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**INTRODUCTION:**

THIS WEB PAGE SUMMARIZES BASIC ENERGY CONCEPTS.

**MOBILE POINTS:**

To a good approximation the universe consists of a large number of discrete mobile point particles. Each point particle is uniquely identified by an integer i. The position of point i with respect to an inertial observer at point **Xo** is **Xi - Xo**.

Since **X** is relative and **Xo** is arbitrary it is customary to choose the observers frame of reference so that:

**Xo** = 0

and

**dXo / dt** = 0

**PARTICLES:**

A discrete mobile point with quantized charge and with velocity:

**Vi** = 0

has energy:

Ei = Eio

where Eio is known as the rest energy of particle i.

**FORCE:**

A fundamental relationship for particle i between a change in energy Ei, a change in time dt, a change in momentum d**Pi** and a change in position d**Xi** is:

dEi dt = **dPi.dXi**

or

dEi = [**dPi** / dt].**dXi**

where:

**Fi** = [**dPi** / dt]

is known as the force on particle i.

Note that:

dEi = **Fi.dXi**

which defines a change in energy of particle i caused by application of a force **Fi**. Note that the measure of a force on particle i is the rate of change in momentum:

d**Pi** / dt

Note that forces arise from interactions between particles. For a single isolated particle:

**Pi** = constant,

Ei = constant,

**dPi** = 0,

dEi = 0,

**Fi** = 0

implying that the energy Ei and momentum **Pi** of particle i are constant and hence are conserved.

**EINSTEIN:**

Einstein found that the relationship between energy Ei and momentum **Pi** that correctly represents physical behavior is:

**Pi** = (Ei / C^2) **Vi**

and

Ei^2 = C^2 [**Pi**]^2 + Eio^2

where:

C = speed of light

and

Eio = Ei|(**Vi = 0**)

where Eio = rest energy of particle i.

Note that differentiation of the expression for Ei gives:

Ei dEi = C^2 **Pi.dPi**

or

dEi = [C^2 / Ei] **Pi.dPi**

= [C^2 /Ei] (Ei / C^2) **Vi.dPi**

= (d**Xi** / dt).**dPi**

Hence:

dEi dt = **dXi.dPi**

as expected.

The essence of quantum mechanics is that for any particle i:

dEi dt = **dXi.dPi** = h

where h is known as the Planck Constant, which leads to:

dEi = h / dt = h F

which is quantization of a change in particle energy.

**PHOTONS:**

A photon is a discrete mobile energy packet with:

Qi = 0

and

Eio = 0

which has energy:

Ep^2 = C^2 **Pp.Pp**

or

Ep = C |**Pp**|

and hence momentum:

|**Pp**| = Ep / C

The universe consists of particles and assemblies of particles moving within a sea of propagating radiation photons. The particle energy takes several forms: Non-Electro-Magnetic (NEM) energy, magnetic field energy, electric field energy, gravitational field energy, kinetic energy, electromagnetic radiation and NEM radiation.

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

Energy might also exist in other forms.

At reference time t = to each particle i has:

1) a quantized net charge Qi,

2) a nominal relative position **(Xi - Xio)**,

3) a nominal relative momentum **(Pi = Pio)** at **(Xi - Xio)**,

and
4) an energy (Ei - Eio). For an isolated particle at rest potential energy is contained in mathematically orthogonal fields that extend to infinity in space but which contain only finite amounts of energy.

**ENERGY PACKETS:**

Thus, energy is a basic constituant of the universe. Everything that exists has some energy. Mathematically the universe is a large collection of point particles, each point particle with energy Ei, nominal relative position (**Xi - Xo**) and a nominal relative velocity:

**Vi** = d(**Xi - Xo**) / dt

at

(**Xi - Xo**).

**TOTAL ENERGY:**

Any real object involves a spacial distribution of N non-zero energy packets Ei where the total energy Et of that real object is given by:

Et = Sum from i = 1 to i = N of:

Ei

**ABSENCE OF ENERGY:**

The energy density of field and particle free empty space is assumed to be zero. Non-existence of energy implies that for all possible "i" values:

Ei = 0.

**TIME:**

Time is measured by an inertial observer counting ticks on his/her own clock. There is no such thing as absolute time. All time measurements are with respect to some initial time to in the observer's frame of reference. Hence:

d(to) = 0

and

d(t - to) = dt

**CONSERVATION OF ENERGY OVER TIME:**

The most fundamental law of physics is the law of conservation of energy. For every isolated energy packet:

dEi / dt = 0

Hence for any isolated energy distribution:

dEt / dt = Sum from i = 1 to i = N of:

dEi / dt

= 0

so for any isolated energy distribution the total energy Et is constant.

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). Energy cannot be either created or destroyed but can change in 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 electromagnetic radiant energy (photons).

**SPACIAL POSITION OF ENERGY PACKETS:**

There is no such thing as absolute position. All positions are relative to a reference position vector:

**Xo**.

Isolated packets of energy can independently move through space.

At any instant in time the nominal position of energy packet Ei is:

(**Xi - Xo**).

where:

**Xo** = **xo + yo + zo**

is a reference position and

**Xi - Xo** = **(xi - xo) + (yi - yo) + (zi - zo)**

is the relative position vector **Xi - Xo**.

**JOINING OF ENERGY PACKETS:**

If at time (t - to) two energy packets 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 ENERGY PACKETS:**

At time (t - to) and position (**Xk - Xo**) an energy packet Ek can split into two energy packets Ei and Ej where:

Ei + Ej = Ek

and at the instant of split:

(**Xk - Xo**)

= (**Xi - Xo**)

= (**Xj - Xo**)

After the split:

dEi / dt = 0

and

dEj / dt = 0

**CENTER OF ENERGY:**

For any isolated cluster of N energy packets the position of the center of energy is:

(**Xce - Xo**)

where **Xce** is defined by:

Sum over all i = 1 to i = N of:

{Ei (**(Xi - Xo) - (Xce - Xo)**)}

= 0

or

Sum over all i = 1 to i = N of:

Ei [**(Xi - Xo)**]

= Sum over all i = 1 to i = N of:

Ei **(Xce - Xo)**

= Et **(Xce - Xo)**

Thus the total energy of an assembly can be thought of as being concentrated at the assembly center of energy.

**DISCRETE POINT MOTION:**

There is no such thing as absolute motion. All discrete point motion occurs at a relative position vector which may itself be changing with respect to the reference position vector.

Hence the velocity vector **Vi** of discrete point i located at position:

**Xi - Xo**

is given by:

**Vi** = d(**Xi - Xo**) / dt

The velocity vector:

d(**Xi - Xo**) / dt

may have a direction and magnitude that are arbitrary with respect to the direction and magnitude of the position vector:

(**Xi - Xo**)

**VELOCITY OF DISCRETE POINT i:**

The velocity **Vi** of discrete point i at position (**Xi - Xo**) is:

**Vi** = d(**Xi - Xo**) / dt.

= d[**(Xi - Xc) + (Xc - Xo)**] / dt

= d(**Xi - Xc**) / dt + d(**Xc - Xo**) / dt.

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

**SPEED OF LIGHT:**

There is no such thing as absolute position or absolute motion. Nature achieves that objective by making the propagation speed of light (photons) the same for all inertial observers.

Since the calculated speed of light is dependent on a measurement of distance and a measurement of time, nature makes these two measurements dependent on the frame of reference of the observer, so that the net effect is that the calculated speed of light is the same for all inertial observers. This behavior of distance and time measurements is the basis of Special Relativity.

**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 the x axis;

Ay = vector component of **A** parallel to the y axis;

Az = vector component of **A** parallel to the z axis;

where the axes are mutually orthogonal.

**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 corresponds to linear motion. The other motion vector component is normal to the radial position vector:

(**Xi - Xc**)

and corresponds to rotation about position **Xc - Xo**.

On the web page titled: Vector Identities it is shown that for two arbitrary vectors **A** and **B** sharing the same position **Xi - Xo**:

|**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 DISCRETE POINT i:**

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**

**ENERGY:**

Discrete point i has energy Ei. The Law of Conservation of Energy is:

d{Sum of all Ei} / dt = 0

**MOMENTUM:**

Momentum is energy motion relative to a reference point. The momentum **Pi** of discrete point i at position **Xi - Xo** is defined by:

**Pi** = (Ei / C^2) **Vi**

= (Ei / C^2) d(**Xi - Xo**) / dt

**EQUATIONS OF MOTION:**

When particles interact with each other there are exchanges of energy and momentum.
However, these exchanges are governed by the family of equations:

d**Pi**.d**Xi** = dEi dt
which must be simultaneously true for all values of i. These equations are sometimes known as the equations of motion. These equations link each particle's changes in energy, time, position and momentum and thus determine the evolution path of the universe.

It is helpful to expand the fundamental equation.

d**Pi**.d**Xi** = dEi dt

Recall that:

**Pi** = (Ei / C^2)d**Xi** / dt

= (Ei / C^2)**Vi**

Hence:

[d[(Ei / C^2)**Vi**] / dt].d**Xi** = dEi

or

[Ei / C^2][d**Vi**].d**Xi** + [(dEi / C^2)**Vi**].d**Xi** = dEi dt

or

[Ei / C^2][d**Vi**].d**Xi** = dEi [ dt - (1 / C^2)**Vi**.d**Xi**]

or

[Ei / C^2][d**Vi**].(d**Xi** /dt)

= dEi [ 1 - (1 / C^2)**Vi**.(d**Xi** / dt)]

or

[Ei / C^2][d**Vi**].(**Vi**)

= dEi [ 1 - (1 / C^2)**Vi**.(**Vi**)]

or

[Ei / C^2][d**Vi**].(**Vi**)

= dEi [ 1 - (|Vi|^2 / C^2)]

or

[d**Vi**].(**Vi**)

/ [C^2 - (|Vi|^2)] = dEi / Ei

Let Ui = C^2 - (**Vi.Vi**)

dUi = - 2 **Vi.dVi**

Hence:

dUi / (- 2 Ui) = dEi / Ei

or

(- 1 / 2) Ln(Uib / Uia) = Ln(Eib / Eia)

or

Ln[(Uia / Uib)^0.5] = Ln(Eib / Eia)

or

[(Uia / Uib)^0.5] = (Eib / Eia)

or

[(C^2 - Via^2) / (C^2 - Vib^2)]^0.5 = (Eib / Eia)

or

[(C^2 - Via^2) / (C^2 - Vib^2)] = (Eib^2 / Eia^2)

or

(C^2 - Via^2) Eia^2 = (C^2 - Vib^2) Eib^2

Note that for a particular mobile point i this equation is true for all a and b values. Hence for mobile point i:

(C^2 - Vi^2) Ei^2 = C^2 Eio^2

where Eio is the mobile point's energy at |Vi| = 0 which is independent of time.

Now assume a particle with an initial condition of Via = 0. Then:

Eia = Eio

giving:

(C^2) Eio^2 = (C^2 - Vib^2) Eib^2

or

Eio^2 = [1 - (Vib / C)^2] Eib^2

or

Eib = Eio / [1 - (Vib / C)^2]^0.5

which is a well known equation from special relativity.

For |Vib| << C this equation simplifies to Newtonian mechanics.

Recall that:

(C^2 - Via^2) Eia^2 = (C^2 - Vib^2) Eib^2

or

C^2 [Eia^2 - Eib^2] = Via^2 Eia^2 - Vib^2 Eib^2

or

[Eia^2 - Eib^2] = [Via^2 Eia^2 / C^2] - [Vib^2 Eib^2 / C^2]

or

= C^2 [Pia^2 - Pib^2]

Rearranging gives:

Eib^2 = Eia^2 + C^2 [Pib^2 - Pia^2]

For an initial condition of **Pia** = 0 so that Eia = Eio this equation gives:

Eib^2 = Eio^2 + C^2 Pib^2

For the special case of a photon:

Eio = 0

which gives:

Eib = C |Pib|

As is shown elsewhere for a photon:

Eib = h F

which gives:

|Pib| = h F / C

Thus we have shown the fundamental nature of the equation of motion:

d**Pi**.d**Xi** = dEi dt

**MOMENTUM OF CLUSTER OF ENERGY PACKETS:**

For cluster of energy packets each point i occurs at a different **Xi** value. In order to find the total momentum of a cluster of energy packets 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

Note that **Xc** may be moving with respect to **Xo**.

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 at **Xc** with respect to **Xo**

+ radial momentum with respect to **Xc**

+ angular momentum about **Xc**

**CONSERVATION OF LINEAR MOMENTUM:**

The law of conservation of linear momentum states that for an isolated system the velocity of the center of momentum 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.

The Law of Conservation of Linear Momentum is a fundamental conservation law of physics.

**CENTER OF MOMENTUM MOTION**

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:

d**Xc** / dt = 0

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

d**Pc**

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).

**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.

**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 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 body:

Net Radial momentum

= Sum from i = 1 to i = N of:

(Ei / 2 C^2) [(**Xi - Xc**) * d(**Xi - Xc**) / dt] / |**(Xi - Xc)**| = 0

The net radial momentum is zero.

**ROTATING RIGID BODY:**

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

**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)**|

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 ROTATING FIGURE SKATER:**

Note that if while in free rotation I is reduced by a rotating 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.

**MOMENTUM SUMMARY:**

Net momentum has two non-zero components, linear momentum at **Xc** and angular momentum about **Xc**. Linear momentum at **Xc** is always conserved. In Newtonian mechanics angular momentum about **Xc** is conserved if and only if kinetic energy is conserved.

**CHARGE, FIELDS, ENERGY:**

Energy is distributed through the universe in the forms of particles, fields and propagating field fluctuations known as radiation. Particles have both electromagnetic energy and non-electromagnetic (NEM) energy and have electric, magnetic and gravitational fields.

The various types of energy and the energy distribution are mathematically intertwined. The existence of a charge quantum at a relative position in space causes a local electric field vector distribution. Motion of the charge quantum around a closed path causes a 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. Note that a gravitational field contains negative energy.

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.

**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 are also energy exchanges between particle energy, static field energy and radiation. Photons (quanta of electromagnetic radiant energy) or gravitons (quanta of grvitational radiant energy) may be emitted from a particle to the radiation environment and/or be absorbed by a particle from the radiation environment.

**POTENTIAL ENERGY WELLS:**

As a free particle or photon falls into a gravitational potential well its potential energy with respect to field free space will become more negative while its kinetic energy becomes more positive. Hence on entering the gravitational potential energy well there is an increase in the particle's 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 sun's gravitational potential energy well while the other planet remains trapped at a lower energy.

Liquid matter is simply a collection of atoms that are mutually bound together in a common potential energy well. In solid matter the atoms are so deeply trapped that they cannot move past each other.

**FORMATION OF MUTUAL POTENTIAL ENERGY WELLS:**

Progressive overlap of static 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 in a low radiation environment such as our local universe there is an overall tendency for the total potential energy of an assembly of particles to become more negative by net emission of radiation which causes the particles to become mutually bound in common potential energy wells.

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

Liquid matter is simply a collection of atoms that are mutually bound together in a common potential energy well. In solid matter the atoms cannot move past each other.

Absorption of sufficient radiation from an external radiation source such as sunlight can in principle reverse this mutual binding process.

In all particle interactions the total isolated system energy, including emitted or absorbed radiant energy, is unchanged. 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 which, when suitably stimulated by neutron or gamma photon absorption, fission by breaking into multiple smaller particles and liberating both kinetic energy and gamma photons. Fission is the main controllable source of nuclear energy available on Earth.

The supply of fission fuel can be enormously increased by transmution of Th-232 into U-233 and by transmutation of U-238 into Pu-239.

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.

Most of known particle energy is non-electromagnetic (NEM). NEM energy has a grvitational field but does not directly interact with electric and magnetic fields.

Dark matter is a term used by astronomers to account for the existence of the invisible NEM energy 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 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 cluster of particles moves from state "a" to state "b" for each particle i the equation of motion requires that:

Eia^2 - Pia^2 C^2 = Eib^2 - Pib^2 C^2

or

Eib^2 - Eia^2 = C^2 (Pib^2 - Pia^2)

Thus a particle that randomly gains energy through interparticle interactions also gains momentum, and vice versa.

When a particle is at rest:

Eia = Eio

and

Pia^2 = 0

which gives:

Eib^2 - Eio^2 = C^2 Pib^2

or

(Eib - Eio)(Eib + Eio) = C^2 Pib^2

or

(Eib - Eio) = C^2 Pib^2 / (Eib + Eio)

= C^2 [Eib Vib / C^2]^2 / (Eib + Eio)

= Eib^2 Vib^2 / [C^2 (Eib + Eio)]

= [Eib / C^2] Vib^2 [Eib / (Eib + Eio)]

For |Vib| << C:

(Eib - Eio) ~ [Eib / C^2] Vib^2 / 2

~ [Eio / C^2] [Vib^2 / 2]

= Mo Vib^2 / 2

= Newtonian Kinetic Energy

**ISOLATED SINGLE PARTICES CANNOT ABSORB OR EMIT PHOTONS:**

Consider a single particle with initial energy Eia.
Assume that it absorbs a photon and changes to energy Eib
Eib - Eia = h F

Pib - Pia = h F / C

Recall that:

Eb^2 - Ea^2 = C^2 (Pb^2 - Pa^2)

or

(Eb - Ea)(Eb + Ea) = C^2 (Pb - Pa)(Pb + Pa)

or

[(Eb - Ea) / (Pb - Pa)] = C^2 (Pb + Pa) / (Eb + Ea)

If energy and momentum are both conserved:

Eb - Ea = h F

and:

Pb - Pa = h F / C

giving:

h F / (h F / C) = C^2 (Pb + Pa) / (Ea + Eb)

or

(1 / C) = (Pb + Pa) / (Ea + Eb)

or

(Ea + Eb) = C (Pa + Pb)

For:

Eb = Ea + h F

and

Pb = Pa + [h F / C]

Then:

2 Ea + h F = C [(2 Pa) + (h F/ C)]

or

Ea = C Pa

= C (h F / C)

= h F

This equation is only true if Ea is a photon.

Therefore isolated single particles are stable and do not absorb or emit photons. Photon absorption and emission primarily occurs as a result of interactions between two or more particles.

**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 weak 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 a small portion of 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.

**GRAVITY:**

Gravitational field energy is a result of the presence and density of energy forms. When all 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. Most of the radiation energy escapes from the 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 cause particles to form mutual potential energy wells. 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 often assumed that particle rest mass energy came from particle pair production by high energy gamma ray photons originating in a "big bang".

**RADIATION MOMENTUM:**

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.

**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 when the center of momentum is in constant linear motion with respect to the observer.

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 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 current 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 turns around the toroid's major axis and Nt 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 Np > 2 Nt and 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 linear 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 (center of momentum).

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 vector 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 that are bound in a mutual potential energy well equals the rate of radiation emission from that potential energy well. 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 = 6.62607015 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 NEM energy.

This web page last updated July 14, 2022.

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