Relativity and modern physical cosmology fundamentally reshaped our conception of time, space, mass, and gravitation: from the invariant speed of light and Minkowski spacetime metrics to Einstein's geometric curvature of gravity, black hole horizons, and the accelerated metric expansion of the cosmos.
This cheat sheet organizes 20 foundational physical laws and governing equations across relativity and cosmology, categorized into four modules: Special Relativity, General Relativity, Black Hole Physics, and Physical Cosmology, complete with mathematical formulations, metric tensors, and experimental verifications.
🔗 Physics Equations Suite: Classical Mechanics · Electromagnetism · Fluid & Thermal · Optics & Acoustics · Nuclear & Particle · Astro & Geophysics · Quantum Mechanics
1. Special Relativity (7 Items)
01 · Constancy of the Speed of Light
First Postulate of Special Relativity
$$c = 299792458\ \text{m/s}$$
- ①Physical Essence: Light in vacuum propagates with speed $c$ in all inertial frames of reference, independent of the motion of the emitting source or the observer.
- ②Mathematical Notation: $c$ is the universal speed limit of causality; legally defined as an exact SI constant ($299,792,458\text{ m/s}$).
- ③Theoretical Significance: Overthrew Galilean velocity addition and the luminiferous aether hypothesis; serves as the definition baseline for the SI meter.
- ④Boundary Conditions & Applications: Verified by the null fringe shift of the Michelson-Morley experiment; essential for GPS clock synchronization and quantum electrodynamics.
02 · Lorentz Factor
Relativistic Kinematic Scaling
$$\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad (\beta \equiv \frac{v}{c})$$
- ①Physical Essence: Dimensionless factor quantifying the magnitude of relativistic kinematics; approaches 1 at low speeds and diverges to infinity as $v \to c$.
- ②Mathematical Notation: $\gamma \ge 1$ is the Lorentz factor, $\beta = v/c$ is velocity normalized to light speed.
- ③Theoretical Significance: The core transformation multiplier across all Lorentz coordinate, momentum, and field transformations between inertial frames.
- ④Boundary Conditions & Applications: At $v = 0.866c$, $\gamma = 2$; in the CERN Large Hadron Collider (LHC), protons reach $\gamma \approx 7460$.
03 · Time Dilation
Moving Clocks Run Slow
$$\Delta t = \gamma \Delta \tau = \frac{\Delta \tau}{\sqrt{1 - \frac{v^2}{c^2}}}$$
- ①Physical Essence: The time interval measured between two events is shortest in the reference frame where the events occur at the same spatial point (proper time $\Delta \tau$).
- ②Mathematical Notation: $\Delta t$ is coordinate elapsed time in an observer frame, $\Delta \tau$ is proper time ($c^2 \mathrm{d}\tau^2 = \mathrm{d}s^2$).
- ③Theoretical Significance: Disproves Newtonian absolute time; confirms time is a dynamic coordinate woven into four-dimensional spacetime.
- ④Boundary Conditions & Applications: Explains why cosmic ray muons with a rest lifetime of $2.2\ \mu\text{s}$ survive to reach sea level; verified by Hafele-Keating atomic clock flights.
04 · Length Contraction
Lorentz-FitzGerald Contraction
$$L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - \frac{v^2}{c^2}}$$
- ①Physical Essence: The spatial length of an object measured along its direction of motion is shortened relative to its rest length ($L_0$). Dimensions perpendicular to motion remain unchanged.
- ②Mathematical Notation: $L$ is measured length in the moving observer frame, $L_0$ is proper length at rest.
- ③Theoretical Significance: Directly stems from the relativity of simultaneity; observers in different frames disagree on spatial boundaries measured simultaneously.
- ④Boundary Conditions & Applications: In high-energy particle storage rings, relativistic gold ions contract into thin pancake charge distributions, modifying electromagnetic wakefields.
05 · Mass-Energy Equivalence
Einstein's Mass-Energy Equation
$$E_0 = m_0 c^2,\qquad E = \gamma m_0 c^2$$
- ①Physical Essence: Rest mass is a concentrated form of latent energy. Inertial mass and energy are mutually convertible through the scaling constant $c^2$.
- ②Mathematical Notation: $E_0$ is rest energy, $m_0$ is invariant rest mass, $c$ is speed of light ($c^2 \approx 9 \times 10^{16}\text{ J/kg}$).
- ③Theoretical Significance: Unifies the classical laws of conservation of mass and conservation of energy into mass-energy conservation.
- ④Boundary Conditions & Applications: Nuclear fission in atomic power plants converts $\sim 0.1\%$ of mass to energy; solar hydrogen fusion converts $\sim 0.7\%$; matter-antimatter annihilation converts $100\%$.
06 · Relativistic Energy-Momentum Relation
Invariant Mass Dispersion
$$E^2 = (pc)^2 + (m_0 c^2)^2$$
- ①Physical Essence: The Minkowski norm of the four-momentum vector $P^\mu = (E/c, \mathbf{p})$ is a Lorentz invariant constant equal to $-m_0^2 c^2$.
- ②Mathematical Notation: $E$ is total energy, $p$ is relativistic spatial momentum ($p = \gamma m_0 v$), $m_0$ is invariant rest mass.
- ③Theoretical Significance: Seamlessly accommodates massless particles ($m_0 = 0 \implies E = pc$, photons) and stationary bodies ($p = 0 \implies E = m_0 c^2$).
- ④Boundary Conditions & Applications: Cornerstone for all high-energy collider kinematics (CERN, Fermilab) and the foundational starting point for the Dirac relativistic electron equation.
07 · Relativistic Velocity Addition
Collinear Velocity Composition
$$u = \frac{u' + v}{1 + \frac{u' v}{c^2}}$$
- ①Physical Essence: Combining velocities in special relativity asymptotically bounds the resultant velocity to strictly less than or equal to $c$, replacing Galilean addition ($u = u' + v$).
- ②Mathematical Notation: $u$ is velocity in frame $S$, $u'$ is velocity relative to frame $S'$, $v$ is relative speed of $S'$ with respect to $S$.
- ③Theoretical Significance: Guarantees that even if two bodies approach each other, each moving at $0.9c$, their relative closing speed cannot exceed light speed ($u \approx 0.9945c$).
- ④Boundary Conditions & Applications: Verified by the Fizeau water drag experiment; essential for calculating apparent superluminal jet motion in astrophysical quasars.
2. General Relativity (5 Items)
08 · Equivalence Principle
Einstein Equivalence Principle
$$m_{\text{inertial}} \equiv m_{\text{gravitational}}$$
- ①Physical Essence: The physical effects of a uniform gravitational field are locally completely indistinguishable from the effects of a linearly accelerated reference frame.
- ②Mathematical Notation: Equivalence of inertial mass (resistance to acceleration) and gravitational mass (coupling to gravity), confirmed to $10^{-15}$ precision by MICROSCOPE.
- ③Theoretical Significance: Transformed gravity from an external Newtonian force into an intrinsic geometric curvature of four-dimensional spacetime.
- ④Boundary Conditions & Applications: Valid locally within infinitesimally small spacetime domains (where tidal gradients vanish). Dictates free-fall weightlessness in orbiting spacecraft.
09 · Einstein Field Equations
Geometric Gravity Field Equations
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$
- ①Physical Essence: *"Spacetime tells matter how to move; matter tells spacetime how to curve."* (John Wheeler). Connects metric curvature tensors directly to the stress-energy tensor.
- ②Mathematical Notation: $R_{\mu\nu}$ is Ricci curvature tensor, $R$ is scalar curvature, $g_{\mu\nu}$ is metric tensor, $\Lambda$ is cosmological constant, $T_{\mu\nu}$ is stress-energy-momentum tensor.
- ③Theoretical Significance: The foundational master equations of modern gravitation, predicting gravitational waves, black holes, and cosmic expansion.
- ④Boundary Conditions & Applications: In weak fields and low velocities, exactly reduces to Poisson's equation $\nabla^2 \Phi = 4\pi G \rho$. Verified by LIGO gravitational wave detections and EHT black hole shadow imaging.
10 · Geodesic Equation
Equations of Motion in Curved Spacetime
$$\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{\mathrm{d}x^\alpha}{\mathrm{d}\tau}\frac{\mathrm{d}x^\beta}{\mathrm{d}\tau} = 0$$
- ①Physical Essence: Free particles under gravity follow the straightest possible paths (geodesics) through curved spacetime; gravity is not an external pull, but free motion through metric geometry.
- ②Mathematical Notation: $x^\mu(\tau)$ are spacetime coordinates parameterized by proper time $\tau$, $\Gamma^\mu_{\alpha\beta}$ are Christoffel symbols of the metric.
- ③Theoretical Significance: Replaces Newton's Second Law ($\mathbf{F} = m\mathbf{a}$) in curved spacetime, proving planetary orbits are natural force-free geodesics.
- ④Boundary Conditions & Applications: Explains the anomalous perihelion precession of Mercury (43 arcseconds per century) without invoking dark mass.
11 · Gravitational Time Dilation & Redshift
Clocks Run Slower in Gravity Wells
$$t(r) = t_\infty \sqrt{1 - \frac{r_s}{r}} \approx t_\infty \left(1 - \frac{GM}{c^2 r}\right)$$
- ①Physical Essence: Clocks deeper inside a gravitational potential well tick slower relative to distant observers; emitted photons lose energy climbing out, causing spectral redshift.
- ②Mathematical Notation: $t_\infty$ is time elapsed at infinite distance, $r_s = 2GM/c^2$ is Schwarzschild radius, $r$ is radial coordinate.
- ③Theoretical Significance: Verified by Pound-Rebka gamma-ray tower experiments (1959); demonstrates that time flow depends on local gravitational depth.
- ④Boundary Conditions & Applications: GPS satellites orbit at higher altitudes where gravity is weaker, causing their clocks to tick faster by $+45\ \mu\text{s/day}$ (combined with $-7\ \mu\text{s/day}$ special relativistic time dilation, requiring a $+38\ \mu\text{s/day}$ offset).
12 · Gravitational Deflection of Light
Gravitational Lensing Angle
$$\theta = \frac{4GM}{c^2 b} = \frac{2r_s}{b}$$
- ①Physical Essence: Photons traveling past massive bodies have their paths bent by curved spacetime; the deflection angle is exactly twice the value predicted by Newtonian corpuscular gravity.
- ②Mathematical Notation: $\theta$ is deflection angle (radians), $b$ is impact parameter distance of closest approach, $M$ is mass of deflecting body.
- ③Theoretical Significance: First confirmed by Arthur Eddington's 1919 solar eclipse expedition ($\theta \approx 1.75''$ at the Sun's limb), catapulting Einstein to global fame.
- ④Boundary Conditions & Applications: Modern astrophysics uses gravitational lensing (Einstein rings, arcs, microlensing) to map invisible dark matter halos across the universe.
3. Black Hole Physics (3 Items)
13 · Schwarzschild Radius
Event Horizon of a Static Black Hole
$$r_s = \frac{2GM}{c^2}$$
- ①Physical Essence: The critical radius at which escape velocity equals light speed; within this sphere, all light and matter are irrevocably trapped.
- ②Mathematical Notation: $r_s$ is event horizon radius, $G$ is gravitational constant, $M$ is black hole mass ($r_s \approx 2.95\text{ km} \times (M/M_\odot)$).
- ③Theoretical Significance: First exact vacuum solution of Einstein's field equations; marks an absolute one-way causal boundary in spacetime.
- ④Boundary Conditions & Applications: For Earth, $r_s \approx 8.87\text{ mm}$; for supermassive black hole Sagittarius A*, $r_s \approx 1.2 \times 10^7\text{ km}$. Imaged directly by the Event Horizon Telescope (EHT).
14 · Hawking Radiation Temperature
Black Hole Thermal Radiation
$$T_H = \frac{\hbar c^3}{8\pi G M k_B}$$
- ①Physical Essence: Quantum vacuum fluctuations near the event horizon result in black holes emitting a steady thermal blackbody spectrum, losing mass and eventually evaporating.
- ②Mathematical Notation: $T_H$ is Hawking temperature, $\hbar$ is reduced Planck constant, $k_B$ is Boltzmann constant, $M$ is black hole mass ($T_H \propto 1/M$).
- ③Theoretical Significance: The historic synthesis bridging General Relativity ($G, c$), Quantum Field Theory ($\hbar$), and Thermodynamics ($k_B$).
- ④Boundary Conditions & Applications: Stellar black holes are colder than the CMB ($T_H \approx 60\text{ nK}$), but primordial micro black holes radiate fiercely at high temperatures, triggering the Black Hole Information Paradox.
15 · Bekenstein-Hawking Black Hole Entropy
Horizon Area Law of Entropy
$$S_{\text{BH}} = \frac{k_B c^3 A}{4 G \hbar} = \frac{k_B A}{4 l_P^2}$$
- ①Physical Essence: The thermodynamic entropy of a black hole is strictly proportional to the surface area of its event horizon ($A$), not its enclosed volume.
- ②Mathematical Notation: $A = 4\pi r_s^2$ is horizon area, $l_P = \sqrt{\hbar G/c^3} \approx 1.6 \times 10^{-35}\text{ m}$ is the Planck length. Every Planck area encodes $1/4$ bit of entropy.
- ③Theoretical Significance: Gave birth to the **Holographic Principle** and AdS/CFT duality; demonstrates that fundamental quantum information lives on bounding horizons.
- ④Boundary Conditions & Applications: Establishes the Generalized Second Law of Thermodynamics: $S_{\text{total}} = S_{\text{matter}} + S_{\text{BH}} \ge 0$.
4. Physical Cosmology (5 Items)
16 · Hubble's Law
Hubble-Lemaître Law of Cosmic Expansion
$$v = H_0 d$$
- ①Physical Essence: Distant galaxies recede from us with apparent recession velocities directly proportional to their proper distance; space itself is uniformly expanding.
- ②Mathematical Notation: $v$ is recession velocity ($\text{km/s}$), $H_0 \approx 67 \sim 73\text{ km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$ is Hubble constant, $d$ is distance ($\text{Mpc}$).
- ③Theoretical Significance: Observational proof that the universe is dynamic, laying the foundation for Big Bang cosmology.
- ④Boundary Conditions & Applications: Valid at low redshifts ($z \ll 1$); current tension between early-universe CMB ($67.4$) and late-universe Cepheid ($73.0$) measurements remains an open cosmological mystery.
17 · Hubble Time & Age of the Universe
Characteristic Expansion Age
$$t_H = \frac{1}{H_0} \approx 13.8 \times 10^9 \text{ years}$$
- ①Physical Essence: The reciprocal of the Hubble constant yields the characteristic expansion timescale of the universe under linear extrapolation.
- ②Mathematical Notation: $t_H = 1/H_0$; when integrated with $\Lambda\text{CDM}$ matter/dark energy ratios, yields an exact age $t_0 \approx 13.787 \pm 0.020\text{ Gyr}$.
- ③Theoretical Significance: Establishes a definitive finite beginning for cosmic time, resolving Olbers' Paradox (why the night sky is dark).
- ④Boundary Conditions & Applications: Constrains stellar evolution models: no globular cluster or radioactive element in the universe can be older than $13.8\text{ Gyr}$.
18 · Critical Density of the Universe
Boundary of Spatial Geometry
$$\rho_c = \frac{3 H_0^2}{8\pi G} \approx 8.5 \sim 9.5 \times 10^{-27}\ \text{kg/m}^3$$
- ①Physical Essence: The critical matter-energy density required for the universe to be spatially flat ($k = 0$). Represents only $\sim 5$ hydrogen atoms per cubic meter.
- ②Mathematical Notation: $\Omega \equiv \rho / \rho_c$; $\Omega > 1$ represents a closed spherical universe, $\Omega < 1$ is open hyperbolic, and $\Omega = 1$ is flat Euclidean.
- ③Theoretical Significance: Dictates cosmic geometric topology and ultimate fate. Planck satellite CMB measurements confirm $\Omega_K = 1 - \Omega \approx 0.0007 \pm 0.0019$ (strictly flat).
- ④Boundary Conditions & Applications: Decomposed in standard $\Lambda\text{CDM}$ cosmology into: baryonic matter $\Omega_b \approx 5\%$, cold dark matter $\Omega_c \approx 27\%$, and dark energy $\Omega_\Lambda \approx 68\%$.
19 · Friedmann Equations
Dynamic Evolution of Cosmic Scale Factor
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}$$
- ①Physical Essence: Governs the expansion rate of a homogeneous, isotropic universe (FLRW metric) driven by matter, radiation, curvature, and dark energy.
- ②Mathematical Notation: $a(t)$ is dimensionless scale factor, $\dot{a}/a \equiv H(t)$ is time-dependent Hubble parameter, $\rho$ is energy density, $k \in \{-1, 0, +1\}$ is spatial curvature.
- ③Theoretical Significance: The master dynamic equation of modern physical cosmology, describing radiation, matter, and dark energy dominated eras.
- ④Boundary Conditions & Applications: Explains cosmic inflation ($a \propto e^{Ht}$) and the current late-time accelerated expansion of the universe (Perlmutter, Schmidt, Riess 2011 Nobel Prize).
20 · Cosmic Microwave Background (CMB)
Primordial Relic Radiation
$$T_{\text{CMB}} = 2.72548 \pm 0.00057\ \text{K}$$
- ①Physical Essence: Relic electromagnetic radiation from 380,000 years after the Big Bang (recombination epoch), redshifted over 13.8 billion years to a cold microwave blackbody bath.
- ②Mathematical Notation: $T \approx 2.725\text{ K}$; exhibits an extraordinary thermal Planck spectrum with temperature fluctuations of only $\sim 10^{-5}$ ($\mu\text{K}$ level).
- ③Theoretical Significance: The definitive empirical confirmation of Big Bang cosmology. Temperature anisotropies map the primordial quantum seeds of galaxies and cosmic web structures.
- ④Boundary Conditions & Applications: Mapped in high precision by COBE, WMAP, and the Planck satellite to extract the 6 fundamental parameters of $\Lambda\text{CDM}$ cosmology.
5. Physical Quantities, Symbols & SI Units
| Physical Quantity | Symbol | SI Base or Derived Unit | Common Astrophysical Unit | Value / Equivalence |
|---|---|---|---|---|
| Speed of Light in Vacuum | $c$ | Meter per second ($\text{m/s}$) | Natural lightspeed scale ($c \equiv 1$) | $c \equiv 299,792,458\ \text{m/s}$ (Exact) |
| Gravitational Constant | $G$ | $\text{m}^3/(\text{kg}\cdot\text{s}^2)$ | Natural gravity scale | $G \approx 6.6743 \times 10^{-11}\ \text{m}^3/(\text{kg}\cdot\text{s}^2)$ |
| Astronomical Unit | $\text{AU}$ | Meter ($\text{m}$) | Mean Earth-Sun distance | $1\text{ AU} \equiv 1.496 \times 10^{11}\ \text{m}$ |
| Light-Year | $\text{ly}$ | Meter ($\text{m}$) | Distance light travels in one year | $1\text{ ly} \approx 9.461 \times 10^{15}\ \text{m} \approx 63,241\text{ AU}$ |
| Parsec | $\text{pc}, \text{Mpc}$ | Meter ($\text{m}$) | Parallax distance of 1 arcsec | $1\text{ pc} \approx 3.086 \times 10^{16}\ \text{m} \approx 3.262\text{ ly}$; $1\text{ Mpc} = 10^6\text{ pc}$ |
| Planck Length | $l_P$ | Meter ($\text{m}$) | $\sqrt{\hbar G / c^3}$ | $l_P \approx 1.616 \times 10^{-35}\ \text{m}$ (Spacetime quantum foam scale) |
| Planck Time | $t_P$ | Second ($\text{s}$) | $l_P / c$ | $t_P \approx 5.391 \times 10^{-44}\ \text{s}$ (Shortest physically meaningful time) |
| Planck Mass | $m_P$ | Kilogram ($\text{kg}$) | $\sqrt{\hbar c / G}$ | $m_P \approx 2.176 \times 10^{-8}\ \text{kg} \approx 1.22 \times 10^{19}\ \text{GeV}/c^2$ |
| Hubble Constant | $H_0$ | $\text{s}^{-1}$ | $\text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$ | $H_0 \approx 67.4 \sim 73.0\ \text{km/s/Mpc}$ |
| Cosmic Critical Density | $\rho_c$ | $\text{kg/m}^3$ | $3H_0^2 / (8\pi G)$ | $\rho_c \approx 9 \times 10^{-27}\ \text{kg/m}^3$ ($\sim 5\text{ protons/m}^3$) |
| CMB Temperature | $T_{\text{CMB}}$ | Kelvin ($\text{K}$) | Relic blackbody temperature | $T_{\text{CMB}} \approx 2.7255\ \text{K}$ |