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# Mass-energy equivalence

## Mass-energy equivalence

The mass-energy equivalence principle states that the mass of a system and its energy are the same property in any physical system.

Intuitively, it means that the total amount of mass in a body would be conserved as energy if the body was destroyed. Conversely, any amount of energy (e.g. in electromagnetic waves) could be transformed into a massive body.

In his theory of special relativity (1905), Albert Einstein expressed the mass-energy equivalence in terms of the formula: $$E=mc^{2}$$

$m$=mass; $c=$speed of light.

The mass-energy equivalence equation can be thought of as similar to liquid water and ice water. They both refer to the same thing but in a different state.

## Kinetic energy in mass-energy equivalence

When particles collide, new particles of matter can be created from the energy released in the collision. The total energy is conserved.

An object with higher kinetic energy will have more mass-energy than an object with lower kinetic energy: $$E_{\text{total}}=E_{\text{kinetic}}+E_{\text{rest}}$$

The rest mass-energy $E_{\text{rest}}$ has an associated rest mass $m_{\text{rest}}$ according to mass-energy equivalence (i.e. $E_{\text{rest}}=m_{\text{rest}}c^{2}$).

The difference in mass due to kinetic energy will only be noticeable at very high speeds/energies because $m=E/c^{2}$ is very small in classical cases.

$m$=mass; $c=$speed of light.

## Rest mass and mass-energy

The rest mass $m_{0}$ is the mass of a body which is at rest relative to the observer. It is an intrinsic property of a body and distinct from its mass $m$ as measured by an observer.

Intuitively, the rest mass is the mass (and hence the energy) stored in the movement and interaction of particles within the object.

The rest mass of an iron atom moving in a gravitational field is primarily determined by the mass of the neutrons, protons and electrons inside it and the interactions among these particles.

Rest mass does not include the mass present in the form of kinetic or potential energy from the movement or position of the object as a whole.

From the mass-energy equivalence principle, we can either talk about rest mass or rest mass-energy $(E_{0})$:$$E_{0}= m_{0}c^{2}$$.

Photons do not have a rest mass-energy since they are never "at rest". The energy carried is purely kinetic and given by the de Broglie relation $E=pc =hf$.

The total energy and mass of a system is conserved. A particle emitting a photon would lose mass corresponding to the mass-energy of the photon. A particle which absorbs a photon gains mass corresponding to the mass-energy of the photon.

$c=$speed of light; $E=$energy; $p=$momentum; $h=$Planck constant.