General relativity Study Guide

📖 Core Concepts General Relativity (GR) – A metric theory that describes gravity as curvature of a four‑dimensional pseudo‑Riemannian spacetime rather than a force. Equivalence Principle – Locally, the effects of a uniform gravitational field are indistinguishable from those of uniform acceleration; this leads to the identification of gravity with spacetime curvature. Metric Tensor \(g{\mu\nu}\) – Encodes distances, time intervals, and causal structure; reduces to the Minkowski metric \(\eta{\mu\nu}\) in locally inertial frames. Einstein Field Equations (EFE) \[ G{\mu\nu}= \frac{8\pi G}{c^{4}}\,T{\mu\nu} \] where \(G{\mu\nu}\) (Einstein tensor) describes curvature and \(T{\mu\nu}\) (stress–energy tensor) describes matter‑energy. Geodesic Equation – Freely falling particles follow paths that extremize proper time: \[ \frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma^{\lambda}{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0 . \] Weak‑field limit – When spacetime is nearly flat, GR predictions reduce to Newton’s law of universal gravitation. --- 📌 Must Remember Schwarzschild radius: \(rs=\dfrac{2GM}{c^{2}}\). Perihelion precession (one orbit): \(\displaystyle \Delta\phi \approx \frac{6\pi GM}{c^{2}a(1-e^{2})}\). Light‑deflection angle (impact parameter \(b\)): \(\displaystyle \alpha = \frac{4GM}{c^{2}b}\). Shapiro time delay (signal passing near mass \(M\) at closest approach \(r\)): extra delay \(\displaystyle \Delta t \simeq \frac{2GM}{c^{3}}\ln\!\left(\frac{4r1 r2}{r^{2}}\right)\) (order‑of‑magnitude form). Gravitational redshift: Frequency ratio \(\displaystyle \frac{\nu{\text{far}}}{\nu{\text{near}}}= \sqrt{\frac{g{00}(\text{far})}{g{00}(\text{near})}}\). Gravitational‑wave strain: \(h\sim\frac{G}{c^{4}}\frac{\ddot{Q}}{r}\) (quadrupole formula). Cosmological constant term: Add \(\Lambda g{\mu\nu}\) to the left‑hand side of the EFE; \(\Lambda>0\) drives accelerated expansion. --- 🔄 Key Processes Solving a GR problem (e.g., planetary orbit) Choose an appropriate metric (Schwarzschild for a static sphere). Compute Christoffel symbols \(\Gamma^{\lambda}{\mu\nu}\). Insert into the geodesic equation to obtain radial and angular equations. Expand in \(GM/(rc^{2})\) (post‑Newtonian) to isolate relativistic corrections (e.g., perihelion precession). Predicting gravitational‑wave emission from a binary Model each body as a point mass; compute the mass quadrupole moment \(Q{ij}\). Differentiate twice to obtain \(\ddot{Q}{ij}\). Insert into the quadrupole formula for strain \(h\). Compare predicted orbital decay \(\dot{P}\) with timing data (Hulse–Taylor pulsar). Testing light deflection Measure apparent star positions during a solar eclipse. Compare observed angular shift with \(\alpha = 4GM/(c^{2}b)\). Use the result to constrain the PPN parameter \(\gamma\). --- 🔍 Key Comparisons GR vs. Newtonian Gravity Force vs. curved spacetime → gravity is geometry, not a vector force. Predicts time dilation, light bending, perihelion precession; Newton does not. Schwarzschild vs. Kerr Metric Schwarzschild: static, non‑rotating, single horizon at \(rs\). Kerr: rotating, exhibits an ergosphere and frame‑dragging; inner/outer horizons. Weak‑field (post‑Newtonian) vs. Strong‑field (exact) solutions Weak‑field: series expansion, suitable for Solar System tests. Strong‑field: full metric (e.g., black holes, early universe) required. Gravitational redshift vs. Doppler shift Redshift from potential difference; Doppler from relative motion. --- ⚠️ Common Misunderstandings “Gravity is a force” – In GR it is the manifestation of spacetime curvature; particles follow geodesics, not experience a force. Equivalence principle ≡ equality of inertial and gravitational mass – The principle is local: it applies only in a sufficiently small region where tidal effects are negligible. All time dilation is due to speed – Gravitational potential also slows clocks (gravitational time dilation). Schwarzschild radius = physical surface – It is a null surface (event horizon); the singularity lies inside it. Cosmological constant is just “extra gravity” – \(\Lambda>0\) produces repulsive acceleration, not an additional attractive force. --- 🧠 Mental Models / Intuition Rubber‑sheet analogy: Massive objects create a depression; test particles roll along the curved surface, mimicking geodesic motion. Elevator thought experiment: Inside a sealed accelerating elevator, you cannot tell whether the weight you feel is due to acceleration or a gravitational field. Spacetime as a fabric: Light follows the straightest possible path (null geodesic); if the fabric is curved, the path appears bent to an external observer. Gravitational waves: Like ripples on a pond; accelerating masses with a changing quadrupole moment create transverse “stretch‑and‑squeeze” distortions that travel at \(c\). --- 🚩 Exceptions & Edge Cases Frame‑dragging: Significant only near rapidly rotating massive bodies (Kerr black holes, Earth‑scale LAGEOS experiments). Cosmic censorship: Not proved; some solutions (e.g., naked singularities) are mathematically possible but believed absent in nature. Post‑Newtonian expansion breaks down for compact binaries in the final inspiral – full numerical relativity required. Light‑deflection formula \(4GM/(c^{2}b)\) assumes a weak field and impact parameter \(b\gg rs\). --- 📍 When to Use Which | Situation | Recommended Tool / Metric | Reason | |-----------|---------------------------|--------| | Static, spherically symmetric mass (planet, non‑rotating star) | Schwarzschild metric | Exact solution, simple form. | | Rotating compact object (Kerr black hole) | Kerr metric | Captures frame‑dragging and ergosphere. | | Large‑scale homogeneous universe | FLRW metric | Enforces isotropy & homogeneity; leads to Friedmann equations. | | Weak gravitational fields (Solar System) | Post‑Newtonian (PN) expansion | Provides systematic corrections to Newtonian predictions. | | Gravitational wave source near merger | Numerical relativity / full Einstein equations | Non‑linear dynamics dominate; PN no longer accurate. | | Small‑scale lensing (microlensing) | Thin‑lens approximation | Treats lens as a projected mass sheet; simplifies calculations. | | Estimating orbital decay of binary pulsar | Quadrupole formula | Gives leading‑order GW energy loss. | --- 👀 Patterns to Recognize \(1/r\) vs. \(1/r^{2}\) corrections: Relativistic perihelion precession and light bending introduce extra terms proportional to \(GM/(rc^{2})\). Redshift ↔ Potential difference: Larger \(|\Phi|\) → larger frequency shift; identical functional form for both clocks and photons. Two polarizations (“+” and “×”) in GW detectors → characteristic quadrupolar strain pattern. Time‑delay ∝ \(\ln\) of distances (Shapiro delay) – appears when a signal passes near a massive body. Scaling of GW strain: \(h \propto \frac{(Mc)^{5/3}}{D}\,f^{2/3}\) (chirp mass \(Mc\), distance \(D\), frequency \(f\)). --- 🗂️ Exam Traps Confusing \(\Delta\phi\) with total orbital angle: The precession formula gives the extra advance per orbit, not the full \(2\pi\). Using Schwarzschild radius for rotating BH: Kerr horizon radius depends on spin; using \(rs\) under‑estimates the outer horizon for high spin. Assuming gravitational redshift = Doppler shift: Redshift due to gravity does not depend on relative velocity; the sign can be opposite to a Doppler shift. Neglecting \(\Lambda\) in cosmology problems: A non‑zero cosmological constant alters the Friedmann equation and expansion history. Treating the metric as “background” – GR is background‑independent; you cannot superimpose a flat metric on a curved one without proper coordinate choice. Misidentifying the PPN parameter \(\gamma\): \(\gamma=1\) in GR; any deviation signals alternative theories, not a different way to compute light deflection. ---