PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Electrodynamics ↓


Special relativity


Special relativity, introduced by Albert Einstein in 1905, revolutionized the way physicists thought about motion and electromagnetism. It arose from the need to reconcile the laws of classical mechanics with the invariability of the speed of light, a fundamental principle of electromagnetism described by Maxwell's equations. In this comprehensive lesson, we will explore the basic principles of special relativity and its profound implications on electrodynamics.

Fundamentals of special relativity

The theory of special relativity is based on two major principles:

  1. Relativity Theory: The laws of physics are the same in all inertial reference systems. This means that no physical experiment can tell whether one is at rest or moving at a constant velocity.
  2. Invariability of the speed of light: the speed of light in a vacuum is the same for all observers, regardless of their speed or the speed of the light source.

Lorentz transformations

To understand how measurements of space and time differ for observers in relative motion, we need to use the Lorentz transformation. This set of equations relates the coordinates of an event measured in one inertial frame to the coordinates measured in another.

x' = γ(x - vt)
t' = γ(t - vx/c²)
y' = y
z' = z where γ = 1/√(1 - v²/c²)

Here, x, y, z and t represent spatial and temporal coordinates in one frame, while x', y', z' and t' are coordinates in a second frame moving with velocity v relative to the first. c is the speed of light, and γ (gamma) is the Lorentz factor.

Visual representation

Consider two reference frames: S and S'. Frame S' moves along the x-axis with a constant velocity v relative to frame S. The following visualization helps to illustrate this relationship:

X Y S S'

This diagram shows two points in space, one in frame S (red) and the other in frame S' (blue). As the frame S' moves forward, the position of an object relative to each frame changes according to the Lorentz transformation.

Implications for electrodynamics

Special relativity theory has important implications for electromagnetism. One of its main effects is that electric and magnetic fields transform into each other when observed from different inertial frames.

Conversion of the E and B regions

The electric field E and magnetic field B measured in one frame can be related to the fields measured in another moving frame by the following transformations:

E'_x = E_x
E'_y = γ(E_y - vB_z)
E'_z = γ(E_z + vB_y)
B'_x = B_x
B'_y = γ(B_y + (v/c²)E_z)
B'_z = γ(B_z - (v/c²)E_y)

These equations show how the electric and magnetic field components mix when moving from one inertial frame to another. Let's look at an example to make this clear.

Example: moving charge and wire

Imagine that an observer in frame S looks at a long straight wire carrying a steady current and a charged particle moving parallel to the wire with velocity v. In frame S, the particle experiences a magnetic force due to the current. However, in frame S', moving with the particle, there is no motion relative to the wire, and thus no magnetic force. Instead, an electric field is produced in frame S', which exerts a force on the charge.

This change in perception from frame S to frame S' shows the relativity of electric and magnetic fields: they are different aspects of a unified electromagnetic field that depends on the motion of the observer.

The concept of synchronicity

An important realization from special relativity is that simultaneity is relative. Events that occur simultaneously in one frame may not be simultaneous in another. Let's use an example to understand this notion further.

Example: train and platform

Consider a train moving at a relative speed on a track with two platforms, each of which is equipped with a flash of light equidistant from an observer standing at the centre of the train. Suppose that these flashes occur simultaneously in the frame of the observer on the train. For an observer standing on one of the platforms, due to the motion of the train, the flash closer to the direction of the train's motion is seen first.

This example demonstrates how simultaneity can depend on the observer's frame of reference, a fundamental departure from classical concepts of time.

Length contraction

Another fascinating result of special relativity is length contraction: objects moving at relativistic speeds are measured to be shorter in the direction of motion by a stationary observer. The compressed length L' is given by the formula:

L' = L√(1 - v²/c²)

where L is the proper length measured in the rest frame of the object. This phenomenon has been confirmed by many experiments and is a cornerstone of relativistic physics.

Example: muon decay

Muons, subatomic particles created high in the Earth's atmosphere, provide compelling evidence for time dilation and length contraction. They travel towards Earth at close to the speed of light and have a short lifetime. In the frame of the muon, the distance traveled is small due to length contraction, which allows the detection of more muons at the Earth's surface than expected from non-relativistic calculations. This experimental observation matches perfectly with the predictions of special relativity.

Time extension

Time dilation refers to the fact that time as measured by an observer moving relative to the clock is stretched or stretched. The time interval Δt' between two events as measured in the frame of a moving observer is longer than the time interval Δt between the same events as measured by an observer stationary relative to him.

Δt' = Δt/√(1 - v²/c²)

This formula shows that, from the point of view of a stationary observer, time moves slower in moving clocks than in stationary clocks.

Example: The twin paradox

The twin paradox is a famous thought experiment in relativity. It involves two twins - one stays on Earth, and the other travels on a spaceship at a significant fraction of the speed of light and then returns. When the traveler returns, they will discover that they are younger than their twin sibling who stayed on Earth. This seemingly paradoxical result is a consequence of time dilation and can be resolved entirely within the framework of special relativity without any paradox.

Conclusion

Special relativity has completely transformed our understanding of the universe by emphasizing that space and time are interconnected and relative to the observer's frame of motion. Its implications extend deep into the field of electrodynamics, where it beautifully unifies electric and magnetic fields and explains their transformation in different frames. As a theory, special relativity passes every experimental test and underpins much of modern physics, laying the groundwork for further explorations into the nature of space, time, and matter.


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Special relativity

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