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 ↓


Relativistic energy and momentum


In the field of modern physics, Einstein's theory of special relativity revolutionized our understanding of the fundamental concepts of energy and momentum. These two quantities, which are conserved in classical mechanics, have fascinating forms and structures when examined from a relativistic perspective. Special relativity delves deeply into how objects behave at high speeds close to the speed of light. At such velocities, conventional Newtonian mechanics is no longer adequate, and we enter the fascinating world of relativistic physics.

Understanding classical momentum and energy

Let us first consider the classical approach as a stepping stone to smoothly transition to the theory of relativity. In classical mechanics, the momentum of an object is given by the product of its mass and velocity.

p = mv

Here, p is momentum, m is mass, and v is velocity. Classical energy, specifically kinetic energy (KE), is expressed as:

KE = 1/2 * mv²

These equations break down at velocities approaching the speed of light (c), which underlines the limitations of classical mechanics.

Relativistic momentum

In special relativity, a new formulation for momentum is needed. The relativistic momentum of an object is given by:

p = γmv

where γ (gamma) is the Lorentz factor, defined as:

γ = 1 / √(1 - v²/c²)

As the velocity of an object approaches the speed of light, the value of γ increases, leading to a substantial increase in speed compared to what would be classically expected. This change has profound implications for understanding how objects behave at relativistic speeds.

Relativistic energy

The concept of energy in relativity combines kinetic and rest energy into a single framework called total energy, which is expressed as:

E = γmc²

The component mc² is known as rest energy, which represents the energy an object has simply by virtue of having mass. The relationship between rest energy and total energy underlies a basic principle: mass itself is a highly concentrated form of energy.

Rest energy perspective

To further explore rest energy, imagine an object at rest (v = 0), so that γ = 1. Thus, the energy simplifies to:

E = mc²

This formula, which is one of the most recognised equations worldwide, demonstrates mass-energy equivalence, and suggests that mass can be converted into energy, and matter into energy – which is crucial in nuclear reactions.

Energy–momentum relation

One of the paramount results in special relativity is the beautiful relation between energy and momentum, given as:

E² = (pc)² + (mc²)²

This equation integrates the contributions of both mass and velocity to energy, reflecting essential properties about the behavior of energy and momentum at relativistic speeds. If you set the momentum p to zero, you recover the rest energy formula.

A journey through relativistic phenomena

To understand how relativistic energy and momentum differ from classical intuition, consider the following scenario: Alice and Bob are both traveling toward each other in spaceships at significant fractions of the speed of light. Bob measures the speed of Alice's spaceship using the formula for relativistic speeds. Even though his prediction is favorable due to their high-velocity travel, the obtained value is greater than the value predicted from classical physics.

Alice Bob

This condition shows that at high velocities, the momentum becomes very large, requiring a very large force to achieve even modest acceleration. This clearly describes real-world phenomena such as the behavior of protons in particle accelerators, where relativistic effects become noticeable.

Conversion of energy and momentum

In special relativity, transformations between reference frames require sophisticated mechanisms. Lorentz transformations govern how different observers in relative motion perceive space and time. For energy and momentum, these transformations adjust values between inertial frames.

Frame S Frame S'

Consider two inertial frames: S and S' moving at relative velocity v. They see the energy and momentum of an object changing as follows:

E' = γ(E - vp_x) p'_x = γ(p_x - (vE/c²)) p'_y = p_y p'_z = p_z

These equations show the working of Lorentz transformations, and preserve conservation laws in the relativistic framework.

Applications in electrodynamics

Understanding relativistic energy and momentum promotes deeper insights into electromagnetic phenomena at high speeds. When charged particles such as electrons move at relativistic speeds, their electromagnetic fields differ from those at non-relativistic speeds due to changes in time dilation and length contraction.

For example, consider synchrotron radiation, a form of electromagnetic radiation that is emitted when charged particles travel in curved paths at relativistic speeds. Changes in energy and momentum underpin important fields such as accelerator physics and astrophysics.

Conclusion

In special relativity, the beautiful conversion and conservation of energy and momentum reveals a grand symmetry of nature, which is in stark contrast to classical explanations. From the quiet journey of rest energy to the complex manifestation of these quantities at high speeds, energy and momentum are unified under the banner of relativity, enriching our understanding of the universe at its most fundamental level.


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Relativistic energy and momentum

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