Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity

Transcription

1 Spacetime Trigonometry: a Cayley-Klein Geomety approach to Special and General Relativity Rob Salgado UW-La Crosse Dept. of Physics

2 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?

3 an infamous puzzle The Clock Effect / Twin Paradox

4 The Clock Effect / Twin Paradox SPACETIME DIAGRAMS But Where do do you place the ticks of of each clock? Spacetime Diagrams are the best way to analyze and interpret Relativity. common sense Galilean Relativity ttruns runs upwards Special Relativity

5 Indeed, Spacetime Geometry has some strange triangles: Too many algebraic formulas, we need more GEOMETRICAL intuition

6 Spacetime Trigonometry GOAL: Teach relativity by developing geometric intuition about spacetime. HOW? Exploit the trigonometric analogies Euclidean space Galilean spacetime Einstein-Minkowski spacetime can its conceptual aspects be slipped into introductory physics??

7 MINKOWSKIAN boost GALILEAN boost EUCLIDEAN rotation

8 But first.the classical Geometries (Cayley-Klein) measure of Distance between Points Initially-parallel lines elliptic Elliptic parabolic Euclidean hyperbolic Hyperbolic EUCLID s FIFTH FIFTH (Playfair) Given Given a line line and and a point point not not on on that that line, line, there there exists exists precisely one one line line through that that point point which which does does not not intersect (i.e., (i.e., ``is ``is parallel to'') to'') the the given given line. line. Zero parallels One parallel Wikipedia Infinitely many parallels

9 the Cayley-Klein Geometries Sommerville uses duality between points and lines Projective Geometry: the geometry of perspective Duality: symmetry between points and lines Any two distinct points are incident with exactly one line. Any two distinct lines are incident with exactly one point. Joined by Meet at

10 the Cayley-Klein Geometries Sommerville uses duality between points and lines measure of Distance between Points elliptic parabolic hyperbolic measure of Angle between Lines elliptic parabolic hyperbolic Elliptic co-euclidean ANTI- NEWTON- HOOKE co-hyperbolic ANTI- DE-SITTER Euclidean doubly- Parabolic GALILEAN RELATIVITY Minkowskian SPECIAL RELATIVITY Hyperbolic MASS-SHELL in Special Relativity co-minkowskian NEWTON- HOOKE doubly- Hyperbolic DE-SITTER

11 measure of Distance between Points elliptic parabolic hyperbolic [Initially parallel lines ] intrinsic Curvature Positive Zero Negative measure of Angle between Lines hyperbolic parabolic elliptic Metric Signature Elliptic co-euclidean ANTI- NEWTON- HOOKE co-hyperbolic ANTI- DE-SITTER Euclidean doubly- Parabolic GALILEAN RELATIVITY Minkowskian SPECIAL RELATIVITY Hyperbolic MASS-SHELL in Special Relativity co-minkowskian NEWTON- HOOKE doubly- Hyperbolic DE-SITTER

12 elliptic Positive measure of Distance between Points parabolic intrinsic Curvature k Zero hyperbolic Negative measure of Angle between Lines hyperbolic parabolic elliptic Metric Signature Elliptic co-euclidean ANTI- NEWTON-HOOKE co-hyperbolic ANTI- DE-SITTER Euclidean doubly-parabolic GALILEAN RELATIVITY Minkowskian SPECIAL RELATIVITY Hyperbolic MASS-SHELL in Special Relativity co-minkowskian NEWTON-HOOKE doubly-hyperbolic DE-SITTER

13 PHYSICS: Trilogy of the Surveyors Euclid s Geometry (300 BC) Galileo s Relativity (1632) Einstein s Relativity (1905) Minkowski s Spacetime Geometry (1908) simultaneity ( same t ) t ) is is absolute simultaneity is is not notabsolute inspired by by the the Parable of of the the Surveyors in in Spacetime Physics by by Taylor and and Wheeler

14 CIRCLES and the METRIC (separation of points) Proper [Wristwatch] time, Space 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

15 HYPERCOMPLEX NUMBERS Maximum Signal Speed y y y t t t

16 HYPERCOMPLEX NUMBERS Maximum Signal Speed Do Do formal formal calculations in in which which is is treated treated algebraically but but never never evaluated until until the the last last step. step. All All physical quantities involve involve alone. alone.

17 ANGLE (separation of lines) Rapidity (Yaglom) Galilean Trig Functions GENERALIZED Trig Functions

18 SLOPE = TANGENT( ANGLE ) Velocity = TANH ( Rapidity ) EUC GAL MIN

19 EULER and TRIGONOMETRIC functions Relativistic factors

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21 Projection onto a line Time dilation EUC GAL MIN

22 an example of applied Spacetime Trigononometry The Clock Effect / Twin Paradox Euclidean Galilean Minkowskian Galilean Galilean Relativity Relativity Special Special Relativity Relativity

23 Law of COSINES Clock Effect

24 Rotations Boost transformations EUC GAL MIN

25 Eigenvectors and Eigenvalues Absolute invariants EUC GAL MIN

26 An interesting trigonometry problem Doppler effect (unified)

27 An interesting trigonometry problem Doppler effect (unified)

28 Curve of constant curvature Uniformly accelerated observer (unified)

29 EUCLID s FIRST Causal Structure of Spacetime EUCLID s FIFTH FIFTH (Playfair) Given Given a line lineand a point pointnot not on onthat line, line, there there exists exists precisely one oneline through that that point point which which does does not not intersect (i.e., (i.e., ``is ``is parallel to'') to'') the the given given line. line. EUCLID s FIRST FIRST To To draw draw a straight straight line line from from any any point point to to any any point. point. EUCLID s FIRST FIRST (like (like Playfair dualized) Given Given a point pointand a line linenot not through that that point, point, there there exists exists no nopoint on onthat line line which which cannot cannot be be joined joined to to (i.e., (i.e., ``is ``is parallel // inaccessible to'') to'') the the given given point point by by an an ordinary line. line. One point Spacetime geometries fail Euclid s First Postulate! Infinitelymany points

30 advanced topic: Visualizing Tensor Algebra The circle is a visualization of its metric tensor g ab! Euclidean metric Galilean metric Minkowskian metric

31 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

32 conclusions Cayley-Klein geometry provides geometrical analogies which can be given kinematical interpretations. starting with the Galilean case, onto the Special Relativistic case, and further onto the desitter spacetimes (the simplest General Relativistic cases) may be an easier approach to learning Relativity Galilean Limits are clarified

33 Energy-Momentum Space Two identical particles with mass m: one at rest in this frame, the other traveling with velocity p E mass-shell p E In component-form, the energy-momentum vector is Note: In the Galilean case, the energy component is always the rest mass.

34 Conservation of Energy-Momentum A particle with rest-mass M decays into two particles with rest-masses m 1 and m 2. Conservation: Geometrically, this is a triangle formed with future-timelike-vectors. In the Galilean case, energy conservation implies conservation of total rest mass.

35 Energy-momentum decay Griffiths (Elementary Particles) Law of Cosines: Law of Sines yields: Multiply by half of the product of the three masses Generalized Heron formula

36 Galilean-invariant Electromagnetism (Jammer and Stachel)