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 ↓


Lorentz transformations


In the field of modern physics, the theory of relativity is one of the most revolutionary concepts. This concept fundamentally changed our understanding of the universe and introduced the world to the fascinating idea of Lorentz transformations. Originally formulated by Dutch physicist Hendrik Lorentz in the late 19th century, these transformations describe how measurements of time and space are changed for observers in different inertial frames of reference.

Understanding inertial reference frames

Before delving deeper into Lorentz transformations, it is essential to understand the concept of inertial reference frame. In simple terms, an inertial reference frame is a frame of reference in which an object either remains at rest or moves at a constant velocity unless a force is applied to it. According to Newton's first law of motion, such frames are non-accelerating.

Background of relativity

Special relativity, proposed by Albert Einstein in 1905, fundamentally changed the way we look at time and space. This theory is based on two assumptions:

  1. The laws of physics are the same in all inertial reference frames.
  2. The speed of light in a vacuum is the same for all observers, regardless of the speed of the light source or the observer.

These theories led to surprising results, especially with regard to time (known as time dilation), length (known as length contraction) and simultaneity.

Introduction to Lorentz transformations

Lorentz transformations mathematically express how measurements of time and space by two observers are related when they are moving at a constant velocity relative to each other. The transformations are important to ensure that the speed of light remains constant in all inertial frames.

Consider two inertial reference frames, ( S ) and ( S' ). Let ( S' ) move along the x-axis with velocity ( v ) relative to ( S ). The Lorentz transformation equations are given by:

        ( t' = gamma (t - frac{vx}{c^2}) )
        ( x' = gamma (x - vt) )
        ( y' = y )
        ( z' = z )

Here, ( gamma ) (gamma) is the Lorentz factor, which is defined as:

        ( gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}} )

Where ( c ) is the speed of light in vacuum.

Visual example: Understanding coordinate transformation

(O) (X') (Y')

This SVG example shows the basic setup for a Lorentz transformation, where the primed frame is rotating at a constant velocity along the (x)-axis with respect to the origin.

Time extension

One of the most interesting consequences of Lorentz transformations is time dilation. Time dilation implies that time passes at different rates for observers in relative motion. An often-cited example of time dilation is the "twin paradox."

Twin paradox: Imagine there are two twins, Alice and Bob. Alice boards a spaceship and travels at a significant fraction of the speed of light, while Bob stays on Earth. When Alice returns, she discovers that less time has passed for her than for Bob. Mathematically, if ( Delta t_0 ) is the proper time interval (time measured by a stationary observer relative to the event), and ( Delta t ) is the time interval measured by a moving observer, then:

        ( Delta t = gamma Delta t_0 )

What this information means is that as the velocity of an observer increases, the time measured by that observer relative to a stationary observer slows down.

Length contraction

The Lorentz transformation also gives rise to the phenomenon of length contraction, according to which the length of an object as measured by an observer in motion relative to the object is shorter than the length as measured by an observer at rest relative to the object.

If ( L_0 ) is the proper length (the length measured by an observer stationary relative to the object), and ( L ) is the length measured by a moving observer, then:

        ( L = frac{L_0}{gamma} )

Therefore, from the perspective of a stationary observer, a moving object appears smaller in the direction of motion.

Rethinking synchronicity

An essential consequence of the Lorentz transformations is the relative nature of simultaneity. In classical mechanics, if two events happen at the same time, they are considered simultaneous regardless of the observer. However, in the framework of special relativity, simultaneity becomes relative - two observers moving relative to each other cannot agree on whether two events are simultaneous or not.

Example: Lightning strike

Suppose two lightning strikes occur simultaneously at points (A) and (B) on the railway track. An observer standing on the embankment (observer (O)) sees both strikes at the same time. Another observer on a train moving relative to the embankment may see the lightning strike at (A) before (B) or vice versa, depending on the direction of motion, thus showing that simultaneity is frame dependent.

Lorentz transformation in electrodynamics

Lorentz transformations extend to the field of electrodynamics, where they serve as the backbone for the unification of electricity and magnetism. James Clerk Maxwell unified these fields by developing Maxwell's equations, which predicted the existence of electromagnetic waves propagating at the speed of light. An important implication of special relativity is that these equations are true in all inertial frames.

The covariance of Maxwell's equations under Lorentz transformations means that the form of these equations is invariant, demonstrating the beauty and consistency of the laws of physics across different frameworks.

Applications and implications

Lorentz transformations have very deep technical and philosophical implications. They are fundamental in technologies such as the Global Positioning System (GPS), where accounting for relativistic effects is crucial to accuracy.

Philosophically, Lorentz transformations challenge our intuitive understanding of time and space, and demand an understanding of the universe that goes beyond classical mechanics.

Conclusion

In short, Lorentz transformations stand as a cornerstone of modern physics. By adding to the understanding of the relationship between time and space for moving observers, they embody the spirit of special relativity. From the perceived slowing of clocks to the shrinking of moving objects, they force a reconsideration of fundamental concepts. As one enters regions where speeds approach that of light, the insights offered by Lorentz transformations become indispensable in navigating this quantum landscape.


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Lorentz transformations

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