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BASIC PHYSICAL CONCEPTS - PART A

RELATIVITY, ENERGY & MOMENTUM

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

INTRODUCTION:
This web page reviews basic physical concepts relating to relative position, time, energy, and momentum.
 

VECTOR NOTATION:
On this web page a bold face parameter like (X - Xo) denotes a vector with orthogonal x, y and z components.
 

RELATIVE POSITION:
All position measurements are of the form (X - Xo) where Xo is a chosen position in the frame of reference of the observer. It is impossible to determine Xo. Only relative positions of the form (X - Xo) can be measured.
 

RELATIVE TIME:
Time is relative. Thus 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 historical event (such as the birth of Jesus Christ). Elapsed time (T - To) is measured by counting ticks of a clock at the location Xo. 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 at significant speeds with respect to each other measure different elapsed times. Similarly clocks are affected by strong gravitational fields.
 

RELATIVE VELOCITY:
Velocity measurements in the frame of reference of an inertial observer with absolute velocity:
Vo = dXo / dT
are of the form:
(V - Vo) = d(X - Xo) / d(T - To)
= d(X - Xo) / dT
It is impossible to determine Vo = dXo / dT. Only relative velocities of the form (V - Vo) can be measured.
 

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 at the observer's velocity Vo given by::
Vo = dXo / dT = constant
so the observer's acceleration is always zero. Such an observer is known as an inertial observer.

Since the observer is inertial, by definition:
dVo / dT = 0
 

RELATIVITY:
Position, time and velocity are relative. There is no means of determining absolute values of reference position Xo, reference time To or reference velocity Vo. Position, time and velocity only have meaning when measured relative to reference position Xo, reference time To, and reference velocity Vo. However, Xo, Vo can only be defined with respect to position Xi and velocity Vi of object i. It is helpful to carry Xo and Vo through calculations to provide insight into the physical meaning of the mathematical equations.
 

ENERGY - AN INTUITIVE DEFINITION:
A high school definition of energy is "capacity to do work". Work is an energy transfer from one system to another system. Historically energy was the amount of food required by men or draft animals or the amount of fuel required by an engine to do a specific amount of work such as lifting a known amount of water a known height. This simple definition of energy is adequate for many commerial transactions involving an exchange of energy but this definition of energy is inadequate for proper description of physical phenomena.

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 open loop transport of water or a combustible fuel, in which case the atmosphere provides the return path for loop closure. Such loops actually deliver a change in energy per particle in the fluid loop. This change is very small compared to the rest mass potential energy of the particles.

Delivery of energy may also be done by unidirectional flow of radiation such as sunlight, a laser beam, a wave guide or an electricity transmission line. In each of these cases partial reflection of radiation reduces the energy delivery capacity.
 

ORTHOGONAL VECTOR COMPONENTS OF POTENTIAL 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.
 

ENERGY - A MORE PRECISE DEFINITION:
Energy is "a conserved constituant of the universe". Everything that we can contemplate has either positive or negative energy. Zero energy implies non-existence.

The potential energy density in an element of volume is the sum of the squares of the contained orthogonal electric, magnetic and gravitational field vectors. When an object is in motion in the observers frame of reference there is an additional kinetic energy component.

In theory energy can be exchanged between the various orthogonal energy components. In practice this energy exchange usually takes place via potential energy converting to kinetic energy and then the kinetic energy converting to another form of potential energy.

Total energy is the volume integral of the energy density (energy per unit volume). The energy density is measured relative to a field free vacuum for which the energy density is usually assumed to be zero. However, it should be noted that astronomical observations of other galaxies suggests that on a cosmological scale the field free vacuum energy level may be non-zero. The issue of "dark energy" is presently unresolved.
 

LAW OF CONSERVATION OF ENERGY:
The law of conservation of energy is one of the most fundamental physical laws. The law of conservation of energy states that energy can neither be created nor destroyed but can be changed in form. From the perspective of an inertial observer, the total energy Ea of any isolated system "a" is constant. Expressed mathematically:
Ea = constant
or
dEa / dT = 0

More generally, the rate of change of energy contained in a finite volume in the frame of reference of an inertial observer is equal to the net flow of energy into or out of that finite volume through the surface of that volume.

If two isolated systems "a" and "b" respectively have energies Ea and Eb in the observers frame of reference then if the two systems are combined without addition or loss of energy to form a new isolated system "c" the total energy Ec of the new isolated system is given by:
Ec = (Ea + Eb)
and if system "c" remains isolated:
dEc / dT = 0
 

PARTICLES, POTENTIAL ENERGY, KINETIC ENERGY:
It is assumed that the known universe is primarily composed of objects known as particles and photons. Each particle i is an entity with a nominal position:
(Xi - Xo),
a velocity:
d(Xi - Xo) / dT,
a net electric charge Qi, a potential energy Eip and a kinetic energy Eik.

The potential energy Eip (rest energy) of object i is its energy component which may depend on the objects position but is independent of the object's velocity:
d(Xi - Xo) / dT
in the observer's frame of reference.

Kinetic energy Eik given by:
Eik = Ei - Eip
is remaining energy which is an explicit function of both rest energy Eip and the velocity:
d(Xi - Xo) / dT
in the observer's frame of reference.

When:
d(Xi - Xo) / dT = 0
then:
Eik = 0

Conservation of energy requires that:
Ei = Eip + Eik

From the perspective of an inertial observer, the total energy Ei of a single isolated particle i is constant, in which case:
dEi / dT = 0

Hence for a single isolated particle with no mechanism for energy exchange:
dEip / dT = 0
and
dEik / dT = 0

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

When there are two or more particles present their extended fields will overlap and there can be exchange of energy between Eip and Eik and interparticle interaction can lead to formation of one or more new particles or photons.
 

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 - Vio)
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 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 accuarately 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 to this author. However, protons appear 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 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 the particle with respect to the observer causes an additional momentum component.
 

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:
F = (G Ma Mb / (R*R)) (R / |R|)
gives satisfactory accuracy where:
F = gravitational force vector between mass Ma and mass Mb;
G = Newton's gravitational constant;
R = relative position vector between center of mass Ma and center of mass Mb.
 

REST ENERGY COMPONENTS:
The rest energy of an isolated particle consists of the sum of its field energy components. In interactions between particles part of the external field energy can converts into kinetic energy which can then convert into radiant energy. The gravitational field energy can do minor work. However, the internal field energy, which is typically over 98% of the total particle energy, cannot do work unless a particle and its corresponding anti-particle annihilate each other to become a radiant energy photon.
 

FIELDS AND FORCES:
Electric charge causes a spherical radial electric field outside 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 locally relatively small and 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 forces arising from field overlap are discussed on the web page titled: FIELD THEORY.
 

FORCE:
A change in potential energy due to a change in particles field overlap 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:
Eik = Ei - Eip
is the portion of energy Ei that is a function of the relative velocity:
(Vi - Vo)
of the particle in the frame of reference of the inertial observer.

If:
(Vi - Vo) = 0
then:
Eik = 0
and
Ei = Eip

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 energy dE of an object in terms of changes in position, time and momentum is:
dE dT = dP*dX
= d(Pi - Poi)*d(Xi - Xoi)
or
dE = [d(Pi - Poi) / d(T - To)]*d(Xi - Xoi)
If the motion is along only one spacial dimension this equation simplifies to:
(Change in Energy) = (Force)*(Distance)
 

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 is properly defined by:
P = (E / C^2) V

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

Hence:
dE [1 - (V / C)^2] = (1 / C^2) E V dV
or
dE / E = V dV / [C^2 - V^2]

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

Choose state a to be the object at rest so Va = 0, Ea = Eo. Then:
Ea^2 = Eb^2 |C^2 - Vb^2| / |C^2|
= Eb^2 - Eb^2 Vb^2 / C^2
or
Eb^2 = (Eb^2 Vb^2 / C^2) + Ea^2
= (Eb^2 Vb^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:
Eik = E - Eo

Hence, provided that the observer is inertial the linear momentum (P - Po) of an object is related to total object energy E and the object's 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 + Eipo^2
where:
Eipo = 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 may involve more orthogonal components.

Combination of these equations gives:
Ei^2 = (Ei / C^2)^2 (Vi - Vo)^2 C^2 + Eipo^2
or
Ei^2 {1 - [(Vi - Vo)^2 / C^2]} = Eipo^2
or
Ei = Eipo / {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 Ei = Eipo and Eki = 0, 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 permits negative rest energy and negative kinetic energy values.

Rearranging this expression gives kinetic energy Eki as:
Eik = Ei - Eip
= 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:
Eik ~ 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:
Eik = (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 with negative energy. 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 POTENTIAL ENERGY:
The potential energy Eip of particle i is defined as:
Eip = Ei - Eik

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

ENERGY OF A PHOTON:
Recall that:
Ei^2 = Eip^2 + C^2 |Pi - Poi|^2

For the special case of a photon which has no rest energy:
Eip = 0
giving:
Ei = Eik = 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.
 

TOTAL ENERGY:
The total 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 Eip and kinetic (motion) energy Eik in the observer's frame of reference.

Thus:
Ei = Eik + Eip

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

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

PAIR PRODUCTION / ANNIHILATION:
Under appropriate circumstances an electromagnetic radiation photon with sufficient energy Ep converts in 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 vacuum state is assumed to be a fully occupied energy level.

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 releasing a photon of energy Ep.

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

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.
 

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

PARTICLE TRAPPED IN A 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^2 = (Pi - Poi)^2 C^2 + Eip^2
Initially while the particle is far away the particle does not interact with other particles. Hence:
Eip = Eipo
and
dEi / dT = 0.
As the particle slides into the potential well Eip^2 decreases and (Pi - Poi)^2 increases. Then while in the potential energy well some of the particle's momentum and energy are lost via emission of photons. While photons are being emitted:
d[Ei^2] / dT < 0
and
d[(Pi - Poi)^2] / dT < 0.

Hence after photon emission:
Ei^2 - (Pi - Poi)^2 < Eipo^2
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 Eipo^2 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 too many particles in the potential energy well Eipo 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.
 

This web page last updated October 17, 2016.

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