Updated December 05, 2021

The equation E=mc2 first appears in Albert Einstein's article "Does the inertia of a body depend on the energy it contains?", published in the journal Annalen der Physik on September 27, 1905. It appears after the article of June 30, 1905 entitled "On the electrodynamics of bodies in motion".

This article from June 1905 contains the basis of what would later be called "The Theory of Special Relativity".

The theory of special relativity is based on two principles: Galilean relativity and the invariance of the speed of light.

Galilean relativity says that the laws of physics remain unchanged in inertial frames of reference (systems of reference in uniform motion) since called Galilean frames of reference. Moreover the speed is relative compared to a reference mark, a point chosen arbitrarily as fixed.

The invariance of the speed of light implies that it is a limit and unsurpassable speed, whatever the frame of reference.

But what exactly does the equation E=mc^{2} mean?

Actually, the original equation is E^{2}=m^{2}c^{4}+p^{2}c^{2}.

(E = energy, m = mass, c = speed of light, p=momentum or momentum).

In other words the total energy of a body is the sum of its mass energy at rest (m^{2}c^{4}) plus its kinetic energy (p^{2}c^{2}).

If a body is motionless, its momentum is zero then E^{2}=m^{2}c^{4} ⇒ E=mc^{2}.

Other forms of the equation m=E/c^{2} or Δm=ΔE/c^{2} (Δ = variation).

This equation E=mc^{2} has enormous consequences on the concepts of space, time, mass and energy but also on the physics of the infinitely large (cosmology) and of the 'infinitely small (quantum physics). *NB: Law of speed composition If V _{1} is the speed of the reference frame R_{1} (for example a train traveling at 100 km/h) If V_{2} is the speed of a walker (R_{2} frame of reference) moving at 10 km/h in the same direction as the train with respect to the train. How fast does V_{0} move the walker in relation to the platform (R_{0} frame of reference)? The answer is V_{0} = V_{1} + V_{2} = 110 km/h From the point of view of the platform the train is moving at 100 km/h and the walker at 110 km/h. From the point of view of the train, the walker moves at 10 km/h and the platform at -100 km/h. From the walker's point of view, the train moves at -10 km/h and the platform at -110 km/h. The speed is relative, it depends on the point of view. No mechanical experiment inside the inertial frame of reference can allow to determine the speed of its own frame of reference. *

In the article of June 30, 1905 published in Annalen der Physik "On the electrodynamics of moving bodies" Einstein postulates that light is not only undulatory but that it is made up of photons (corpuscles of light). This will allow him to question the existence of the ether.

The light redefined by Maxwell, is considered since the 17th century, as a wave. To propagate sound, it needs a medium (air) which vibrates as the sound passes. Similarly, a wave to propagate needs a medium (water).

The physicists of the 18th century then invented the luminiferous ether, an intangible substrate which serves as a support in a vacuum to convey light. The ether which vibrates with the passage of light is a consensus in the scientific world.

For Einstein, if light is not only wavy but also made up of photons then there is no need for luminiferous ether.

Einstein vigorously rejected the stationary nature of the ether defended by Lorentz because it was contrary to the principle of relativity. Light can propagate in a vacuum like corpuscles. *NB: According to the equations of James Clerk Maxwell (1831-1879), light is a self-propagating electromagnetic transverse wave with electric and magnetic components where electric and magnetic fields oscillate at right angles to each other and propagate perpendicularly to the direction in which they move indefinitely unless absorbed by the intermediate matter. In other words, each type of field - electric and magnetic - generates the other in order to propagate the whole of the composite structure in empty space at the finite speed of light, c (from the Latin celeritas "speed"). *

This immobile ether was a landmark in which electromagnetic phenomena took place. Einstein vigorously rejected the stationary nature of the ether defended by Lorentz because it was contrary to the principle of relativity.

The elimination of the ether tells us that the notion of absolute speed does not exist, the speed is relative. This implies that if an object has a speed c in a frame of reference, then it has a speed c in all the frames of reference.

How to accept this oddity?

The following assumptions are necessary to understand the special theory of relativity.

• There is no privileged point in space and time, space-time is homogeneous. If we do an experiment, the measurements will remain valid here and there, yesterday and tomorrow.

• There is no privileged direction in space (space is isotropic). If we carry out an experiment, the measurements will remain valid in all directions of space, unlike time which always goes in the same direction (from the past to the future).

• The same transformation laws make it possible to link the measurements of the experiments seen by observers located in different inertia frames of reference.

• If event A is the cause of event B, regardless of the observer, event A will always be the cause of event B. Causality must be respected, that is to say that the order of events will never be reversed whatever the movement.

If we accept these postulates, the Lorentz transformations (1853-1928) modify the law of the composition of velocities.

V=V_{1}+V_{2} becomes V=V_{1} + V_{2}/1 + V_{1}V_{2}/c^{2}

We prove that if V_{2} = c ⇒ V=c.

The energy of V_{2} being maximum, the speed V_{1} is zero!

V=V_{1}+c/1+V_{1}c/c^{2}=V_{1}+c/1+V_{1}/c=cV_{1}+c/c+V_{1}=c

c has become a limit constant and objects that move at speed c do not depend on the frame of reference. If an object is propagating at the speed of light, then it will propagate at the speed of light for all observers. The experiment gives c = 299,792,458 m/s.

On our scale, V_{1} and V_{2} are small compared to c then the Lorentz factor (γ=1/√1-v^{2}/c^{2}) very close to 1, allows us to find V≈V_{1}+V_{2}.

The Lorentz factor is the proportionality factor by which relativistic time, lengths, and mass change for an object while that object is in motion. This factor takes on an infinite value when the speed of the object reaches the speed of light.

γ=1/√1-v^{2}/c^{2} ⇒ γ=1/√1-c^{2}/c^{2} that is γ=1/0 =∞

The curve representative of the Lorentz factor shows an asymptote. This asymptote suddenly rises vertically as v approaches c.

Here are some values of the Lorentz factor corresponding to different values of the speed.

At 10% of c the factor γ = 1.005

At 20% of c the factor γ = 1.021

At 50% of c the factor γ = 1.16

At 99% of c the factor γ ≈ 7

At 99.9% of c the factor γ ≈ 22

At 99.99% of c the factor γ ≈ 71

At 100% of c the gamma factor is infinite.

The Lorentz factor (γ=1/√1-v^{2}/c^{2}) applies to all concepts in a relativistic framework and of course time is concerned.

• A laser beam that is reflected on the Moon takes 1.3 s to go and 1.3 s to return. The back and forth movement of the laser beam traveling between 2 mirrors can be seen as a clock.

A pilot passing at very high speed alongside the Earth observes a strange phenomenon. He sees the light moving back and forth, not vertically but in a zigzag fashion. The ray travels a longer distance for him than for an observer on Earth. The direction of the laser seen by the observer on the ground is vertical so the distance is shorter than that seen by the pilot.

Now v = dt and the speed of light is invariant.

If v is invariant and the distance d is greater then the time is shorter. From the pilot's point of view the pendulum of light oscillates more slowly than for the observer on Earth.

In the spacecraft, time expands by a factor γ which is a function of the speed of the vessel. The movement causes time to slow down. This has been practically demonstrated with atomic clocks which allow extremely precise measurements. Two atomic clocks, one installed in an airplane in flight and the other placed on earth, obtain different measurements of time. There is a desynchronization of the clocks.

The airplane clock is a few nanoseconds slower, but the difference is real and measurable. This tiny difference is due to the low speed of the plane compared to the speed of light.

The Lorentz factor (γ=1/√1-v^{2}/c^{2}) applies to all concepts within a relativistic framework. Lengths are also concerned.

• To measure the length of a moving object, you have to measure its ends simultaneously (exactly at the same time). This is not possible for an object going at the speed of light. To be able to measure it, it must be stationary. It is therefore necessary to be in its own frame of reference where the object is stationary.

In its frame of reference, the length of moving objects is reduced by a gamma factor.

To understand this phenomenon of length contraction, an example will be more explicit than equations.

Muons are unstable, negatively electrically charged elementary particles produced in the upper atmosphere at an altitude of 35 km. Cosmic rays are streams of relativistic high energy atomic nuclei (protons) from space. When cosmic rays strike the oxygen or nitrogen atoms in the upper atmosphere, they produce showers of particles including muons. Muons have an average lifespan of 2.2 μs, and travel very close to the speed of light. That is to say that during their entire life they cover only 660 m on average.

How is it possible to find muons on the ground (35 km lower) when they can only travel 660 m?

Muons have high energy, so the time dilation effect described by special relativity makes them observable on the surface of the Earth.

Indeed from the point of view of the observer on the ground, the lifetime of the muon which moves at the speed of light is dilated by a factor γ (2.2 γ). From his point of view, the muon will live 75 times longer and it will be possible to find it on the ground.

From the muon's point of view, it is the Earth moving towards it at the speed of light. He will see the size of the Earth reduced by the same factor γ. For the muon, the 35 km of atmosphere will only be 660 m and it will be able to reach the ground before disappearing.

The two points of view arrive at the same physical result. From the point of view of the observer on the ground there is a dilation of time. From the muon point of view, there is a contraction of the lengths.

Just as with time dilation, we do not observe the contraction of lengths in our daily life because our speeds of movement are very low compared to that of light.

In E=mc^{2}, the enormity of the factor c^{2} tells us that an object has gigantic energy just because it has mass.

• If we could extract all the energy contained in 1 g of matter we would obtain 90,000 GJ [E=0.001x(3x10^{8})^{2}=9x10^{16} joules] or 25 GWh, ie the daily output of a 1 GW nuclear power plant. The mass energy contained in 1 mg of matter is equal to the chemical energy of 2 tons of petroleum (1 ton of petroleum = 41.868 GJ).

The mass energy which is hidden from us is truly gigantic !!!

• A massive body emits electromagnetic radiation and therefore light (visible in particular when heating a metal). A massive body loses mass when it loses energy. In other words, a body can lose mass without losing corpuscles. This means that mass no longer measures the quantity of matter in a body but its quantity of energy.

For example, the mass of a hydrogen nucleus is not equal to the sum of the mass of the proton plus that of the electron. When the proton and the electron are linked, the nucleus tends towards a parsimonious ground state (lowest potential energy). To reach this ground state, a quantity of energy ΔE is released thus the energy of the system decreases. By losing this extremely low energy, the nucleus also loses mass (ΔE=Δmc^{2}). Conversely, when the atom recovers energy, the system increases its mass. This is valid for all atoms of matter. For example, the mass of the helium nucleus (2 protons + 2 neutrons) = 6.646 yg (1 yoctogram=10^{−24} gram). The mass of the proton = 1.673 yg. The mass of the neutron = 1.675 yg. The mass of the 2 protons + the mass of the 2 neutrons = 6,696 yg.

It can be seen that the mass of the nucleus is less than the mass of its constituents. When protons and neutrons bond with each other, the mass of the system decreases by 1% Δm=ΔE/c^{2} (lowest potential energy). A ΔE is released. It's not huge, but this nuclear fusion energy allows the stars to shine.

• The mass of a proton (2 u quarks + 1 d quark) is 1.673 yg or 1.673x10^{-24} g. The mass of a neutron (1 u quark + 2 d quarks) is 1.675 yg. The mass of a u quark is 0.004 yg. The mass of a d quark is 0.009 yg.

Why is the mass of quarks only about 2 thousandths of the mass of nucleons?

The kinetic energy and the strong interaction energy stir and hold the quarks together. Which means that the mass contains only energy. The deep nature of the mass is the energy due to the frantic dance of the elementary particles.

• If a particle that emits light has no mass (m^{2}c^{4}=E^{2}-p^{2}c^{2}) then its total energy is equal to its kinetic energy (E^{2}-p^{2}c^{2}⇒E=pc). Thus, the mass is equal to the inertia as in Newton's equations (f = ma and f = mv).

• If a particle is not at rest, the momentum must be added (E^{2}=m^{2}c^{4}+p^{2}c^{2}) thus special relativity makes possible the existence of particles of zero mass. A particle can have energy without having mass, it suffices that its momentum is non-zero. This is the case with photons. Massless particles cannot know rest, they are forced to go at the speed of light.

• If a particle has a speed equal to c its energy is maximum, therefore, its mass is zero m^{2}c^{4}=E^{2}-p^{2}c^{2}=0.

• Mass is converted into energy (m=E/c^{2} or Δm=ΔE/c^{2}) in the production of nuclear energy. Very heavy atomic nuclei (uranium 235, plutonium 239) contain a lot of protons, and are unstable. If one of these atoms captures a neutron, it turns into an even more unstable nucleus (236U or 240Pu), and recovers energy. The resulting nucleus divides into two nuclei and then releases two or three neutrons available for other nucleus fissions (principle of chain reaction).

In this reaction, the mass of the two nuclei plus the mass of the released neutrons is lower than the mass of the initial uranium 235 nucleus. Mass has been converted into kinetic energy. It is this energy produced by the fission that we will recover. The fission of a uranium 235 atom releases approximately 193.2 MeV of recoverable energy. The energy released by nuclear fuels is a million times greater than that of chemical fossil fuels.

• Chemical reactions also transform mass into energy, but at a much lower level of the order of one billionth.

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The Lorentz factor (γ=1/√1-v^{2}/c^{2}) applies to all concepts in a relativistic framework and of course the energy is concerned.

• In the equation E=mc^{2}/√1-v^{2}/c^{2}-mc^{2}, if the speed of the body is very small compared to the speed of light (up to 100,000 km / s) we find the kinetic energy of the macroscopic world E_{c}=mc^{2}/√1-v^{2}/c^{2}-mc^{2} ⇒ ½mv^{2}

• If the speed approaches the speed of light, kinetic energy tends towards infinity. This means that we cannot exceed the speed of light since to communicate to an object a speed equal to c we would need an infinite quantity of energy E_{c}=∞ therefore impossible. In other words, the more massive a body, the greater its resistance to movement. The more its speed increases, the more difficult it is to modify its movement. We gradually reach a speed limit impossible to exceed. At that moment the inertia is so high that all the energy in the universe would not be enough to make it go faster. We understand why the speed of light is a limit speed throughout the observable Universe.

• E_{c}=mc^{2}/√1-v^{2}/c^{2}-mc^{2} in this equation, kinetic energy is converted to mass, which is what we do in particle accelerators.

The beams of very energetic particles which circulate in opposite directions and collide cause the appearance of a very large number of particles which can be detected thanks to high-energy particle detectors. When we measure the total mass of all the particles created by the collision, and compare it to the mass of the original particles, we find a value up to 200,000 times higher. In other words, the mass does not conserve and the kinetic energy of the initial particles has materialized. It turned into new particles.

Kinetic energy has transformed in mass!

• Δm=ΔE/c^{2}, in thermonuclear fusion it is the mass which is converted into energy. A deuterium nucleus or a positively charged tritium nucleus cannot bind naturally. But in tokamak under the influence of extreme temperature and pressure, hydrogen atoms can merge and generate energy. The mass balance shows that the mass after the fusion reaction is smaller than the mass before the reaction. The difference in mass turned into energy.

Antimatter is also a consequence of the equation E=mc^{2}. Matter and antimatter on meeting each other annihilate and transform 100% of their mass into pure energy.

• For example, an electron (negative) and a positron (positive) can fuse and transform into very high energy gamma photons. Thus, the conversion of matter into energy is total.

Each time a particle encounters an antiparticle it immediately annihilates itself into pure energy as in the functional nuclear medicine imaging technique used in PET (positron emission tomography).

The patient is injected with a radioactive isotope (fluorine 18) in microdose incorporated into a so-called "precursor" molecule. The product will diffuse in the organs and from time to time the 18F-fluoro-deoxy-D-glucose (18FDG) disintegrates, emitting positrons.

In the presence of nearby electrons, these positrons will decay by emitting beta radiation. These photons will pass through the patient's body and be detected by the machine. We can thus tell where the glucose is going.

The isotopes most commonly used in PET imaging have a generally short half-life (time required for the radioactivity to decrease by half). Therefore, radiotracers are produced by a cyclotron a few hours before they are used.

Positron emission tomography is a technique that uses antimatter. *NB: beta radiation is a form of radioactivity in which a nucleus emits an electron and an antineutrino (beta minus ray) or a positron and a neutrino (beta plus ray). This process gives rise to another nucleus with one less neutron and one more proton than the initial nucleus. *

Space-time appeared with special relativity and its geometric representation was taken up by Hermann Minkowski (1864-1909) in 1908.

The space-time continuum has four dimensions. Three dimensions for space (x, y, z) and one for time (ct) that we transform into distance by associating it with the constant c (d = ct). In space-time, time is geometrized, which allows all events to be positioned in time and space by their ct, x, y, z coordinates which all depend on the frame of reference because time does not unfold not in the same way depending on the repository.

The measure of time transformed into a measure of distance can be associated with the three other coordinates of space. So all measurements are in units of distance. Time has become space!

All one-off events are related to the speed of light.

The set of all events is called "space-time".

The boundary defined by the cone of light is called the "cosmological horizon".

The universe lines defining the cone can be described by a particle moving at the speed of light which defines the boundary of the cone. The representation of the lines of Universe in the lower part (inverted cone) comes from the fact that an event can also have a past.

If the object is stationary in space-time, it does not move in space but moves in time (vertical axis).

If it is in motion, necessarily at a speed lower than c, the displacement will take place inside the cone of light having the origin for its vertex (the present).

The universe line of a moving character has an angle in the cone which depends on its speed, it moves in space and time. The edges of the cone define the speed limit, that of the particles of light. The surface of the cone is reserved for the representation of the propagation of the light signals emitted by the object.

The inside of the top light cone (positive side) represents "the future".

The top of the cone of light represents the "here and now".

The inside of the bottom light cone (negative side) represents "the past".

The outside of the cone is not accessible by a light signal because it would go faster than the speed of light. The exterior represents "the elsewhere".

A real world body has a universe line in space-time. The whole of his history (all the events of his life) draws this line.

In Minkowski's four-dimensional Euclidean Universe, the trajectories of objects in space-time are always straight lines.

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Understanding special relativity is far from immediate, it takes a lot of imagination to understand the concepts behind the equation E=mc^{2}. Many physicists have favored the emergence of this formula in Einstein's mind.

- Galileo (1564-1642) for Galilean relativity and the system of reference in uniform motion.

- Isaac Newton (1642-1727) for his laws of motion.

- Michael Faraday (1791-1867) for his work on electricity and magnetism.

- James Clerk Maxwell (1831-1879) for unifying electricity and magnetism and integrating light into the electromagnetic phenomenon.

- George Fitzgerald (1851-1901) for length contraction.

- Hendrik Antoon Lorentz (1853-1928) for length contraction and time dilation.

- Henri Poincaré (1854-1912) for the group structure of Lorentz transformations.

- Albert Abraham Michelson (1852-1931) and Edward Williams Morley (1838-1923) for the experiment on the invariance of the speed of light in the luminiferous aether.

What you must remember:

• For Einstein, light is made up of photons there is no need for luminiferous aether.

• There is a speed limit that objects cannot exceed. This speed connects mass and energy.

• it is no longer the speed of an object without mass but becomes a universal constant of physics that we will find in all phenomena.

• At the speed of light time no longer passes. There is no 0 speed, there is no absolute frame of reference. The idea of stillness does not exist.

• The simple application of the Lorentz factor explains the invariance of the speed of light and its limit, time dilation and length contraction.

• Energy and matter are of the same nature. For energy to become mass it takes a lot of energy, for mass to become energy it takes very little mass. The mass of a body represents the energy container.

• Matter and antimatter on meeting each other annihilate and transform 100% of their mass into pure energy.

• The space-time continuum has four dimensions. Three dimensions for space (x, y, z) and one for time (ct) that we transform into distance by associating it with the constant c (d = ct). In space-time, time is geometrized.

• With this mysterious formula E=mc^{2} which defies common sense, Albert Einstein equated the entire universe, from the infinitely large to the infinitely small. E=mc^{2} brings together cosmology and quantum physics. If this equation were to fail, all of our physics would collapse.

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