The Mathematics of Special Relativity Jared Ruiz Advised by Dr. - PowerPoint PPT Presentation

Essentials of Relativity The Space of Relativity The Gamma Factor 2 c ( k 2 − 1) t 2 ( k 2 + 1) t = c ( k 2 − 1) 1 v = d B = ( k 2 + 1) . 1 t B Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Gamma Factor 2 c ( k 2 − 1) t 2 ( k 2 + 1) t = c ( k 2 − 1) 1 v = d B = ( k 2 + 1) . 1 t B vk 2 + v ck 2 − c = ck 2 − vk 2 c + v = k 2 ( c − v ) c + v = � c + v k = c − v > 1 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Gamma Factor 2 c ( k 2 − 1) t 2 ( k 2 + 1) t = c ( k 2 − 1) 1 v = d B = ( k 2 + 1) . 1 t B vk 2 + v ck 2 − c = ck 2 − vk 2 c + v = k 2 ( c − v ) c + v = � c + v k = c − v > 1 . kt = ( k 2 + 1) t time E to B measured by O ′ = t B time E to B measured by O = γ ( v ) 2 kt Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Gamma Factor 2 c ( k 2 − 1) t 2 ( k 2 + 1) t = c ( k 2 − 1) 1 v = d B = ( k 2 + 1) . 1 t B vk 2 + v ck 2 − c = ck 2 − vk 2 c + v = k 2 ( c − v ) c + v = � c + v k = c − v > 1 . kt = ( k 2 + 1) t time E to B measured by O ′ = t B time E to B measured by O = γ ( v ) 2 kt γ ( v ) = ( k 2 + 1) t 1 = 2 kt � 1 − v 2 / c 2 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Lorentz Transformation Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Lorentz Transformation Theorem The inertial coordinate systems set up by two observers are related by: t = t ′ + ( vx ′ / c 2 )    � 1 − ( v / c ) 2  x ′ + vt ′ x =    � 1 − ( v / c ) 2 We can write this more concisely as � ct � � ct ′ � � � 1 v / c = γ ( v ) (1) x v / c 1 x ′ where v is the relative velocity. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Proof Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Proof First consider two observers moving at constant speeds. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Proof First consider two observers moving at constant speeds. We express an event in terms of both coordinate systems. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Proof First consider two observers moving at constant speeds. We express an event in terms of both coordinate systems. We substitute in for Bondi’s k -factor � ct � k + k − 1 � � ct ′ k − k − 1 � = 1 � k − k − 1 k + k − 1 x ′ x 2 After some algebra and simplification, we have our result ✷ . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Proof First consider two observers moving at constant speeds. We express an event in terms of both coordinate systems. We substitute in for Bondi’s k -factor � ct � k + k − 1 � � ct ′ k − k − 1 � = 1 � k − k − 1 k + k − 1 x ′ x 2 After some algebra and simplification, we have our result ✷ . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Velocity Boost In four dimensions,    γ γ v / c 0 0   ct ′  ct x γ v / c γ 0 0 x ′        = (2)       0 0 1 0 y ′ y      z 0 0 0 1 z ′ where v is the relative velocity and γ = γ ( v ). Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Velocity Boost In four dimensions,    γ γ v / c 0 0   ct ′  ct x γ v / c γ 0 0 x ′        = (2)       0 0 1 0 y ′ y      z 0 0 0 1 z ′ where v is the relative velocity and γ = γ ( v ). Definition The 4 × 4 matrix in (2) is known as the boost , denoted L v . With the properties of the boost, we can look at how an observer would judge a moving particle’s velocity. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Theorem A particle cannot travel faster than c , the velocity of light. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Theorem A particle cannot travel faster than c , the velocity of light. Proof: � ct ′ � � � � � 1 − u / c ct = γ ( u ) . x ′ − u / c 1 − vt + b Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Theorem A particle cannot travel faster than c , the velocity of light. Proof: � ct ′ � � � � � 1 − u / c ct = γ ( u ) . x ′ − u / c 1 − vt + b w = − dx ′ v + u dt ′ = 1 + uv / c 2 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity v + u w = 1 + uv / c 2 . Letting | u | < c and | v | < c , we see that | w | < c since Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity v + u w = 1 + uv / c 2 . Letting | u | < c and | v | < c , we see that | w | < c since − c 2 − uv ( c − u )( c − v ) > 0 ⇐ ⇒ − ( u + v ) c > 1 + uv � � u + v < c c 2 w < c . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity v + u w = 1 + uv / c 2 . Letting | u | < c and | v | < c , we see that | w | < c since − c 2 − uv ( c − u )( c − v ) > 0 ⇐ ⇒ − ( u + v ) c > 1 + uv � � u + v < c c 2 w < c . − c 2 − uv ( c + u )( c + v ) > 0 ⇐ ⇒ ( u + v ) c > 1 + uv � � u + v > − c c 2 w > − c . ✷ Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity In four-dimensions, we can say that Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity In four-dimensions, we can say that    ct ′  ct x x ′      = L (3)     y ′ y    z ′ z � 1 � 1 � � 0 0 where L = L v and H , K are 3 × 3 K T 0 H 0 orthogonal matrices. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity In four-dimensions, we can say that    ct ′  ct x x ′      = L (3)     y ′ y    z ′ z � 1 � 1 � � 0 0 where L = L v and H , K are 3 × 3 K T 0 H 0 orthogonal matrices. Definition The matrix L in (3) is a Lorentz transformation if Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity In four-dimensions, we can say that    ct ′  ct x x ′      = L (3)     y ′ y    z ′ z � 1 � 1 � � 0 0 where L = L v and H , K are 3 × 3 K T 0 H 0 orthogonal matrices. Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity In four-dimensions, we can say that    ct ′  ct x x ′      = L (3)     y ′ y    z ′ z � 1 � 1 � � 0 0 where L = L v and H , K are 3 × 3 K T 0 H 0 orthogonal matrices. Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  . L is orthochronous if l 1 , 1 > 0,   0 0 − 1 0  0 0 0 − 1 where l 1 , 1 is the first entry of the first row of L Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. x 2 + y 2 + z 2 . � The distance D from a point to the origin is D = Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. x 2 + y 2 + z 2 . � The distance D from a point to the origin is D = Emit a light pulse when t = x = y = z = 0. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. x 2 + y 2 + z 2 . � The distance D from a point to the origin is D = Emit a light pulse when t = x = y = z = 0. x 2 + y 2 + z 2 = D . � It arrives at ( ct , x , y , z ) if ct = Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. x 2 + y 2 + z 2 . � The distance D from a point to the origin is D = Emit a light pulse when t = x = y = z = 0. x 2 + y 2 + z 2 = D . � It arrives at ( ct , x , y , z ) if ct = c 2 t 2 = D 2 , and c 2 t 2 − x 2 − y 2 − z 2 = 0. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition In special relativity, the space we live in is called Minkowski Space , denoted by M = R × R 3 where R = R × 0 is the time axes, and R 3 = 0 × R 3 the space axes. x 2 + y 2 + z 2 . � The distance D from a point to the origin is D = Emit a light pulse when t = x = y = z = 0. x 2 + y 2 + z 2 = D . � It arrives at ( ct , x , y , z ) if ct = c 2 t 2 = D 2 , and c 2 t 2 − x 2 − y 2 − z 2 = 0. Definition In Minkowski space, the interval between any two events x = ( t 1 , x 1 , y 1 , z 1 ) and y = ( t 2 , x 2 , y 2 , z 2 ) is defined to be c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Invariance of the Interval If c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = 0 for an observer O , Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Invariance of the Interval If c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = 0 for an observer O , then 1 ) 2 − ( x ′ 1 ) 2 − ( y ′ 1 ) 2 − ( z ′ 1 ) 2 = 0 c 2 ( t ′ 2 − t ′ 2 − x ′ 2 − y ′ 2 − z ′ for an observer O ′ , moving with constant velocity relative to O . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Invariance of the Interval If c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = 0 for an observer O , then 1 ) 2 − ( x ′ 1 ) 2 − ( y ′ 1 ) 2 − ( z ′ 1 ) 2 = 0 c 2 ( t ′ 2 − t ′ 2 − x ′ 2 − y ′ 2 − z ′ for an observer O ′ , moving with constant velocity relative to O . Because of this, we say the interval is invariant . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Inner Product Definition In M , two objects X = ( X 0 , X 1 , X 2 , X 3 ) and X ′ = ( X ′ 0 , X ′ 1 , X ′ 2 , X ′ 3 ) are called four-vectors if X = LX ′ where L is the general Lorentz transformation. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Inner Product Definition In M , two objects X = ( X 0 , X 1 , X 2 , X 3 ) and X ′ = ( X ′ 0 , X ′ 1 , X ′ 2 , X ′ 3 ) are called four-vectors if X = LX ′ where L is the general Lorentz transformation. For x = ( ct 1 , x 1 , y 1 , z 1 ) , y = ( ct 2 , x 2 , y 2 , z 2 ) ∈ M , the displacement four-vector X = y − x = c ( t 2 − t 1 ) + ( x 2 − x 1 ) + ( y 2 − y 1 ) + ( z 2 − z 1 ) . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Inner Product Definition In M , two objects X = ( X 0 , X 1 , X 2 , X 3 ) and X ′ = ( X ′ 0 , X ′ 1 , X ′ 2 , X ′ 3 ) are called four-vectors if X = LX ′ where L is the general Lorentz transformation. For x = ( ct 1 , x 1 , y 1 , z 1 ) , y = ( ct 2 , x 2 , y 2 , z 2 ) ∈ M , the displacement four-vector X = y − x = c ( t 2 − t 1 ) + ( x 2 − x 1 ) + ( y 2 − y 1 ) + ( z 2 − z 1 ) . Definition The inner product between two four-vectors X , Y ∈ M is: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Predictions Definition The four-velocity of a particle, ( V 0 , V 1 , V 2 , V 3 ), is given by: V 0 = c dt V 1 = dx V 2 = dy V 3 = dz d τ , d τ , d τ , d τ . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Predictions Definition The four-velocity of a particle, ( V 0 , V 1 , V 2 , V 3 ), is given by: V 0 = c dt V 1 = dx V 2 = dy V 3 = dz d τ , d τ , d τ , d τ . Theorem Let an observer O be moving with constant velocity V . Then O sees two events A and B as being simultaneous if and only if the displacement g ( X , V ) = 0, where X = B − A . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Predictions Definition The four-velocity of a particle, ( V 0 , V 1 , V 2 , V 3 ), is given by: V 0 = c dt V 1 = dx V 2 = dy V 3 = dz d τ , d τ , d τ , d τ . Theorem Let an observer O be moving with constant velocity V . Then O sees two events A and B as being simultaneous if and only if the displacement g ( X , V ) = 0, where X = B − A . Corollary Two events which are simultaneous for one observer may not be simultaneous for another observer moving at constant velocity with the respect to the first observer. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Lorentz Contraction Theorem If a rod has length R 0 in a rest frame, then in an inertial coordinate system oriented in the direction of the unit vector e and moving with respect to the rod with velocity v , the length of the rod is: Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity The Lorentz Contraction Theorem If a rod has length R 0 in a rest frame, then in an inertial coordinate system oriented in the direction of the unit vector e and moving with respect to the rod with velocity v , the length of the rod is: √ c 2 − v 2 R 0 R = � c 2 − v 2 sin 2 ( θ ) where θ is the angle between e and v . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The rest mass m 0 = m (0) of a body is the mass of a body measured in an inertial coordinate system in which the body is at rest. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The rest mass m 0 = m (0) of a body is the mass of a body measured in an inertial coordinate system in which the body is at rest. Theorem A body’s inertial mass m is a function of v , and can be rewritten as m = m ( v ) = γ ( v ) m 0 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The rest mass m 0 = m (0) of a body is the mass of a body measured in an inertial coordinate system in which the body is at rest. Theorem A body’s inertial mass m is a function of v , and can be rewritten as m = m ( v ) = γ ( v ) m 0 . Theorem E = mc 2 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : g ( X , Y ) = g ( Y , X ). Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : g ( X , Y ) = g ( Y , X ). g ( α X + β Y , Z ) = α g ( X , Z ) + β g ( Y , Z ). Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : g ( X , Y ) = g ( Y , X ). g ( α X + β Y , Z ) = α g ( X , Z ) + β g ( Y , Z ). But g ( X , Y ) < 0 if X 0 Y 0 < X 1 Y 1 + X 2 Y 2 + X 3 Y 3 . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : g ( X , Y ) = g ( Y , X ). g ( α X + β Y , Z ) = α g ( X , Z ) + β g ( Y , Z ). But g ( X , Y ) < 0 if X 0 Y 0 < X 1 Y 1 + X 2 Y 2 + X 3 Y 3 . So g is an indefinite inner product. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Back to the Inner Product Recall: g ( X , Y ) = X 0 Y 0 − X 1 Y 1 − X 2 Y 2 − X 3 Y 3 . For any three four-vectors X , Y , Z ∈ M , and α ∈ R : g ( X , Y ) = g ( Y , X ). g ( α X + β Y , Z ) = α g ( X , Z ) + β g ( Y , Z ). But g ( X , Y ) < 0 if X 0 Y 0 < X 1 Y 1 + X 2 Y 2 + X 3 Y 3 . So g is an indefinite inner product. Thus g ( X , X ) (or the interval) can be used to split Minkowski space into cones. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For an event x ∈ M , we have: Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For an event x ∈ M , we have: Light-Cone C L ( x ) = { y : g ( X , X ) = 0 } Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For an event x ∈ M , we have: Light-Cone C L ( x ) = { y : g ( X , X ) = 0 } Time-Cone C T ( x ) = { y : g ( X , X ) > 0 } Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For an event x ∈ M , we have: Light-Cone C L ( x ) = { y : g ( X , X ) = 0 } Time-Cone C T ( x ) = { y : g ( X , X ) > 0 } Space-Cone C S ( x ) = { y : g ( X , X ) < 0 } where y ∈ M , and X is the displacement vector from x to y Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For events x , y ∈ M we define a partial ordering < on M by x < y if the displacement vector X from x to y lies in the future light-cone Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition For events x , y ∈ M we define a partial ordering < on M by x < y if the displacement vector X from x to y lies in the future light-cone That is, x < y if t y > t x , and g ( X , X ) > 0. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) = g ( LX , LY ) Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) = g ( LX , LY ) = g ( λ X , λ Y ) Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  .   0 0 − 1 0  0 0 0 − 1 Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) = g ( LX , LY ) = g ( λ X , λ Y ) = g ( X , Y ) . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  . L is orthochronous if l 1 , 1 > 0,   0 0 − 1 0  0 0 0 − 1 where l 1 , 1 is the first entry of the first row of L . Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) = g ( LX , LY ) = g ( λ X , λ Y ) = g ( X , Y ) . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Definition The matrix L in (3) is a Lorentz transformation if L − 1 = gL T g ,  1 0 0 0  0 − 1 0 0   where g =  . L is orthochronous if l 1 , 1 > 0,   0 0 − 1 0  0 0 0 − 1 where l 1 , 1 is the first entry of the first row of L . Definition Define the Lorentz group L = { λ : M → M : ∀ X , Y ∈ M , g ( λ X , λ Y ) = g ( X , Y ) } g ( X ′ , Y ′ ) = g ( LX , LY ) = g ( λ X , λ Y ) = g ( X , Y ) . Definition The orthochronous Lorentz group L + is the subgroup of L whose elements preserve the partial ordering < on M . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Usually we think of the topology on R 4 as the standard Euclidean topology T . Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Usually we think of the topology on R 4 as the standard Euclidean topology T . Physically, this topology is not useful because: 1. The 4-dimensional Euclidean topology is locally homogeneous, yet M is not (the light cone separates timelike and spacelike events). Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Essentials of Relativity The Space of Relativity Usually we think of the topology on R 4 as the standard Euclidean topology T . Physically, this topology is not useful because: 1. The 4-dimensional Euclidean topology is locally homogeneous, yet M is not (the light cone separates timelike and spacelike events). 2. The group of all homeomorphisms of R 4 include mappings of no physical significance. Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity