Gravity is
geometry.Gravity is geometry.
General relativity,
from the inside out.General relativity, from the inside out.
Begin with a clock and a falling object.
Build your way to the shape of spacetime.Begin with a clock and a falling object. Build your way to the shape of spacetime.
Basic calculus. Linear algebra. Curiosity.Basic calculus. Linear algebra. Curiosity.
Already know some relativity? Explore Einstein’s equationAlready know some relativity? Explore Einstein’s equation
EINSTEIN’S FIELD EQUATIONContentsContents
Choose a learning path →Measurements and motion
How can a few measurements help us predict motion?
Gravity, free fall, and clocks
Why does a scale read zero in free fall?
Vectors, covectors, and tensors
How can components change while the physical object stays the same?
What a clock actually measures
Why do two reunited clocks record different elapsed times?
Manifolds, maps, and metrics
Why can a position-dependent metric describe a perfectly flat plane?
Free fall and the straightest path
What equation replaces Newton’s force law for free fall?
Differentiating vector fields
What is missing when we differentiate vector components?
The connection and parallel transport
Where does the Christoffel formula come from?
Curvature and parallel transport
What does the Riemann tensor actually measure?
Ricci, Weyl, and Einstein curvature
How can empty spacetime be curved?
Tides: curvature you can measure
How does curvature change the separation of falling objects?
Energy, momentum, and stress
Why is gravity’s source a tensor rather than just mass density?
Einstein’s field equation
Why subtract half the curvature trace?
Variations and stationary action
How do we differentiate an action with respect to a path or a field?
Deriving Einstein’s equation
How does one scalar action produce the full tensor equation?
Symmetry and conservation
When can we compare energy at different events?
Clocks, light, and Mercury
What does a real experiment read out from the metric?
Black holes: horizons and orbits
What is special about a black-hole horizon?
Gravitational waves
How can a passing gravitational wave change a detector’s reading?
Cosmology: an expanding universe
What controls the expansion and acceleration of the universe?
Initial data and numerical relativity
What must be specified before Einstein’s equation can predict a future?
Tetrads and differential forms
Can we calculate curvature with fewer indices?
Singularities and black-hole thermodynamics
How can a statement about converging paths lead to black-hole entropy?
General relativity as an effective theory
How can GR be incomplete and still be an excellent theory?
From geometry to measurements
Can I reconstruct the theory without memorizing a wall of equations?