Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Relativity → Special relativity ↓


Relativistic energy and momentum


In the field of physics, Albert Einstein's theory of special relativity introduced an important way to understand the behavior of energy and momentum by taking into account the effects of relativity. Special relativity, developed in 1905, revolutionized the conceptual basis of physics by describing how physical quantities change when objects approach the speed of light. One of the main aspects of this theory is understanding relativistic energy and momentum. This subject explores how energy and momentum change for objects moving at very high speeds.

Energy in classical physics

Before diving into relativistic concepts, it is necessary to revisit how energy is considered in classical physics. Energy is generally classified into two primary forms: kinetic energy and potential energy. Kinetic energy is the energy an object has due to its motion, given by the formula:

E_k = frac{1}{2} mv^2

Where m is the mass of the object and v is its velocity. Potential energy, on the other hand, is the energy stored in an object due to its position or arrangement.

The total mechanical energy in a classical system is given by:

E_{text{total}} = E_k + E_p

However, as objects approach the speed of light, these classical definitions begin to break down, leading us to relativistic energies.

Relativistic energy

In special relativity, the total energy of a particle is not simply the sum of kinetic and potential energy. Instead, energy is related to the object's mass and velocity in a more complicated way. Einstein proposed that the energy of an object moving at high speed is related to its mass not only through velocity, but also through its position in spacetime. The famous equation that shows this relationship is:

E = gamma mc^2

Where:

  • E is the total energy.
  • m is the rest mass of the object.
  • c is the speed of light in vacuum.
  • gamma is the Lorentz factor, defined as gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}}.

This equation shows the relationship between energy, mass, and velocity. Importantly, it implies that as an object approaches the speed of light, its energy increases dramatically, which effectively prevents it from reaching or surpassing the speed of light.

Example: Calculating relativistic energy

Let us consider a particle with a rest mass of 1 kg moving at 80% of the speed of light (0.8c). To find the relativistic energy:

m = 1 text{ kg}, quad v = 0.8c, quad c = 3 times 10^8 text{ m/s} gamma = frac{1}{sqrt{1 - (0.8)^2}} = frac{1}{sqrt{1 - 0.64}} = frac{1}{0.6} approx 1.667 E = gamma mc^2 = 1.667 times 1 times (3 times 10^8)^2 E approx 1.67 times 9 times 10^{16} = 1.503 times 10^{17} text{ Joules}

This example shows how energy increases significantly with speed in relativistic scenarios.

Relativistic momentum

Classical physics describes momentum as the product of mass and velocity:

p = mv

However, in the field of relativity, this definition is not sufficient. Instead, the relativistic speed takes the Lorentz factor into account and is given as:

p = gamma mv

This adjustment ensures that as an object's velocity approaches the speed of light, its momentum also approaches infinity, which is consistent with the concept that an object with mass cannot reach or exceed the speed of light.

Example: Relative momentum calculation

Consider a particle with a rest mass of 1 kg moving at a speed 0.8 times the speed of light. Calculate its relativistic momentum:

m = 1 text{ kg}, quad v = 0.8c, quad c = 3 times 10^8 text{ m/s} gamma = frac{1}{sqrt{1 - (0.8)^2}} = frac{1}{0.6} approx 1.667 p = gamma mv = 1.667 times 1 times 0.8 times 3 times 10^8 p approx 4 times 10^8 text{ kg m/s}

Again, this shows how momentum in relativity differs significantly from classical mechanics.

Energy–momentum relation

In the special theory of relativity, energy and momentum are closely related. This relationship is known as the energy-momentum relation, which is expressed as follows:

E^2 = (pc)^2 + (mc^2)^2

This formula beautifully unifies energy and momentum, showing that both quantities are intertwined when considering high-speed particles. This equation also reduces directly to the well-known E = mc^2 (where p = 0) for stationary particles.

Example: Verification of the energy–momentum relation

Using the previous example, where a particle has energy 1.503 times 10^{17} joules and momentum 4 times 10^8 text{ kg m/s}, we can verify the energy-momentum relation:

E = 1.503 times 10^{17} text{ J} p = 4 times 10^8 text{ kg m/s} c = 3 times 10^8 text{ m/s} E^2 = (1.503 times 10^{17})^2 (pc)^2 = (4 times 10^8 times 3 times 10^8)^2 (mc^2)^2 = (1 times (3 times 10^8)^2)^2 E^2 approx (pc)^2 + (mc^2)^2

Visualization of relativistic effects

To understand relativistic effects, one can benefit from a visual model. Consider a train moving at high speed where each observer measures time intervals differently due to their relative motion:

Supervisor A Supervisor B

In this model, observer A inside the train and observer B outside see events and signals differently because of their relative velocities. The relative momentum and energy changes ensure that physical laws remain consistent regardless of their relative speed, which reflects the theory of relativity.

Implications of relativistic energy and momentum

Recognizing relativistic energy and momentum leads to several important implications:

  • Mass-energy equivalence: This principle implies that energy and mass are interchangeable. For example, a small mass can be converted into a lot of energy, which is the fundamental concept behind nuclear energy.
  • Speed limit: Relativistic momentum ensures that momentum increases without limit as velocity approaches the speed of light, which effectively imposes a universal speed limit, c.
  • Opposite effect: Time and length contraction affects objects traveling at high speeds, causing time to dilate and distances to change when viewed by stationary observers.
  • Cosmological models: Relativistic theories guide understanding cosmic phenomena, including the behavior of galaxies, black holes, and the expanding universe.

Conclusion

Understanding relativistic energy and momentum enriches our understanding of how the universe functions when considering objects moving at high velocities. From Einstein's equations, we draw underlying relationships between mass, energy, and the limits imposed by the ultimate speed, the speed of light. This theoretical framework drives scientific exploration, increasing our understanding of the complexity of the universe.


Undergraduate → 6.1.3


U
username
0%
completed in Undergraduate

← Prev (6.1.2)
Time expansion and length contraction
Next (6.2) →
General relativity

Comments 



Relativistic energy and momentum

https://www.physicsflow.com/ug/6.1.3?pfal=en