Caption: Shown is the only physics formula everyone knows: E=mc**2, where E is energy, m is mass and the vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s =3*10**5 km/s ≅ 1 ft/ns.
E=mc**2 is, of course, a basic result of special relativity and is obtained in the physicsy derivation of special relativity from the special relativity postulates. A physicsy derivation is one where you start with basic axioms, but introduce extra ones has you go along as seems reasonable to physical intuition (i.e., educated guessing) or by clairvoyance (i.e., you believe you know where you have to arrive). Pure mathematics it isn't.
In fact, the derivation of E=mc**2 is remarkably simple though well beyond the scope of this file: i.e., e_mc2.html.
But what does E=mc**2 mean?
Actually, E=mc**2 means two things:
All energy has mass: i.e., exerts a gravitational field and has a resistance to acceleration.
To emphasize this sameness, one often uses the unified term mass-energy in relativity speak rather than either of energy or mass. In fact, in the appropriate contexts, mass-energy, energy, and mass are all synonyms. However, one often uses energy or mass depending on which aspect of mass-energy is being emphasized.
As an example of Meaning 1, if you add/subtract kinetic energy or heat energy E to/from a system, you add/subtract mass E/c**2 to/from the system.
Such changes in mass are NOT noticed in everyday life NOR were they observed experimentally prior to the advent of special relativity in 1905 (when people started looking for them) because ΔE/c**2 is so small compared to a system's rest mass (see the Meaning 2 of E=mc**2 below). Nowadays, they have been super well verified at least by indirect means.
Some further points:
In modern formulation, the laws of conservation of mass and conservation of energy are fundamentally the same thing and are considered as separate laws only as emergent laws in cases where changes in rest mass are below notice (see the Meaning 2 of E=mc**2 below and Wikipedia: Relativistic mechanics: Rest mass and relativistic mass). However, it many applications scientific and technological those changes are below notice.
Actually, yours truly believes that a measurement of mass of an object is the only direct measurement of energy and it is only a measurement of total energy. All other energy measurements are indirect measurements since you measure some other quantities and calculate energy from a formula. For example, kinetic energy KE=(1/2)mv**2 (the energy of motion in the classical limit) is measured by measuring m and v and using the just given formula.
It does NOT strictly hold for
the energy
of the gravitational field.
In fact, the
gravitational field
is NOT like other
force fields
(the electromagnetic field
and the force fields
of the strong nuclear force
and weak nuclear force).
It is an emergent field
from the
curvature of space
and it does NOT have localizable energy that itself gravitates.
Albert Einstein (1879--1955)
ruled that out in developing
general relativity
since it led to the
paradox
of mass-energy
creating a gravitational field
that had its own
mass-energy
that created its own
gravitational field, and so on
ad infinitum.
The
curvature of space is what
creates gravity itself
and that curvature of space
encodes energy implicitly
in a non-local fashion that does NOT in itself have anymore gravitating effect.
The encoded implicit energy does NOT appear in the
energy-momentum tensor T_ij
which dictates the
curvature of space in the
Einstein field equations
(which along with the
geodesic equation
are the core formulae of
general relativity).
Hence, general relativity
avoids the ad infinitum
paradox discussed above.
Note by "non-local", we mean there is NOT so much
energy here and
so much
energy there
and there is NO
density of energy.
This is unlike the other
force fields.
Consider gravitational waves again.
When they are emitted from a
physical system
(e.g., slowly from
a binary pulsar or in
sudden blast from
a binary black hole merger)
energy
(or, if you prefer,
mass-energy)
is lost from the
physical system,
but as the
gravitational waves
propagate they propagate they make NO
contribution to the
energy-momentum tensor T_ij
of the space
they propagate through.
In fact, the
energy-momentum tensor T_ij
would be all zeros
if there were nothing else in
space.
Yet when the gravitational waves
interact with other
physical systems
they deposit
energy.
Where was that energy
between emission and interaction of the
gravitational waves?
It wasn't locally in the
energy-momentum tensor T_ij.
As mentioned above,
general relativity
does NOT guarantee that the amount
of energy emitted will be
recovered ever from
the gravitational waves
(see
Roger Penrose, The Road to Reality, 2004,
p. 467--468).
To give an imperfect
analogy, say you wanted
to get a car
to Reno, Nevada.
You could drive it and then it was a
car all the time.
Or you could disassemble the
car and send the parts
separately to Reno
and then reassemble the
car.
Did the car exist when in parts?
Non-locally.
In fact,
general relativity
does NOT guarantee the "car"
will be the same size when reassembled: it could be smaller or bigger.
For more on the tricky point of non-local
energy,
see
Roger Penrose, The Road to Reality, 2004,
p. 464--469.
There is such a thing as rest mass energy (usually just called just rest mass for simplicity) since massive particles have intrinsic mass NOT due to any other energy form.
This is the energy calculated from E=mc**2 for an object of mass m when the object is at rest in an inertial frame: hence the name rest mass.
Since any energy form can be converted into other all other energy forms, rest mass can be converted into all other energy forms. But note conversions are NOT always easy to do in practice.
Some further points:
Exotic dark matter particles (if they actually exist) also have rest mass and are massive particles, but they are NOT baryonic matter.
Massive particles can be observed when at rest.
Note that massless particles always move at vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s =3*10**5 km/s ≅ 1 ft/ns relative to any local inertial frame, and so are NEVER at rest in a local inertial frame. But could they still have rest mass in some sense? NO. If they had rest mass, then the formula
But E=mc**2 means that massless particles do have mass since they have energy (finite energy). To be super precise, we should call massless particles rest-massless particles, but that's NOT the convention.
But it can be done in principle.
For example, just rapidly collide 0.5 kg of
matter with 0.5 kg of
antimatter and both will
annihilate to form gamma rays
(and maybe other stuff ???) which will give explosive heating with the
above calculated amount of 25
megatons TNT.
But there is NO practical way to accumulate any macroscopic amounts of
antimatter.
Pair creation is actually
inverse annihilation since you create
massive particles, NOT
destroy them.
At the microscopic level, the mutual annihilation of
matter
and antimatter goes on
all the time at a very low level including wherever you are.
For example, positrons
(the antiparticles
of electrons) are
products of certain
radioactive decay processes
(specifically some beta decay processes)
that occur at a low level just about everywhere.
But the rate of positron production
is very low usually.
The produced positrons run into
electrons pretty quickly and
mutually annihilate to create
gamma rays.
The rate of heating from this natural and common process is relatively low.
Ever since the
advent of special relativity
in 1905
people have been mesmerized by how much
energy is available
in principle in rest mass.
But the practical ways of getting a significant fraction of it from
matter are limited
to nuclear reactors
and nuclear bombs.
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By the by,
the photon is its own
antiparticle
and can mutually annihilate with another
photon
in pair creation
(of massive particles)
when it has enough energy,
but having enough energy
for pair creation
requires the
photons
to be gamma rays.
Creating even large microscopic amounts of
antimatter is challenging in the lab.
   
P = mc**2/t=(1 kg)*(3*10**8 m/s)**2/(1 s) ≅ 10**17 W = 10**5 terawatts
which is a lot of terawatts
relative to societal needs
since
world commercial power
usage is currently ∼ 20 terawatts
(see
Wikipedia: World energy supply and consumption:
Primary energy production (2021); see also
Physics file:
energy_units.html).
Our energy resources would be endless if we could easily
convert rest mass
into other forms of energy.