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

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

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

All position measurements are of the form (X - Xo) where Xo is a chosen reference 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.

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
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.

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.

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 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

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, even though Xo and Vo cannot be determined it is sometimes helpful to carry Xo and Vo through calculations to provide insight into the physical meaning of the mathematical equations.

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 describing changes in energy but is inadequate for proper description of physical phenomena.

Energy is "a conserved constituant of the universe". Everything that we can contemplate has non-zero energy. Zero energy generally implies non-existence.

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
dEa / dT = 0

More generally, the rate of change of energy contained within a closed surface is equal to the net flow of energy through that closed surface.

If two isolated systems "i" and "j" respectively have energies Ei and Ej 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 the total energy E of the new isolated system is given by:
E = (Ei + Ej)
and if the new system remains isolated:
dE / dT = 0

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 generally 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.

Each of the aforementioned concepts of energy is really a change in energy (Eb - Ea) between states a and b and is not an absolute measure of 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)

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
Ek = 0

(Vi - Vo) = 0
Ek = 0
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 a field free vacuum might have a non-zero energy density. The issue of "dark energy" is presently not well understood.

In practice most energy exchange is between Ek and Eb. However, in nuclear reactions changes in Eo become important.

Conservation of energy requires that total particle energy Ei is:
Ei = Eoi + Eki + Ebi

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

d(Eoi + Eki + Ebi)/ dT = 0

However by definition:
dEoi / dT = 0

d(Eki + Ebi) = 0
dEki = - dEbi

For a single isolated particle since:

dEbi / dT = 0
dEki / dT = 0

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 will overlap and there can be exchange of energy between Ebi and Eki. In some cases the interparticle interaction can lead to formation of a new particle or the emission or absorption of a photon.

In many practical situations there is ongoing energy exchange between Ek and Eb so that:
(Ek + Eb) = constant
dEk + dEb = 0

(Ek + Eb) < 0
then multiple particles are bound in a mutual potential energy well.

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.

Ei = Eki + Eoi

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

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

For any isolated object total linear momentum P is conserved.

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.

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.
(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.

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.

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).

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.

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.

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.

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.

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.

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.

(Vi - Vo) = 0
Eki = 0
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.

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)
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.


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.

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
dEo = 0

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

dE [1 - ((V - Vo) / C)^2] = (1 / C^2) E (V - Vo) dV
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|
Ln[(Ea / Eb)^2] = Ln(|C^2 - (Vb - Vo)^2| / |C^2 - (Va - Vo)^2|)
(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
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
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.

Application of the Einstein relationship to particle i gives:
Ei^2 = (Pi - Poi)^2 C^2 + Eoi^2
Eoi = potential energy of particle i at rest
(Pi - Poi) = (Ei / C^2)(Vi - Vo)
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
(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
Ei^2 {1 - [(Vi - Vo)^2 / C^2]} = Eoi^2
Ei = Eoi / {1 - [(Vi - Vo)^2 / C^2]}^0.5
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.

The rest energy Eoi of particle i is defined as:
Eoi = Ei - Eki

If |Vi - Vo| << C
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.

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
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.

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

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.

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 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)

This web page last updated March 11, 2018.

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