YOUR LEARNING PATH

One idea opens
the next.

Start with what you know. Follow the dependencies. Return to the exact point where a new idea becomes useful.

Find your starting point

Try a calculation before reading its explanation. If it is familiar, continue; if it catches you, a short lesson is one click away. These checks suggest preparation, not a placement grade.

One distance, two ways to count it

A rectangle is 2 metres long and 30 centimetres wide. What is its area in square metres?

Compare your reasoning

$30\,\mathrm{cm}=0.30\,\mathrm m$ , so the area is $(2\,\mathrm m)(0.30\,\mathrm m)=0.60\,\mathrm{m^2}$ .

Work through the method and a different example →

A law needs a starting state

Use seconds and metres. For $\ddot x+4x=0$ , $x(0)=0$ , $\dot x(0)=6$ , find $x(\pi/4)$ .

Compare your reasoning

$x=3\sin(2t)$ , so $x(\pi/4)=3\,\mathrm m$ .

Work through the method and a different example →

Motion, stored energy, and pressure work

A $2\,\mathrm{kg}$ particle falls from rest through $5\,\mathrm m$ in constant $g=10\,\mathrm{m/s^2}$ , without drag. Find its final speed.

Compare your reasoning

$v=\sqrt{2gh}=\sqrt{100}=10\,\mathrm{m/s}$ . The mass cancels.

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Build a boost from exponentials

If $e^\eta=2$ , calculate $v/c=\tanh\eta$ .

Compare your reasoning

$\tanh\eta=(2-1/2)/(2+1/2)=3/5$ .

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Units belong to coordinates, too

In $dl^2=dr^2+r^2d\theta^2$ , $r=3\,\mathrm m$ . What is $g_{\theta\theta}$ , in square metres?

Compare your reasoning

$g_{\theta\theta}=3^2=9\,\mathrm{m^2}$ .

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Do the two moves in the other order

For $X=\partial_x$ and $Y=x^2\partial_y$ , find the y component of $[X,Y]$ at $x=3$ .

Compare your reasoning

$[X,Y]=2x\partial_y$ , so the component is 6.

Work through the method and a different example →

A field has motion and stored energy at every point

A vacuum plane electromagnetic wave has electric energy density $2\,\mathrm{J/m^3}$ . Find its total instantaneous energy density.

Compare your reasoning

$u=u_E+u_B=2+2=4\,\mathrm{J/m^3}$ .

Work through the method and a different example →

Find the source by enclosing it

Inside a uniform-density ball, write $\Phi=ar^2+b$ . Since $\nabla^2r^2=6$ , what is $a/(\pi G_N\rho)$ ?

Compare your reasoning

$a=2\pi G_N\rho/3$ , so the requested ratio is $2/3$ .

Work through the method and a different example →

Say what is allowed to move

For $L=m\dot x^2/2-kx^2/2$ , take $m=2\,\mathrm{kg}$ , $k=8\,\mathrm{N/m}$ and $x=3\,\mathrm m$ . Find $\ddot x$ .

Compare your reasoning

$\ddot x=-(8/2)3=-12\,\mathrm{m/s^2}$ .

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Which initial data can determine this event?

For initial data on $[-5,5]$ at $t=0$ with $c=1$ , what is the half-width in x of $D^+(S)$ at $t=2$ ?

Compare your reasoning

$L-t=5-2=3$ . The available interval is $[-3,3]$ .

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From an initial ripple to a retarded field

In dimensionless units $c=1$ , take $f(x)=x^2$ and $g(x)=0$ . Use d’Alembert’s formula to find $u(0,2)$ .

Compare your reasoning

$u(0,2)=(4+4)/2=4$ . The solution is $u=x^2+t^2$ and satisfies $u_{tt}=u_{xx}=2$ .

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What temperature adds to a theory · deeper trail

A system has four equally likely states. Find $S/(k_B\ln2)$ .

Compare your reasoning

$\ln4/\ln2=2$ .

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A pure whole can have an uncertain part · deeper trail

For $\varrho_A=I_2/2$ , calculate the purity $\operatorname{tr}(\varrho_A^2)$ .

Compare your reasoning

$1/4+1/4=1/2$ . A normalized pure state instead has purity 1.

Work through the method and a different example →

Measurements and motion

Read a partial derivative, interpret a differential equation, and explain force, energy, and flux.

Starting point · basic calculus and linear algebra

Differentiate, integrate, and use the chain rule ↗For f(x,y)=x²y along x=t, y=2t, find df/dt at t=1.
Multiply matrices and read a bilinear form ↗For G=diag(1,4) and v=(3,2), calculate vᵀGv.

One distance, two ways to count itExplore & test A law needs a starting stateExplore & test Motion, stored energy, and pressure workExplore & test

Gravity, free fall, and clocks

Explain weightlessness and distinguish an accelerometer reading from acceleration relative to the ground.

Builds on00 Measurements and motion

Motion, stored energy, and pressure work ↗A $2\,\mathrm{kg}$ particle falls from rest through $5\,\mathrm m$ in constant $g=10\,\mathrm{m/s^2}$ , without drag. Find its final speed.

Release two pebbles, not just oneExplore & test

Vectors, covectors, and tensors

Transform a vector and a covector and check that their pairing is unchanged.

Builds on00 Measurements and motion

Differentiate, integrate, and use the chain rule ↗For f(x,y)=x²y along x=t, y=2t, find df/dt at t=1.
Multiply matrices and read a bilinear form ↗For G=diag(1,4) and v=(3,2), calculate vᵀGv.

Change the coordinates. Keep the measurement.Explore & test

What a clock actually measures

Calculate proper time and distinguish timelike, null, and spacelike separation.

Builds on00 Measurements and motion 01 Gravity, free fall, and clocks 02 Vectors, covectors, and tensors

One distance, two ways to count it ↗A rectangle is 2 metres long and 30 centimetres wide. What is its area in square metres?
Motion, stored energy, and pressure work ↗A $2\,\mathrm{kg}$ particle falls from rest through $5\,\mathrm m$ in constant $g=10\,\mathrm{m/s^2}$ , without drag. Find its final speed.
Transform vectors and covectors ↗Change coordinates in a directional derivative and recover the same scalar.

Build a boost from exponentialsExplore & test Chase a photon without slowing it downExplore & test

Manifolds, maps, and metrics

Read a metric, invert it, and calculate the size of a coordinate cell.

Builds on02 Vectors, covectors, and tensors 03 What a clock actually measures

Transform vectors and covectors ↗Change coordinates in a directional derivative and recover the same scalar.
Calculate a proper-time interval ↗Compare two timelike paths between the same pair of events.

Give one sphere two reliable addressesExplore & test Teach a coordinate grid how to measureExplore & test Units belong to coordinates, tooExplore & test

Free fall and the straightest path

Follow the worldline-action derivation of the geodesic equation.

Builds on04 Manifolds, maps, and metrics

Use a metric to turn components into measurements ↗Find a speed from radial and angular coordinate rates.
A law needs a starting state ↗Use seconds and metres. For $\ddot x+4x=0$ , $x(0)=0$ , $\dot x(0)=6$ , find $x(\pi/4)$ .
Build a boost from exponentials ↗If $e^\eta=2$ , calculate $v/c=\tanh\eta$ .

A straight path can have an accelerating labelExplore & test

Differentiating vector fields

Explain the connection correction and why lower indices acquire a minus sign.

Builds on02 Vectors, covectors, and tensors 04 Manifolds, maps, and metrics

Use a metric to turn components into measurements ↗Find a speed from radial and angular coordinate rates.
Transform vectors and covectors ↗Change coordinates in a directional derivative and recover the same scalar.

An unchanged arrow can have changing componentsExplore & test

The connection and parallel transport

Derive the Levi-Civita connection and test it on the flat polar plane.

Builds on05 Free fall and the straightest path 06 Differentiating vector fields

Differentiate the vector and its basis ↗Explain a nonzero component derivative for a constant Cartesian arrow.
A law needs a starting state ↗Use seconds and metres. For $\ddot x+4x=0$ , $x(0)=0$ , $\dot x(0)=6$ , find $x(\pi/4)$ .

Do the two moves in the other orderExplore & test

Curvature and parallel transport

Connect a transport loop, a derivative commutator, and the curvature of a sphere.

Builds on06 Differentiating vector fields 07 The connection and parallel transport

Derive and use the Levi-Civita connection ↗Calculate both nonzero types of polar Christoffel coefficient.
Do the two moves in the other order ↗For $X=\partial_x$ and $Y=x^2\partial_y$ , find the y component of $[X,Y]$ at $x=3$ .

Carry a direction home—and find it changedExplore & test Measure curvature without leaving the surfaceExplore & test

Ricci, Weyl, and Einstein curvature

Distinguish Ricci curvature, scalar curvature, Weyl curvature, and the Einstein tensor.

Builds on08 Curvature and parallel transport

Compare transport around a loop ↗Infer a sphere’s curvature from holonomy or a small-circle deficit.

A trace loses information you can nameExplore & test

Tides: curvature you can measure

Read geodesic deviation, check its Newtonian sign, and estimate terrestrial tides.

Builds on07 The connection and parallel transport 09 Ricci, Weyl, and Einstein curvature

Distinguish Riemann, Ricci, scalar and Weyl curvature ↗Explain why zero Ricci curvature need not remove tides.
Derive and use the Levi-Civita connection ↗Calculate both nonzero types of polar Christoffel coefficient.

Read the tidal matrix as an instrumentExplore & test

Energy, momentum, and stress

Read each region of the stress-energy matrix and derive the perfect-fluid form.

Builds on02 Vectors, covectors, and tensors 03 What a clock actually measures 07 The connection and parallel transport

Calculate a proper-time interval ↗Compare two timelike paths between the same pair of events.
Motion, stored energy, and pressure work ↗A $2\,\mathrm{kg}$ particle falls from rest through $5\,\mathrm m$ in constant $g=10\,\mathrm{m/s^2}$ , without drag. Find its final speed.

A field has motion and stored energy at every pointExplore & test

Einstein’s field equation

Trace-reverse the equation and recover the coefficient $8\pi G_N/c^4$ .

Builds on09 Ricci, Weyl, and Einstein curvature 10 Tides: curvature you can measure 11 Energy, momentum, and stress

Distinguish Riemann, Ricci, scalar and Weyl curvature ↗Explain why zero Ricci curvature need not remove tides.
Measure energy and momentum flux ↗Compare the energy density of dust in its rest frame and a boosted frame.

Find the source by enclosing itExplore & test

Variations and stationary action

Derive Euler–Lagrange equations and vary inverse metrics and determinants.

Builds on05 Free fall and the straightest path 11 Energy, momentum, and stress 12 Einstein’s field equation

A law needs a starting state ↗Use seconds and metres. For $\ddot x+4x=0$ , $x(0)=0$ , $\dot x(0)=6$ , find $x(\pi/4)$ .
A field has motion and stored energy at every point ↗A vacuum plane electromagnetic wave has electric energy density $2\,\mathrm{J/m^3}$ . Find its total instantaneous energy density.

Say what is allowed to moveExplore & test

Deriving Einstein’s equation

Follow the Einstein–Hilbert variation, including its boundary term.

Builds on12 Einstein’s field equation 13 Variations and stationary action

Vary a path and a field with stated boundary data ↗Derive a field Euler–Lagrange equation and name the boundary variation.
Trace reverse and identify the Newtonian source ↗Recover the factor of one half in the trace-reversed source.

A boundary term that refuses to disappearExplore & test

Symmetry and conservation

Construct a current using a Killing vector and interpret vacuum stress-energy.

Builds on07 The connection and parallel transport 12 Einstein’s field equation 14 Deriving Einstein’s equation

Derive and use the Levi-Civita connection ↗Calculate both nonzero types of polar Christoffel coefficient.
Vary a path and a field with stated boundary data ↗Derive a field Euler–Lagrange equation and name the boundary variation.

Earn a conserved quantity from a symmetryExplore & test

Clocks, light, and Mercury

Calculate redshift and GPS clock corrections, and explain light bending and perihelion advance.

Builds on03 What a clock actually measures 07 The connection and parallel transport 15 Symmetry and conservation

Calculate a proper-time interval ↗Compare two timelike paths between the same pair of events.
Use a spacetime symmetry to find a conserved quantity ↗Differentiate ξ·u along a geodesic and identify the Killing cancellation.
Build a boost from exponentials ↗If $e^\eta=2$ , calculate $v/c=\tanh\eta$ .

Why an orbit keeps its angular momentumExplore & test A tiny frequency change can rotate an entire orbitExplore & test

Black holes: horizons and orbits

Read the Schwarzschild geometry through its horizon using regular coordinates.

Builds on03 What a clock actually measures 07 The connection and parallel transport 12 Einstein’s field equation 16 Clocks, light, and Mercury

Trace reverse and identify the Newtonian source ↗Recover the factor of one half in the trace-reversed source.
Use a spacetime symmetry to find a conserved quantity ↗Differentiate ξ·u along a geodesic and identify the Killing cancellation.

At a horizon, ask which directions lead into the futureExplore & test Build a global causal map one coordinate change at a timeExplore & test Which initial data can determine this event?Explore & test

Gravitational waves

Connect the linearized field equation to detector strain and quadrupole radiation.

Builds on03 What a clock actually measures 09 Ricci, Weyl, and Einstein curvature 12 Einstein’s field equation

Distinguish Riemann, Ricci, scalar and Weyl curvature ↗Explain why zero Ricci curvature need not remove tides.
A field has motion and stored energy at every point ↗A vacuum plane electromagnetic wave has electric energy density $2\,\mathrm{J/m^3}$ . Find its total instantaneous energy density.
Find the source by enclosing it ↗Inside a uniform-density ball, write $\Phi=ar^2+b$ . Since $\nabla^2r^2=6$ , what is $a/(\pi G_N\rho)$ ?

Read a wave before reading its complex notationExplore & test From an initial ripple to a retarded fieldExplore & test

Cosmology: an expanding universe

Derive Friedmann evolution and distinguish redshift, distance, and horizons.

Builds on11 Energy, momentum, and stress 12 Einstein’s field equation 15 Symmetry and conservation

Measure energy and momentum flux ↗Compare the energy density of dust in its rest frame and a boosted frame.
Trace reverse and identify the Newtonian source ↗Recover the factor of one half in the trace-reversed source.
A law needs a starting state ↗Use seconds and metres. For $\ddot x+4x=0$ , $x(0)=0$ , $\dot x(0)=6$ , find $x(\pi/4)$ .

Calculate three distances to one galaxyExplore & test

Initial data and numerical relativity

Distinguish four constraints from evolution equations and count the physical degrees of freedom.

Builds on15 Symmetry and conservation 18 Gravitational waves 19 Cosmology: an expanding universe

Solve the Friedmann and fluid equations ↗Find a(t) for a spatially flat matter-only model.
Trace reverse and identify the Newtonian source ↗Recover the factor of one half in the trace-reversed source.
Which initial data can determine this event? ↗For initial data on $[-5,5]$ at $t=0$ with $c=1$ , what is the half-width in x of $D^+(S)$ at $t=2$ ?
Follow a state in phase space ↗Derive Hamilton’s equations from a Lagrangian and interpret a momentum-zero turning point.

Let the constraint audit your simulationExplore & test

Tetrads and differential forms

Use Cartan’s equations on the polar plane and the round sphere.

Builds on07 The connection and parallel transport 08 Curvature and parallel transport 14 Deriving Einstein’s equation 15 Symmetry and conservation

Derive and use the Levi-Civita connection ↗Calculate both nonzero types of polar Christoffel coefficient.
Compare transport around a loop ↗Infer a sphere’s curvature from holonomy or a small-circle deficit.
Vary a path and a field with stated boundary data ↗Derive a field Euler–Lagrange equation and name the boundary variation.

Calculate both sides of Stokes’ theoremExplore & test

Singularities and black-hole thermodynamics

Read the focusing argument and distinguish classical horizon laws from semiclassical radiation.

Builds on17 Black holes: horizons and orbits 18 Gravitational waves 21 Tetrads and differential forms

Which initial data can determine this event? ↗For initial data on $[-5,5]$ at $t=0$ with $c=1$ , what is the half-width in x of $D^+(S)$ at $t=2$ ?
Integrate an oriented differential form ↗Reverse an oriented boundary integral and check Stokes’ theorem.
From an initial ripple to a retarded field ↗In dimensionless units $c=1$ , take $f(x)=x^2$ and $g(x)=0$ . Use d’Alembert’s formula to find $u(0,2)$ .

What temperature adds to a theoryExplore & test A pure whole can have an uncertain partExplore & test Separate the geometry from the quantum assumptionExplore & test

General relativity as an effective theory

State what is known, what is an assumption, and what quantum gravity must explain.

Builds on12 Einstein’s field equation 19 Cosmology: an expanding universe 22 Singularities and black-hole thermodynamics

Trace reverse and identify the Newtonian source ↗Recover the factor of one half in the trace-reversed source.
A pure whole can have an uncertain part ↗For $\varrho_A=I_2/2$ , calculate the purity $\operatorname{tr}(\varrho_A^2)$ .
Solve the Friedmann and fluid equations ↗Find a(t) for a spatially flat matter-only model.

Make the approximation promise numericalExplore & test

From geometry to measurements

Organize a metric-to-measurement calculation and explain the Einstein equation in plain language.

Builds on10 Tides: curvature you can measure 16 Clocks, light, and Mercury 17 Black holes: horizons and orbits 18 Gravitational waves 19 Cosmology: an expanding universe

Distinguish Riemann, Ricci, scalar and Weyl curvature ↗Explain why zero Ricci curvature need not remove tides.
Use Schwarzschild constants and regular horizon coordinates ↗Calculate outgoing null directions on both sides of the horizon.
Solve the Friedmann and fluid equations ↗Find a(t) for a spatially flat matter-only model.

Build a universe with matter, then observe itExplore & test

Scope, notation, and how to use this route

The core route reaches observations and cosmology. Chapters 20–23 are optional extensions. The black-hole route includes their geometric prerequisites before Chapter 22; Hawking radiation uses explicitly stated quantum input. A solved example is evidence for that particular skill, not a claim of mastery of a chapter.

Reading guide & conventions · Color & notation · Synthesis and further study

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