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

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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

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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

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SLIDE 2

Minkowski space

Pseudo inner product space R2

1 = f(t, x) : t, x 2 Rg

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

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SLIDE 3

Minkowski space

Pseudo inner product space R2

1 = f(t, x) : t, x 2 Rg

Pseudo inner product h(t1, x1) , (t2, x2)i = t1t2 x1x2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

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SLIDE 4

Minkowski space

Pseudo inner product space R2

1 = f(t, x) : t, x 2 Rg

Pseudo inner product h(t1, x1) , (t2, x2)i = t1t2 x1x2 Minkowski norm k(t, x)k = p t2 x2 ranges over all non-negative real and positive imaginary values

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 2 / 30

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SLIDE 5

Curves of constant Minkowski norm

Points on curves of constant Minkowski norm satisfy t2 x2 = a2 (1)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

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SLIDE 6

Curves of constant Minkowski norm

Points on curves of constant Minkowski norm satisfy t2 x2 = a2 (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

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SLIDE 7

Curves of constant Minkowski norm

Points on curves of constant Minkowski norm satisfy t2 x2 = a2 (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

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SLIDE 8

Curves of constant Minkowski norm

Points on curves of constant Minkowski norm satisfy t2 x2 = a2 (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 t2 x2 = a2 inside the light cone)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

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Curves of constant Minkowski norm

Points on curves of constant Minkowski norm satisfy t2 x2 = a2 (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 t2 x2 = a2 inside the light cone) a = ib 2 iR+ de…nes an imaginary hyperbolic circle of radius ib (the hyperbola x2 t2 = b2 outside the light cone)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 3 / 30

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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

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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

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SLIDE 12

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

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Euclidean Isometries

Rotations ρθ about the origin through angle θ

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

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Euclidean Isometries

Rotations ρθ about the origin through angle θ Re‡ections σθ in lines through the origin with inclination θ

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

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Euclidean Isometries

Rotations ρθ about the origin through angle θ Re‡ections σθ in lines through the origin with inclination θ Rotations …x circles centered at the origin and send lines through the

  • rigin to lines through the origin

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 5 / 30

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Euclidean Isometries

Rotations ρθ about the origin through angle θ Re‡ections σθ in lines through the origin with inclination θ Rotations …x circles centered at the origin and send lines through the

  • rigin 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

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Matrix Representation

Rotations represented by orthogonal matrices with determinant +1 : [ρθ] = cos θ sin θ sin θ cos θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 6 / 30

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SLIDE 18

Matrix Representation

Rotations represented by orthogonal matrices with determinant +1 : [ρθ] = cos θ sin θ sin θ cos θ

  • Re‡ections represented by orthogonal matrices with determinant 1 :

[σθ] =

  • cos 2θ

sin 2θ sin 2θ cos 2θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 6 / 30

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SLIDE 19

Matrix Representation

Rotations represented by orthogonal matrices with determinant +1 : [ρθ] = cos θ sin θ sin θ cos θ

  • Re‡ections represented by orthogonal matrices with determinant 1 :

[σθ] =

  • cos 2θ

sin 2θ sin 2θ cos 2θ

  • These form the orthogonal group O (2) , which has two components:

[ρθ] = cos θ sin θ sin θ cos θ

  • and [ρθ] [σ0] =

cos θ sin θ sin θ cos θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 6 / 30

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The Orthogonal Group O(2)

The component cos θ sin θ sin θ cos θ

  • = cos θ

1 1

  • + sin θ

1 1

  • parametrizes the circle

C1 : u2

1 + u2 2 = 2

in the 2-plane spanned by B1 = ( u1 = "

1 p 2 1 p 2

# , u2 = " 1

p 2 1 p 2

#)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 7 / 30

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SLIDE 21

The Orthogonal Group O(2)

The component cos θ sin θ sin θ cos θ

  • = cos θ

1 1

  • + sin θ

1 1

  • parametrizes the circle

C1 : u2

1 + u2 2 = 2

in the 2-plane spanned by B1 = ( u1 = "

1 p 2 1 p 2

# , u2 = " 1

p 2 1 p 2

#) The trivial rotation [ρ0] = 1 1

  • =

p 2u1 + 0u2 is on this circle

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 7 / 30

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SLIDE 22

The Orthogonal Group O(2)

The component cos θ sin θ sin θ cos θ

  • = cos θ

1 1

  • + sin θ

1 1

  • parametrize the circle

C2 : u2

3 + u2 4 = 2

in the 2-plane spanned by B2 = ( u3 = "

1 p 2

1

p 2

# , u4 = "

1 p 2 1 p 2

#)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 8 / 30

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SLIDE 23

The Orthogonal Group O(2)

The component cos θ sin θ sin θ cos θ

  • = cos θ

1 1

  • + sin θ

1 1

  • parametrize the circle

C2 : u2

3 + u2 4 = 2

in the 2-plane spanned by B2 = ( u3 = "

1 p 2

1

p 2

# , u4 = "

1 p 2 1 p 2

#) B1 [ B2 linearly independent ) C1 \ C2 = ? and C1 [ C2 = O (2)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 8 / 30

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Minkowski Isometries

Hyperbolic rotations represented by the matrices [Rθ] = cosh θ sinh θ sinh θ cosh θ

  • r in coordinates by

Rθ (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

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Minkowski Isometries

Hyperbolic rotations represented by the matrices [Rθ] = cosh θ sinh θ sinh θ cosh θ

  • r in coordinates by

Rθ (t, x) = (t cosh θ + x sinh θ, t sinh θ + x cosh θ) Rθ …xes hyperbolic circles: (¯ t)2 (¯ x)2 = (t cosh θ + x sinh θ)2 (t sinh θ + x cosh θ)2 = t2 x2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 9 / 30

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Minkowski Isometries

Hyperbolic rotations represented by the matrices [Rθ] = cosh θ sinh θ sinh θ cosh θ

  • r in coordinates by

Rθ (t, x) = (t cosh θ + x sinh θ, t sinh θ + x cosh θ) Rθ …xes hyperbolic circles: (¯ t)2 (¯ x)2 = (t cosh θ + x sinh θ)2 (t sinh θ + x cosh θ)2 = t2 x2 Rθ sends lines through the origin to lines through the origin: Rθ (a, 0) = a (cosh θ, sinh θ)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 9 / 30

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Minkowski Isometries

Re‡ections S0 in the t-axis and S∞ in the x-axis are represented by [S0] = 1 1

  • and

[S∞] = 1 1

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 10 / 30

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Minkowski Isometries

Re‡ections S0 in the t-axis and S∞ in the x-axis are represented by [S0] = 1 1

  • and

[S∞] = 1 1

  • Hyperbolic re‡ection in line with inclination θ 6= π

4 is represented by

[Sm] = 8 > > > > < > > > > :

  • cosh 2θ

sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < π/4 cosh 2θ sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < 3π/4

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 10 / 30

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Minkowski Isometries

Re‡ections S0 in the t-axis and S∞ in the x-axis are represented by [S0] = 1 1

  • and

[S∞] = 1 1

  • Hyperbolic re‡ection in line with inclination θ 6= π

4 is represented by

[Sm] = 8 > > > > < > > > > :

  • cosh 2θ

sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < π/4 cosh 2θ sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < 3π/4 Hyperbolic re‡ections …x hyperbolic circles

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 10 / 30

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Minkowski Isometries

Re‡ections S0 in the t-axis and S∞ in the x-axis are represented by [S0] = 1 1

  • and

[S∞] = 1 1

  • Hyperbolic re‡ection in line with inclination θ 6= π

4 is represented by

[Sm] = 8 > > > > < > > > > :

  • cosh 2θ

sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < π/4 cosh 2θ sinh 2θ sinh 2θ cosh 2θ

  • ,

if π/4 < θ < 3π/4 Hyperbolic re‡ections …x hyperbolic circles Hyperbolic re‡ections …x their re‡ecting lines point-wise

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 10 / 30

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The Poincaré Group O(1,1)

Hyperbolic rotation and re‡ections form the group O(1, 1)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 11 / 30

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The Poincaré Group O(1,1)

Hyperbolic rotation and re‡ections form the group O(1, 1) O(1, 1) has four disjoint components: [Rθ] = cosh θ sinh θ sinh θ cosh θ

  • [Rθ] [S0] [S∞] =

cosh θ sinh θ sinh θ cosh θ

  • [Rθ] [S0] =

cosh θ sinh θ sinh θ cosh θ

  • [Rθ] [S∞] =

cosh θ sinh θ sinh θ cosh θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 11 / 30

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The Poincaré Group O(1,1)

The components cosh θ sinh θ sinh θ cosh θ

  • = cosh θ

1 1

  • + sinh θ

1 1

  • and

cosh θ sinh θ sinh θ cosh θ

  • = cosh θ

1 1

  • sinh θ

1 1

  • form the two branches of the hyperbola

H1 : u2

1 u2 4 = 2

in the 2-plane spanned by B0

1 =

( u1 = "

1 p 2 1 p 2

# , u4 = "

1 p 2 1 p 2

#)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 12 / 30

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The Poincaré Group O(1,1)

The components cosh θ sinh θ sinh θ cosh θ

  • = cosh θ

1 1

  • + sinh θ

1 1

  • and

cosh θ sinh θ sinh θ cosh θ

  • = cosh θ

1 1

  • sinh θ

1 1

  • form the two branches of the hyperbola

H2 : u2

3 u2 2 = 2

in the 2-plane spanned by B0

2 =

( u3 = "

1 p 2

1

p 2

# , u2 = " 1

p 2 1 p 2

#)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 13 / 30

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The Poincaré Group O(1,1)

The trivial hyperbolic rotation [R0] = 1 1

  • =

p 2u1 + 0u4 lies in the component of cosh θ sinh θ sinh θ cosh θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 14 / 30

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The Poincaré Group O(1,1)

The trivial hyperbolic rotation [R0] = 1 1

  • =

p 2u1 + 0u4 lies in the component of cosh θ sinh θ sinh θ cosh θ

  • B0

1 [ B0 2 linearly independent ) H1 \ H2 = ? and H1 [ H2 = O (1, 1)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 14 / 30

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SLIDE 37

Special Relativity

The speed of light c 3 108 m/sec

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time The world-line of a particle is a curve r (t) = (ct, x (t))

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time The world-line of a particle is a curve r (t) = (ct, x (t)) The relative velocity of a particle P is v (t) = (c, x0 (t))

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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SLIDE 41

Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time The world-line of a particle is a curve r (t) = (ct, x (t)) The relative velocity of a particle P is v (t) = (c, x0 (t)) The ordinary velocity of P is x0 (t)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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SLIDE 42

Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time The world-line of a particle is a curve r (t) = (ct, x (t)) The relative velocity of a particle P is v (t) = (c, x0 (t)) The ordinary velocity of P is x0 (t) The relative speed of P is kvk = q c2 (x0)2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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SLIDE 43

Special Relativity

The speed of light c 3 108 m/sec An event is a point (t, x) in space-time The world-line of a particle is a curve r (t) = (ct, x (t)) The relative velocity of a particle P is v (t) = (c, x0 (t)) The ordinary velocity of P is x0 (t) The relative speed of P is kvk = q c2 (x0)2 The ordinary speed of P is jx0j

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 15 / 30

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Physical Assumptions

The ordinary speed of a particle cannot exceed the speed of light

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 16 / 30

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SLIDE 45

Physical Assumptions

The ordinary speed of a particle cannot exceed the speed of light A particle traveling at the speed of light has zero mass

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 16 / 30

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SLIDE 46

Physical Assumptions

The ordinary speed of a particle cannot exceed the speed of light A particle traveling at the speed of light has zero mass The speed of light is the same in every frame of reference

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 16 / 30

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SLIDE 47

The World Line of a ...

particle at rest is a horizontal line inside the light cone

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 17 / 30

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SLIDE 48

The World Line of a ...

particle at rest is a horizontal line inside the light cone particle with non-zero mass is a curve inside the light cone

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 17 / 30

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SLIDE 49

The World Line of a ...

particle at rest is a horizontal line inside the light cone particle with non-zero mass is a curve inside the light cone particle with non-zero mass and constant speed is a line inside the light cone

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 17 / 30

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SLIDE 50

The World Line of a ...

particle at rest is a horizontal line inside the light cone particle with non-zero mass is a curve inside the light cone particle with non-zero mass and constant speed is a line inside the light cone photon is a line on the light cone

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 17 / 30

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SLIDE 51

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction.

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 52

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction. The relative velocity of P is (c, v) and its relative speed is p c2 v2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 53

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction. The relative velocity of P is (c, v) and its relative speed is p c2 v2 The arc length function s for the world-line of P is s (t) =

Z t

0 k(c, v)k du =

p c2 v2

Z t

0 du = t

p c2 v2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 54

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction. The relative velocity of P is (c, v) and its relative speed is p c2 v2 The arc length function s for the world-line of P is s (t) =

Z t

0 k(c, v)k du =

p c2 v2

Z t

0 du = t

p c2 v2 t = s p c2 v2 = s/c p 1 v2/c2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 55

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction. The relative velocity of P is (c, v) and its relative speed is p c2 v2 The arc length function s for the world-line of P is s (t) =

Z t

0 k(c, v)k du =

p c2 v2

Z t

0 du = t

p c2 v2 t = s p c2 v2 = s/c p 1 v2/c2 The proper elapsed time of P is: s c = t p 1 v2/c2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 56

Proper Ellapsed TIme

Consider a particle P with non-zero mass and constant ordinary velocity v in the positive x-direction. The relative velocity of P is (c, v) and its relative speed is p c2 v2 The arc length function s for the world-line of P is s (t) =

Z t

0 k(c, v)k du =

p c2 v2

Z t

0 du = t

p c2 v2 t = s p c2 v2 = s/c p 1 v2/c2 The proper elapsed time of P is: s c = t p 1 v2/c2 If P is at rest, for example, its proper elapsed time is t.

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 18 / 30

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SLIDE 57

Lorentz Transformations

Hyperbolic change of coordinates ct x

  • =

α β γ δ c¯ t ¯ x

  • .

(2)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 19 / 30

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SLIDE 58

Lorentz Transformations

Hyperbolic change of coordinates ct x

  • =

α β γ δ c¯ t ¯ x

  • .

(2) A = α β γ δ

  • ! I as v ! 0

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 19 / 30

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SLIDE 59

Lorentz Transformations

Hyperbolic change of coordinates ct x

  • =

α β γ δ c¯ t ¯ x

  • .

(2) A = α β γ δ

  • ! I as v ! 0

A is a hyperbolic rotation cosh θ sinh θ sinh θ cosh θ

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 19 / 30

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SLIDE 60

Lorentz Transformations

Hyperbolic change of coordinates ct x

  • =

α β γ δ c¯ t ¯ x

  • .

(2) A = α β γ δ

  • ! I as v ! 0

A is a hyperbolic rotation cosh θ sinh θ sinh θ cosh θ

  • ct = c¯

t cosh θ + ¯ x sinh θ x = c¯ t sinh θ + ¯ x cosh θ

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 19 / 30

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SLIDE 61

Example

When θ = ln 2, cosh θ = 5

4 and sinh θ = 3 4

5 3

  • =

5/4 3/4 3/4 5/4 4

  • and

4

  • =

5/4 3/4 3/4 5/4 3 5

  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 20 / 30

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SLIDE 62

Lorentz Transformations

change coordinates of time-like vectors from ¯ K to K-coordinates

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 21 / 30

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SLIDE 63

Lorentz Transformations

change coordinates of time-like vectors from ¯ K to K-coordinates change coordinates of space-like vectors from K to ¯ K-coordinates

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 21 / 30

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SLIDE 64

Lorentz Transformations

change coordinates of time-like vectors from ¯ K to K-coordinates change coordinates of space-like vectors from K to ¯ K-coordinates The world line of a particle P at the origin of the moving frame ¯ K is the horizontal line (c¯ t, 0)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 21 / 30

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SLIDE 65

Lorentz Transformations

change coordinates of time-like vectors from ¯ K to K-coordinates change coordinates of space-like vectors from K to ¯ K-coordinates The world line of a particle P at the origin of the moving frame ¯ K is the horizontal line (c¯ t, 0) When viewed from frame K, this world line has positive slope and is parameterized by (ct, x) = (c¯ t cosh θ, c¯ t sinh θ)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 21 / 30

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SLIDE 66

Lorentz Transformations

Dividing second components of (ct, x) = (c¯ t cosh θ, c¯ t sinh θ) by …rst components gives tanh θ = x ct = vt ct = v c

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 22 / 30

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SLIDE 67

Lorentz Transformations

Dividing second components of (ct, x) = (c¯ t cosh θ, c¯ t sinh θ) by …rst components gives tanh θ = x ct = vt ct = v c 1 = cosh2 θ sinh2 θ = cosh2 θ

  • 1 tanh2 θ
  • (MU/F&M Mathematics Colloquium )

The Linear Algebra of Space-Time November 18, 2010 22 / 30

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SLIDE 68

Lorentz Transformations

Dividing second components of (ct, x) = (c¯ t cosh θ, c¯ t sinh θ) by …rst components gives tanh θ = x ct = vt ct = v c 1 = cosh2 θ sinh2 θ = cosh2 θ

  • 1 tanh2 θ
  • cosh θ =

1 p 1 tanh2 θ = 1 p 1 v2/c2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 22 / 30

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SLIDE 69

Lorentz Transformations

Dividing second components of (ct, x) = (c¯ t cosh θ, c¯ t sinh θ) by …rst components gives tanh θ = x ct = vt ct = v c 1 = cosh2 θ sinh2 θ = cosh2 θ

  • 1 tanh2 θ
  • cosh θ =

1 p 1 tanh2 θ = 1 p 1 v2/c2 sinh θ = v/c p 1 v2/c2

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 22 / 30

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SLIDE 70

Lorentz Transformations

Substituting gives t = 1 p 1 v2/c2 ¯ t +

  • v/c2

¯ x

  • x =

1 p 1 v2/c2 (v ¯ t + ¯ x)

  • r in matrix form

t x

  • =

1 p 1 v2/c2 1 v/c2 v 1 ¯ t ¯ x

  • (3)

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 23 / 30

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SLIDE 71

Lorentz length contraction

A spaceship ‡ying through space along a line with constant speed v ‡ies by the international space station

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 24 / 30

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SLIDE 72

Lorentz length contraction

A spaceship ‡ying through space along a line with constant speed v ‡ies by the international space station At instant ¯ t in the moving frame ¯ K of the spaceship, the ship’s captain observes that the endpoints of the space station are positioned at ¯ x1 and ¯ x2 on the ¯ x-axis

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 24 / 30

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SLIDE 73

Lorentz length contraction

A spaceship ‡ying through space along a line with constant speed v ‡ies by the international space station At instant ¯ t in the moving frame ¯ K of the spaceship, the ship’s captain observes that the endpoints of the space station are positioned at ¯ x1 and ¯ x2 on the ¯ x-axis The ordinary length of the space station as measured by the ship’s captain is ∆¯ x = ¯ x2 ¯ x1

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 24 / 30

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SLIDE 74

Lorentz length contraction

A spaceship ‡ying through space along a line with constant speed v ‡ies by the international space station At instant ¯ t in the moving frame ¯ K of the spaceship, the ship’s captain observes that the endpoints of the space station are positioned at ¯ x1 and ¯ x2 on the ¯ x-axis The ordinary length of the space station as measured by the ship’s captain is ∆¯ x = ¯ x2 ¯ x1 Thinking of ¯ x1 and ¯ x2 as events (¯ t, ¯ x1) and (¯ t, ¯ x2) , we use (3) to change coordinates and calculate the ordinary length ∆x at instant ¯ t : ∆x = x2 x1 = v ¯ t + ¯ x2 p 1 v2/c2 v ¯ t + ¯ x1 p 1 v2/c2 = 1 p 1 v2/c2 ∆¯ x

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 24 / 30

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SLIDE 75

Lorentz length contraction

A spaceship ‡ying through space along a line with constant speed v ‡ies by the international space station At instant ¯ t in the moving frame ¯ K of the spaceship, the ship’s captain observes that the endpoints of the space station are positioned at ¯ x1 and ¯ x2 on the ¯ x-axis The ordinary length of the space station as measured by the ship’s captain is ∆¯ x = ¯ x2 ¯ x1 Thinking of ¯ x1 and ¯ x2 as events (¯ t, ¯ x1) and (¯ t, ¯ x2) , we use (3) to change coordinates and calculate the ordinary length ∆x at instant ¯ t : ∆x = x2 x1 = v ¯ t + ¯ x2 p 1 v2/c2 v ¯ t + ¯ x1 p 1 v2/c2 = 1 p 1 v2/c2 ∆¯ x Equivalently, ∆¯ x = p 1 v2/c2∆x

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 24 / 30

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SLIDE 76

Example

If v = .73c, then p 1 .732 p .47 .69

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 25 / 30

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SLIDE 77

Example

If v = .73c, then p 1 .732 p .47 .69 If ∆x = p 34 5.83, then ∆¯ x (.69) (5.83) 4

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 25 / 30

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SLIDE 78

Conclusion

To an observer in a reference frame moving along a straight line with constant speed v relative to a …xed reference frame, the ordinary length of an object at rest in the …xed frame appears to be shorter than it does to an observer in the …xed frame by a factor of p 1 v2/c2.

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 26 / 30

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SLIDE 79

Time dilation

A clock on board the spaceship is positioned at the origin ¯ O in the moving frame ¯ K

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 27 / 30

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SLIDE 80

Time dilation

A clock on board the spaceship is positioned at the origin ¯ O in the moving frame ¯ K As the spaceship passes the space station, the captain takes two clock readings ¯ t1 and ¯ t2 and determines the elapsed time to be ∆¯ t = ¯ t2 ¯ t1

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 27 / 30

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SLIDE 81

Time dilation

A clock on board the spaceship is positioned at the origin ¯ O in the moving frame ¯ K As the spaceship passes the space station, the captain takes two clock readings ¯ t1 and ¯ t2 and determines the elapsed time to be ∆¯ t = ¯ t2 ¯ t1 Thinking of ¯ t1 and ¯ t2 as events (¯ t1, 0) and (¯ t2, 0), the relationship between ∆¯ t and ∆t is ∆t = ¯ t2 p 1 v2/c2 ¯ t1 p 1 v2/c2 = 1 p 1 v2/c2 ∆¯ t

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 27 / 30

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SLIDE 82

Example

If v = .73c, then 1 p 1 .732 1 p .47 1.46

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 28 / 30

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SLIDE 83

Example

If v = .73c, then 1 p 1 .732 1 p .47 1.46 If ∆¯ t = 4, then ∆t (1.46) (4) = 5.84

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 28 / 30

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SLIDE 84

Conclusion

To an observer in a …xed reference frame, the elapsed time measured in a reference frame moving along a straight line with constant speed v appears to dilate by a factor of 1/ p 1 v2/c2.

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 29 / 30

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SLIDE 85

The End

Moral: Live fast; live long (relatively speaking...)!

(MU/F&M Mathematics Colloquium ) The Linear Algebra of Space-Time November 18, 2010 30 / 30