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Mechanics

Physics 151

Lecture 15

Special Relativity

(Chapter 7)

What We Did Last Time „

Defined Lorentz transformation^ „

Linear transformation of 4-vectors that conserve the lengthin Minkowski space

„

Derived general form of homogeneous Lorentztransformation^ „

Product of rotation and proper Lorentz transformation „

Found explicit matrix expression of PLT „

HLTs form a group, PLTs don’t

„

HLT × HLT = HLT

„

PLT × PLT = HLT

Fun With Paradox „

How do you know two objects passed the line“simultaneously”?^ „

By sending light and receiving reflection „

What happens if the observer was moving?

ll l

Light is sent at

l

t^

c

=

Reflections come back at

l

t^

c

=

Took same time

Æ

same distance

Came back simultaneously

Æ

reflected

at the same time

Fun With Paradox

„

For a moving observer

„

Definition of “simultaneous”depends on the observer^ „

Causing this effect

ll l

A B

Light is sent at the same momentReflection from A comes backearlier than from B Æ

A must have passed earlier!

rotation!

4-Vectors „

We write 4-vectors as

x

μ

„

It seems confusing, but you’ll get used to it

„

Let’s follow a particle traveling in 4-space^ „

Trajectory is given by „

is a parameter that varies monotonously along the curve „

Proper time

is a convenient possibility for

„

At any point on the curve, we can define a tangent 4-vector

0 x

ct

1 x

x

2 x

y

3 x

z

Greek index = 0…

0

1

2

3

(^

(^

(^

(^

x

x

x

x

dx

u

d

μ

μ^

Call it 4-velocity

4-Velocity

„

S

is observer’s frame.

S’

is the particles rest frame

„

Particle’s 3-velocity in

S

is

v

„

We define 4-velocity^ „

“Length” is „

4-velocity is the relativistic extension of

v

„

(^0) It’s timelike, by the way

0

x

u

c

x

ct

c

Space part is proportional

to the 3-velocity

0

0

2

2

2

2

(^

i^

i

u u

u u

c

v

c

Lorentz invariant

i^

i^

i

x

c

v

Roman index = 1…

i

i^

i

x

u

v

Lorentz Tensor „

Proper Lorentz transformation turns a 4-vector intoanother 4-vector^ „

Consider it a linear function of 4-vector „

Express it as

„

You can define a whole bunch of quantities using thisconvention

Æ

Call them tensors

„

We’ll find their physical meanings as we go

x

L x

μ^

μ^

ν ν

′^

Upper index = 4-vector

Lower index = function that accepts 4-vector

X

μ^

X

μ^

X

μ ν^

X

μν

X

μν

X

Tensor Product „

Tensor product of two 4-vectors is defined by^ „

T

αβ

is a tensor of rank 2

„

You can repeat this to define tensors of rank

n

„

Lorentz transformation of

T

αβ

can be easily found

„

Use as many Lorentz tensor as necessary to convert allindices

T

u

v

T

u v

αβ

α^

β

Write this as

T

u

v

L u L v

L L T

αβ

α^

β^

α^

μ^

β^

ν^

α^

β^

μν

μ^

ν

μ^

ν

′^

′^

This (irresponsible-sounding) rule worksfor Lorentz transformation of any tensor

Metric Tensor „

A coordinate system in general has basis vectors^ „

Scalar product

can be written as

„

That’s what “metric” means

0

1

2

3

0

1

2

3

u

u

u

u

u

u

μ^

μ

e

e

e

e

e

u v

u v

u

v

μ

ν

μ

ν

⋅^

e

e

g

μν

μ^

ν

e

e

Metric tensor defines the scalar products of the basis vectors

Æ

Lengths of and angles between the basis vectors

General Metric Tensor „

Metric

g

μν

in Minkowski space is diagonal

„

Coordinate axes are always orthogonal

„

Formalism of tensors allows more flexibility^ „

Useful in curved space coordinates „

General relativity makes full use of this

„

Let’s not get into it for now…

Gradient „

Consider a scalar function^ „

A particle goes along a curve „

Rate of increase of

is given by

„

Gradient operates on velocity to make a scalar

Æ

1-form

„

Gradient operator is defined by^ „

Also known as

d

„

But I’ll avoid this notation

(^

f^

x

μ

x

x

μ^

μ^

(^

f^

x

μ^

(^

)^

(^

f^

x

dx

f^

x

v

x

d

x

μ

μ

μ^

μ

μ

μ

gradient of

f

x

μ

μ ∂

Lower index shows

it’s a 1-form

4-Vector and 1-Form „

4-vector can be turned into its 1-form by^ „

Obviously you can do the reverse „

g

μν

looks identical to

g

μν

in Minkowski space

„

This gives us Lorentz transformation for 1-form^ „

Works just the same as 4-vector, as it should

u

u g

μ

ν

μν

u

g

u

μ

μν

ν

where

g

g αβ

α

βγ

γ δ =

u

g

u

g

L u

g

L g

u

L u

ν

ν^

α

ν^

αβ

β

μ^

μν

μν

α

μν

α

β^

μ^

β

′^

Lorentz Transformation „

We can find Lorentz transformation for any tensor^ „

Transform all indices using Lorentz tensor „

Example:

„

We now know all the rules for^ „

Lorentz transformation „

Moving indices up and down

„

They aren’t even all that complicated

scalar

T a b

α^

β

β^

α^

T

a b

T

L a L b

T

a b

α

β

α^

μ^

β^

ν

μ^

ν

β^

α

β^

α^

μ^

ν

ν^

μ

′^

′^

′^

Transform this to get

T

L L T

α^

α^

ν^

μ

β

μ^

β^

ν

′^

Force „

Newton’s laws must be correct if the velocity is zero^ „

in the rest frame of the object

„

Momentum transforms as a 4-vector „

Time dilation changes the time derivative

„

Natural extension would be^ „

K

μ^

must be a 4-vector

„

τ is proper time. Connected with

t

by

„

How do we find the 4-force

K

μ

d dt

p

F

 p

dp

K

d

μ^

μ

τ

dt

d