What is the physical meaning of Einstein’s famous equation 𝐸=π‘šπ‘Β²?

Einstein’s iconic equation 𝐸=π‘šπ‘Β² is perhaps the most recognized equation of all time. It is famously simple and profound. Ask anyone familiar with the formula and they’ll probably tell you that mass and energy are two different forms of the same thing. They might even tell you that the conversion rate between them is the speed of light squared.

Most explanations of Einstein’s formula acknowledge the math but stop short of explaining the physical meaning of 𝑐². Perhaps you’re wondering what it means to square the speed of light?

To understand the physical meaning of Einstein’s equation, you need to know a couple of things about elementary particles. The first is that particles once described as tiny bits of matter are actually wave-like vibrations of a quantum field. Even massive particles have wave-like attributes including wavelength, amplitude, and frequency.

Second, you need to know what it means for wave-like particles to have kinetic energy. For any type of particle, including photons and electrons, kinetic energy is generated from two factors. The first factor is a particle’s wavelength. The shorter the wavelength, the greater its potential for producing kinetic energy. A particle’s energy potential due to wavelength is called momentum.

The second factor that determines a particle’s kinetic energy is velocity. The faster a particle moves, the more kinetic energy it has.

For elementary particles, Einstein’s formula is a recipe for translating the rest mass of particles like electrons into the kinetic energy of massless particles like photons. The recipe calls for the aforementioned kinetic energy factors which give Einstein’s formula it’s shape. If we apply the formula to an electron, the first term on the right-hand side of the equation π‘š describes the particle’s rest massβ€”also written as π‘šβ‚€. This is the quantity we will convert into energy.

Working backwards, the second factor of kinetic energy is 𝑐, the speed of light. This term represents the particle’s velocity, or rate of displacement. The formula says that to convert rest mass into energy, we need to maximize the particle’s velocity.

The first factor of kinetic energy is confusing; it is also 𝑐 but does not represent the particle’s velocity. Rather, it represents the energy potential of the particle’s wavelength. The formula says that to convert rest mass into energy, we need to maximize the particle’s energy potential by minimizing its wavelength. In traditional units of momentum, this is done by multiplying rest mass by 𝑐.

The combination of rest mass and the speed of light is puzzling, but don’t be discouraged if it doesn’t make sense. Mass times velocity isn’t a physical description of the particle, but a mathematical tool for calculating the potential of a massive particle’s wavelength. Here’s why the math works:

A massive particle’s wavelength is proportional to its velocity. At one end of the scale is the particle’s minimum wavelength which corresponds with the maximum limit of its velocityβ€”the speed of light. Moving down the scale, as velocity decreases relative to 𝑐, the particle’s wavelength increases in the same ratio. The result is a squared quantity of kinetic energy compared to changes in either the wavelength or velocity. The combined magnitude of wavelength and velocity potentials is 𝑣²/𝑐².

Since a massive particle’s minimum wavelength coincides with its maximum velocity, momentum is traditionally counted in unit dimensions of mass times velocity, or LMT-1.

Now we can summarize Einstein’s iconic equation. The formula allows us to convert a latent quantity of rest energy, π‘š0, into a comparable amount of kinetic energy following a two-step process. Step one quantifies the particle’s kinetic energy potential which is found at its minimum wavelength. Mathematically, this is accomplished by multiplying rest mass by the speed of light. Step two converts the particle’s velocity at rest into the maximum velocityβ€”the speed of light.

Einstein’s formula is applicable to more than just elementary particles. The rest mass of composite particlesβ€”in which multiple elementary particles are boundβ€”includes additional rest energy from the strong and electromagnetic interactions. Converting the rest mass of composite particles into energy also takes binding energy into account.

Einstein’s formula reveals the structure of massive and massless particles

An elementary particle is sometimes called a quantum because its mass and energy are conserved by the particle’s wavelength and velocity. At the maximum extreme, Einstein’s formula relates the Planck mass (π‘šπ‘ƒ) and Planck energy (𝐸𝑃)

𝐸𝑃 = π‘šπ‘ƒπ‘Β²

Any reduction in energy from the Planck scale must be accounted for by an increase in the particle’s wavelengthβ€”and for massive particles, by a decrease in velocity. This structure gives a simple, natural formula for the kinetic energy of massive and massless particles that incorporates classical and quantum mechanical formulas.


Natural formula for kinetic energy applied to particles with no rest mass; the spin, wavelength, and velocity operators multiplied by Planck energy


Natural formula for kinetic energy applied to particles with rest mass; the spin, wavelength, and velocity operators multiplied by Planck energy

The illustrations convey two distinct parts of every natural formula. One part is the maximum potential of the quantity we’re solving forβ€”in this case, Planck energy. The second part consists of dimensionless proportionality operators that describe the spatial and temporal distributions of particles and fields. These operators reduce the maximum potential in the correct proportions.

Applying the formula to a photon shows that changes in kinetic energy are due entirely to wavelength. The ratio between the photon’s wavelength Ζ› and the shortest possible wavelength 𝑙𝑃 (called the Planck length), is the correct ratio for quantifying the photon’s energy. Since photons move at the speed of light, there is no energy reduction due to velocity.

It is easy to show that this natural formula is equal to the historical formula


when we convert Planck’s constant into natural Planck units.

Applying the same structure to an electron shows three reductions to the Planck energy. Two reductions are the wavelength and velocity factors already discussed. The third factor is a Β½ reduction in energy that can be accounted for by an electron’s spin attribute. A spin of one-half doubles the electron’s oscillation period, decreasing its energy by half.

The natural formula is equal to the classical formula for kinetic energy


because π‘šβ‚€π‘£ is always equal to (𝑙𝑃 / Ζ›) 𝑐

The natural kinetic energy formula depicts a standard quantity of Planck energy that is diluted by an elementary particle over space and time. In fact, we can consolidate the second and third operators into one temporal operator, giving the following simple kinetic energy formula.


Natural formula for kinetic energy; the ratio of Planck time to oscillation period multiplied by Planck energy


Natural formula for kinetic energy; one-half multiplied by the ratio of Planck time to oscillation period, multiplied by Planck energy

The New Foundation Model of physics explains abstract mathematical quantities in terms of natural quantities of length, mass, and time. It reformulates traditional physics in a way that is mathematically equivalent, but reveals new meaning.

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