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 ↓


Minkowski spacetime


Minkowski spacetime provides a profound framework for understanding the nature of space and time in the field of special relativity. Introduced by Hermann Minkowski, this concept unifies space and time into a four-dimensional continuum that accurately describes the physics of high-speed phenomena, especially phenomena approaching the speed of light. It also provides important insights into the behavior of electric and magnetic fields described by Maxwell's equations in electrodynamics.

Understanding the basics: what is Minkowski spacetime?

In classical physics, space and time were traditionally viewed as separate entities. Space has three dimensions, while time is a separate, universal scale. However, Albert Einstein's special theory of relativity challenged these assumptions by demonstrating that space and time are fundamentally intertwined. Minkowski spacetime represents this merging of three-dimensional space and one-dimensional time into a single four-dimensional manifold.

Four dimensional universe

In Minkowski spacetime, any point is described by a set of four coordinates: three for space (x, y, and z) and one for time (t). This four-tuple, often denoted as (x, y, z, ct), where c is the speed of light, is referred to as an event.

(x, y, z, ct)

Visualization of Minkowski spacetime

whistle X Event A

The above diagram represents a simplified view, showing a 2D slice of 4D Minkowski spacetime, focusing on only one spatial dimension and time. Points such as event A represent specific coordinates in this spacetime. The vertical axis represents time (ct) while the horizontal axis represents location (x).

Interval: invariant measurement

A central concept in Minkowski spacetime is the invariant spacetime interval. While distances in space and time may change under different frames of reference, the spacetime interval between two events remains constant. Mathematically, the interval s is defined as:

s² = (ct₂ – ct₁)² – ((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²)

This equation takes a deeper form. Positive s² defines time-like separation, implying a cause-effect relationship between events. Negative s² indicates space-like separation, suggesting no possible causal relationship since they require more than the speed of light to be connected. Zero interval corresponds to light-like or zero separation curated by photons.

Illustrative text example

To understand this better, imagine two events: pressing a light switch and the bulb lighting up. The interval between them is like time - there is a definite cause-effect sequence. Compare this to two simultaneously shining stars seen from Earth, where the interval is like space: one cannot affect the other because of the vast distance.

Minkowski diagram

Minkowski diagrams help to better understand relative motion and time in spacetime. These diagrams typically combine a space and a time dimension on orthogonal axes. Worldlines representing the trajectories of objects over time show how different observers see the same event depending on their frame.

whistle X Worldline

Consider the diagram above. The tilted red line represents the world line, which depicts an object's travel through spacetime. The slope of this line relates to speed; steeper lines indicate slower speeds, while the 45-degree line corresponds to the speed of light.

Special relativity in electrodynamics

Maxwell's equations describe the fundamental interaction of electromagnetism. However, their formulation in purely 3-dimensional space does not naturally incorporate relativistic effects. Minkowski spacetime provides a natural framework for expanding these equations in a manner consistent with special relativity theories.

Reformulating Maxwell's equations in Minkowski spacetime involves introducing the electromagnetic field tensor, a rank 2 tensor that beautifully encapsulates electric and magnetic fields in a relativistically invariant form. The tensor, denoted as Fμν, captures electric fields as time-space components and magnetic fields as spatial components:

F = | 0 -E_x -E_y -E_z |
        | E_x 0 -B_z B_y |
        | E_y B_z 0 -B_x |
        | E_z -B_y B_x 0 |

Here, Ex, Ey and Ez represent the components of the electric field, while Bx, By and Bz represent the components of the magnetic field. These components are naturally combined into a single tensor.

Invariance and covariance in electrodynamics

The main strength of using Minkowski spacetime in electrodynamics is the ability to express physical laws in covariant forms. This means that physical laws remain unchanged under Lorentz transformations, ensuring that they are true in any inertial reference frame.

For example, the Lorentz force law, which describes the force exerted on a charged particle, can be effectively expressed in terms of the electromagnetic field tensor. The force F acting on a particle of charge q moving with velocity v in an electromagnetic field is given by:

Fμ = qFμνUν

In this form, the velocity is expressed as a four-velocity Uν, which includes time-like and space-like components, ensuring that force calculations remain consistent in different inertial frames.

Worldlines and causality in electrodynamics

The merging of spacetime into a four-dimensional construct enables a specific understanding of causality in electromagnetics. The interactions between charges and fields, represented in the world lines of charged particles, incorporate the invariant nature of the speed of light.

whistle X Charge Path Field Path

In the diagram, the blue line represents the path of the charged particle, while the green line represents the field propagation. The causal nature is highlighted: any interaction within the space-time determined by field effects propagates no faster than the speed of light, which is underlined by the predicted structure of the geometry.

Conclusion

Minkowski spacetime remains central to modern physics, offering a consistent view of space and time. It beautifully accommodates the demands of special relativity, allowing a comprehensive exploration of electromagnetism while maintaining relativistic consistency through covariant formulas. From ensuring invariant gaps to providing visual tools such as Minkowski diagrams, its influence extends across diverse physics fields, deepening the understanding of our universe at higher dimensions.


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Minkowski spacetime

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