What does E = mc² state, and what is 'rest energy'?

E = mc² means energy equals mass — they are exactly the same thing E₀ = mc² is the energy equivalent of mass at rest — even a stationary object contains enormous energy; c² ≈ 9 × 10¹⁶ J/kg means 1 kg of mass = 9 × 10¹⁶ joules E = mc² only applies to nuclear weapons — it doesn't describe ordinary matter E = mc² means objects gain mass when they gain energy — this is why moving objects are heavier

What is the full energy-momentum relation for a relativistic particle?

E = mc² + ½mv² — rest energy plus kinetic energy E² = (pc)² + (mc²)² — for massless particles (m=0): E = pc; for particles at rest (p=0): E = mc² E = pc for all particles — momentum times speed of light E = γmc — energy equals relativistic mass times speed of light

What is the mass defect of a nucleus, and why does it correspond to binding energy?

The mass defect is the error in measuring nuclear masses — it represents experimental uncertainty The mass defect Δm = (sum of constituent masses) − (nucleus mass) is the mass converted to binding energy E_binding = Δmc² when the nucleus formed — bound nuclei are lighter than their parts Mass defect means nuclei are heavier than their parts — the extra mass comes from nuclear binding force Mass defect only applies to hydrogen — all other nuclei have exactly the summed mass of their parts

In nuclear fission of uranium-235, about 0.1% of the mass is converted to energy. How much energy is released per kilogram of U-235?

9 × 10⁷ J — the same as burning high-quality coal About 9 × 10¹³ J — E = 0.001 × 1 × c² = 0.001 × 9×10¹⁶ ≈ 9×10¹³ J, equivalent to about 21 kilotons of TNT 9 × 10¹⁶ J — the full rest mass energy of 1 kg 9 × 10¹³ eV — in particle physics units

In hydrogen fusion (4H → He-4 + energy), about 0.7% of mass is converted to energy. Why does this power the Sun?

The Sun burns hydrogen chemically — 0.7% of mass converted to heat, as in combustion The Sun converts ~600 million tonnes of hydrogen per second into helium — 0.7% of that mass (≈ 4 million tonnes/s) becomes energy (3.8 × 10²⁶ W) The Sun loses mass by gravitational collapse — the energy is gravitational potential energy, not E = mc² The Sun converts 100% of hydrogen to energy — 0.7% just represents experimental uncertainty

An electron and a positron (antielectron) annihilate. What is produced, and how much energy?

One gamma ray photon with energy 2 MeV — all energy goes into one photon Two gamma ray photons, each with energy 0.511 MeV — the rest mass energy of one electron (mc² = 0.511 MeV), so total energy = 1.022 MeV = 100% of rest mass converted to radiation A burst of heat — the electron and positron neutralise each other and release thermal energy Nothing — matter and antimatter cancel to zero energy

Does E = mc² mean that mass and energy are the same thing?

No — mass and energy are always distinct; E = mc² is just a conversion formula like miles to kilometres In a deep sense, yes — mass is a form of energy (rest energy), and energy has inertia (adding energy increases rest mass); the distinction between mass and energy breaks down in relativity E = mc² applies only at the speed of light — at lower speeds, mass and energy are separate Mass and energy are the same for photons only — for massive particles, they remain distinct

Why is it significant that proton rest mass (938 MeV/c²) is much greater than the sum of its quark rest masses (~few MeV/c²)?

The discrepancy shows that the quark model is wrong — protons must contain other unknown particles Most of the proton's mass comes from the kinetic and binding energies of quarks and gluons (via E = mc²) — confirming that mass is stored energy, not an intrinsic property of matter The proton mass is entirely from the Higgs field — quarks contribute no mass at all The extra mass comes from dark matter inside protons — this is why dark matter is so hard to detect

What is pair production, and what energy is required?

A proton and antiproton are created from a gamma ray — minimum energy = 938 MeV A high-energy photon creates an electron-positron pair (γ → e⁻ + e⁺) — minimum energy required = 2 × 0.511 MeV = 1.022 MeV (two electron rest masses) A neutron and proton are created from a photon — minimum energy = 1876 MeV Pair production only happens for photons with exactly 0.511 MeV — any other energy is absorbed differently

Why is E = mc² described as 'the most important equation in physics'?

E = mc² is only important for nuclear weapons — it has no application in everyday physics It unified mass and energy as a single conserved quantity, explained nuclear energy and stellar power, predicted antimatter, and underpins particle physics, nuclear engineering, PET scanners, and our understanding of the early universe E = mc² is less important than Newton's laws — it only applies at extreme speeds The equation is important only historically — modern physics has moved beyond it