The Linear Algebra of Space-Time: Length Contraction and Time - PowerPoint PPT Presentation

The Linear Algebra of Space-Time: Length Contraction and Time Dilation Near the Speed of Light Ron Umble, speaker Millersville Univ of Pennsylvania MU/F&M Mathematics Colloquium November 18, 2010 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 1 / 30

Minkowski space Pseudo inner product space R 2 1 = f ( t , x ) : t , x 2 R g (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

Minkowski space Pseudo inner product space R 2 1 = f ( t , x ) : t , x 2 R g Pseudo inner product h ( t 1 , x 1 ) , ( t 2 , x 2 ) i = t 1 t 2 � x 1 x 2 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

Minkowski space Pseudo inner product space R 2 1 = f ( t , x ) : t , x 2 R g Pseudo inner product h ( t 1 , x 1 ) , ( t 2 , x 2 ) i = t 1 t 2 � x 1 x 2 p t 2 � x 2 ranges over all non-negative Minkowski norm k ( t , x ) k = real and positive imaginary values (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

Curves of constant Minkowski norm Points on curves of constant Minkowski norm satisfy t 2 � x 2 = a 2 (1) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

Curves of constant Minkowski norm Points on curves of constant Minkowski norm satisfy t 2 � x 2 = a 2 (1) The parameter a determines three families of curves (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

Curves of constant Minkowski norm Points on curves of constant Minkowski norm satisfy t 2 � x 2 = a 2 (1) The parameter a determines three families of curves a = 0 de…nes the light cone x = � t (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

Curves of constant Minkowski norm Points on curves of constant Minkowski norm satisfy t 2 � x 2 = a 2 (1) The parameter a determines three families of curves a = 0 de…nes the light cone x = � t a 2 R + de…nes a real hyperbolic circle of radius a (the hyperbola t 2 � x 2 = a 2 inside the light cone) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

Curves of constant Minkowski norm Points on curves of constant Minkowski norm satisfy t 2 � x 2 = a 2 (1) The parameter a determines three families of curves a = 0 de…nes the light cone x = � t a 2 R + de…nes a real hyperbolic circle of radius a (the hyperbola t 2 � x 2 = a 2 inside the light cone) a = ib 2 i R + de…nes an imaginary hyperbolic circle of radius ib (the hyperbola x 2 � t 2 = b 2 outside the light cone) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

Three kinds of vectors Isotropic vectors: zero Minkowski norm; live on the light-cone (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 4 / 30

Three kinds of vectors Isotropic vectors: zero Minkowski norm; live on the light-cone Time-like vectors: positive real Minkowski norm; live inside the light-cone (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 4 / 30

Three kinds of vectors Isotropic vectors: zero Minkowski norm; live on the light-cone Time-like vectors: positive real Minkowski norm; live inside the light-cone Space-like vectors: positive imaginary Minkowski norm; live outside the light-cone (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 4 / 30

Euclidean Isometries Rotations ρ θ about the origin through angle θ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

Euclidean Isometries Rotations ρ θ about the origin through angle θ Re‡ections σ θ in lines through the origin with inclination θ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

Euclidean Isometries Rotations ρ θ about the origin through angle θ Re‡ections σ θ in lines through the origin with inclination θ Rotations …x circles centered at the origin and send lines through the origin to lines through the origin (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

Euclidean Isometries Rotations ρ θ about the origin through angle θ Re‡ections σ θ in lines through the origin with inclination θ Rotations …x circles centered at the origin and send lines through the origin to lines through the origin Re‡ections …x their re‡ecting lines pointwise (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

Matrix Representation Rotations represented by orthogonal matrices with determinant + 1 : � cos θ � � sin θ [ ρ θ ] = sin θ cos θ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 6 / 30

Matrix Representation Rotations represented by orthogonal matrices with determinant + 1 : � cos θ � � sin θ [ ρ θ ] = sin θ cos θ Re‡ections represented by orthogonal matrices with determinant � 1 : � � cos 2 θ � sin 2 θ [ σ θ ] = � sin 2 θ � cos 2 θ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 6 / 30

Matrix Representation Rotations represented by orthogonal matrices with determinant + 1 : � cos θ � � sin θ [ ρ θ ] = sin θ cos θ Re‡ections represented by orthogonal matrices with determinant � 1 : � � cos 2 θ � sin 2 θ [ σ θ ] = � sin 2 θ � cos 2 θ These form the orthogonal group O ( 2 ) , which has two components: � cos θ � � cos θ � � sin θ sin θ [ ρ θ ] = and [ ρ θ ] [ σ 0 ] = sin θ cos θ sin θ � cos θ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 6 / 30

The Orthogonal Group O(2) The component � cos θ � � 1 � � 0 � � sin θ 0 � 1 = cos θ + sin θ sin θ cos θ 0 1 1 0 parametrizes the circle C 1 : u 2 1 + u 2 2 = 2 in the 2-plane spanned by ( " # " #) 1 � 1 0 0 p p 2 2 B 1 = u 1 = , u 2 = 1 1 0 0 p p 2 2 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 7 / 30

The Orthogonal Group O(2) The component � cos θ � � 1 � � 0 � � sin θ 0 � 1 = cos θ + sin θ sin θ cos θ 0 1 1 0 parametrizes the circle C 1 : u 2 1 + u 2 2 = 2 in the 2-plane spanned by ( " # " #) 1 � 1 0 0 p p 2 2 B 1 = u 1 = , u 2 = 1 1 0 0 p p 2 2 � 1 � p 0 The trivial rotation [ ρ 0 ] = = 2 u 1 + 0 u 2 is on this circle 0 1 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 7 / 30

The Orthogonal Group O(2) The component � cos θ � � 1 � � 0 � sin θ 0 1 = cos θ + sin θ sin θ � cos θ 0 � 1 1 0 parametrize the circle C 2 : u 2 3 + u 2 4 = 2 in the 2-plane spanned by ( " # " #) 1 1 0 0 p p 2 2 B 2 = u 3 = , u 4 = � 1 1 0 0 p p 2 2 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 8 / 30

The Orthogonal Group O(2) The component � cos θ � � 1 � � 0 � sin θ 0 1 = cos θ + sin θ sin θ � cos θ 0 � 1 1 0 parametrize the circle C 2 : u 2 3 + u 2 4 = 2 in the 2-plane spanned by ( " # " #) 1 1 0 0 p p 2 2 B 2 = u 3 = , u 4 = � 1 1 0 0 p p 2 2 B 1 [ B 2 linearly independent ) C 1 \ C 2 = ? and C 1 [ C 2 = O ( 2 ) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 8 / 30

Minkowski Isometries Hyperbolic rotations represented by the matrices � cosh θ � sinh θ [ R θ ] = sinh θ cosh θ or in coordinates by R θ ( t , x ) = ( t cosh θ + x sinh θ , t sinh θ + x cosh θ ) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 9 / 30

Minkowski Isometries Hyperbolic rotations represented by the matrices � cosh θ � sinh θ [ R θ ] = sinh θ cosh θ or in coordinates by R θ ( t , x ) = ( t cosh θ + x sinh θ , t sinh θ + x cosh θ ) R θ …xes hyperbolic circles: t ) 2 � ( ¯ x ) 2 = ( t cosh θ + x sinh θ ) 2 � ( t sinh θ + x cosh θ ) 2 = t 2 � x 2 ( ¯ (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 9 / 30

Minkowski Isometries Hyperbolic rotations represented by the matrices � cosh θ � sinh θ [ R θ ] = sinh θ cosh θ or in coordinates by R θ ( t , x ) = ( t cosh θ + x sinh θ , t sinh θ + x cosh θ ) R θ …xes hyperbolic circles: t ) 2 � ( ¯ x ) 2 = ( t cosh θ + x sinh θ ) 2 � ( t sinh θ + x cosh θ ) 2 = t 2 � x 2 ( ¯ R θ sends lines through the origin to lines through the origin: R θ ( a , 0 ) = a ( cosh θ , sinh θ ) (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 9 / 30

Minkowski Isometries Re‡ections S 0 in the t -axis and S ∞ in the x -axis are represented by � 1 � � � 1 � 0 0 [ S 0 ] = and [ S ∞ ] = 0 � 1 0 1 (MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 10 / 30