribution. At any point in space and time the sum of the squares of the net electric, net magnetic and net gravitational field vector magnitudes is the local potential energy density. There is also a relative motion component to energy known as kinetic energy. Total energy content is computed by integrating energy density over volume. The total energy integrated over all space is a constant invariant over time.

Everything that exists contains some energy. The energy density of field free space is zero. A gravitational field contains negative energy.

Radiation consists of vector field fluctuations which propagate at the speed of light. Radiation conveys both energy and momentum (energy motion) without conveying net quantum charge.

PARTICLES:
The universe consists of particles and assemblies of particles within a sea of propagating radiation photons. At reference time T = To each particle has a quantized net charge Q, a nominal relative position (X = Xo), a nominal relative momentum (P = Po) and a relative energy (E = Eo). The particle energy has magnetic, electric, gravitational and motion components. For an isolated particle at rest potential energy is contained in mathematically orthogonal fields that extend to infinity but for each orthogonal field type the field energy density decreases sufficiently quickly with increasing radial distance from Xo that the total energy is finite.

PARTICLE INTERACTION:
Particles interact with each other at a distance via overlap of their extended fields. Field overlap causes the net field vector at each point in space to change which in turn causes potential energy to convert into kinetic energy (momentum related energy) or vice versa. During interparticle interactions there is also an energy exchange between particle energy and radiation. Photons (quanta of radiant energy) may be emitted from a particle to the radiation environment and/or absorbed by a particle from the radiation environment.

FORMATION OF MUTUAL POTENTIAL ENERGY WELLS:
Progressive overlap of fields may cause an assembly of particles to gain more positive kinetic energy by acquiring more negative potential energy via field overlap. Then net emission of radiation may cause the assembly of particles to lose kinetic energy. Hence there is an overall tendency for the total potential energy of an assembly of particles to become more negative which causes the particles to become mutually bound to each other in a common potential energy well.

Absorption of radiation from an external radiation source such as sunlight at a sufficient rate can in principle reverse this particle binding process.

In all particle interactions the total isolated system energy, including emitted radiant energy, is unchanged during a particle interaction. This principle is known as the law of conservation of energy.

In all particle interactions the total isolated system energy motion vector (linear momentum), including emitted photons, is unchanged during a particle interaction. This principle is known as the law of conservation of linear momentum.

In the case of four hydrogen atoms going through a succession of interactions to form one helium atom the particle aggregation process is known as fusion. The photons emitted during fusion reactions are the main source of energy emitted by the sun and the stars.

High atomic weight nuclei result from end-of-life stellar explosions. There are a few high atomic weight atomic nuclei such as U-233, U-235 and Pu-239 that, when suitably stimulated by neutron or gamma photon absorption, fission by breaking into smaller particles and liberating both kinetic energy and gamma photons. Fission is the main controllable source of nuclear energy available on Earth.

There are a large number of semi-stable atomic nuclei that over time gradually break down into more stable nuclei by random spontaneous emission of electrons, positrons or alpha particles (He-4 nuclei), as well as gamma photons. These semi-stable nuclei are known as radio isotopes.

Dark matter is a term used by astronomers to account for the apparent existence of a non-luminous gravitational mass component of galaxies.

Dark energy is a term used by astronomers to account for the apparent accelerating expansion of the visible universe.

ENERGY AND MOMENTUM OF A PARTICLE:
An issue of fundamental importance is the tradeoff between potential energy and momentum (kinetic energy) when particles interact via field overlap. The total energy remains constant but part of the potential energy is converted into kinetic energy. As a particle moves from state "a" to state "b" energy is conserved by the particle following:
Ea^2 + Pa^2 C^2 = Eb^2 + Pb^2 C^2

Typically a free particle has an initial potential energy Ea and Pa ~ 0. As the particle loses potential energy it gains momentum. Hence:
Eb < Ea
and
Pb > Pa

The energy conservation equation gives:
Ea^2 - Eb^2 = C^2 ( Pb^2 - Pa^2)

The corresponding differential equation is:
E dE = - C^2 P dP
This differential equation precisely describes the tradeoff between potential energy and momentum based kinetic energy.

Integrating both sides of the differential equation gives:
Eb^2 - Ea^2 = - C^2 (Pb^2 - Pa^2) or
Ea^2 + C^2 Pa^2 = Eb^2 + C^2 Pb^2
as expected.

Hence the differential equation which relates the change in energy E of a particle to its change in linear momentum P is:
E dE = - C^2 P dP
where:
C = speed of light.

For particles with rest mass it is often convenient to choose the initial condition reference point (to = ta, Xo = Xa, Po = Pa, Eo = Ea) as the state when the particle is at rest in the observers frame of reference. Then:
Pa = 0 at E = Ea to get:
Ea^2 = Eb^2 + C^2 Pb^2

Einstein recognized that:
Ea = Mo C^2
where Mo = rest mass
and that:
Pb = (Eb Vb) / C^2
= Mb Vb
where Mb = moving mass.

Alternatively:
Mb = Eb / C^2
or
Eb = Mb C^2
which is one of the most famous equations in physics.

NEWTONIAN KINETIC ENERGY:
Recall that:
Ea^2 = Eb^2 + C^2 Pb^2
= Eb^2 + C^2 (Eb / C^2)^2 Vb^2
= Eb^2 [1 + Vb^2 / C^2]
or
Eb = Ea / [1 + (Vb^2 / C^2)]^0.5

The Newtonian kinetic energy is:
KE = Ea - Eb
= Ea - {Ea / [1 + (Vb^2 / C^2)]^0.5}
= Ea {1 - [1 / [1 + (Vb^2 / C^2)]^0.5]}

For (Vb / C) << 1:
KE ~ Ea {1 - [1 - (Vb^2 / 2 C^2)]}
= Ea Vb^2 / 2 C^2
= Mo C^2 (Vb^2 / 2 C^2)
= Mo Vb^2 / 2
which is the Newtonian kinetic energy.

QUANTUM CHANGES IN MOMENTUM AND ENERGY FOR A SINGLE PARTICLE:
It is shown in this section that quantization due to radiative emission/absorption cannot work for a single particle. Recall that:
Ea^2 + C^2 Pa^2 = Eb^2 + C^2 Pb^2
or
Ea^2 - Eb^2 = C^2 (Pb^2 - Pa^2)
or
(Ea - Eb)(Ea + Eb) = C^2 (Pb - Pa)(Pb + Pa)
or
[(Ea - Eb) / (Pb - Pa)] = C^2 (Pb + Pa) / (Ea + Eb)

If energy and momentum are both constrained by quantization for radiative absoption/emaiison then:
Ea - Eb = h F
and:
Pb - Pa = h F / C
giving:
C = C^2 (Pb + Pa) / (Ea + Eb)
which rearrranged gives:
(Ea + Eb) = C (Pa + Pb)

For:
Ea = Eb + h F
and
Pb = Pa + [h F / C]
Then:
2 Eb + h F = C (2 Pb) - h F
or
Eb = C Pb - h F

However:
h F = Ea - Eb
which gives:
Eb = C Pb - h F = C Pb - (Ea - Eb)
or
Ea = C Pb
which does not work, implying that if quantum radiative emission or absorption is involved other particles must also be involved.

POTENTIAL ENERGY WELLS:
As a particle following a constant energy path with respect to field free space enters a gravitational potential well its rest potential energy Ea with respect to field free space will decrease or become more negative. However, Eb remains unchanged as required to maintain a constant particle energy with respect to field free space. Hence on entering the gravitational potential energy well there is an increase in the particle's (Eb - Ea) value and hence an increase in particle momentum P. This momentum increase causes a decrease in the wavelength of a photon and hence a deflection of a photon following a path tangential to a mass concentration.

Within a potential energy well a particle's Ea value is lower than outside the potential energy well. If while within the potential energy well a particle loses sufficient energy that the particle's kinetic energy becomes less than than the potential energy well depth:
(Ea|outside - Ea|inside)
then the particle will be trapped within the potential energy well. A simple physical example of such trapping is the presence of planets orbiting around our sun. The planets cannot escape from our sun because they are trapped in our sun's gravitational potential energy well. However, if two orbiting planets interact in a manner that transfers energy from one planet to the other the planet with the higher energy might be able to escape the energy well.

Solid matter is simply a collection of atoms that are mutually bound together in a common potential energy well.

FORCES:
Recall the differential equation:
E dE = - C^2 P dP
can be rewitten as:
dE = - C^2 (P / E) dP
= - C^2 (E V) / (C^2 E) dP
= - V dP

However:
V = - dX / dt

Hence:
dE = [dX / dt] dP
or
dE dt = dX dP
which equation links a particle's changes in energy, time, position and momentum.

Alternatively:
(dE / dX) = dP / dt
= d(M V) / dt
= [M (dV / dt) + V (dM / dt)]

For V << C the term V (dM / dt) is negligibly small giving:
(dE / dX) = M (dV / dt)
or
(force) = (mass) X (acceleration)

A force is simply a change in particle potential energy with respect to a change in particle position with respect to the other particles with which it is interacting.

Thus the differential equation:
E dE = - C^2 P dP
incorporates many aspects of particle motion.

MODERN PHYSICS:
Electromagnetic energy propagates through space via radiation. The energy density with respect to a field free vacuum at every point in space and time causes a gravitational field that extends to infinity. Gravitational fields form negative potential energy wells that cause interactions between widely separated particles.

During the first half of the 20th century it was shown that every large mass consists of an aggregation of stable atomic particles with quantized charges and corresponding discrete energies. Subject to structural constraints the atomic particles vibrate or move randomly with thermal kinetic energy. Planck showed that the vibrating atomic particles constantly emit and absorb photons which are quanta of thermal electromagnetic radiation.

Later during the 20th century it was shown that a sufficient concentration of mass would cause formation of a black hole that absorbs both mass and radiant energy from its surroundings. Smaller mass concentrations cause local changes in photon frequency and photon propagation direction.

In the early 21st century it became apparent that the electromagnetic energy of a charged particle is stored in a spheromak formed by massless quantized charge that circulates around a closed spiral path at the speed of light. A spheromak has associated with it electric and magnetic fields that contain potential energy. This energy is the particle's rest mass. The fields give the spheromak's charge motion path geometrical stability. Changes in spheromak electromagnetic energy dE caused by emission or absorption of a photon follow the equation:
dE = h dF
where:
dE = photon energy
h = Planck constant
dF = change in spheromak natural frequency

RELATIVITY:
The natural physical laws are such that it is impossible to determine absolute position, absolute velocity or absolute time. Position, momentum and time are relative quantites with respect to an observer or a system center of momentum. All inertial (non-accelerating) observers measure the same speed of light. As a consequence the experience of time, distance, momentum and energy are different for observers in relative motion. The change in particle energy with respect to a change in particle momentum gives rise to the concept of kinetic energy.

CONSERVATION OF ENERGY:
The law of conservation of energy requires that the amount of energy inside a closed surfce at time Tb equals the amount of energy inside that closed surface at time Ta plus the net amount of energy that flows into that enclosed space during the time interval (Tb - Ta). Hence energy cannot be either created or destroyed but can change form and position and can be transferred between particles.

In an isolated system total energy is always conserved. For an isolated system a decrease in potential energy causes a corresponding increase in radiant or kinetic energy and vice-versa. The energy of a non-isolated system can change via energy absorption from another system or via energy emission to another system. Often these energy exchanges occur via emission or absorption of quanta of radiant energy (photons).

Energy is a result of the existence of atomic particles and the existence of radiation. Energy takes several forms: nuclear energy, magnetic field energy, electric field energy, gravitational field energy, kinetic energy and radiation. Energy may possibly exist in other forms. The energy density of field free empty space is assumed to be zero.

GRAVITY:
Gravitational field energy is a result of the presence and density of other energy forms. When other forms of energy are zero there is also zero gravitional field energy. Unlike most other energy forms which are normally positive, the gravitational field energy density is normally negative.

Gravitational fields have an imaginary unit vector which causes a gravitational field to contain negative potential energy. Gravity becomes important when large numbers of quantized negative and positive charges are in nearly exact balance. The exact nature of propagating gravitational field energy quanta is a subject of current research.

Assume that isolated particles are initially widely separated and not in relative motion. Initially these particles have zero kinetic energy. Thus gravity causes gradual conversion of isolated particle potential energy into kinetic energy. The kinetic energy then often partially converts into photon energy that is emitted into deep space. Hence the negative energy content of a gravitational field is a result of long term emission of positive energy photons into deep space.

The effect of an energy concentration is to form a local gravitational potential energy well. Far from the potential well the gravitational field energy density approaches zero. Closer to the center of the potential well the gravitational field energy density becomes increasingly more negative. The gravitational field geometry and hence the gravitational field energy and hence the gravitational force changes with relative particle position. Gravitational forces are a result of the change in total gravitational field energy with respect to a change in particle position. The apparent depth of a gravitational field potential energy well is proportional to the amount of energy carried by the sensing particle. Increased overlap of gravitational fields makes the total gravitational potential energy of a collection of particles more negative.

Local concentrations of energy are often mathematically approximated by point masses.

Under circumstances of low incident electromagnetic radiation density kinetic energy can be emitted as electromagnetic radiant energy which then propagates away at the speed of light. Under circumstances of high incident electromagnetic radiation density electromagnetic radiation can be absorbed and converted into kinetic energy and then potential energy. In our local universe the radiation density in deep outer space is very low compared to the thermal radiation density near the sun. Hence emission of solar radiation gradually increases the depth of the solar system's gravitational potential energy well.

STAR EVOLUTION:
At present the local universe primarily evolves by gradual aggregation of nearly isolated neutral hydrogen molecules. This aggregation causes formation of negative gravitational field potential energy and positive kinetic energy at star locations. Part of the kinetic energy converts into positive energy radiation which is emitted into deep space. The remaining kinetic energy raises stars to their fusion ignition temperature.

GRAVITY AND LIGHT:
If a light beam wave front passes tangentially through a gravitational field gradient the portion of the wave front closest to the gravitation source passes through a region of lower potential energy than the portion of the wave front further from the gravitation source. Within the wave the law of conservation of energy applies. When the photon's potential energy Ea becomes more negative its energy difference:
(Eb - Ea) = h Fp
increases to maintain a constant photon energy with respect to field free space. Since h (the Planck constant) is constant within the gravitational energy well this equation forces the photon frequency Fp to increase and hence its wavelength:
Lamdap = C / Fp
to decrease. Hence from a wave front propagation perspective the photon path slightly bends toward the gravitation source. This bending has been experimentally observed via bending of the path of star light as it passes tangentially past the sun or a black hole.

A photon wave front entering a gravitational energy well gains momentum and hence from the perspective of an external observer increases in frequency. A photon wave front exiting a gravitational energy well loses momentum and hence from the perspective of an external observer decreases in frequency. Hence, if a photon originates from inside a gravitational energy well, on exiting the well the photon decreases in momentum and frequency. This effect contributes to observed astronomical red shifts.

DIRECTION OF TIME:
Our local universe ages by gradual aggregation of isolated particles into mutual potential energy wells. During the aggregation process radiation is emitted into deep space. Most of the radiation photons escape from the local mutual gravitational potential energy well. Thus there is an apparent ongoing decrease in the average energy density of the local universe, which might be interpreted as local universe expansion. This process is known as an increase in entropy and establishes the direction of energy exchanges over time.

BLACK HOLES:
Gravitational aggregation of particles eventually leads to formation of a deep gravitational potential energy well known as a black hole from which radiation photons cannot easily escape. A gravitational black hole acts as an energy sink rather than an energy source. The issues of what happens inside a gravitational black hole are beyond the scope of this web site.

Black holes perform an important life enabling function of absorbing radiation, which cools the space around them. Life processes on Earth rely on emission of thermal infrared radiation into a cold universe for temperature maintenance. The issue of whether emitted thermal radiation is absorbed just by black holes or is absorbed by a physically expanding universe or by both is beyond the scope of this web site.

COSMOLOGY:
Gravitational fields are believed to be a result of concentrations of energy affecting the structure of space-time. The spacial energy density at each point in space and time causes a gravitational field vector distribution that slightly modifies the potential energy density at other points in space at later times. The relationship between energy and the structure of space-time is the subject of general relativity.

The exact manner in which the energy of galaxies affects the structure of space-time is a subject of current astronomical research. For the purpose of this web site gravity is treated as an imaginary field causing negative potential energy. This treatment might be imperfect, but it is adequate for most practical engineering purposes.

The history of the universe prior to the formation of neutral hydrogen atoms is highly speculative. It is usually assumed that particle rest mass energy came from particle pair production by high energy gamma ray photons originating in a "big bang".

Radiation propagates linearly according to the Poynting vector which is the vector cross product of the electric and magnetic fields. Radiation photons with energy Ep convey both energy given by:
Ep = h Fp
and linear momentum Pp given by:
Pp = [Ep / C^2] C
= [h Fp / C^2]C
where
h = Planck Constant

ISOLATED SYSTEM EVOLUTION:
A system can only evolve along a path that is consistent with both conservation of energy and conservation of linear momentum.

CENTER OF MOMENTUM
An isolated assembly of particles each with energy Ei has a center of momentum location Xc at which point the vector sum of all:
Ei (d((Xi - Xc) / dt) = 0
where:
(Xi - Xc) is a vector from the center of momentum to the ith energy bit Ei:
and
t = time as measured at the center of momentum.

From the perspective of an observer at the center of momentum:
dXc / dt = 0

From the perspective of an external inertial observer the center of momentum has a momentum:
dPc

If an energy packet is isolated from external fields its center of momentum moves at a constant velocity in the frame of reference of an external inertial observer (an observer not subject to acceleration).

MEASURES OF ENERGY:
Cosmology is concerned with absolute energy. However, most energy measurements and engineering calculations are expressions of energy differences with respect to some reference energy level. The reference energy level depends on the application and the position and motion of the observer.

It is convenient to distinguish between energy seen by an observer at the center of momentum and energy seen by an observer in constant linear motion with respect to the center of momentum.

Potential energy in the frame of reference of the center of momentum is usually chemical potential energy and/or nuclear potential energy. Kinetic energy in the frame of reference of the center of momentum is usually heat or kinetic energy of rotation.

Potential energy related to the position of the center of momentum is commonly referred to as potential energy of position. Energy related to motion of the center of momentum with respect to the observer is commonly referred to as kinetic energy.

An isolated stable concentration of potential energy with a nominal relative position, relative linear motion and heat content is often treated as a particle. Each particle has radial fields that extend to infinity but that contain finite amounts of energy.

PARTICLE INTERACTIONS:
Interactions between fields of different particles cause part of the field potential energy of the particles to convert to kinetic energy (energy of motion) in the frame of reference of the center of momentum or vice versa. This energy form change causes particles to accelerate or decelerate. However the net energy motion (linear momentum) of the center of momentum of the entire cluster of isolated particles remains unchanged. This principle is known as conservation of linear momentum.

Radiation transports quanta of energy away at the speed of light. A radiation quantum may be an electromagnetic photon, a neutrino or a graviton.

FIELD INTERACTIONS:
Each isolated free particle has an energy consisting of rest (potential) energy and kinetic (motion) energy in the external observer's frame of reference. Each bound particle has an additional binding energy (negative potential energy) component relating to the local interaction of that particle with other particles.

CHARGE AND CHARGE MOTION DISTRIBUTION IN THE UNIVERSE:
The universe can be viewed as consisting of a particle distribution in a sea of radiation. Each particle has an energy, motion and charge. Electric charge is a conserved parameter. The integral of the net electric charge density over all space is believed to be close to zero. A significant net electric charge would cause rapid expansion of the universe. Cosmologists usually assume that the net charge in the universe is zero. Some cosmologists also assume that the net angular momentum of the universe is zero.

Each element of electric charge causes a radial vector electric field. A sheet charge causes a step change in electric field. Charge motion causes a vector magnetic field. At every point in space and time the prevailing static three dimensional electric and magnetic field vectors separately add causing an electromagnetic field energy density distribution.

FIELD DISTRIBUTION IN THE UNIVERSE:
It is helpful to represent the energy contained in confined radiation by a nuclear field vector. Then at every point in space and time there is a characteristic net static nuclear field vector, electric field vector, magnetic field vector and gravitational field vector. These net vectors are mathematically orthogonal to each other and are the result of the sums of vector fields arising from nuclear density, charge density, charge motion and energy density at other points in space at previous times.

ELECTRIC CHARGE, FIELDS AND TIME:
Electric charge, electric charge motion and electric and magnetic field vectors and electromagnetic energy densities are mathematically intertwined. Electric charge causes an electric field. Electric charge motion (current) causes a magnetic field. A change in magnetic field with time causes an induced electric field. A change in electric field with time corresponds to charge motion which causes a magnetic field. Thus the spacial electric charge distribution over time defines the electric and magnetic vector field distributions and vice-versa. The sum of the squares of these orthogonal net field vectors is the local electromagnetic field energy density.

Positive potential energy is contained in static electric and magnetic fields. These fields occur as a result of the existence and closed path motion of electric charge. At the microscopic level the mathematical equations that determine the spacial distribution of energy may have multiple real solutions. This issue leads to a branch of physics known as quantum mechanics. Atomic particles have characteristic natural frequencies and exhibit electromagnetic wave like properties. Due to the multiple real solutions there is always uncertainty in simultaneous measurements of particle energy and time and in simultaneous measurements of particle position and particle momentum.

POTENTIAL ENERGY DENSITY DISTRIBUTION IN THE UNIVERSE:
The potential energy density at any point in space and time is a function of:
a) The nuclear field vector at that point and time;
b) The net electric field vector at that point and time;
c) The net magnetic field vector at that point and time;
d) The net gravitational field vector at that point and time;
These four energy components are mathematically orthogonal.

The total field energy density at any position in space at any instant in time is the sum of the squares of the instantaneous values of the four mathematically orthogonal vector field components (nuclear, electric, magnetic, gravity). Each vector field component has three orthogonal dimension components.

FIELD ENERGY DEFINITION:
A modern definition of field energy density U at any point Xo and time To is:
U = [C1 E^2 + C2 H^2 + C3 (i G)^2] where mutually orthogonal vectors E, B, (i G) are defined by:
E = net electric field vector at Xo, To
H = net magnetic field vector at Xo, To
(i G) = net gravitational field vector at Xo, To
C1 = (Epsilono / 2),
C2 = (Muo / 2),
C3 = _______
Note that i^2 = -1

For electric charge:
Q = charge;
E = Q / (Epsilono 4 Pi R^2)
= (Q / Epsilono)(1 / 4 Pi R^2)
U = (Epsilono / 2)[Q / (4 Pi Epsilono R^2)]^2

For gravity:

F = (G i M i m) / R^2
(F /i m) = G i M / R^2
= (4 Pi G) i M / (4 Pi R^2)
= i M / [(1 / 4 Pi G)(4 Pi R^2)]

i M takes the role of Q

(1 / 4 Pi G) takes the role of Epsilono

i M / [(1 / 4 Pi G)(4 Pi R^2)] takes the role of the spherical electric field

[i M / (1 / 4 Pi G)] takes the role of [Q / Epsilono]

Hence:
Ug = [1 / (8 Pi G)]{i M / [(1 / 4 Pi G)(4 Pi R^2)]}^2
where the gavitational field vector is:
GRAV = {i M / [(1 / 4 Pi G)(4 Pi R^2)]}
= {i M G / R^2}

Then the gravitational field energy density Ug due to point mass M is:
Ug = [1 / (8 Pi G)][GRAV]^2
= [1 / (8 Pi G)][{i M G / R^2}]^2
= [1 / (8 Pi G)][{- M^2 G^2 / R^4}]
= [- M^2 G / 8 Pi R^4]

In summary point mass M causes a gravitational vector:
GRAV = i M G / R^2
and an energy density:
Ug = [- M^2 G / 8 Pi R^4]

Hence the total local energy density at any point is given by:

U = Ue + Um + Ug
= (Epsilono / 2) E^2 + (Muo / 2) H^2 + [1 / (8 Pi G)] [GRAV]^2

Thus field energy density arises from the squares of the net electric, magnetic and gravitational vector field terms. Alternatively the field energy distribution can be viewed as arising from the charge, charge motion and energy distribution. Note that this formulation is only valid for fields that are static with respect to an inertial observer. The general case of an accelerating observer and/or propagating field changes is more complex. The local electric, magnetic and gravitational fields are functions of the spacial distributions of particles, charge, charge motion and energy elsewhere at earlier times.

SPHEROMAKS:
The electromagnetic energy of elementary charged particles such as electrons and protons is held in spheromaks. A charged particle spheromak contains a quantum of charge that circulates around a stable closed path at the speed of light. A spheromak has associated vector electric and vector magnetic fields that each contain potential energy. Generally the static field energy density is high near the nominal particle position (on the spheromak symmetry axis) and diminishes rapidly with increasing radial distance from the nominal particle position. The total energy contained in a vector field is finite even though the field radially extends to infinity. The field energy forms part of the particle's total energy. Much of a particle's energy lies in a confined photon. Motion of a particle in the frame of reference of an inertial observer gives the particle kinetic energy and linear momentum.

The extended electric and magnetic fields of different spheromaks interact. This interaction converts field rest potential energy into kinetic (motion) energy. In circumstances of low external radiation density kinetic energy can convert into photons that are radiated away, leaving the interacting particles bound together in a mutual potential energy well. The emitted photon energy and frequency are a result of the electromagnetic spheromak structures of elementary particles.

SPHEROMAK STRUCTURE:
If counterflowing uniform strings of positive and negative electric charge (and hence an electric current) having a net charge follows the path of a closed spiral the result is a physically stable electromagnetic structure known as a spheromak. Inside the closed spiral known as the spheromak wall there is a toroidal magnetic field. Outside the spheromak wall there is a poloidal magnetic field. Due to the net charge on the spiral within the spheromak wall there is a cylindrically radial electric field and in the far field outside the spheromak wall there is a spherically radial electric field. A requirement for geometrical stability is that at the toroidal surface, known as the spheromak wall, formed by the closed spiral path the total static field energy density is equal on both sides of the spheromak wall. There is additional dynamic energy within a confined photon that exists inside the spheromak wall.

To realize a stable spheromak the geometry of the closed spiral path must correspond to a static spheromak total energy minimum. For quantum charged atomic particles that energy minimum occurs at number of poloidal charge path turns Np = 222 and at number of toroidal charge path turns Nt = 305. This integer pair leads to the Planck Constant.

The stable spheromak structure enables the existence of highly stable elementary atomic particles such as electrons and protons as well as semi-stable particles and plasmas. The electric and magnetic fields associated with stable particles contain energy that contributes to the atomic particles rest mass. The external fields caused by these particles radially extend out to infinity. However, the total energy of an isolated particle is finite.

The static spheromak that dominates the particles electromagnetic behavior only accounts for a portion of the particle's rest mass. Most of the rest mass of quantum charged particles is carried by the confined photon.

CLOSED SPIRAL PATH:
A closed spiral path forming a spheromak is like a a uniform single layer wire winding on a toroid with the two ends of the winding connected together. The spiral path makes Np = 222 turns around the toroid's major axis and Nt = 305 turns around the toroid's minor axis before retracing its path. Thus an electric current following a closed spiral path causes both toroidal and poloidal magnetic fields. In order for a charged particle to be stable the path must precisely repeat itself.

QUANTIZED CHARGE:
In stable charged particles the electric charge is quantized. The mechanism of charge quantization is unknown. An element of charge has associated with it a radial vector electric field. An element of charge motion (current) has associated with it a vector magnetic field.

SPHEROMAK FORMATION:
If uniformly distributed electric charge continuously circulates around a closed spiral path with no change in spacial charge distribution and no change in current, then the electric and magnetic fields are static and there is no absorption or emission of radiation. Hence there is no change in energy. The result is a spheromak forming a stable charged particle. The electric and magnetic fields of stable charged particles at rest contain the particle's spheromak potential energy. A stable charged atomic particle at rest has a distribution of energy which is spacially constant over time.

At any instant in time, every stable spheromak can be characterized by its nominal position Xo with respect to the observer (known as its center of momentum), its rest energy Ett, its momentum vector P, its charge Q, its poloidal magnetic field vector M (angular momentum), its toroidal magnetic field vector (spin) S. Note that for a particular M there are two possible S vector values.

STABLE PARTICLES:
The known universe is primarily an assembly of stable particles known as electrons, protons and photons. All electrons seem to exhibit exactly the same net charge and characteristic isolated rest energy. All protons seem to exhibit an exactly equal but opposite net charge and a different characteristic isolated rest energy.

Neutrons can be viewed as being composed of an electron-proton assembly with zero net charge plus a small amount of additional energy.

Electromagnetic radiation photons have an oscillating electric field vector and an oscillating magnetic field vector but have no net charge and no rest energy.

The total energy of an atomic particle has a potential energy component and a kinetic energy component. If the particle is in an external field there may also be potential energy of position. This position dependent potential energy results from overlap of the particle's electric, magnetic and gravitational fields with the corresponding external fields.

The kinetic energy component is the energy component due to motion of the particle's nominal position.

Kinetic energy of free rotation is kinetic energy due to rotation of a rigid body about an axis through the body's nominal center of momentum. For reasons of mathematical simplicity it is often convenient to treat kinetic energy of rotation as a component of potential (rest) energy rather than as kinetic energy. Kinetic energy of free rotation can be important in both large rotating rigid bodies and in gases with multi-atomic molecules.

INTERACTIONS BETWEEN STABLE PARTICLES AND PHOTON EMISSION / ABSORPTION:
Stable particles have external fields. Particles interact with each other via their extended vector fields. At each point in space field vectors of a particular type vectorially add. Each orthogonal net field vector (nuclear field, electric field, magnetic field, gravitational field) squares to yield a field energy density component. Progressive vector field overlap causes a change in total potential energy and a corresponding change in kinetic energy. Since the vector fields extend from an object's nominal position to infinity, objects that are widely separated still weakly interact. The apparent force between distant objects is really the change in the total system potential energy with respect to a change in an object's position relative to the other objects in the system. Conservation of total energy requires that the change in potential energy either become an equal change in kinetic energy or be converted into emitted/absorbed radiation.

In a low external radiation environment part of the molecular kinetic energy may be lost to outer space via net emission of radiation photons, leaving the stable particles more closely bound in a mutual potential energy well. By this process in a low radiation environment particles tend to aggregate to form stable atomic nuclei, atoms, molecules, liquids, crystals, rocks, planets and stars.

At steady state the rate of energy absorption by particles bound in a mutual potential energy well equals the rate of radiation emission. At steady state in a high external radiation density environment the rate of photon emission must be high to equal the relatively high rate of photon capture. Thus in a high radiation density (high temperature) environment particle aggregations are less stable than in a low radiation density (low temperature) environment.

At steady state the thermal photon density indicates the temperature. Thus the infrared radiation spectrum emitted by matter into a vacuum with low radiation density indicates the temperature of the matter.

When an object is in thermal equilibrium with its environment the rates of photon emission and photon absorption are identical.

Most chemical reactions occur in low radiation environments in which the reactants shift from a high energy state to a lower energy state by net emission of infrared photons. An exception is the photosynthesis reaction which occurs in a high radiation environment (sunlight) in which the reactants shift from a low energy state to a higher energy state by net absorption of solar photons. Another exception is electrolysis driven chemical reactions in which the reactants gain energy from an externally applied electric field.

Absorption of high energy ultra-violet photons causes breakup of plastic hydrocarbon polymers by shifting the polymer components from a low energy bound state to a higher energy unbound state. Absorption of still higher energy X-ray photons and gamma photons can cause destruction of biological tissue compounds such as DNA.

In most spontaneous nuclear decay reactions the reactants shift from an unstable high energy state to a more stable lower energy state by emission of kinetic energy and x-ray or gamma photons. However, there are some important nuclear reactions such as gamma initiated fission that are triggered by absorption of gamma photons.

An important physical state change is absorption of solar photons by fine wind blown sea water droplets at ambient temperature to form water vapor. The inter-molecular binding energy per molecule is the latent heat of vaporization. However, this binding energy per molecule is less than the energy carried by a solar photon.

The free water molecules in the atmosphere exchange energy with free N2 and O2 molecules. The water molecules condense int liquid droplets over a wide range of temperatures and pressures.

Another important physical state change is freezing of liquid water droplets in lower temperature clouds which converts the molecular vibration energy into far infrared radiation. This freezing occurs at exactly 0 degrees C over a wide range of pressures.

The sun is constantly emitting solar photons into deep space. Since the sun's energy is finite the potential energy contained in the sun is decreasing and hence the period during which the sun can support life on Earth is finite.

The Earth is constantly emitting thermal infrared photons into deep space. Absent daily warming by solar radiation Earth's surface would soon cool.

Temperature is an indication of average kinetic energy per particle degree of freedom. Temperature is also related to the steady state infrared radiant energy density spectrum within a material.

The temperature at the Earth's surface is nearly constant over prolonged time indicating that the Earth's average rate of energy loss via infrared radiation emission is close to the Earth's average rate of energy gain via solar radiation absorption plus heat gain via radio isotope decay.

The flow of energy which is absorbed by the Earth from the sun and then emitted by Earth into deep space can be tapped do useful work. eg To grow plants and to produce hydroelectric, solar and wind power.

A difference between the flow of energy absorbed by Earth from the sun and the flow of energy emitted by Earth into deep space causes changes in stored thermal energy on Earth which in turn causes formation or melting of polar ice and/or a gradual change in ocean surface temperature. Changing the flow of solar energy absorbed by Earth or the flow of infrared energy emitted from Earth leads to long term climate change.

ENERGY QUANTIZATION:
The field interaction equations involving stable atomic particles with quantized charge often have multiple real solutions corresponding to discrete energy states that are separated by energy gaps. A transition from one such energy state to another such energy state is usually accompanied by absorption or emission of a photon and/or by emission /absorption of a particle carrying kinetic energy equal to the energy difference between the two separated energy states.

PHOTON-SPHEROMAK INTERACTIONS:
Under suitable circumstances charged particles can absorb or emit electromagnetic radiation quanta known as a photons. The energy and momentum carried by a photon changes the absorbing or emitting particle's total energy and total momentum. The relationship between the amount of energy Ep contained in a photon and the photon's frequency Fp is:
Ep = h Fp
where:
h = Planck constant = 6.62607004 X 10-34 m2 kg / s

The origin of the Planck constant arises from the structure of electromagnetic spheromaks. Every stable atomic particle spheromak at rest has a characteristic natural frequency Fh. The relationship between spheromak energy E and frequency Fh is:
E = h Fh
The factor h arises from the manner in which energy is stored in a stable charged particle spheromak. The Planck constant and related units have recently been slightly changed due to redefinition of a standard kilogram.

An alternative definition of the Planck Constant is:
h = dE / dFh

In the presence of an external magnetic field atomic particles gain or lose energy via absorption or emission of quanta of radiant energy known as photons. Photons have no rest energy and propagate at the speed of light. Thus when a stable particle in state a with energy Ea emits a photon and hence shifts to state b with energy Eb the change in energy (dE) is given by:
(dE) = (Ea - Eb)
~ h (Fa - Fb)
= h (Fp)
where:
(dE) = change in particle energy [(dE) is positive for photon emission, (dE) is negative for photon absorption];
|Fp| = photon frequency
Thus to the extent that stable particles exhibit discrete energy states photon energies are also discrete.
Photon categories in order of increasing frequency |Fp| are:
AC power, audio, radio, microwave, infrared, optical, ultra-violet, x-ray and gamma ray.

Note that Ep is not precisely equal to the change in spheromak rest potential energy (Ettb - Etta) due to the photon's momentum that causes recoil kinetic energy in the emitting or absorbing particles.

PHOTON ENERGY AND MOMENTUM QUANTIZATION:
Since photons result from quantum energy changes in atomic particles, photon energy and photon linear momentum are also quantized.
Photon energy = Ep = h F
Photon momentum = Pp = h F / C

EFFECT OF RADIATION DENSITY:
Under circumstances of charge separation and low electromagnetic radiation density kinetic energy can become electromagnetic radiant energy which then propagates away from the center of momentum at the speed of light. Under circumstances of charge separation and high electromagnetic radiation density electromagnetic radiation can be absorbed and converted into kinetic energy and then potential energy. In our local universe the radiation density in outer space is very low compared to the thermal radiation density on Earth which means that Earth constantly emits thermal infrared radiation. Radiation emission by our sun gradually increases the depth of our solar system's potential energy well.

CHARGED PARTICLE BEAMS:
Charged particle beams also exhibit wave like behaviour. A beam of electrons incident upon two parallel slits forms an interference pattern. The effective wavelength Lamda of the electron beam is set by the electron's linear momentum P by the equation:
Wavelength: Lamda = h / P

Rearranging this equation gives:
P = (h / Lamda)
= (h F / C)
which is the same momentum versus frequency expression as for a photon.

The dominant source of rest mass energy for a charged particle is the particle's fields. The fields also give a charged particle its wave like behavior.

ATOMIC NUCLEI:
Atomic nucleons behave as if composed of mathematical sub-units known as quarks, but quarks have never been observed in isolation. In the "standard model" a hydrogen nucleus (proton) is composed of 3 quarks. A deuterium nucleus may involve six quarks. A helium-4 nucleus may be assembled from two deuterium nuclei containing 12 quarks. Larger nuclei involve a collection of particles bound together in a common mutual potential energy well. Weak nuclear binding occurs via the spheromak electromagnetic fields. Strong nuclear binding generally involves merging or destruction of spheromaks.

LARGE PARTICLE AGGREGATIONS:
For many practical engineering calculations involving assemblies of large numbers of aggregated particles (such as planets) the energy content of the net external electric and magnetic fields is negligible compared to the energy content of the net gravitational field. For these cases the field complexity of the universe can be ignored and the universe can instead be represented as a time dependent spacial energy (or mass) distribution, for which both energy and linear momentum are conserved parameters.

Chemical reactions generally involve changes in the spheromak field overlaps.

Strong nuclear reactions generally involve changes in confined photons.

This web page last updated September 18, 2020.