Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity Rob Salgado UW-La Crosse Dept. of Physics
outline • motivations • the Cayley-Klein Geometries (various starting points) • Relativity (Trilogy of the Surveyors) • development of “Spacetime Trigonometry” as a unified approach to the geometry of Galilean and Special Relativity • [affine] Cayley-Klein Geometries (tour of more starting points) • Spacetime Trigonometry : “geometry of the Galilean spacetime as a bridge to Special Relativity” (How can some of the ideas be introduced to a physics student without all of the machinery that is available?
The Clock Effect / Twin Paradox an infamous puzzle
The Clock Effect / Twin Paradox SPACETIME DIAGRAMS But…Where do you place But…Where do you place the ticks of each clock? the ticks of each clock? Spacetime Diagrams are the best way to analyze and interpret Relativity . “common sense” Galilean Relativity Special Relativity t runs upwards t runs upwards
• Indeed, Spacetime Geometry has some strange triangles: Too many “algebraic formulas”, we need more GEOMETRICAL intuition
Spacetime Trigonometry Trigonometry Spacetime GOAL: Teach relativity by developing geometric intuition about spacetime. HOW? Exploit the trigonometric analogies can its conceptual •Euclidean space aspects be slipped into introductory •Galilean spacetime physics?? •Einstein-Minkowski spacetime
rotation boost boost EUCLIDEAN GALILEAN MINKOWSKIAN
But first…. the classical Geometries (Cayley-Klein) measure of Distance between Points “elliptic” “parabolic” “hyperbolic” Elliptic Euclidean Hyperbolic Initially-parallel lines… EUCLID’s FIFTH http://astronomy.swin.edu.au/cosmos/C/Critical+Density EUCLID’s FIFTH (Playfair) (Playfair) Wikipedia Given a line and Given a line and a point not on that line, a point not on that line, there exists there exists precisely one line precisely one line through that point through that point which does not intersect which does not intersect (i.e., `` is parallel to '') Zero One Infinitely many (i.e., `` is parallel to '') the given line. parallels parallel parallels the given line.
the Cayley-Klein Geometries Sommerville uses “duality” between points and lines Projective Geometry: the geometry of perspective Duality: symmetry between points and lines Joined Any two distinct points are incident by with exactly one line. Meet Any two distinct lines are incident at with exactly one point.
the Cayley-Klein Geometries Sommerville measure of Distance between Points uses “duality” between points “elliptic” “parabolic” “hyperbolic” and lines Elliptic Euclidean Hyperbolic measure of Angle between Lines MASS-SHELL in “elliptic” Special Relativity co-Euclidean doubly- co-Minkowskian Parabolic ANTI- “parabolic” NEWTON- GALILEAN NEWTON- HOOKE RELATIVITY HOOKE co-Hyperbolic Minkowskian doubly- Hyperbolic ANTI- SPECIAL “hyperbolic” DE-SITTER RELATIVITY DE-SITTER
measure of Distance between Points elliptic parabolic hyperbolic [Initially parallel lines…] intrinsic Curvature Positive Zero Negative Elliptic Euclidean Hyperbolic measure of Angle between Lines elliptic MASS-SHELL in Special Metric Signature Relativity co-Euclidean doubly- co-Minkowskian parabolic Parabolic ANTI- NEWTON- GALILEAN NEWTON- HOOKE RELATIVITY HOOKE hyperbolic co-Hyperbolic Minkowskian doubly- Hyperbolic ANTI- SPECIAL DE-SITTER RELATIVITY DE-SITTER
measure of Distance between Points elliptic parabolic hyperbolic intrinsic Curvature k Positive Zero Negative Elliptic Euclidean Hyperbolic measure of Angle between Lines elliptic MASS-SHELL in Special Relativity Metric Signature co-Euclidean doubly-Parabolic co-Minkowskian parabolic ANTI- GALILEAN NEWTON-HOOKE RELATIVITY NEWTON-HOOKE hyperbolic co-Hyperbolic Minkowskian doubly-Hyperbolic ANTI- DE-SITTER SPECIAL RELATIVITY DE-SITTER
PHYSICS: Trilogy of the Surveyors PHYSICS: Trilogy of the Surveyors Euclid’s Galileo’s Einstein’s Minkowski’s Geometry Relativity Relativity Spacetime (300 BC) (1632) (1905) Geometry (1908) simultaneity (“same t ”) simultaneity (“same t ”) is absolute is absolute simultaneity is not absolute simultaneity is not absolute inspired by the “Parable of the Surveyors” in inspired by the “Parable of the Surveyors” in Spacetime Physics by Taylor and Wheeler Spacetime Physics by Taylor and Wheeler
CIRCLES and the METRIC (separation of points) CIRCLES and the METRIC (separation of points) Proper [Wristwatch] time, “ “Space Space” ” Proper [Wristwatch] time, spatial distance radius vector is a timelike-vector spacelike-vector is tangent to the circle, perpendicular to timelike null-vector has y y y t t t
HYPERCOMPLEX NUMBERS HYPERCOMPLEX NUMBERS Maximum Signal Speed Maximum Signal Speed y y y t t t
HYPERCOMPLEX NUMBERS HYPERCOMPLEX NUMBERS Maximum Signal Speed Maximum Signal Speed Do formal calculations Do formal calculations in which is treated in which is treated algebraically but algebraically but never evaluated never evaluated until the last step. until the last step. All physical quantities All physical quantities involve alone. involve alone.
ANGLE (separation of lines) ANGLE (separation of lines) Rapidity Rapidity (Yaglom) Galilean Trig Functions GENERALIZED Trig Functions
SLOPE = TANGENT( ANGLE ) SLOPE = TANGENT( ANGLE ) Velocity = TANH ( Rapidity ) Velocity = TANH ( Rapidity ) EUC GAL MIN
EULER and TRIGONOMETRIC functions EULER and TRIGONOMETRIC functions Relativistic “ “factors factors” ” Relativistic
Projection onto a line Projection onto a line Time dilation Time dilation EUC GAL MIN
an example of applied Spacetime Trigononometry The Clock Effect / Twin Paradox Euclidean Galilean Minkowskian Galilean Relativity Special Relativity Galilean Relativity Special Relativity
Law of COSINES Law of COSINES Clock Effect Clock Effect
EUC GAL MIN Boost transformations Boost transformations Rotations Rotations
Eigenvectors and Eigenvalues Eigenvalues Eigenvectors and “Absolute Absolute” ” invariants invariants “ EUC GAL MIN
An interesting trigonometry problem An interesting trigonometry problem Doppler effect (unified) Doppler effect (unified)
An interesting trigonometry problem An interesting trigonometry problem Doppler effect (unified) Doppler effect (unified)
Curve of constant curvature Curve of constant curvature Uniformly accelerated observer (unified) Uniformly accelerated observer (unified)
EUCLID’ ’s s FIRST FIRST EUCLID Causal Structure of Spacetime Spacetime Causal Structure of EUCLID’s FIRST EUCLID’s FIRST “To draw a straight line from any point to any point.” “To draw a straight line from any point to any point.” EUCLID’s FIFTH EUCLID’s FIRST EUCLID’s FIFTH EUCLID’s FIRST Spacetime (Playfair) (like Playfair dualized) (Playfair) (like Playfair dualized) Given a line and Given a point and geometries fail Given a line and Given a point and a point not on that line, a line not through that point, a point not on that line, a line not through that point, Euclid’s First there exists there exists there exists there exists precisely one line no point Postulate! precisely one line no point through that point on that line through that point on that line which does not intersect which cannot be joined to Infinitely- which does not intersect which cannot be joined to (i.e., `` is parallel to '') (i.e., `` is parallel / inaccessible to '') many (i.e., `` is parallel to '') (i.e., `` is parallel / inaccessible to '') the given line. the given point by an ordinary line. points the given line. the given point by an ordinary line. One point
Visualizing Tensor Algebra advanced topic: The “circle” is a visualization of its metric tensor g ab ! Galilean Euclidean Minkowskian metric metric metric
Some problems I am working on: Interpret the “Law of Sines” physically. (Interpret a result from [Euclidean] geometry in terms of a physical situation in spacetime.) Collisions (in Galilean and Special Relativity) – elastic collisions, inelastic collisions, coefficient of restitution; energy (Kinetic and Rest energy), spatial-momentum Hypercomplex numbers – do Geometry as one does with Complex Numbers (Dual numbers are used in robotics. How?) Differential Geometry with “degenerate metrics” (Galilean limits) Connection to Norman Wildberger’s Universal Hyperbolic Trigonometry? Electromagnetism (Maxwell’s Equations) Galilean-invariant version (Jammer and Stachel) “If Maxwell had worked between Ampere and Faraday?” De Sitter spacetimes as analogues of Elliptic and Hyperbolic Geometries
Recommend
More recommend
