A FIELD GUIDE TO SPACETIME
THE VISUAL ATLAS / AN EXPLORABLE COLLECTION

Give the symbols
some space.

Turn a surface. Follow a worldline. Watch a direction change. Geometry becomes clearer when you can move around it.

SPACETIME LAB / 01

Compare two falling objects

Follow the rose points as they fall toward Earth. Watch the gap between points on the same radial line, then compare points side by side.

EarthA still moment in a freely falling reference gridInteractive geometry is loading.
drdt=−2GNMr\frac{dr}{dt}=-\sqrt{\frac{2G_NM}{r}}

Read the scene. The teal grid joins freely falling observers; rose points make their motion visible. This model treats Earth as a nonrotating sphere. Each observer has fallen from rest very far away and is already moving when it enters the picture. New points enter from beyond the view, and the motion stops at the surface. The grid lines simply join observers at the same time.

How this animation is calculated

Each rose point follows the speed rule derived in §1.10. Integrating that rule tells us where to draw the point at a later time:

drdt=−2GNMrr(t)=[r03/2−322GNM t]2/3\begin{gathered}\frac{dr}{dt}=-\sqrt{\frac{2G_NM}{r}}\\r(t)=\left[r_0^{3/2}-\frac32\sqrt{2G_NM}\,t\right]^{2/3}\end{gathered}

r0r_0 is the distance from Earth’s center when the point enters, and tt is the time since entry. MM is Earth’s mass; GNG_N is Newton’s gravitational constant. The formula applies until the point reaches the surface.

The points show a continuous stream of falls with these starting conditions. The connecting lines help you compare their positions. They are neither material threads nor the paths of individual objects.

Motion is shown at 110 times real time. The model uses Earth’s mean radius and gravitational parameter, giving an incoming surface speed of about 11.2 km/s11.2\,\mathrm{km/s}. It omits rotation, the atmosphere’s resistance, and Earth’s interior. The glow and lighting identify the globe; they carry no gravity data.

This speed rule also occurs in a useful relativistic description of spherical gravity. Chapter 17 develops the geometry needed to understand that description. The technical source is Hamilton and Lisle’s river model.

Presentation inspired by ScienceClic’s visualization. Earth texture: NASA, Blue Marble: Next Generation. Sources and image rights ↗

SURFACES, ORBITS, WAVES & CURVATURE

Enter the spacetime labs

CALCULATE & COMPARE

Build a star from its centre

Choose a central density. Follow the pressure outward until it reaches zero: that is the surface of your star.

Chapter 17 · Numerical experimentCALCULATE & COMPARE

Ask the universe three distance questions

Which distance did the observation actually measure?

Chapter 19 · Numerical experimentCALCULATE & COMPARE

Follow the photon. Ask each observer.

Send light between two observers. Change their heights and motion, and compare the frequencies they measure.

Chapter 17 · Numerical experimentCALCULATE & COMPARE

Make a universe—and audit the answer

Will a smaller time step repair the apparent motion, the constraint, or both?

Chapter 20 · Numerical experimentEXPLORE

A curved world. A flat address.

Link a smooth surface to two overlapping coordinate charts.

Chapter 4 · Interactive modelEXPLORE

Twenty ways to curve

Follow the symmetries that turn 256 slots into 20 independent components.

Chapter 8 · Interactive modelEXPLORE

Volume and shape

Compare all three tidal directions in a freely falling cloud.

Chapter 9 · Interactive modelEXPLORE

Mercury remembers

Watch the closest approach advance from orbit to orbit.

Chapter 16 · Interactive modelLAB 01

Compare two falling objects

Follow the rose points as they fall toward Earth. Watch the gap between points on the same radial line, then compare points side by side.

Chapter 01 · Interactive 3DLAB 02

An arrow meets a family of planes

A covector measures how many level intervals a displacement crosses. Rotate the scene to see that this is a pairing, not a length.

Chapter 02 · Interactive 3DLAB 03

The shape of a possible future

A light cone becomes a surface when we restore a second space direction. Change the speed and watch the observer’s worldline tilt.

Chapter 03 · Interactive 3DLAB 04

Carry a direction. Discover curvature.

Move an arrow around three great-circle arcs. It stays parallel within each local tangent plane, yet returns pointing a quarter-turn away.

Chapter 08 · Interactive 3DLAB 05

Gravity changes a cloud’s shape

A small freely falling cloud stretches radially and squeezes in two independent sideways directions. Rotate to find the dimension a flat diagram hides.

Chapter 10 · Interactive 3DLAB 06

The same interval. A longer ruler.

Keep two coordinate radii fixed. Increase the black hole’s mass and measure how much farther apart they become on this spatial slice.

Chapter 17 · Interactive 3DLAB 07

One wave. Two ways to stretch.

Watch one detector ring. Compare plus and cross polarization, then separate what strain and frequency change.

Chapter 18 · Interactive 3DLAB 08

More distance. The same cosmic grid.

Choose a scale factor. Every comoving separation grows together; no point in the grid becomes a preferred center of the expansion.

Chapter 19 · Interactive 3DLAB 09

One step in time. Two different choices.

Lapse carries you along the normal. Shift slides the coordinate grid sideways. Rotate the slices to separate these two pieces.

Chapter 20 · Interactive 3D
SPACETIME LAB / 04

Carry a direction. Discover curvature.

Move an arrow around three great-circle arcs. It stays parallel within each local tangent plane, yet returns pointing a quarter-turn away.

A sphere remembers the loopAn orthographic spherical octant connects the north pole with two equatorial points ninety degrees apart. The returned tangent arrow differs from the initial arrow. Great-circle transport around this octant rotates a tangent vector by 90°. Projection changes apparent angles on the page; the right angles and 90° rotation are intrinsic to the sphere.13 / A SPHERE REMEMBERS THE LOOPThree right-angle turnsReturn: perpendicular to startThe arrow never twists within its tangent plane.Interactive geometry is loading.
Start at the poleReturn to the pole
Δα=Aa2=π2\Delta\alpha=\frac{A}{a^2}=\frac{\pi}{2}

Read the scene. A sphere is an analogy for intrinsic curvature, not a picture of four-dimensional spacetime. The octant loop encloses one eighth of its area.

40 STUDIES IN GEOMETRY & PHYSICS

The vector collection

LaTeX throughout

40 figures

A derivative predicts the next small stepA parabola and its tangent at t=2. They meet with equal slope but differ away from the contact point. At two seconds the cart is 12 metres from the start and moving at 12 metres per second. The straight line predicts its next small displacement; the curve also includes its acceleration.01 / A DERIVATIVE PREDICTS THE NEXT SMALL STEPThe cart at two secondsThe line matches value and slope.The cart continues to speed up.
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A derivative predicts the next small step. At two seconds the cart is 12 metres from the start and moving at 12 metres per second. The straight line predicts its next small displacement; the curve also includes its acceleration.
Read Chapter 0
Comparing two free fallsOne worldline is compared with a pair of approaching free-fall worldlines, with time increasing upward. The paired experiment measures relative acceleration. Curved drawn paths are coordinate schematics; their appearance alone is not evidence of curvature.02 / COMPARING TWO FREE FALLSOne test bodyAn accelerometer reads zero.Two test bodiesTheir separation can change.positiontimepositiontimeinitial gapsmaller gapA schematic of tangential convergence near Earth. Neither observer feels a local force.
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Comparing two free falls. The paired experiment measures relative acceleration. Curved drawn paths are coordinate schematics; their appearance alone is not evidence of curvature.
Read Chapter 1
A century of consequencesTen dated milestones progress from Newton’s Principia to the first direct gravitational-wave detection. Dates label selected developments, not a complete priority history. The 2015 LIGO event was publicly reported in 2016. The text discusses the collaborative and iterative development of the field equations.03 / A CENTURY OF CONSEQUENCESA route through discoveryThe teaching sequence is not the historical sequence.Newtononelaw of gravityEinsteinspecialrelativityFree fallequivalenceprincipleMinkowskispacetimegeometry1912–13GrossmanngeometriccollaborationField equationtheNovember papersSchwarzschildaspherical solutionKerrarotating solutionHawkingblack-holeradiationLIGOfirstdirect GW detection
03 /
A century of consequences. Dates label selected developments, not a complete priority history. The 2015 LIGO event was publicly reported in 2016. The text discusses the collaborative and iterative development of the field equations.
Read Chapter 1
Arrows and the questions they answerA diagonal vector crosses three evenly spaced level intervals of the function x. The covector dx returns the change in x, not the Euclidean length of the arrow. Changing the coordinate scale changes both components so their pairing stays the same.04 / ARROWS AND THE QUESTIONS THEY ANSWERA vector supplies a displacementA covector asks a linear questionThree level intervals crossed → answer 3
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Arrows and the questions they answer. The covector dx returns the change in x, not the Euclidean length of the arrow. Changing the coordinate scale changes both components so their pairing stays the same.
Read Chapter 2
The minus sign creates a light coneTwo light rays bound the possible future of an event. A slower-than-light path lies inside, and a spacelike displacement lies outside. The graph uses the same scale for x and ct. A massive observer follows a timelike worldline; no rest frame exists for a light ray.05 / THE MINUS SIGN CREATES A LIGHT CONEtimelikespacelikeFUTURE LIGHT CONESignals from the starting event remain on or inside this cone. One space dimension is shown.
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The minus sign creates a light cone. The graph uses the same scale for x and ct. A massive observer follows a timelike worldline; no rest frame exists for a light ray.
Read Chapter 3
Two histories between the same eventsA home clock follows a vertical worldline for ten years. A traveller reaches three light-years in five years and returns in another five. The idealized travelling clock accumulates 10√(1 − 0.6²) = 8 years. The sharp turnaround is an approximation; the path integral, not a local feeling of slow time, gives the age difference.06 / TWO HISTORIES BETWEEN THE SAME EVENTSdistance (light-years)time (years)Home clockTravelling clockSame departure. Same reunion.Different lengths in spacetime.
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Two histories between the same events. The idealized travelling clock accumulates 101−0.62=810\sqrt{1-0.6^2}=8 years. The sharp turnaround is an approximation; the path integral, not a local feeling of slow time, gives the age difference.
Read Chapter 3
Coordinates are not rulersA polar grid has an annular sector highlighted; its radial and angular edges have different measuring factors. For an infinitesimal cell, the angular edge is r dθ and the radial edge is dr. The finite cell is enlarged for visibility; the plane is intrinsically flat.07 / COORDINATES ARE NOT RULERSA flat plane, with polar labelsA grid can bend without the surface curving.Two physical edge lengths
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Coordinates are not rulers. For an infinitesimal cell, the angular edge is r dθr\,d\theta and the radial edge is drdr. The finite cell is enlarged for visibility; the plane is intrinsically flat.
Read Chapter 4
Why a determinant belongs in the volumeA square transforms into a parallelogram under a linear map; the absolute determinant is its area scaling. This Euclidean two-dimensional example explains the square root of the metric determinant. In four-dimensional Lorentzian geometry the positive measure uses √(−g). The drawn transformation is illustrative.08 / WHY A DETERMINANT BELONGS IN THE VOLUMEThe determinant measures area scalingA unit coordinate square maps to a parallelogram.
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Why a determinant belongs in the volume. This Euclidean two-dimensional example explains the square root of the metric determinant. In four-dimensional Lorentzian geometry the positive measure uses −g\sqrt{-g}. The drawn transformation is illustrative.
Read Chapter 4
Acceleration without curvatureThree Rindler hyperbolae lie inside a lightlike asymptote in an inertial spacetime diagram. Each curve is a stationary rocket-frame observer. In a rigid accelerated laboratory, different heights require different proper accelerations. These worldlines occupy flat Minkowski spacetime.09 / ACCELERATION WITHOUT CURVATUREStationary in the rocketAccelerated in inertial coordinates.A varying clock rate is not proof of tides.
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Acceleration without curvature. Each curve is a stationary rocket-frame observer. In a rigid accelerated laboratory, different heights require different proper accelerations. These worldlines occupy flat Minkowski spacetime.
Read Chapter 5
Do not confuse a route with its parameterTwo identical straight paths have respectively equally spaced and nonuniformly spaced parameter labels. This flat straight-line illustration isolates parameter choice. Affine parameters are related by λ ↦ aλ + b; an arbitrary nonlinear relabeling changes the standard coordinate form of the geodesic equation.10 / DO NOT CONFUSE A ROUTE WITH ITS PARAMETERThe same path can wear different parametersThe curve stays fixed; the labels along it can speed up or slow down.Affine parameterNon-affine parameterA non-affine parameter can add a tangent-proportional term to the geodesic equation.
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Do not confuse a route with its parameter. This flat straight-line illustration isolates parameter choice. Affine parameters are related by λ↦aλ+b\lambda\mapsto a\lambda+b; an arbitrary nonlinear relabeling changes the standard coordinate form of the geodesic equation.
Read Chapter 5
Different components can describe the same arrowAt three points on a circle the eastward vector stays horizontal while radial and angular basis directions rotate. The ordinary component derivatives see changing numbers. The connection correction accounts for the basis change. Together they report that this field is constant.11 / DIFFERENT COMPONENTS CAN DESCRIBE THE SAME ARROWSame arrow. Changing local basis.Each teal arrow points east in the same flat plane.eastward fieldradial directionangular directionComponent change + basis change = zero geometric change
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Different components can describe the same arrow. The ordinary component derivatives see changing numbers. The connection correction accounts for the basis change. Together they report that this field is constant.
Read Chapter 6
The connection cancels a false changeThe coordinate components of a fixed eastward unit vector are followed in the angular direction. The explicit Christoffel symbol supplies the opposite change. On the flat polar plane, the radial component of the covariant derivative is zero. The derivative direction is θ, and Γ^r_{θθ} = −r. These are coordinate-basis components, not components in a unit basis.12 / THE CONNECTION CANCELS A FALSE CHANGEThe arrow stays fixed. Its address changes.Differentiate in the angular direction, at fixed radius.COMPONENTS IN THE COORDINATE BASISTHE BASIS CORRECTIONThe indices select the term we need.Changing components and a changing basis cancel exactly.
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The connection cancels a false change. On the flat polar plane, the radial component of the covariant derivative is zero. The derivative direction is θ\theta, and Γrθθ=−r\Gamma^r{}_{\theta\theta}=-r. These are coordinate-basis components, not components in a unit basis.
Read Chapter 7
A sphere remembers the loopAn orthographic spherical octant connects the north pole with two equatorial points ninety degrees apart. The returned tangent arrow differs from the initial arrow. Great-circle transport around this octant rotates a tangent vector by 90°. Projection changes apparent angles on the page; the right angles and 90° rotation are intrinsic to the sphere.13 / A SPHERE REMEMBERS THE LOOPThree right-angle turnsReturn: perpendicular to startThe arrow never twists within its tangent plane.
13 /
A sphere remembers the loop. Great-circle transport around this octant rotates a tangent vector by 90∘90^\circ. Projection changes apparent angles on the page; the right angles and 90∘90^\circ rotation are intrinsic to the sphere.
Read Chapter 8
From 256 slots to 20 independent entriesA six-by-six matrix of antisymmetric pairs repeats each off-diagonal symbol in its reflected tile. The upper triangle and diagonal contain twenty-one symbols. Riemann antisymmetry creates six pair labels; pair-exchange symmetry leaves 21 entries. The algebraic Bianchi identity removes one in four dimensions. This counts local tensor components, not propagating gravitational degrees of freedom.14 / FROM 256 SLOTS TO 20 INDEPENDENT ENTRIESTwo antisymmetric index pairsIn four dimensions there are six possible independent pairs.Pair-exchange symmetryDiagonal + mirrored pairsThen one cyclic relationindependent components
14 /
From 256 slots to 20 independent entries. Riemann antisymmetry creates six pair labels; pair-exchange symmetry leaves 21 entries. The algebraic Bianchi identity removes one in four dimensions. This counts local tensor components, not propagating gravitational degrees of freedom.
Read Chapter 8
Volume and shape ask different questionsAn initial circular section is compared with a smaller circle and with a stretched ellipse. These are exaggerated cross-sections of infinitesimal clouds, not exact finite volume-preserving motions. The trace controls initial volume acceleration for an initially comoving cloud; shear can later change the volume.15 / VOLUME AND SHAPE ASK DIFFERENT QUESTIONSIsotropic initial squeezeThe trace is nonzero.Trace-free initial distortionStretch one way; squeeze another.Volume acceleration is negativeInitial volume acceleration can be zero
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Volume and shape ask different questions. These are exaggerated cross-sections of infinitesimal clouds, not exact finite volume-preserving motions. The trace controls initial volume acceleration for an initially comoving cloud; shear can later change the volume.
Read Chapter 9
Stretching and squeezing near EarthRadial particles accelerate apart; tangential particles accelerate toward one another. The three tidal acceleration eigenvalues sum to zero. The labels give eigenvalues of the Newtonian relative-acceleration matrix, not curvature components: divide by c² for the corresponding curvature scale, with the convention-dependent sign tracked in the text.16 / STRETCHING AND SQUEEZING NEAR EARTHVacuum tides near a spherical massRadial stretching balances two tangential squeezes.radialRelative acceleration / separation
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Stretching and squeezing near Earth. The labels give eigenvalues of the Newtonian relative-acceleration matrix, not curvature components: divide by c2c^2 for the corresponding curvature scale, with the convention-dependent sign tracked in the text.
Read Chapter 10
The source has more than one kind of entryA four-by-four stress-energy matrix colors energy density, mixed energy-momentum entries, and spatial stresses differently. S is energy flux, π is momentum density, and σ denotes the spatial stress entries using the momentum-flux convention. The tensor is symmetric in ordinary metric GR.17 / THE SOURCE HAS MORE THAN ONE KIND OF ENTRYEnergy and momentum, in one local inertial frameEnergy densityEnergy flow ↔ momentum densityStressDiagonal pressure; off-diagonal shear.
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The source has more than one kind of entry. SS is energy flux, π\pi is momentum density, and σ\sigma denotes the spatial stress entries using the momentum-flux convention. The tensor is symmetric in ordinary metric GR.
Read Chapter 11
Read the equation as a relationshipThe Einstein equation is followed by separate definitions of the Ricci tensor, Ricci scalar, metric tensor, cosmological constant, Einstein constant, and stress-energy tensor. Curvature is violet, the metric teal, matter amber, and constants neutral. Each symbol has its own job. The equation constrains spacetime geometry and matter together; the matter must also obey its dynamics and conservation laws.18 / READ THE EQUATION AS A RELATIONSHIPRicci tensorA contraction ofspacetime curvature.Ricci scalarThe metric trace ofthe Ricci tensor.Metric tensorTurns displacementsinto intervals.Cosmological constantMultiplies the metricin the equation.Einstein constantSets the strengthof the coupling.Stress–energy tensorEnergy, momentum,and stress.
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Read the equation as a relationship. Each symbol has its own job. The equation constrains spacetime geometry and matter together; the matter must also obey its dynamics and conservation laws.
Read Chapter 12
Where the factor of eight comes fromGeometry, the trace-reversed dust source, and Poisson’s equation combine to fix the coupling constant. The displayed component estimates assume weak, stationary fields and slow pressureless matter, with x⁰=ct. The matching is performed in the chapter with the full index conventions.19 / WHERE THE FACTOR OF EIGHT COMES FROMThe Newtonian calibrationUse the trace-reversed equation for slowly moving dust.GeometryMatterNewtonMatch the two
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Where the factor of eight comes from. The displayed component estimates assume weak, stationary fields and slow pressureless matter, with x0=ctx^0=ct. The matching is performed in the chapter with the full index conventions.
Read Chapter 12
A derivative of whole pathsThree curves share fixed initial and final positions; a smooth variation changes their interiors. A variation is a mathematical comparison with nearby candidate histories. The physical stationary path is found by requiring the first action change to vanish for every allowed variation.20 / A DERIVATIVE OF WHOLE PATHStimepositionVary the route, not its endpointsThe varied paths need not obey the motion law.
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A derivative of whole paths. A variation is a mathematical comparison with nearby candidate histories. The physical stationary path is found by requiring the first action change to vanish for every allowed variation.
Read Chapter 13
Two variations make one field equationA two-branch flowchart separates the variation of scalar curvature from the variation of the invariant volume. The boundary divergence requires its own treatment. After that treatment, the bulk coefficient of the arbitrary inverse-metric variation is the Einstein tensor.21 / TWO VARIATIONS MAKE ONE FIELD EQUATIONThe product rule behind Einstein’s equationCurvature changesVolume changesRicci + a boundary divergenceTogether: the Einstein tensor
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Two variations make one field equation. The boundary divergence requires its own treatment. After that treatment, the bulk coefficient of the arbitrary inverse-metric variation is the Einstein tensor.
Read Chapter 14
Fixing the value does not fix the slopeA variation vanishes at both endpoints while having nonzero slopes there. The scalar graph is an analogy for each metric variation component along a normal direction. It explains why the Einstein–Hilbert action needs appropriate boundary treatment even when the boundary metric is fixed.22 / FIXING THE VALUE DOES NOT FIX THE SLOPEcoordinate normal to boundaryallowed variationZero value at the boundaryDoes not imply zero normal derivative.A boundary term may therefore survive.The variational problem needs a boundary policy.
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Fixing the value does not fix the slope. The scalar graph is an analogy for each metric variation component along a normal direction. It explains why the Einstein–Hilbert action needs appropriate boundary treatment even when the boundary metric is fixed.
Read Chapter 14
A symmetry supplies an energy comparisonRepeated geometric slices are connected by a time-translation vector, next to the conserved pairing minus p dot xi. The drawing is schematic. A timelike Killing vector provides a stationary energy; normalization and the observer’s local energy measurement must still be specified.23 / A SYMMETRY SUPPLIES AN ENERGY COMPARISONA time-translation symmetryThe geometry repeats under a shift in time.spacetimeA direction along which the metric is unchanged.Constant along a free geodesic.No such symmetry in general?Then no automatic conserved energy of this form.
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A symmetry supplies an energy comparison. The drawing is schematic. A timelike Killing vector provides a stationary energy; normalization and the observer’s local energy measurement must still be specified.
Read Chapter 15
A ray samples more than the clock rateA light ray bends toward a mass relative to its straight incoming reference line; the impact parameter is marked. This is the shape from a first-order weak-field deflection profile, with its amplitude enlarged. The book derives why keeping only the clock term misses half the leading GR deflection.24 / A RAY SAMPLES MORE THAN THE CLOCK RATEBoth clock and spatial terms affect lightA weak-field ray past a spherical mass; deflection exaggerated.incoming direction
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A ray samples more than the clock rate. This is the shape from a first-order weak-field deflection profile, with its amplitude enlarged. The book derives why keeping only the clock term misses half the leading GR deflection.
Read Chapter 16
A slowly turning orbitThree aligned-focus ellipses have progressively rotated perihelia. Their offset is greatly exaggerated. The curves are schematic successive Kepler ellipses, compressed vertically for layout, not a numerical integration of an exact relativistic orbit. The text derives the small secular advance and states its regime.25 / A SLOWLY TURNING ORBITAn ellipse whose closest point slowly rotatesA leading-order orbit picture, with precession greatly enlarged.Perihelion = closest approachMercury’s extra advance:about 43 arcseconds per century
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A slowly turning orbit. The curves are schematic successive Kepler ellipses, compressed vertically for layout, not a numerical integration of an exact relativistic orbit. The text derives the small secular advance and states its regime.
Read Chapter 16
Altitude and motion competeA net orbital-clock gain curve crosses zero near 3186 km and is positive at GPS altitude. Calculated for circular orbits around a nonrotating spherical Earth and compared with a stationary surface clock. Real GPS also models rotation, eccentricity, the geoid, and signal propagation.26 / ALTITUDE AND MOTION COMPETEaltitude (thousand km)About 3,186 km above the surface.Above: orbital clock gains time.Below: orbital clock loses time.
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Altitude and motion compete. Calculated for circular orbits around a nonrotating spherical Earth and compared with a stationary surface clock. Real GPS also models rotation, eccentricity, the geoid, and signal propagation.
Read Chapter 16
The tossed clock records more timeA parabolic height-time trajectory returns to the shelf after one second; its altitude and motion clock effects are compared. This calculation assumes uniform g, weak gravity, slow motion, and negligible launch/catch durations. The shelf clock and tossed clock share both comparison events.27 / THE TOSSED CLOCK RECORDS MORE TIMEHeight wins over speed lossFor a one-second toss near Earth:
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The tossed clock records more time. This calculation assumes uniform g, weak gravity, slow motion, and negligible launch/catch durations. The shelf clock and tossed clock share both comparison events.
Read Chapter 16
Do not merge these three radiiThree concentric circles have coordinate radii in the ratio two to three to six. Circle radii are proportional to the Schwarzschild areal coordinate r. The drawing is a radial coordinate guide, not an isometric picture of spatial proper distances.28 / DO NOT MERGE THESE THREE RADIIThree radii, three questionsFor a nonrotating, uncharged black hole.Boundary of causal escapeUnstable circular light orbitsInnermost stable circular timelike orbit
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Do not merge these three radii. Circle radii are proportional to the Schwarzschild areal coordinate rr. The drawing is a radial coordinate guide, not an isometric picture of spatial proper distances.
Read Chapter 17
A horizon changes which way the future goesAt four radii, future radial light cones tilt inward. The outward generator is vertical at the horizon and points toward smaller radius inside. The diagram uses regular ingoing coordinates, not singular Schwarzschild time. Cone slopes follow dr/dT = −1 and (r − rₛ)/(r + rₛ). T is a drawing coordinate, not a freely falling clock reading.29 / A HORIZON CHANGES WHICH WAY THE FUTURE GOESRegular horizon coordinatesOutside: an outward ray can escape.
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A horizon changes which way the future goes. The diagram uses regular ingoing coordinates, not singular Schwarzschild time. Cone slopes follow dr/dT = −1 and (r − rₛ)/(r + rₛ). T is a drawing coordinate, not a freely falling clock reading.
Read Chapter 17
Two independent patterns of strainTwo test-particle rings are distorted into ellipses along axes separated by forty-five degrees. The figure shows a local detector-frame interpretation at one phase, to first order in strain. Real astrophysical strains at Earth are vastly smaller; the interactive figure lets you move through a cycle.30 / TWO INDEPENDENT PATTERNS OF STRAINPlus polarizationAxes stretch and squeeze in alternation.Cross polarizationReference ring: gray. One exaggerated wave phase: colored. Propagation is perpendicular to the page.
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Two independent patterns of strain. The figure shows a local detector-frame interpretation at one phase, to first order in strain. Real astrophysical strains at Earth are vastly smaller; the interactive figure lets you move through a cycle.
Read Chapter 18
Expansion is not the same as accelerationRadiation and dust scale factors grow with downward curvature; a positive cosmological-constant model grows exponentially. Each ideal flat model is normalized to a(1)=1. The exponential curve uses H=2/3 in these plot units; it does not begin with a finite-time big bang in this slicing. These are separate single-component universes.31 / EXPANSION IS NOT THE SAME AS ACCELERATIONtime / reference timeExpands, but decelerates.Expands, but decelerates.Accelerated expansion.
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Expansion is not the same as acceleration. Each ideal flat model is normalized to a(1)=1a(1)=1. The exponential curve uses H=2/3H=2/3 in these plot units; it does not begin with a finite-time big bang in this slicing. These are separate single-component universes.
Read Chapter 19
A causal horizon is not just a distance scalePast and future light rays in conformal coordinates meet initial and finite future boundaries. The diagram depicts a model with finite past and future conformal intervals. Their lengths determine particle and event horizons. Other expansion histories can lack one or both boundaries; c/H is a different construction.32 / A CAUSAL HORIZON IS NOT JUST A DISTANCE SCALEThree quantities that should not share one namenowinitial conformal boundaryfinite future conformal boundaryParticle horizonHow far light has reached us since the start.Event horizonHow far a signal sent now can ever reach.An expansion scale; not generally a causal horizon.
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A causal horizon is not just a distance scale. The diagram depicts a model with finite past and future conformal intervals. Their lengths determine particle and event horizons. Other expansion histories can lack one or both boundaries; c/H is a different construction.
Read Chapter 19
Lapse and shift separate two choicesA diagonal coordinate-time step is decomposed into a normal step between slices and a tangential shift. Lapse controls normal proper-time separation; shift controls the tangential relabeling. Arrow lengths are schematic and do not represent an ordinary Euclidean decomposition of a Lorentzian norm.33 / LAPSE AND SHIFT SEPARATE TWO CHOICESA coordinate step between spatial slicesThe split is bookkeeping, not a preferred cosmic clock.next sliceinitial sliceshiftcoordinate-time step
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Lapse and shift separate two choices. Lapse controls normal proper-time separation; shift controls the tangential relabeling. Arrow lengths are schematic and do not represent an ordinary Euclidean decomposition of a Lorentzian norm.
Read Chapter 20
Two physical degrees of freedom, counted honestlyTwelve initial-data functions lose four constraints and four gauge directions, leaving four phase-space functions. This is the local canonical count for ordinary four-dimensional GR, with first-class constraints. Four remaining phase-space functions describe two propagating configuration degrees of freedom.34 / TWO PHYSICAL DEGREES OF FREEDOM, COUNTED HONESTLYCount initial data in phase spacePositions and their conjugate momenta are counted separately.constraint equationsgauge directions
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Two physical degrees of freedom, counted honestly. This is the local canonical count for ordinary four-dimensional GR, with first-class constraints. Four remaining phase-space functions describe two propagating configuration degrees of freedom.
Read Chapter 20
Same method, different curvatureA closed blue transport circuit lies on a flat polar plane and another on a round sphere. Local coframes translate coordinate steps into ruler readings. At each circuit’s starting point, an inset compares the initial and returned arrows in one orthonormal tangent plane. The plane returns the arrow unchanged. On the sphere of radius a, a positively oriented loop encloses area A and returns the arrow rotated by A/a², modulo full turns. The pictured sphere patch has area πa²/6, so its tangent-plane inset shows a 30-degree return rotation. The surface perspective does not measure that angle. These are local coframes away from the polar origin and sphere poles; the figure illustrates a consequence of Cartan’s equations, not their full derivation.35 / SAME METHOD, DIFFERENT CURVATUREFlat planeA polar grid rotates; the plane stays flat.Ruler readings from coordinate stepsinitialreturnedNo return rotation: the plane is flat.Round sphereA closed loop reveals intrinsic curvature.Ruler readings from coordinate stepsinitialreturnedReturn rotation measures the enclosed curvature.
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Same method, different curvature. The plane returns the arrow unchanged. On the sphere of radius a\text{radius }a, a positively oriented loop encloses area A\text{area }A and returns the arrow rotated by A/a2A/a^2, modulo full turns. The pictured sphere patch has area πa2/6\pi a^2/6, so its tangent-plane inset shows a 30-degree return rotation. The surface perspective does not measure that angle. These are local coframes away from the polar origin and sphere poles; the figure illustrates a consequence of Cartan’s equations, not their full derivation.
Read Chapter 21
Focusing is not yet a singularityFive straight timelike paths converge to a caustic, alongside the focusing time bound. This flat-spacetime schematic intentionally has no curvature. Raychaudhuri can force a congruence to focus, but singularity theorems require additional causal and global assumptions to conclude geodesic incompleteness.36 / FOCUSING IS NOT YET A SINGULARITYpositionaffine timecausticConvergence can have a deadlineBut straight lines can cross in flat spacetime.Global hypotheses do the extra work.
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Focusing is not yet a singularity. This flat-spacetime schematic intentionally has no curvature. Raychaudhuri can force a congruence to focus, but singularity theorems require additional causal and global assumptions to conclude geodesic incompleteness.
Read Chapter 22
Two areas constrain one remnantTwo equal initial Schwarzschild horizons are compared with a limiting final circle whose area equals their sum. The equality illustration is the bound’s limiting case, not an achievable merger prediction. Initial binding energy is neglected, and all holes are assumed nonspinning; Kerr horizons require a different area formula.37 / TWO AREAS CONSTRAIN ONE REMNANTArea increase is a bound, not an efficiency predictionAn ideal comparison of nonspinning black holes.Minimum final mass from area alone
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Two areas constrain one remnant. The equality illustration is the bound’s limiting case, not an achievable merger prediction. Initial binding energy is neglected, and all holes are assumed nonspinning; Kerr horizons require a different area formula.
Read Chapter 22
The information question has a shapeA schematic entropy curve rises and returns to zero, while a dashed comparison curve continues to rise. Axes are qualitative and the curves are not a quantitative evaporation solution. A final pure radiation state has zero fine-grained entropy for the whole radiation system; individual portions can remain mixed.38 / THE INFORMATION QUESTION HAS A SHAPEfraction of evaporation elapsedradiation entropy (schematic)Unitary expectationRise, then fall as information is recovered.Uncorrected semiclassical trendKeeps growing in the leading approximation.A thermal-looking spectrum can stillcontain correlations between quanta.
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The information question has a shape. Axes are qualitative and the curves are not a quantitative evaporation solution. A final pure radiation state has zero fine-grained entropy for the whole radiation system; individual portions can remain mixed.
Read Chapter 22
A theory has a resolution scaleA scale axis progresses from long wavelengths with controlled corrections toward a cutoff with no small expansion parameter. This is a hierarchy diagram, not a measured error curve. The cutoff depends on the theory and physical setting; the text distinguishes gravitational scales from other possible new-physics scales.39 / A THEORY HAS A RESOLUTION SCALEAsk a theory questions at the scale it resolvesThe expansion parameter is energy / cutoff, or microscopic length / wavelength.long wavelengthshorter wavelengthnear the cutoffcontrolled correctionsno small parameterLeading theory + smaller terms + smaller terms + …The coefficients encode unresolved physics; the ordering makes low-energy calculations useful.
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A theory has a resolution scale. This is a hierarchy diagram, not a measured error curve. The cutoff depends on the theory and physical setting; the text distinguishes gravitational scales from other possible new-physics scales.
Read Chapter 23
From a question to a measurementSix stages organize a general relativity calculation, beginning with the measurement you want to predict. A clock comparison may need only the metric; a trajectory can need the connection; a tidal measurement needs curvature. The chosen observer and a familiar limit make the prediction physically interpretable.40 / FROM A QUESTION TO A MEASUREMENTChoose a measurementClock, ray, orbit, or tideSpecify the metricCoordinates and unitsChoose the observerFour-velocity or tetradUse the needed geometryConnection, curvature, or neitherEvaluate the predictionA physical comparisonCheck a familiar limitSigns, units, and physicsCompute only what your measurement needs.
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From a question to a measurement. A clock comparison may need only the metric; a trajectory can need the connection; a tidal measurement needs curvature. The chosen observer and a familiar limit make the prediction physically interpretable.
Read Chapter 24

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