Skip to content

Relativity

1. The Speed of Light & Ether Hypothesis

Historical Measurements of \(c\) (\(c = \frac{\Delta L}{\Delta t}\))

  • Galileo (1638): Attempted to measure \(c\) using lanterns over a distance \(L\), finding the time delay practically zero for \(L \le\) 1 mile.

  • Based on modern values, Galileo's expected delay was \(\approx 10^{-5}\) s, which is too fast for human reaction times.

  • Ole Rømer (1644–1710): Studied Jupiter's moon Io (orbital period \(\approx\) 42h) and proved \(c\) is finite by observing eclipse periods shorten or lengthen due to the Doppler effect as Earth moved.

  • James Bradley (1729): Used stellar aberration (\(\tan\theta = \frac{v}{c}\)) to calculate \(c \approx 3.01 \times 10^8\) m/s, yielding a 0.4% error.

  • Léon Foucault (1862): Bounced light off a rotating mirror over distance \(h\) and used \(c = \frac{2\omega h}{\theta}\) to find \(c \approx 2.98 \times 10^8\) m/s (0.6% error).

  • Albert Michelson (1926): Improved Foucault's apparatus with a larger distance (\(h \approx\) 35 km) to refine \(c\) to \(2.99796 \times 10^8\) m/s, dropping the error to 0.001%.

The Ether Paradox

  • Visible light is an electromagnetic wave with wavelengths from 400 nm (Violet) to 700 nm (Red) and frequencies from \(7.5 \times 10^{14}\) Hz to \(4.3 \times 10^{14}\) Hz.

  • The "luminiferous aether" was a hypothesized space-filling medium for light, but a transverse wave traveling at \(c\) would require an impossibly hard medium, contradicting planetary motion.

  • Fizeau Experiment (1851): Measured light in moving water and found a partial ether drag factor of \((1 - \frac{1}{n^2}) \approx\) 0.44.

  • Michelson-Morley Experiment (1887): Failed to detect an ether wind (expected 0.4 fringes drift, yielded \(\le\) 0.01 fringes), implying complete ether drag.

  • The direct contradiction between Fizeau (partial drag) and Michelson-Morley (complete drag) proved the ether hypothesis was self-contradicting.

2. Special Relativity: Kinematics

Einstein's Postulates & Lorentz Transformation

  • Alväger et al. (1964) proved that photons emitted from Pion particles moving at \(v \sim 0.99c\) still travel at precisely \(c\).

  • Relativity Principle: Physical laws are identical in all inertial frames, rendering a static ether unnecessary.

  • Invariance of \(c\): The speed of light is identical in all inertial frames, regardless of relative motion.

  • Standard Transformation from track frame \(K\) to moving train frame \(K'\) uses factor \(\gamma = \frac{1}{\sqrt{1-\beta^2}}\) where \(\beta = v/c\).

  • Position transformations: \(x' = \frac{x - vt}{\sqrt{1 - v^2/c^2}}\), \(y' = y\), \(z' = z\).

  • Time transformation: \(t' = \frac{t - \frac{v}{c^2}x}{\sqrt{1 - v^2/c^2}}\). \(t' = \gamma(t - \frac{v}{c^2}x)\), \(t = \gamma(t' + \frac{v}{c^2}x')\)

  • Relativistic Velocity Addition generalizes to \(w = \frac{u + v}{1 + \frac{uv}{c^2}}\), proving that if \(u=c\), the combined speed \(w\) remains strictly \(c\).

  • \(\Phi = kx-\omega t\) is constant in any inertial frames

Time Dilation & Length Contraction

  • Relativity of Simultaneity: Events simultaneous in one frame project to different times in another frame via \(T = \frac{v}{c^2}D\), where \(D\) is spatial difference.

  • Proper Length (\(L_0\)): Length measured at rest; moving objects experience Length Contraction (\(L = \gamma^{-1} L_0\)).

  • Proper Time (\(\Delta t_0\)): Time measured by a single clock at rest; moving clocks experience Time Dilation (\(\Delta t = \gamma \Delta t_0\)).

  • Muon Decay Application: Muons traveling at \(v = 0.98c\) have a proper half-life of \(1.52\ \mu\text{s}\).

  • To an Earth observer, the muon's time dilates (\(t_{1/2} \approx 7.6\ \mu\text{s}\)); to the muon, the atmosphere's length contracts (from 2000 m to \(\approx\) 398 m).

  • Both frames mathematically agree on a 54% survival rate reaching the surface.

Minkowski Space & The Invariant Interval

  • The spacetime interval is mathematically invariant across all frames: \(s^2 \equiv (ct)^2 - x^2 - y^2 - z^2\).

  • Timelike Separation (\(s^2 > 0\)): Events inside the light cone connectable by sub-light travel, defining Proper Time (\(\Delta t_0 = s/c\)).

  • Spacelike Separation (\(s^2 < 0\)): Events outside the light cone, cannot influence each other, defining Proper Length (\(L_0 = \sqrt{-s^2}\)).

  • Lightlike Separation (\(s^2 = 0\)): Events exactly on the light cone, connected only by massless particles.

3. Special Relativity: Dynamics

Energy, Momentum & Mass Equivalence

  • Relativistic Momentum: \(\vec{p} = \frac{m_0}{\sqrt{1 - v^2/c^2}} \vec{v}\) modifies classical mechanics to preserve momentum conservation in all frames.

  • Newton's second law (\(\vec{F} = m\vec{a}\)) is invalid; Force must be defined as \(\vec{F} = \frac{d\vec{p}}{dt}\).

  • Rest Energy: \(E_0 = m_0 c^2\).

  • Total Energy: \(E = \frac{m_0}{\sqrt{1 - (v/c)^2}} c^2\).

  • Kinetic Energy: \(E_k = m c^2 - m_0 c^2\).

  • Energy and Momentum unify into a 4-Vector with an invariant magnitude: \(\left( \frac{E}{c} \right)^2 - p^2 = m_0^2 c^2\) / \(E^2 = (pc)^2 + (m_0 c^2)^2\).

  • \(v = \frac{p c^2}{E}\)

  • Mass-Energy Equivalence (\(E = mc^2\)): Confirmed by Uranium-235 fission, where a mass defect of \(\approx 3 \times 10^{-28}\) kg releases \(\approx 1.6 \times 10^{13}\) J per mole.

The Twin Paradox

  • Scenario: Twin A stays on Earth; Twin B travels to \(\alpha\)-Centauri at \(v = 0.8c\).

  • Resolution via Frame Shifts: Twin B must accelerate to turn around, instantaneously jumping inertial frames, which shifts the line of simultaneity and accounts for Twin A's rapid aging.

  • Resolution via Doppler Effect: Using \(f = \sqrt{\frac{1 \mp v/c}{1 \pm v/c}} f_0\), both twins count physical light signals emitted yearly.

  • Twin B receives signals for 3 years redshifted and 3 years blueshifted (totaling 6 years).

  • Twin A (Earth) receives redshifted signals for 9 years and blueshifted for 1 year (totaling 10 years), providing perfect observational agreement without paradox.

4. General Relativity

Core Principles & The Equivalence Principle

  • General Relativity defines gravity not as a force, but as the curvature of space and time.

  • Principle of Equivalence: Gravitational mass (\(m_g\)) is strictly equal to Inertial mass (\(m_i\)).

  • An arbitrary gravitational field is physically equivalent to an accelerated reference frame at any local space-time point.

.

Predictions & Evidence

  • Light Deflection: Light traveling through an accelerating elevator (or gravitational field) bends downward.

  • Starlight grazing the Sun deflects by \(\Delta\theta = \frac{4GM_{\odot}}{R_{\odot}}\), which equals 1.75'' arcseconds and was confirmed by Eddington in 1919.

  • Gravitational Time Dilation: Clocks deeper in a gravitational potential \(\varphi\) run slower according to \(\Delta t = \frac{\Delta t_0}{\sqrt{1 + 2\varphi/c^2}}\).

  • GPS Navigation: Requires constant correction because satellite clocks lose 7 μs/day (Special Relativity velocity effect) but gain 46 μs/day (General Relativity weak gravity effect).

  • Gravitational Redshift: Photons expend energy to escape massive objects, increasing their period and decreasing their frequency \(\nu\).

  • Spacetime curvature governs matter, evidenced by light following null geodesics, Mercury's perihelion precession, and LIGO's 2016 detection of Gravitational Waves.

Thermal Physics

1. Fundamental Thermodynamic Concepts & Temperature

  • Thermodynamic Systems: Macroscopic bodies separated from surroundings; classified as Open (exchanges energy and matter), Closed (exchanges only energy), or Isolated (exchanges neither).

  • Thermal Equilibrium: A state where macroscopic properties do not change with time, though microscopic properties remain dynamic.

  • The Zeroth Law: If systems A and B are each in thermal equilibrium with system C, they are in thermal equilibrium with each other.

  • Absolute Temperature: Gases at constant volume extrapolate to zero pressure at -273.15°C. The Kelvin scale conversion is \(T^{(C)} = T^{(K)} - 273.15\).

  • Thermal Expansion: Linear expansion is \(\Delta L = \alpha L_0 \Delta T\) and volume expansion is \(\Delta V = \beta V_0 \Delta T\), where \(\beta = 3\alpha\).

  • Water's Expansion Anomaly: Between 0°C and 4°C, water has \(\beta < 0\), meaning it contracts when heated and reaches maximum density at 4°C.

2. Heat & Transport Phenomena

  • Heat Capacity (\(C\)): Dictates temperature change upon absorbing heat: \(C = \frac{\Delta Q}{\Delta T}\).

  • Latent Heat (\(L\)): Heat required for a phase transition where temperature remains constant, calculated as \(Q = LM\).

  • Fourier's Law of Heat Conduction: Heat current \(j_q = -k \frac{dT}{dx}\), where \(k\) is thermal conductivity and the negative sign indicates heat flows opposite to the temperature gradient.

  • Mean Free Path (\(l\)): Average distance traveled between molecular collisions: \(l = \frac{1}{\sqrt{2} \pi d^2 (N/V)}\) or \(l = \frac{k_B T}{\sqrt{2} \pi d^2 p}\) for ideal gases.

  • Microscopic Thermal Conductivity: Derived as \(k = \frac{1}{3} \rho \overline{v} l c_v\).

3. Kinetic Theory of Gases

  • Ideal Gas Law: \(PV = nRT\) or \(PV = N k_B T\).

  • Van der Waals Equation (Real Gas): \((P + \frac{aN^2}{V^2})(V - Nb) = N k_B T\), where \(a\) corrects for intermolecular forces and \(b\) corrects for molecular volume.

  • Kinetic Derivation of Pressure: \(P = \frac{n N_A m}{3V} \overline{v^2}\).

  • Equipartition Theorem: The average kinetic energy per degree of freedom is \(\frac{1}{2}k_B T\) per molecule. Degrees of freedom (\(f\)) equals \(t + r + 2s\) (translational + rotational + 2 \(\times\) vibrational).

  • Internal Energy (Monoatomic Ideal Gas): \(U = \frac{3}{2} n R T\).

  • Molar Heat Capacities: Constant volume \(C_V = \frac{1}{n}\left(\frac{dU}{dT}\right)_V\) and constant pressure \(C_P = C_V + R\) (Mayer's relation).

  • Maxwell Velocity Distribution characteristic speeds: Order is \(v_p < \overline{v} < v_{rms}\).

  • Most Probable Speed (\(v_p\)): \(v_p = \sqrt{\frac{2 k_B T}{m}}\).

  • Average Speed (\(\overline{v}\)): \(\overline{v} = \sqrt{\frac{8 k_B T}{\pi m}}\).

  • Root-Mean-Square Speed (\(v_{rms}\)): \(v_{rms} = \sqrt{\frac{3 k_B T}{m}}\).

4. The First Law of Thermodynamics

  • First Law Equation: \(\Delta U = \Delta Q - W\), or in differential form, \(dU = \delta Q - \delta W\).

  • Work Done (\(W\)): \(W = \int P dV\), representing the path-dependent area under a P-V curve.

  • Quasi-static Processes: Isobaric (constant \(P\)), Isochoric (constant \(V\)), Isothermal (constant \(T\)), and Adiabatic (\(Q = 0\)).

  • Adiabatic Process Relation: \(P V^\gamma = \text{constant}\), where \(\gamma = \frac{C_P}{C_V}\). \(\Rightarrow\) \(T V^{\gamma-1} = \text{constant}\) for ideal gases.

  • Heat Engine Efficiency: \(\varepsilon = 1 - \frac{Q_L}{Q_H}\).

  • Otto Cycle (Petrol): Uses isochoric heat absorption/rejection. Efficiency is \(\varepsilon_{\text{Otto}} = 1 - \left(\frac{V_2}{V_1}\right)^{\gamma-1}\).

  • Diesel Cycle: Uses isobaric heat absorption and isochoric heat rejection.

5. The Second Law of Thermodynamics

  • Carnot Cycle: Reversible cycle consisting of two isothermal and two adiabatic processes.

  • Carnot Efficiency: \(\varepsilon_{\text{Carnot}} = 1 - \frac{T_L}{T_H}\).

  • Refrigerator (Reverse Carnot) COP: \(\varepsilon^{(r)} = \frac{Q_L}{W} = \frac{T_L}{T_H - T_L}\).

  • Clausius Statement: Heat cannot spontaneously flow from cold to hot without external work.

  • Kelvin-Planck Statement: A cyclic heat engine cannot completely convert absorbed heat into work without rejecting some heat.

  • Carnot's Theorem: No irreversible engine can be more efficient than a reversible (Carnot) engine operating between the same reservoirs (\(\varepsilon_I \leq \varepsilon_{\text{Carnot}}\)).

6. Entropy & The Third Law

  • Clausius Inequality: \(\oint \frac{\delta Q}{T} \leq 0\), where equality holds only for reversible processes.

  • Entropy Definition (\(S\)): A state function defined by the exact differential \(dS = \frac{\delta Q}{T}\) for a reversible path.

  • Second Law (Entropy Statement): For any isolated system, \(\Delta S \geq 0\) (entropy never decreases).

  • Ideal Gas Entropy Change: \(S = S_0 + n C_V \ln\frac{T}{T_0} + nR \ln\frac{V}{V_0}\).

  • T-S Diagrams: Area under a T-S curve represents the total heat transferred (\(Q = \int T dS\)).

  • Third Law (Nernst Statement): As \(T \to 0\), the change in entropy for an isothermal process approaches zero (\(\lim_{T \to 0} \Delta S = 0\)).

  • Unattainability Principle: It is impossible to reduce a system's temperature to absolute zero in a finite number of steps.

  • Boltzmann's Entropy (Statistical): \(S = k_B \ln W\), where \(W\) is multiplicity (number of microscopic configurations).

  • Stirling's Approximation: For large \(N\), \(\ln N! \approx N \ln N - N\), used to prove macroscopic entropy from statistical mechanics.