Premed · Premed · Physics 2
Lecture 20: Special Relativity
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- State Einstein's two postulates of special relativity
- Explain and calculate time dilation and length contraction
- Apply the relativistic addition of velocities
- Derive and apply the relativistic energy-momentum relation
- Explain the equivalence of mass and energy (E = mc^2) and its significance
Lecture Content
I. Einstein's Postulates (1905)
Classical (Galilean) relativity works well at low speeds but fails as speeds approach the speed of light. The Michelson-Morley experiment of 1887 found no evidence for the "luminiferous ether" that was thought to be the medium through which light propagated, setting the stage for a radical rethinking of space and time.
Einstein resolved the puzzle with two remarkably simple postulates. The principle of relativity states that the laws of physics are the same in all inertial reference frames. The constancy of the speed of light states that the speed of light in vacuum (c = 3 x 10^8 m/s) is the same for all observers, regardless of the motion of the source or the observer. An inertial reference frame is one that moves at constant velocity with no acceleration. From these two postulates flow consequences that profoundly reshape our understanding of space, time, energy, and mass.
II. Time Dilation
Time intervals are not absolute; they depend on the observer's frame of reference. The proper time (Delta t_0) is the time interval measured by a clock at rest relative to the events, where both events occur at the same location. The dilated time (Delta t) is the time interval measured by an observer in relative motion: Delta t = gamma Delta t_0 = Delta t_0 / sqrt(1 - v^2/c^2), where gamma = 1/sqrt(1 - v^2/c^2) is the Lorentz factor. The Lorentz factor is always greater than or equal to 1: it equals 1 when v = 0 and approaches infinity as v approaches c.
The implication is that "moving clocks run slow": a clock in motion relative to an observer ticks more slowly than an identical clock at rest. This is not a mechanical effect but a fundamental property of time itself. Muon decay provides compelling evidence: muons created in the upper atmosphere at v approximately 0.998c (gamma approximately 15.8) survive long enough to reach Earth's surface because their decay time is dilated. GPS satellites require corrections of about 38 microseconds per day to account for both speed-related and gravitational time dilation. The twin paradox illustrates that a twin who travels at high speed ages less than the twin who remains on Earth.
III. Length Contraction
Lengths, like time intervals, are not absolute. The proper length (L_0) is the length measured in the frame where the object is at rest. The contracted length (L) measured by an observer relative to whom the object is moving is L = L_0 / gamma = L_0 sqrt(1 - v^2/c^2). Length contraction occurs only along the direction of motion; perpendicular dimensions are unaffected.
The statement is often summarized as "moving objects are shorter" in the direction of motion. At everyday speeds where v << c, the contraction is imperceptibly small. The muon example can be reinterpreted from the muon's rest frame: in that frame, the muon lives for its proper lifetime, but the atmosphere is length-contracted, making the distance to the ground shorter and consistent with the muon surviving the trip.
<image>A two-panel diagram illustrating time dilation and length contraction. Panel A (Time Dilation): A light clock (light bouncing vertically between two mirrors) is shown at rest and in motion. At rest, the light path is vertical with period Delta t_0 = 2d/c. In motion (moving to the right at speed v), the light follows a diagonal path, traveling a longer distance, so the period Delta t = gamma Delta t_0 is greater. The geometry forming a right triangle (vertical side d, horizontal side v Delta t/2, hypotenuse c Delta t/2) is shown. Panel B (Length Contraction): A rocket of proper length L_0 at rest is compared with the same rocket moving at speed v, which appears contracted to length L = L_0/gamma in the direction of motion. Perpendicular dimensions remain unchanged.</image>
IV. Relativistic Velocity Addition
Classical velocity addition (v_total = v_1 + v_2) fails at high speeds because it allows exceeding c. The correct relativistic velocity addition formula is u' = (u + v) / (1 + uv/c^2), where u is the velocity of an object in frame S, v is the velocity of frame S' relative to frame S, and u' is the velocity of the object in frame S'.
This formula has several important consequences. If u = c, then u' = c regardless of v, confirming that the speed of light is invariant, consistent with Einstein's second postulate. When u << c and v << c, the formula reduces to u' approximately equals u + v, recovering the classical result. No combination of subluminal velocities can ever produce a velocity exceeding c. For example, if two spaceships approach each other, each traveling at 0.9c relative to Earth, their relative velocity is not 1.8c but rather u' = (0.9c + 0.9c) / (1 + 0.81) = 1.8c/1.81 = 0.994c.
V. Relativistic Momentum and Energy
Relativistic momentum is p = gamma mv, which reduces to the classical p = mv at low speeds. As v approaches c, the momentum approaches infinity, making it impossible to accelerate a massive object to the speed of light.
Relativistic kinetic energy is KE = (gamma - 1)mc^2, which approximates (1/2)mv^2 at low speeds (using the binomial expansion). The total relativistic energy is E = gamma mc^2 = KE + mc^2, which includes the rest energy E_0 = mc^2. This is Einstein's famous equation, expressing the equivalence of mass and energy: mass is a form of energy, and energy has inertia. The amount of energy contained in even a small mass is enormous: 1 kg of mass is equivalent to 9 x 10^16 J, roughly 21 megatons of TNT.
The energy-momentum relation E^2 = (pc)^2 + (mc^2)^2 is a Lorentz-invariant quantity. For a massless particle such as a photon, this reduces to E = pc (and the particle always travels at c). For a particle at rest, it reduces to E = mc^2.
VI. Implications and Applications
Mass-energy equivalence manifests in many physical processes. In nuclear fission, a heavy nucleus splits and the products have less total mass than the original; the mass difference is converted to energy. In nuclear fusion, light nuclei combine with a similar mass deficit converted to energy, powering the sun. In matter-antimatter annihilation, all mass converts to energy (E = 2mc^2 for a particle-antiparticle pair). PET scans exploit positron-electron annihilation, which produces two 511 keV gamma rays.
Nothing with mass can reach the speed of light, because gamma approaches infinity as v approaches c, requiring infinite energy. Only massless particles such as photons and gluons travel at c.
Simultaneity is relative: events that are simultaneous in one reference frame are generally not simultaneous in another. There is no absolute "now"; simultaneity depends on the observer's frame of reference.
<image>A graph showing how the Lorentz factor gamma varies with speed v (expressed as a fraction of c). The horizontal axis goes from 0 to c, and the vertical axis shows gamma from 1 to about 10. The curve is nearly flat (gamma ≈ 1) for v < 0.5c, then rises steeply as v approaches c, reaching gamma = 7.09 at v = 0.99c. Key values are marked: gamma = 1.15 at v = 0.5c, gamma = 2.29 at v = 0.9c, gamma = 7.09 at v = 0.99c. A dashed vertical asymptote at v = c indicates gamma approaches infinity. Below the graph, the key formulas are summarized: time dilation, length contraction, relativistic momentum, total energy, and the energy-momentum relation.</image>
