E=mc² Calculator - Einstein's Mass-Energy Equivalence & Relativity Equations
Comprehensive calculator for Einstein's famous equation E=mc² and other relativity formulas. Calculate mass-energy equivalence, relativistic effects, and explore special and general relativity. Complete guide to Albert Einstein's revolutionary equations that changed physics forever.
E=mc² Mass-Energy Calculator
Special Relativity Calculator
Lorentz Factor (γ) Calculator
Photoelectric Effect Calculator
Understanding E=mc²: The World's Most Famous Equation
E=mc² is arguably the most famous equation in physics, formulated by Albert Einstein in 1905 as part of his Special Theory of Relativity. This deceptively simple equation reveals a profound truth: mass and energy are interchangeable—they are two forms of the same thing. The equation states that energy (E) equals mass (m) multiplied by the speed of light squared (c²). Since the speed of light is enormous (approximately 300,000,000 m/s), even tiny amounts of mass contain astronomical amounts of energy.
This mass-energy equivalence revolutionized physics and enabled understanding of nuclear reactions, stellar energy production, particle physics, and countless modern technologies. From nuclear power plants and atomic weapons to PET scanners and our understanding of the universe's origin, E=mc² fundamentally changed how we comprehend matter and energy.
Einstein's Major Equations
Mass-Energy Equivalence (E=mc²)
The complete equation in its most famous form:
\[ E = mc^2 \]
Where:
- \( E \) = Energy (Joules)
- \( m \) = Mass (kilograms)
- \( c \) = Speed of light in vacuum = 299,792,458 m/s ≈ 3 × 10⁸ m/s
Full relativistic energy equation:
\[ E^2 = (pc)^2 + (m_0c^2)^2 \]
Where \( p \) is momentum and \( m_0 \) is rest mass.
Special Relativity: Lorentz Factor
The Lorentz factor governs time dilation and length contraction:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
Time Dilation:
\[ \Delta t = \gamma \Delta t_0 \]
Length Contraction:
\[ L = \frac{L_0}{\gamma} \]
Relativistic Mass:
\[ m = \gamma m_0 \]
Einstein Field Equations (General Relativity)
The foundation of general relativity describing spacetime curvature:
\[ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]
Or more compactly:
\[ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \]
Where:
- \( G_{\mu\nu} \) = Einstein tensor (describes spacetime curvature)
- \( g_{\mu\nu} \) = Metric tensor (describes spacetime geometry)
- \( R_{\mu\nu} \) = Ricci curvature tensor
- \( R \) = Ricci scalar curvature
- \( \Lambda \) = Cosmological constant
- \( T_{\mu\nu} \) = Stress-energy tensor (describes matter/energy distribution)
- \( G \) = Gravitational constant
This represents 10 nonlinear partial differential equations (due to tensor symmetries).
Photoelectric Effect
Einstein's 1905 equation explaining the photoelectric effect (Nobel Prize 1921):
\[ E_k = h\nu - \Phi \]
Where:
- \( E_k \) = Maximum kinetic energy of ejected electron
- \( h \) = Planck's constant = 6.626 × 10⁻³⁴ J·s
- \( \nu \) = Frequency of incident light
- \( \Phi \) = Work function of material (threshold energy)
Photon energy: \( E = h\nu = \frac{hc}{\lambda} \)
E=mc² Explained for Beginners
What Does E=mc² Really Mean?
E (Energy): The total energy contained in or released by converting mass. Measured in Joules.
m (Mass): The amount of matter in an object. Even a tiny mass contains enormous energy.
c² (Speed of Light Squared): The conversion factor. c = 299,792,458 m/s, so c² ≈ 9 × 10¹⁶ m²/s². This huge number explains why small masses yield enormous energies.
Simple interpretation: If you could convert 1 kg of matter completely to energy, you'd get 90,000,000,000,000,000 Joules—enough to power a 100-watt light bulb for 28 million years!
Example 1: Energy from 1 Gram of Matter
Problem: How much energy is contained in 1 gram of matter?
Step 1: Convert to SI units: m = 0.001 kg
Step 2: Apply E=mc²
\[ E = 0.001 \times (3 \times 10^8)^2 = 0.001 \times 9 \times 10^{16} = 9 \times 10^{13} \text{ J} \]
Answer: 90 trillion Joules—equivalent to:
- 21,500 tons of TNT
- Energy released by atomic bomb dropped on Hiroshima
- Burning 2,000 tons of coal
Example 2: Mass Defect in Nuclear Reactions
Problem: In nuclear fusion, four hydrogen nuclei (mass 1.0078 u each) fuse to form helium (mass 4.0026 u). Calculate energy released.
Step 1: Calculate mass defect
\[ \Delta m = (4 \times 1.0078) - 4.0026 = 0.0286 \text{ u} \]
Step 2: Convert to kg (1 u = 1.66054 × 10⁻²⁷ kg)
\[ \Delta m = 0.0286 \times 1.66054 \times 10^{-27} = 4.75 \times 10^{-29} \text{ kg} \]
Step 3: Calculate energy
\[ E = 4.75 \times 10^{-29} \times 9 \times 10^{16} = 4.27 \times 10^{-12} \text{ J} = 26.7 \text{ MeV} \]
Answer: This is the energy released per helium nucleus formed—the process powering the Sun!
Einstein's Equations: Complete List
| Equation | Formula | Theory | Significance |
|---|---|---|---|
| Mass-Energy | \(E = mc^2\) | Special Relativity (1905) | Mass and energy are equivalent |
| Time Dilation | \(\Delta t = \gamma \Delta t_0\) | Special Relativity | Moving clocks run slower |
| Length Contraction | \(L = L_0/\gamma\) | Special Relativity | Moving objects contract |
| Relativistic Energy | \(E = \gamma m_0 c^2\) | Special Relativity | Total energy of moving object |
| Photoelectric Effect | \(E_k = h\nu - \Phi\) | Quantum Theory (1905) | Light has particle properties |
| Field Equations | \(G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\) | General Relativity (1915) | Gravity is curved spacetime |
| Schwarzschild Radius | \(r_s = \frac{2GM}{c^2}\) | General Relativity | Black hole event horizon |
| Gravitational Time Dilation | \(\frac{t_0}{t_f} = \sqrt{1 - \frac{2GM}{rc^2}}\) | General Relativity | Time slower in gravity wells |
Energy Conversion Examples
| Mass | Energy (Joules) | Energy (Tons TNT) | Real-World Equivalent |
|---|---|---|---|
| 1 mg | 9 × 10¹⁰ J | 21.5 tons | Small tactical nuke |
| 1 g | 9 × 10¹³ J | 21,500 tons | Hiroshima bomb |
| 1 kg | 9 × 10¹⁶ J | 21.5 megatons | Large thermonuclear bomb |
| 1 proton mass | 1.503 × 10⁻¹⁰ J | - | 938 MeV (particle physics) |
Applications of E=mc²
Nuclear Energy and Weapons
E=mc² explains how nuclear reactions release vast energy. In fission (uranium/plutonium splitting), about 0.1% of mass converts to energy. In fusion (hydrogen combining), about 0.7% converts. Nuclear power plants harness fission; hydrogen bombs use fusion. The Sun's energy comes from fusion converting 4 million tons of mass to energy every second, powering Earth for billions of years.
Particle Physics and Accelerators
Particle accelerators like the Large Hadron Collider create new particles by converting kinetic energy to mass. When protons collide at 99.9999% light speed, their enormous kinetic energy creates exotic particles like Higgs bosons. This demonstrates E=mc² in reverse: energy creating mass. Mass of created particles equals energy divided by c².
Medical Imaging: PET Scans
Positron Emission Tomography (PET) uses E=mc² for medical imaging. Positrons from radioactive tracers annihilate with electrons, converting their combined mass entirely to gamma rays (photons). Detecting these photons reveals metabolic activity, diagnosing cancers and brain disorders. Each positron-electron pair produces exactly 1.022 MeV from their rest mass.
Stellar Evolution and Astronomy
E=mc² governs stellar lifecycles. Stars fuse hydrogen to helium, converting mass to energy via nuclear fusion. Eventually, massive stars explode as supernovae, creating heavier elements. The mass difference between reactants and products powers these cosmic engines. Neutron stars and black holes represent extreme mass-energy concentrations where general relativity dominates.
GPS and Satellite Technology
GPS satellites require both special and general relativity corrections. Special relativity (velocity-based time dilation) slows satellite clocks by 7 microseconds/day. General relativity (gravitational time dilation) speeds them up by 45 microseconds/day. Net +38 microseconds/day would cause 10 km/day errors without correction. Einstein's equations make GPS accurate to meters.
Common Misconceptions About E=mc²
E=mc² Doesn't Enable Time Travel
While special relativity shows time dilation at high velocities, this doesn't enable traveling backward in time. Moving clocks run slower relative to stationary observers (time dilation), allowing "time travel" to the future—astronauts age slightly slower than Earth-bound people. However, backward time travel violates causality and isn't supported by E=mc² or any verified physics. General relativity allows theoretical solutions (wormholes, closed timelike curves) requiring exotic matter not known to exist.
We Can't Convert All Mass to Energy
Complete mass-energy conversion requires matter-antimatter annihilation. Nuclear reactions convert only small fractions: fission ~0.1%, fusion ~0.7%. Burning fossil fuels converts zero mass—chemical reactions rearrange atoms without changing mass (energy comes from electron orbital changes). Only particle-antiparticle annihilation achieves 100% conversion, but antimatter is extremely difficult to produce and store.
Mass Doesn't Increase with Velocity (Modern Interpretation)
Older textbooks spoke of "relativistic mass" increasing with velocity. Modern physics prefers "rest mass" (invariant mass) remaining constant while energy increases. The equation E = γm₀c² shows total energy increases with velocity via Lorentz factor γ, but rest mass m₀ stays constant. This avoids confusion—mass is an intrinsic property, energy depends on reference frame.
Frequently Asked Questions
What does E=mc² actually mean in simple terms?
E=mc² means mass and energy are two forms of the same thing and can convert between each other. A small amount of mass contains enormous energy—multiply mass by the speed of light squared (a huge number). Example: converting 1 gram completely to energy yields 90 trillion Joules, enough to power a city for hours. This explains nuclear energy, stars, and why the universe exists as both matter and energy.
Did Einstein derive E=mc² from general relativity?
No, E=mc² comes from Special Relativity (1905), published 10 years before General Relativity (1915). Einstein derived it from considering the energy-momentum relationship in special relativity. General relativity extends gravity and spacetime curvature but builds on special relativity's foundation. E=mc² remains valid in both theories as a fundamental principle of mass-energy equivalence.
What is the complete equation beyond E=mc²?
The complete relativistic energy equation is E² = (pc)² + (m₀c²)², where p is momentum and m₀ is rest mass. For stationary objects (p=0), this reduces to E=m₀c². For massless particles like photons (m₀=0), E=pc. For moving objects, total energy includes both rest mass energy and kinetic energy. This full equation reconciles mass, energy, and momentum in all reference frames.
How does E=mc² relate to nuclear weapons?
Nuclear weapons release energy by converting small amounts of mass via fission (splitting heavy nuclei) or fusion (combining light nuclei). The mass of products is slightly less than reactants—this "mass defect" converts to energy per E=mc². In fission bombs (uranium/plutonium), about 0.7 kg of the 50 kg undergoes fission, releasing 60 trillion Joules. Fusion bombs (hydrogen bombs) are more efficient, converting about 0.7% of fuel mass to energy.
Can we use E=mc² to create energy from matter?
Yes, but inefficiently in practice. Nuclear fission reactors convert ~0.1% of uranium mass to energy—still producing millions of times more energy than chemical reactions. Fusion reactors (under development) could convert ~0.7% of hydrogen. Complete conversion requires matter-antimatter annihilation, but producing and storing antimatter is prohibitively expensive. Stars naturally perform fusion, converting mass to energy that powers life on Earth.
What are Einstein's field equations of general relativity?
Einstein's field equations describe how matter and energy curve spacetime, creating gravity. Written Gμν = (8πG/c⁴)Tμν, they relate spacetime curvature (left side) to matter-energy distribution (right side). These are 10 coupled nonlinear partial differential equations (due to tensor symmetry). Solutions describe black holes, expanding universe, gravitational waves, and GPS corrections. They're among physics' most mathematically complex but experimentally verified equations, confirmed by phenomena from planetary orbits to gravitational wave detection.
Historical Context: Einstein's Miracle Year (1905)
Einstein published four groundbreaking papers in 1905, his "Annus Mirabilis" (miracle year):
- Photoelectric Effect: Explained light's particle nature, won 1921 Nobel Prize
- Brownian Motion: Proved atoms exist through statistical mechanics
- Special Relativity: Revolutionized concepts of space and time
- E=mc²: Derived mass-energy equivalence as corollary to relativity
At age 26, while working as a patent clerk, Einstein fundamentally transformed physics. General Relativity followed in 1915, completing his revolutionary contribution to understanding the universe.
Important Disclaimer
These calculators implement Einstein's equations for educational purposes and conceptual understanding. Relativity calculations at extreme velocities, strong gravitational fields, or quantum scales require sophisticated numerical methods and quantum field theory. Results assume idealized conditions and classical interpretations. Nuclear energy calculations are theoretical—handling radioactive materials requires specialized training, licensing, and safety protocols. This educational tool does not replace formal physics education, experimental validation, or professional consultation. Discussions of nuclear technology are educational only and do not constitute instructions for weapon or reactor design. Always consult qualified physicists, regulatory authorities, and adhere to all applicable laws and safety standards.