Schwarzschild metric definition

Static spacetime

General form of Schwarzschild metric

Reversal of the radial coordinate

Invariants of the line interval

Spherical polar coordinates

Schwarzschild position vector

Schwarzschild line element

Schwarzschild metric in diagonal form

Weak-field limit

Administrative Announcements

The Schwarzschild Solution of General Relativity

Solution to this Equation of Motion

Spherical Coordinates

Einstein's Equations

The Schwarzschild Space-Time

Singularity

The Schwarzschild Solution Is the Unique Solution of the Equations of General Relativity

Equations of Motion of General Relativity

Gravity Waves

Einstein Field Equations

The Einstein Field Equations

Solve the Einstein Field Equations

Contravariant Metric

Metric Tensor

Nonzero Component of the Ricci Curvature Tensor

Radial Acceleration

Time Component

Zero Component

Newton's Law of Gravitation

Characteristics of Schwarzschild Metric

Spherical Symmetric

Signature Remains Unaltered

Inverse Metric Tensor

General Form

Coordinate Transformation

Reduced Field Equations

Summary

Uniqueness proof

Conclusion

Spherically symmetric solution

Gauss' Law

The Schwarzschild solution in spherical coordinates

Source freeness

Spherical polar coordinates

Spherical Exfoliation

Schwarzschild Metric

Lorentzian Signature Metric

Stationary versus Static

Summary

Introduction

Geodesics Review

Newtonian Orbits Review

Constants of Motion (Geodesics)

Tangent Vector Length (Geodesics)

Timelike Geodesics (massive bodies)

Graphing Effective Potentials

Lightlike Geodesics (photon sphere)

Perihelion Advance of Mercury

Summary

Introduction

Gauss' Law

The Schwarzschild Radius

The Schwarzschild solution

N-Space Equations

Assuming a foliation of spacetime

Radial trajectories

Schwarzschild Metric

Signature of spacetime

Reduced metric

Introduction

Gravitational Time Dilation

r-coordinate interpretation (radius)

r-coordinate interpretation (circumference)

Singularities

Schwarzschild Radius + Event Horizon

Light-like geodesics

Light rays on spacetime diagrams

Summary

Deflection of Light around a Massive Body

The Effective Potential

Asymptotically Flat Region

Impact Parameter

Solving the Differential Equation

Change of Integration Variables

Deflection Angle

The Schwarzschild Spacetime is Static

Null geodesics

Geodesic in Schwarzschild Slice

Introduction

What is the Schwarzschild spacetime?

The Schwarzschild metric

Analyzing the Schwarzschild metric

The connection coefficient catalog

The geodesic equation

Geodesic equations for R, theta, and phi

Substitute Schwarzschild coordinates into Schwarzschild differential equations

The Schwarzschild geodesic equation

Conclusion

Birkhov's Theorem

Laplace equation for Schwarzschild metric

Dimensional analysis of the Schwarzschild metric

Dimensional analysis of Schwarzschild radius

Forward velocity of a stationary observer

Gravitational Redshift

Gravitational Relationship of the Photon

Gravitational Time Dilation

Gravitational Pressure

Common misconceptions

The Line Element in Schwarzschild Coordinates

Time Dilation Effect

Relationship between Schwarzschild Time

History of Schwarzschild Spacetime

Tangent Space Vectors

What is a Spacetime?

Einstein Equation

Minkowski metric in Cartesian coordinates

Other forms of a spherically symmetric metric

What is a Neutron Star?

Tensors on spacetime

Ernst Spacetime

Schwarzschild solution setup

Black Holes

The Schwarzschild radius

Singularities

Event horizon

Introduction to Schwarzschild metric

Spherical Coordinates Review

Schwarzschild Metric Assumptions

Connection Coefficient Calculation

Ricci Tensor Calculation

Solving for A(r) and B(r)

Solving for Schwarzschild Radius

Warning + Conclusion