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 → General relativity ↓


Schwarzschild metric


The Schwarzschild metric is a solution to Albert Einstein's field equations in general relativity. It describes the gravitational field outside a spherical mass, assuming the mass has no charge and is not rotating. This solution was found by Karl Schwarzschild in 1916, shortly after Einstein introduced general relativity. Understanding the Schwarzschild metric is important for analyzing phenomena such as black holes and the distortion of space-time around massive objects.

Basics of general relativity

Before diving into the Schwarzschild metric, it is important to have a basic understanding of general relativity. This theory proposed by Einstein explains gravity not as a force but as a curvature of space-time due to mass and energy. Objects follow paths or geodesics determined by this curvature, which alters their trajectories compared to the way they would move in flat space-time.

The fundamental relation in general relativity is represented by the Einstein field equations:

G μν + Λg μν = 8πGT μν

Where:

  • G μν is the Einstein tensor, which describes the curvature of space-time.
  • Λ is the cosmological constant, which measures the energy density of empty space.
  • g μν is the metric tensor, which describes how distances are measured in space-time.
  • T μν is the stress–energy tensor, which describes the distribution and flow of energy and momentum.
  • G is the gravitational constant.

Schwarzschild solution

The Schwarzschild solution satisfies the Einstein field equations for a point mass at the origin of a coordinate system, assuming vacuum conditions prevail outside the mass. The solution represents a non-rotating, uncharged black hole.

Schwarzschild metric equation

The Schwarzschild metric is expressed as:

ds² = -(1 - 2GM/rc²) c²dt² + (1 - 2GM/rc²) -1 dr² + r²dθ² + r²sin²θ dφ²

Where:

  • ds² is the line element or interval, indicating the squared distance in space-time between events.
  • c is the speed of light.
  • G is the gravitational constant.
  • M is the mass of the object creating the gravitational field.
  • r is the radial coordinate, the distance from the mass.
  • θ and φ are angular coordinates (latitude and longitude).

Each term contributes to understanding the geometry of space-time around a spherical object. The exciting part of this metric is how it incorporates both time and spatial geometry, showing how time is affected by gravity.

Understanding the components

1. **Time Dilation:** The term -(1 - 2GM/rc²) c²dt² indicates time dilation near a massive object. Closer to the object, time moves slower relative to an observer located at a greater distance.

2. **Spatial Curvature:** The term (1 - 2GM/rc²) -1 dr² represents the radial part of space curvature. As you approach a massive object, space itself is distorted, affecting the path of light and matter.

3. **Angular Part:** r²dθ² + r²sin²θ dφ² describes the spherical nature of space, similar to polar coordinates on the sphere.

Visualization and effects

Examples: bending light

Imagine a beam of light passing a star. According to the Schwarzschild metric, the curvature of space-time around the star bends the light path:

Ray before curvature Ray after curvature star
(Mass)

These lines show the path of light approaching the massive star, which is being bent due to the distortion of space-time, a phenomenon known as gravitational lensing.

Example: Event Horizon and Black Holes

The Schwarzschild metric explains the concept of the black hole's event horizon, which is the point of no return. When r is equal to the Schwarzschild radius (r s = 2GM/c²), the escape velocity is equal to the speed of light:

r s = 2GM/c²
Black holes Event horizon (r = r s )

The inner circle represents the event horizon, beyond which nothing, not even light, can escape. Outside the event horizon, paths in space-time can still curve and move away from the mass; once an object crosses the horizon, it is destined for the central singularity.

Importance and applications

The Schwarzschild metric is important in astrophysics and cosmology. It provides a theoretical basis for understanding stellar dynamics, black hole properties, and gravitational waves. The metric helps estimate the time variations around massive objects such as planets and stars, providing information for satellite orbit corrections and Global Positioning Systems (GPS).

Black hole studies

Astrophysicists use the Schwarzschild metric to study black holes, revealing properties such as temperature, entropy, and information loss. This forms the basis for discussing Hawking radiation and black hole thermodynamics.

Observational astronomy

This metric helps explain the observation of light bending seen as gravitational lensing. It also helps map the mass distribution in galaxies and clusters through astrophysics and study the effects of dark matter on visible matter.

Global Positioning System

On Earth, GPS must take into account the time dilation effects predicted by the Schwarzschild metric. Corrections ensure that satellites provide accurate positional data, incorporating both the time dilation from the Earth's mass due to general relativity and the velocity-based time dilation from their orbital motion.

Conclusion

The Schwarzschild metric goes beyond a theoretical exercise; it is a vital tool in modern science, facilitating advances in astrophysics, cosmology, and practical technologies. Its discovery highlights the profound connection between geometry and gravity, reshaping our understanding of the universe.


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Schwarzschild metric

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