Special Relativity
Special relativity is a physical theory proposed by Albert Einstein in 1905 that describes the relationship between space and time for objects moving at constant velocities, particularly near the speed of light. It is founded on two postulates: the laws of physics are identical in all inertial frames of reference, and the speed of light in a vacuum is constant for all observers regardless of their motion. The theory reveals that time, length, and mass are not absolute but depend on the relative motion between observer and object, unifying space and time into a single four-dimensional continuum called spacetime.
Key Formula
E² = (pc)² + (m₀c²)²
LaTeX: E^2 = (pc)^2 + (m_0 c^2)^2
| Symbol | Meaning | Unit |
|---|---|---|
| E | Total energy of the particle | Joule (J) |
| p | Relativistic momentum | kg·m/s |
| m_0 | Rest mass of the particle | kilogram (kg) |
| c | Speed of light in vacuum | m/s |
Worked Example
Problem
An electron (rest mass 9.11 × 10⁻³¹ kg) travels at 0.8c. Calculate its total relativistic energy.
Solution
Step 1: Find the Lorentz factor γ = 1/√(1 − v²/c²) = 1/√(1 − 0.64) = 1/√0.36 = 1/0.6 = 5/3 ≈ 1.667. Step 2: Rest energy E₀ = m₀c² = (9.11 × 10⁻³¹)(3 × 10⁸)² = 8.199 × 10⁻¹⁴ J. Step 3: Total energy E = γm₀c² = 1.667 × 8.199 × 10⁻¹⁴ J = 1.367 × 10⁻¹³ J.
Answer
E ≈ 1.37 × 10⁻¹³ J (approximately 0.854 MeV)
Relativistic Effects at Various Speeds (fraction of c)
| Speed (v/c) | Lorentz Factor γ | Time Dilation Factor | Length Contraction Factor | Mass Increase Factor |
|---|---|---|---|---|
| 0.10 | 1.005 | 0.5% | 0.5% | 0.5% |
| 0.50 | 1.155 | 15.5% | 13.4% | 15.5% |
| 0.80 | 1.667 | 66.7% | 40.0% | 66.7% |
| 0.90 | 2.294 | 129.4% | 56.4% | 129.4% |
| 0.99 | 7.089 | 608.9% | 85.9% | 608.9% |
| 0.9999 | 70.71 | 6971% | 98.6% | 6971% |
Interactive Tools
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Related Terms
Time Dilation
Time dilation is the phenomenon predicted by Einstein's relativity theories whereby time passes at different rates for observers in different states of motion (velocity-based) or in different gravitational fields (gravitational). A clock moving relative to an observer ticks more slowly than a stationary clock, and a clock in a stronger gravitational field ticks more slowly than one in a weaker field. This effect has been confirmed experimentally using atomic clocks on aircraft and satellites, and it must be corrected for in the GPS navigation system to maintain centimeter-level accuracy.
Mass-Energy Equivalence
Mass-energy equivalence is the principle, derived from Einstein's special theory of relativity, stating that mass and energy are two forms of the same physical quantity and can be converted into each other. Expressed by the famous equation E = mc², it reveals that even a small amount of mass corresponds to an enormous amount of energy, since c² (the square of the speed of light) is approximately 9 × 10¹⁶ m²/s². This principle underlies the energy released in nuclear fission and fusion reactions, and explains the origin of stars' energy output.
Spacetime
Spacetime is the four-dimensional continuum that combines the three dimensions of space (x, y, z) with the one dimension of time (t) into a single mathematical framework, first described by Hermann Minkowski in 1908 based on Einstein's special relativity. In this framework, events are described by four coordinates, and the separation between events is measured by the spacetime interval, which remains invariant under Lorentz transformations. In general relativity, spacetime is not flat but can be curved by mass and energy, and this curvature is what we experience as gravity.
From Latin "relativus" (having reference or relation), coined by Henri Poincaré in 1904 and fully developed by Albert Einstein in his 1905 paper "Zur Elektrodynamik bewegter Körper" (On the Electrodynamics of Moving Bodies). The term "special" distinguishes it from general relativity (1915), which covers accelerated frames.