Theory of relativity Study Guide

📖 Core Concepts Spacetime continuum – Space and time are unified into a four‑dimensional fabric; events are described by coordinates \((ct, x, y, z)\). Two postulates of Special Relativity (SR) Physical laws are identical in every inertial frame. Light in vacuum travels at the constant speed \(c\) for all observers, independent of source motion. Relativity of simultaneity – “Simultaneous” in one frame need not be simultaneous in another moving frame. Time dilation – Moving clocks tick slower: \(\Delta t = \gamma\,\Delta\tau\) where \(\gamma = 1/\sqrt{1-v^{2}/c^{2}}\). Length contraction – Moving objects are shortened along the motion direction: \(L = L{0}/\gamma\). Mass–energy equivalence – \(E = mc^{2}\); mass can be viewed as condensed energy. Equivalence principle (GR) – Uniform acceleration is locally indistinguishable from a uniform gravitational field. Curved spacetime – Gravity is geometry: mass–energy tells spacetime how to curve, curvature tells matter how to move (Einstein’s field equations). Gravitational time dilation – Clocks deeper in a potential run slower: \(\Delta t{\text{far}} = \Delta t{\text{near}}/\sqrt{1-2GM/(rc^{2})}\). --- 📌 Must Remember Lorentz factor \(\displaystyle \gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\). Lorentz transformations (frame moving at velocity \(v\) along \(x\)): \[ \begin{aligned} t' &= \gamma\!\left(t-\frac{v x}{c^{2}}\right),\\ x' &= \gamma\!\left(x - vt\right),\\ y' &= y,\; z' = z. \end{aligned} \] Invariant spacetime interval: \(\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}\) (same in all inertial frames). \(E=mc^{2}\) – Energy of a body at rest; for moving bodies \(E = \gamma mc^{2}\). Einstein field equations (schematic): \(\displaystyle G{\mu\nu}= \frac{8\pi G}{c^{4}}\,T{\mu\nu}\). Classic experimental confirmations Michelson–Morley → no aether wind ⇒ isotropic \(c\). Ives–Stilwell → transverse Doppler shift ⇒ time dilation. Gravitational redshift & light‑deflection → GR predictions verified. GPS corrections: SR contributes \(\approx -7\;\mu\text{s/day}\); GR contributes \(+45\;\mu\text{s/day}\). --- 🔄 Key Processes Compute SR effects for a moving clock Find velocity \(v\). Calculate \(\gamma = 1/\sqrt{1-v^{2}/c^{2}}\). Proper time interval \(\Delta\tau\) → observed interval \(\Delta t = \gamma\Delta\tau\). Apply Lorentz transformation to an event Write coordinates \((t, x, y, z)\) in the “stationary” frame. Plug into the formulas for \(t'\) and \(x'\) using the same \(\gamma\). Determine gravitational time dilation Identify the gravitational potential \(\Phi = -GM/r\). Use \(\displaystyle \Delta t{\text{far}} = \Delta t{\text{near}}\sqrt{1+\frac{2\Phi}{c^{2}}}\). Use the equivalence principle to replace acceleration with gravity In a uniformly accelerating rocket with acceleration \(a\), treat \(a\) as a “gravitational field” of strength \(g=a\). Solve a simple GR problem (e.g., light deflection near the Sun) Use the weak‑field approximation: \(\displaystyle \delta\theta \approx \frac{4GM}{c^{2}R}\) where \(R\) is the impact parameter. --- 🔍 Key Comparisons SR vs. GR SR: flat spacetime, inertial frames only, constant \(c\). GR: curved spacetime, all (including accelerated) frames, gravity = curvature. Proper time \(\Delta\tau\) vs. Coordinate time \(\Delta t\) \(\Delta\tau\): time measured by a clock moving with the event (always smallest). \(\Delta t\): time measured by a distant observer; \(\Delta t = \gamma\Delta\tau\). Length contraction vs. Gravitational length dilation Length contraction: \(L = L{0}/\gamma\) for moving objects. Gravitational “stretching”: radial distances appear larger to a distant observer due to spacetime curvature (no simple linear formula). Mass increase vs. Relativistic energy “Relativistic mass” \(m{\text{rel}} = \gamma m\) (historical). Modern view: keep invariant mass \(m\) and account for kinetic energy via total energy \(E = \gamma mc^{2}\). --- ⚠️ Common Misunderstandings “Time dilation is symmetric” – It is symmetric only between inertial observers; acceleration (or gravity) breaks the symmetry (e.g., twin paradox). “Objects can appear longer” – Length contraction always shortens the dimension parallel to motion; transverse dimensions are unchanged. “\(E = mc^{2}\) applies only to nuclear reactions” – It is universal; any change in mass corresponds to an energy change, however small. “Gravity is a force in GR” – In GR, gravity is not a force but the manifestation of spacetime curvature; free‑falling objects feel no force. “All clocks run at the same rate in orbit” – GPS satellites experience both SR (slower) and GR (faster) effects; net correction is essential. --- 🧠 Mental Models / Intuition Spacetime as a stretched rubber sheet – Masses create dents; objects follow the straightest possible paths (geodesics) in the deformed sheet. Light‑clock thought experiment – Visualize a photon bouncing between mirrors; moving the clock sideways lengthens the light’s path → time appears dilated. “Train‑platform” simultaneity picture – Imagine lightning striking both ends of a moving train; observers on the platform see non‑simultaneous strikes because light must travel different distances. --- 🚩 Exceptions & Edge Cases Massless particles (\(m=0\)) travel exactly at \(c\); \(\gamma\) is undefined but the invariant interval \(s^{2}=0\). Ultra‑relativistic limit (\(v\to c\)) – \(\gamma\) → ∞, time dilation and length contraction become extreme; classical approximations break down. Strong‑field GR – Weak‑field formulas (e.g., \(\delta\theta\approx 4GM/c^{2}R\)) fail near black holes; full Schwarzschild metric needed. Non‑inertial frames in SR – Pure SR formulas assume inertial frames; accelerated frames require GR’s equivalence principle. --- 📍 When to Use Which Use SR (Lorentz formulas) when Velocities are a sizable fraction of \(c\) and gravitational potentials are negligible. Switch to GR when Gravitational potential \(GM/rc^{2}\) ≳ \(10^{-6}\) (e.g., near Earth’s surface for precision timing, near massive stars). Apply the equivalence principle to replace a uniform acceleration problem with a uniform gravitational field problem (e.g., elevator thought experiments). Choose the invariant interval \(s^{2}\) to check whether two events can be causally connected (if \(s^{2}>0\) → timelike). Select GPS relativistic corrections: SR correction = \(-\frac{v^{2}}{2c^{2}}\) per day. GR correction = \(+\frac{GM}{rc^{2}}\) per day. --- 👀 Patterns to Recognize Whenever you see “same speed of light for all observers,” think Lorentz factor \(\gamma\). If a problem mentions clocks at different heights or speeds, look for time‑dilation or gravitational‑redshift formulas. Presence of “orbit,” “precession,” or “bending of light” → GR curvature effects, likely need Einstein field equation concepts or weak‑field approximations. Experimental setups mentioning interferometers, moving atoms, or atomic clocks → classic SR tests (Michelson–Morley, Ives–Stilwell). --- 🗂️ Exam Traps Mistaking coordinate time for proper time – Answer choices that use \(\Delta t\) where \(\Delta\tau\) is required (or vice‑versa). Forgetting the \(\gamma\) factor in length‑contraction or time‑dilation formulas; a common distractor drops the denominator or numerator. Assuming “mass increase” is a physical force – Some options claim a “new force” appears; correct answer emphasizes geometry, not force. Mixing up SR and GR predictions – E.g., attributing gravitational redshift to SR time dilation; the correct explanation involves spacetime curvature. Neglecting the sign of gravitational potential – Light‑deflection formulas require a positive angle; a trap may present a negative result. GPS problem pitfalls – Using only SR or only GR correction leads to a net error of 38 µs/day; the exam expects both contributions summed. ---