Mass from Energy

The Mass from Energy formula is a variation of Einstein's Mass-energy equivalence formula solved for mass.

INSTRUCTIONS: Choose units and enter the following:

• (E) Energy

Mass from Energy (m): The calculator returns the mass equivalent in atomic units (u).  However, this can be converted to compatible units via the pull-down menu.

The Math / Science

Mass is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass when it is measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum.

Under such circumstances, and as described by this equation, the invariant mass is equal to the relativistic mass. The relativistic mass computed by this equation is the total energy of the system divided by c (the speed of light) squared.

It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by c²) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed to all influences (The technical term is isolated system meaning that an idealized boundary is drawn around the system, and no mass/energy is allowed across it).

The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c²). As seen from the center of momentum frame, the relativistic mass is also the invariant mass(just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the center of mass of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity.

For a discussion of mass in general relativity, see mass in general relativity. For a general discussion including mass in Newtonian mechanics, see the article on mass.

Mass-Energy-Light Calculators

• E = m•c² (mass/energy equivalent)
• m = E/c² (mass from energy)
• E = m•γ•c² (mass/energy equivalent not at rest)
• E = h•ν (Quantum Energy)
• p = m•γ•v (Relativistic Momentum)
• c (Speed of light) = 2.99792458E8 m/s
• c² (Speed of Light Squared) = 931.46 MeV/u
• Lorentz factor (`gamma`)
• KE ≈ m(1+v²/(2c²)-1)c² (Relativistic Kinetic Energy)

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