mass and Energy
-------------
Einstein and others showed that mass and energy are equivalent and
can be exchanged with each other in physics calculations. We can see
how this is possible by examining an electromagnetic wave flowing past
an observer. The wave imparts to the observer a momentum and energy,
which can be converted to useful work.
EMR travels at lightspeed c. The wave conveys energy E, according
as its frequency or wavelength. The momentum imparted to the observer
is
p = E / c
From all experiments, this momentum is entirely equivalent to that
delivered by a material particle, of mass m, travelling at speed c.
Thus
p = E / c
= m * c
E / c = m * c
E = m * c^2
This is Einstein's most famous of all his equations, but it is
also one of the more misunderstood principles of his physics.
For one point, we did NOT demonstrate that EMR has mass. EMR is
identicly massless. There are two interpretations of this equation.
The first is that we can express mass as an equivalent amount of
energy or vice versa. In a formula calling for a quantity in mass
units, we can freely replace it with the equal amount energy. Or the
other way round. You may think of this equivalence like the way we use
the identities (in(a)^2 = 1-(cos(a))^2and meter/second^2 =
newton/kilogram.
The other way is that mass and energy are convertible from a one
into the other. This is by far the most popular way to see the E =
m*c^2 relation. By various processes we can indeed extract energy from
a material body and its mass will decrease. Or we can add energy to a
material and increase its mass.
Altho we used electromagnetic waves to show this equivalence, it
is general for all forms of energy and mass.
+---------------------------+
| MASS-ENERGY EQUIVALENCE |
| |
| E = m * c^2 |
| m = E / c^2 |
+---------------------------+
dePretto's Ideas
--------------
dePretto in 1903 developed an equivalence of energy and mass in a
crude manner, based on the aether theory. In it he figured that
matter, in order to conduct light and EMR, had to contain energy,
which was dependent on the speed of light in that material. For
aether, lightspeed was the maximum value, c, and its material (of
thoroly unknown substance) had an equivalent energy content of E =
m*c^2.
He was not a mainstream physicist, altho he published his findings
in science journals. He could not see a way to progress farther in his
inquiries. He did speculate about the enormous leverage the square
of lightspeed had for a given quantity of mass. Perhaps, he
wondered, this energy could be the source of geothermal heat and the
light and heat of the Sun?
Rest Mass
-------
An early common feature of Einstein physics is that of the rest
mass of a particle. This is merely the regular mass, as we understand
it in Newton physics, when the particle is at rest in our own frame of
reference. It's what we measure thru experiments of inertia in a
laboratory.
However, in the case of a photon, the term 'rest mass' is
misleading. The 'mass' we found above applies to a particle which is
absolutely without mass and can only travel at speed c in any
platform. Yet it is the mass of a fictitious particle, moving at speed
c, that carries the same momentum as the photon.
Note well that since the energy, and therefore momentum, of a
photon is a function of its wavelength, the mass equivalent is also a
function of wavelength. Shorter wavelength, higher frequency, photons
carry more energy than those with longer wavelength, lower frequency.
The more addiurnate concept of rest mass is to avoid the term
'rest mass' completely but to recognize that there is associated with
a real body or a photon a mass that is invariant against motion. The
'mass' of the photon, because a photon's speed is always c, is the
same for any observer. Likewise, the mass of a material thing at rest
in an observer's platform, is the same for all such platforms.
Therefore, there is really only one parameter of mass, now called
just 'mass' without 'rest' or other qualifier, for each body or
quantum.
Conservation Law
--------------
The establishment of equivalence between mass and energy leads to
a unification of the two separate conservation laws. In place of one
for mass and one for energy, we now have a single one for the
combination of mass-energy. Altho it is now possible to change from
mass to energy and vice versa, the sum of the two as ingredients to a
reaction must equal the sum of the two as egredients.
mi + Ei = mo + Eo
The first to demonstrate this were Cockcroft and Walton in 1932.
They measured the mass and kinetic energy of a lithium-7 nucleus and a
proton. When the proton interacted with the lithium it was absorbed
and the combined nucleus, boron-8, decayed into two helium nuclei, or
alpha particles. They measured the mass and kinetic energy of these
heliums.
The mass of the heliums was a trifle less than that of the lithium
and proton; the kinetic energy of the heliums was a trifle more than
those of the proton and lithium. The decrease of mass was exactly
equal, by E = m*c^2, to the increase of energy
Mass into Energy
--------------
Under Einstein physics, energy is produced by converting mass into
energy thru the E=m*c^2 relation. When a fire burns fuel, some of the
mass of the fuel is indeed lost and is replaced by the heat, light,
and kinetic motion of gases in the fire. The loss loss is so small as
to be undetectable because the factor c^2 is so large. It takes only
the merest bit of mass to yield a substantial amount of energy.
One kilogram of mass yields 9e16 joules of energy. This is a
stupendous quantity, equal in electricity to about 2.5e10 kilowatt-
hours. This is quite the total electrical energy requirements of New
York City for one year, estimated from 1990s data!
In principle we could weigh all the ingredient fuel at the power
plants supplying New York. (This is in fact done by the power
companies to insure they are getting the quantity and quality of fuel
they pay for.)
Then collect and weigh all the exhaust gases, sludge, soot, ash.
The two should differ by but one kilogram. In practice, the exhaust
material is not actually collected but is calculated for care and
control of the environment. An error of many THOUSAND of kilograms is
well within acceptable tolerances. So the one kilo of mass converted
into energy is utterly beyond detection.
In actuality the mass loss is about four kilograms because only
some 1/4 of the energy released by burning the fuel ends up in the
electricity. The rest is thrown into the environment as waste heat.
This is a brutal feature of energy conversion via heat engines, as
limited by Carnot's Law andt he second law of thermodynamics.
Energy into Mass
--------------
There is no way so far to produce bulk amounts of mass from
energy. In atomic labs individual nuclear particles can be created
from interaction beams of radiation but these have no practical value
outside of the experiment.
In ordinary life, when ever we add internal energy to a body, like
by heating it, the mass of that body does increase ever so slightly.
The increment is undiscernible save by the most heroic of effort.
In experiments in atomic labs, the concept of mass in itself is of
minimal value. The movement of atomic particles is governed by the
energy we put into them, like in an accelerator. Hence, it is quite
common to speak of the 'mass' of a particle in terms of its equivalent
energy. A proton is said to have a 'mass' of 1.494e-12 joule rather
than 1.673e-27 kilogram.
When the proton is accelerated to high speed, it acquires energy,
in joules, which is added to this base energy to get the new total
energy equivalent of the proton.
Electron-Volt
-----------
A very common unit of energy in atomics is the electron-volt. This
is the energy acquired by an electron when impelled between a voltage
difference of one volt. One electron-volt, abbreved eV, is 1.602e-19
joule. On this scale, the mass of an electron has energy equivalent of
5.110e5eV and a proton has energy of 9.324e8eV.
+-----------------------------+
| ENERGY OF ELECTRON-VOLT |
| |
| 1 eV = 1.602e-19 joule |
| |
| electron energy = 5.110e5eV |
| |
| proton energy = 9.324e8eV |
+-----------------------------+
In comparison, a neutrino, which is almost massless, has energy no
greater than 5eV.
Metric prefixes are used to make the kiloelectron-volt, keV, and
higher multiples. The electron energy is 0.511MeV; proton, 932MeV.
These multiples are prevalently pronounced 'kevv', mevv', 'gevv',
'tevV, and so on. It is rare to hear 'megaelectron-volt' and the like.
There is almost no use for the submultiples like microelectron-volt.
Critiques
-------
As simple and direct as the Einstein mass-energy equation is, it
generated over the decades severe criticism and debate. Some modern
physicists and philosophers argue that Einstein somehow fudged the
original derivation in 1905. Others argue that this derivation was
proper and sound.
Over the years, right thru the end of the 20th century, new angles
of interpreting this formula were offered. Some ideas are that mass
and energy are really only one substance with different names. Mass is
'frozen' or 'static' energy. The equation is really based on
experiment and can not be formally derived. Mass and energy are two
facets of a new other entity as yet understood. The equation is all
baloney and mass-energy can not be merged.
Exactly from such lack of a definitive and consistent treatment of
this Einstein principle, comes the glossing over of a derivation in
almost all popularizations of relativity. The trick of the author is
to say 'Einstein showed that E = mc2', 'By complex math Einstein
derived E = mc2', 'E = mc2 is one of the features of Einstein's theory
of relativity', and similar glosses.
We did no better here due to this odd history in Einstein physics,
but at least there is good sense to it as showed by actual atomic
experiments. For home astronomy purposes it's easiest to allow that
mass and energy are two entities but they can be commuted from the one
to the other.
Atomic Bombs
----------
The allure of extracting energy from atoms comes from the vastly
greater fraction of an atom's mass that is converted into energy when
atoms decay. The mass of the fission or daughter atoms is less by
order 1/5 percent of the original atom. This may not seem like much
but the factor of c^2 makes for a lot of energy release.
In the 1920s there arose the general realization that the atom
locked up stupendous reserves of energy which conceivably could
substitute for fossil fuels. Fossil fuels -- coal, oil, gas -- were
dirty substances, dangerous to mine and purify. They released during
combustion noxious and even poisonous byproducts. Besides, there were
fewer and fewer easily mined fuels as the coal and oil fields were
depleted.
In the case of petroleum, the world's supply was controlled by the
Middle East, whose stability, politics, and ideology clashed decade
after decade with the western nations. There was always the threat of
an oil stoppage, throwing the world's industry, commerce, and
transport into chaos.
By World War II atomic power was under intense study but its first
large scale use was for atomic weapons. The atom bomb was a runaway
uncontrolled release of energy by the instantaneous decay of, typicly,
uranium atoms. Neutrons ejected from the initial set of atoms strike
and induce fission in neighboring ones. The reaction concatenates
thruout the mass of uranium to create an instantaneous issuance of
heat and shock and radiation.
However, if the mass is too small, most of the initial neutrons
leave it without hitting other atoms and the reaction remains subdued
and slow. By assembling a certain critical mass of uranium, the
majority of neutrons do hit other atoms and the reaction 'goes
critical' or self perpetuates. The atom bomb in essence is two chunks
of uranium in separate chambers, each being a little more than half
of the critical mass. A chemical bomb and mechanical mechanism at
triggering force and hold the two chunks together to make one of a
little more than the critical mass and the bomb detonates.
Atomic Power
----------qThe atom bomb released the nuclear energy almost
instantaneously. if this everhy could be released slowly in a
deliberate way its heat could be used to generate electriciry. An
atomic, or nulclear, power plant does this. The urnaium totalling
amore that the critical mass, is placed in rods set up in a chamber.
The rods are close enough for neutrons from one to reach the others
and start the chain reaction.
To control the strength of the reaction a second set of rods is
place among the fuel rods. Thesse, the control rods, are movanle into
and out off the fuel rod grid and are filled with a neutron-anosrbing
material. Boron and graphite are common materials.
When the power plant is ready to start operations the control tods
are fully in place. They soak up neutrons, preventing them from
triggering the chain reaction. To begin heat production the control
rodsa re pulled out. Neutrons can strike other uranium fuel rods and
the power plant 'goes critical'. The released heat vboils circulating
water around the rods for turbine stram.
Mind well that regardless of the degree of steam generationm even
noe for the control rods shutting off the reaction, the uranium atoms
continue to decay and spit out neutrons. To maximize the use of
nuclear heat, a nuclear power plant tupicly runs full output 24/7. it
is a 'base load' project supplying electric for all hours of the day.
Other fossil fuel or eater, projects run on and off to follow the
need for electric above the base amount furnihed by the atomic plant.,
Recycling of Fuel
---------------
After a couple years the uranium in the fuel rods is too 'cold' to
make turbine steam. The uranium isotopes are used up, and the rods are
laced with stable or weakly radioactive daughter atoms. The rods are
removed and replaced with new fuel rods.
In some countries, like the United States, the old rods are
stored in deep pools of water, for shielding, on the plant premises.
They are taken away by an agency or firm to be buried in deep rock
caves, probably for many decades. All-new rods are built with fresh
uranium for the power plant. a
In other countries the firm or agency removes the old uranium and
daughter products from the rods. New uranium is loaded and the
filled rods are sent back to the power plant. The collects material is
packaged for industrial or medical uses.
Each country with atomic power plants works out its own program
for handling the spent fuel rods. Some built a recycling program with
the development of their nuclear power system. Others delayed many
years, maybe after the first round of reloading of fuel rods, to start
recycling. Others, like the United States, as at end of the 20th
century have no recycling program in the works.
New York Els
----------
In about 1900 the company running the els in New York (Brooklyn had
a separate system) was under severe social and political pressure to
do something about its coal-fired locomotives. The company itself was
moved to seek alternate power due to the hazards of handling coal. For
starts, the supply was interrupted by continual rail and mine labor
unrest and was subjected to erratic price swings.
One alternative was electricity, then offered by nascent electric
utilities, too small and weak to furnish the power for a vast network
of els. More over, converting to electric would call for junking the
entire fleet of coal-fired engines, wiring a hundred kilometers of
road, hiring and training allnew crews, and running onsite electric
generating stations. The cost would excede the all-time investment in
the existing el system!
The Manhattan el company came across a news item about a magic
rock found by a lady scientist in France. This rock effused heat
without flame or smoke or soot. This heat flux was constant,
apparently immune to weather or other circumstances. This heat was so
vigorous that a pebble of this material could boil a barrel of water
within an hour.
Could not this rock, like a hunk of coal, be loaded into a
locomotive? A quick calculation, in modern measure, showed that a
kilogram of this stuff had the heating equivalent of 5,000 TONS of
coal! By placing just a kilo in the engine's boiler, the engine could
run without refueling for the rest of its mechanical life!
The company wired the discoverer in France to buy a few kilos for
testing. The word came back that in all the world there only mere
milligrams of this material. The 'rock' was the raw ore. And to get
the special substance out in pure form cost thousands of dollars per
gram. A kilo, if it could even be accumulated, would coast over a
million dollars, a vast sum in the early 1900s. The samples asked
would equal in cost the all-time investment in the whole el system!
The material was the newly found radium, discovered by Marie
Curie. Gritting their teeth, the els converted to electric and the
rest is history. Never the less it is incredible that at the start of
the 20th century there was serious, if wholly impossible, thought of
using atomic power for large-scale civic purposes.
Mass Increase
-----------
Among the most misunderstood aspects of Einstein physics is that
of the increase of a body's mass with increase in speed relative to
the observer. This is of particular concern for the home astronomer.
In his early days in astronomy he may read popular works on Einstein
physics. These almost universally describe this feature of mass and
speed. As he moves along in astronomy he starts to read the technical
articles and books, intended for within th profession. There he finds
that the whole concept of mass increase under relative motion is
almost completely absent!
As a matter of fact, among physicists the notion that the actual
matter of a body enlarges because of motion vanished in the 1950s.
Einstein himself in 1948 urged that it not to promulgated anymore for
being an outdated and erroneous concept!
What happened?
Early Ideas
---------
To see how the idea that a body materially increases its mass with
motion, consider a spaceship with an engine that exerts a constant
known thrust. The spaceship is its own motional platform has mass
m[m/m]. When the rocket is turned on, the spaceship suffers an
acceleration a[m/m] = delV[m/m]/delT[m/m]. Recall the notion that the
first subscript, in the 'numerator', is the platform whose parameter
is under examination. The second, in the 'demoniator', is the platform
where the examination is carried out. 'm' is the motional frame; 's',
stational.
Thus we have
F = m[m/m] * delV[m/m] / delT[m/m].
= m[m/m] * delX[m/m] / (delT[m/m] * delT[m/m])
We now examine this acceleration from the stational frame with
the relative speed V[m/s]
F = m[m/s] * delX[m/s] / (delT[m/s] * delT[m/s])
But space and time are distorted as perceived from the stational
platform by the factor beta = (1-(V[m/m]/c)^2)^(1/2)
delT[m/s] = delT[m/m] / beta
delX[m/s] = delX[m/m] / beta
F = m[m/s] * (delX[m/m] / beta) / ((delT[m/m]
/ beta) * (delT[m/m] / beta))
= m[m/s] * delX[m/m] / (delT[m/m] * delT[m/m])
* (1 / beta) / ((1 / beta) * (1 / beta))
= (m[m/s] * delX[m/m] / (delT[m/m] * delT[m/m]))
* 1 / (1 / beta))
= (m[m/s] * delX[m /m] / (delT[m/m] * delT[m/m])) *beta
Which is to say, from the stational frame the rocket undergoes a
lesser acceleration for the given force compared to that experienced
in the rocket itself.
Now, because the force is the same in the both frames,
F = m[m/m] * delX[m/m] / (delT[m/m] * delT[m/m])
= (m[m/s] * delX[m/m] / (delT[m/m] * delT[m/m]))*beta
m[m/m] * delX[m/m] / (delT[m/m] * delT[m/m]) =
(m[m/s] * delX[m/m] / (delT[m/m] * delT[m/m])) * beta
m[m/m] = m[m/s] * beta
m[m/s] = m[m/m] / beta
Hence, at first blush it does indeed seem that the mass of the
rocket sensed on the stational frame is greater than that sensed within
the rocket. Mass increased with motion.
Relativistic Mass
---------------
Popular treatises on relativity speak of 'relativistic mass' and
'rest mass'. The rest mass of a body is the material content assayed
in that body when it is at rest in the observer's frame of reference.
This is m[m/m] or m[s/s]. Rest mass is commonly symboled by m0.
The mass of the body when observed from an other frame and
augmented by the factor beta is the relativistic mass, synboled by m
or M. This is m[m/s] or m[s/m]. m[s/m] = m[s/s]/beta or m[m/s] =
m[m/m]/beta.
These terms come from the early era of relativity, mainly before
experience was accumulated from atomic experiments.
The present practice of physics discourages the use of 'mass' in
relativity and atomics except to mean the rest mass. In fact, the
practice nowayears is to use plain m for this mass, without any
distinction with a subscript 0.
Any 'increase' in mass, what made the rest mass become the
relativistic mass, is now always expressed as energy. Because so much
of the experience with changes of mass with velocity comes from atomic
experiments, this energy is prevalently stated in electron-volts.
That's why when you read litterature within physics or astronomy,
rather than that for the general public, the notion of mass increase
is just about completely absent. This lapse is a source of great
consternation among home astronomers!
Mass-Energy Equation
------------------
We can see this by starting with the rocket, where we found
m[m/s] = m[m/m] / beta
m[m/s] * c^2 = m[m/m] * c^2 / beta
m[m/s] * (c^2) * beta = m[m/m]] * c^2
m[m/s] * (c^2) * (1 - (V^^2/c^2)^(1/2) = m[m/m] * c^2
(m[m/s] * (c^2))^2 * (1 - (V^2 / c^2) = (m[m/m] * c^2)^2
(m[m/s] * (c^2))^2 - (m[m/s] * (c^2))^2 * (V^2 / c^2)
= (m[m/m] * c^2)^2
(m[m/s] * (c^2))^2 - (m[m/s]^2 * V^2 * c^2) = (m[m/m] * c^2)^2
(m[m/s] * (c^2))^2 = (m[m/m] * c^2)^2 + (m[m/s]^2 * V^2 * c^2)
E^2 = (m[m/m] * c^2)^2 + (m[m/s]^2 * V^2 * c^2)
The total energy of the body is the sum of the energy equivalent
of its rest mass plus the energy equivalent of its momentum. The more
usual statement of this relation is
E^2 = (m[m/m] * c^2)^2 + p[m/s]^2 * c^2
In simplified notation,
E^2 = (m0 * c^2)^2 + (p * c)^2
+--------------------------------+
| MASS-ENERGY EQUATION |
| |
| E^2 = (m0 * c^2)^2 + (p * c)^2 |
+--------------------------------+
No Real Mass Increase
-------------------
In the present scheme of relativity, the old rest mass,
the current simple mass, is invariant with motion. There is NO actual
enlargement of the matter in the body with motion. All the increase is
in the form of momentum. In the stead of citing just this increment in
energy units, the entire package of the rest mass plus the energy
increment is expressed as an energy.
The proper way to see this is to note that the total energy of a
body in motion is the combination of its rest mass, expressed as
energy thru E = m0*c^2, and its momentum, also expressed as energy
thru E = p*c.
When an object is in motion relative to the stational frame, it
acquired energy of motion but its real material content does not
change at all. The flub comes from the naive conversion of the entire
energy of the body, mass plus momentum, back to a mass by dividing by
c^2. We get the original equation based on the rocket experiment
m[m/s] = m[m/m] / beta
This manipulation can lead to really silly explanations. One is
that the observer on the motional frame, the spaceship, will some how
actually feel heavier than when he's at rest! This is ridiculous once
you realize that relative to some, even unknown, place in the universe
we ourselfs are moving at near lightspeed. Do we sense ourselfs to be
far more massive than if we are at rest? Of course not.
We are in relative motion with many places out there at various
large fractions of lightspeed. For which of them do we respond with
acquiring more mass? How does -- or can? -- nature choose which place
knows about us to trigger the 'correct' mass increase?
Invariant Mass
------------
The proper interpretation is that in the mass-energy relation. The
'rest mass' remains the same for all platforms. That's the m[m/m] or
m0. The 'increase' of inertia or mass is entirely due to the mistaken
mass equivalent of the increase in momentum. That's the p*c part.
It's wrong to convert this -- or the sum of it and the rest
energy -- back to a 'mass' with 1/(c^2). In fact, leaving the two
components separate helps us to see better why a photon, while having
energy and momentum, can be massless. The rest mass of a photon is
identicly zero,; it is pure radiant energy. In the mass-energy
formula,
E^2 = (m0 * c^2)^2 + (p * c)^2
= (0 * c^2)^2 + (p * c)^2
= 0^2 + (p * c)^2
= 0 + (p * c)^2
= (p * c)^2
E = p * c
which is just what is demonstrated by Maxwell physics.