Relativistic Classical Mechanics

65
Relativistic Classical Mechanics

description

Relativistic Classical Mechanics. Albert Abraham Michelson (1852 – 1931). Edward Williams Morley (1838 – 1923). James Clerk Maxwell (1831-1879). XIX century crisis in physics: some facts Maxwell : equations of electromagnetism are not invariant under Galilean transformations - PowerPoint PPT Presentation

Transcript of Relativistic Classical Mechanics

Page 1: Relativistic Classical Mechanics

Relativistic Classical Mechanics

Page 2: Relativistic Classical Mechanics

XIX century crisis in physics:some facts

• Maxwell: equations of electromagnetism are not invariant under Galilean transformations

• Michelson and Morley: the speed of light is the same in all inertial systems

James ClerkMaxwell

(1831-1879)

Edward Williams Morley

(1838 – 1923)

Albert Abraham Michelson

(1852 – 1931)

Page 3: Relativistic Classical Mechanics

Postulates of the special theory

• 1) The laws of physics are the same to all inertial observers

• 2) The speed of light is the same to all inertial observers

• Formulation of physics that explicitly incorporates these two postulates is called covariant

• The space and time comprise a single entity: spacetime

• A point in spacetime is called event

• Metric of spacetime is non-Euclidean

7.1

Page 4: Relativistic Classical Mechanics

Tensors

• Tensor of rank n is a collection of elements grouped through a set of n indices

• Scalar is a tensor of rank 0• Vector is a tensor of rank 1• Matrix is a tensor of rank 2• Etc.

• Tensor product of two tensors of ranks m and n is a tensor of rank (m + n)

• Sum over a coincidental index in a tensor product of two tensors of ranks m and n is a tensor of rank (m + + n – 2)

7.5

n

ijA ..

AiA

ijA

mnmn

ijij CBA

........

2

........

mnmn

ikj

jijk DBA

Page 5: Relativistic Classical Mechanics

Tensors

• Tensor product of two vectors is a matrix

• Sum over a coincidental index in a tensor product of two tensors of ranks 1 and 1 (two vectors) is a tensor of rank 1 + 1 – 2 = 0 (scalar): scalar product of two vectors

• Sum over a coincidental index in a tensor product of two tensors of ranks 2 and 1 (a matrix and a vector) is a tensor of rank 2 + 1 – 2 = 1 (vector)

• Sum over a coincidental index in a tensor product of two tensors of ranks 2 and 2 (two matrices) is a tensor of rank 2 + 2 – 2 = 2 (matrix)

7.5

ijji CBA

DBAi

ii

ij

jij DBA

ikj

jkij DBA

Page 6: Relativistic Classical Mechanics

Metrics, covariant and contravariant vectors

• Vectors, which describe physical quantities, are called contravariant vectors and are marked with superscripts instead of a subscripts

• For a given space of dimension N, we introduce a concept of a metric – N x N matrix uniquely defining the symmetry of the space (marked with subscripts)

• Sum over a coincidental index in a product of a metric and a contravariant vecor is a covariant vector or a 1-form (marked with subscripts)

• Magnitude: square root of the scalar product of a contravariant vector and its covariant counterpart

7.47.5

iA

ijg

ij

jij AAg

Page 7: Relativistic Classical Mechanics

3D Euclidian Cartesian coordinates

• Contravariant infinitesimal coordinate vector:

• Metric

• Covariant infinitesimal coordinate vector:

• Magnitude:

7.47.5

dz

dy

dx

dr i

3

1j

jiji drgdr

100

010

001

g

dz

dy

dx

dz

dy

dx

100

010

001i

i drdr

3

1i

iidrdr 222 )()()( dzdydx

Page 8: Relativistic Classical Mechanics

3D Euclidian spherical coordinates

• Contravariant infinitesimal coordinate vector:

• Metric

• Covariant infinitesimal coordinate vector:

• Magnitude:

7.47.5

d

d

dr

dr i

3

1j

jiji drgdr

22

2

sin00

00

001

r

rg

dr

dr

dr

d

d

dr

r

r22

2

22

2

sinsin00

00

001i

i drdr

3

1i

iidrdr 222222 sin drdrdr

Page 9: Relativistic Classical Mechanics

Hilbert space of quantum-mechanical wavefunctions

• Contravariant vector (ket):

• Covariant vector (bra):

• Magnitude:

• Metric:

7.47.5

David Hilbert(1862 – 1943)

Page 10: Relativistic Classical Mechanics

4D spacetime

• Contravariant infinitesimal coordinate 4-vector:

• Metric

• Covariant infinitesimal coordinate vector:

7.47.5

dz

dy

dx

cdt

dx

3

0

dxgdx

1000

0100

0010

0001

g

dz

dy

dx

cdt

dz

dy

dx

cdt

1000

0100

0010

0001

Page 11: Relativistic Classical Mechanics

4D spacetime

• Magnitude:

• This magnitude is called differential interval

• Interval (magnitude of a 4-vector connecting two events in spacetime):

• Interval should be the same in all inertial reference frames

• The simplest set of transformations that preserve the invariance of the interval relative to a transition from one inertial reference frame to another: Lorentz transformations

7.47.5

3

0

dxdx 22222 dzdydxdtc

ds

22222 zyxtcs

Page 12: Relativistic Classical Mechanics

Lorentz transformations

• We consider two inertial reference frames S and S’; relative velocity as measured in S is v :

• Then Lorentz transformations are:

• Lorentz transformations can be written in a matrix form

7.2

ctr

rrrctct

2

)1()(');('

Hendrik AntoonLorentz

(1853 – 1928)

21

1;

c

v

3

0

'

xLx

Page 13: Relativistic Classical Mechanics

Lorentz transformations7.2

3

0

'

xLx

2

22

2

2

2

22

2

)1(1)1()1(

)1()1(1)1(

)1()1()1(1

v

v

v

vv

v

vv

v

vv

v

v

v

vv

v

vv

v

vv

v

v

zzyzxz

zyyyxy

zxyxxx

zyx

L

Page 14: Relativistic Classical Mechanics

Lorentz transformations

• If the reference frame S‘ moves parallel to the x axis of the reference frame S:

• If two events happen at the same location in S:

• Time dilation

7.2

1000

0100

00

00

L

)0,0,(

zz

yy

ctxx

xctct

'

'

)('

)('

)('

)('

222

111

xc

tt

xc

tt

))(('' 121212 xxc

tttt

12 xx

)('' 1212 tttt

vtxc

vctxctxx

ttc

v

'

;';1;0

Page 15: Relativistic Classical Mechanics

Lorentz transformations

• If the reference frame S‘ moves parallel to the x axis of the reference frame S:

• If two events happen at the same time in S:

• Length contraction

7.2

1000

0100

00

00

L

)0,0,(

zz

yy

ctxx

xctct

'

'

)('

)('

)('

)('

222

111

ctxx

ctxx

))(('' 121212 ttcxxxx

12 tt

)('' 1212 xxxx

Page 16: Relativistic Classical Mechanics

Velocity addition

• If the reference frame S‘ moves parallel to the x axis of the reference frame S:

• If the reference frame S‘‘ moves parallel to the x axis of the reference frame S‘:

7.3

1000

0100

00'''

00'''

'''

SSL

1000

0100

00

00

'

SSL

Page 17: Relativistic Classical Mechanics

Velocity addition

• The Lorentz transformation from the reference frame S to the reference frame S‘‘:

• On the other hand:

7.3

1000

0100

00)'1(')'('

00)'(')'1('

''''''

SSSSSS LLL

1000

0100

00''''''

00''''''

''

SSL

)'('''''

)'1('''

'1

'''

Page 18: Relativistic Classical Mechanics

Four-velocity

• Proper time is time measured in the system where the clock is at rest

• For an object moving relative to a laboratory system, we define a contravariant vector of four-velocity:

7.4

d

dxu

dctd

u)(0

d

cd )( c

ddx

u 1

d

dx

)(

d

dx

dt

dx xv

zy vuvu 32 ;

z

y

x

v

v

v

c

Page 19: Relativistic Classical Mechanics

Four-velocity

• Magnitude of four-velocity

7.4

z

y

x

z

y

x

v

v

v

c

v

v

v

c

1000

0100

0010

0001

3

0

uu

3

0

3

0

uug 2222 )()()()( zyx vvvc

c

v

;

1

12

2

222 )()()(1

c

vvvc

zyx

2

2

1c

vc

1

c c

23

0

cuu

Page 20: Relativistic Classical Mechanics

HermannMinkowski

(1864 - 1909)

Minkowski spacetime

• Lorentz transformations for parallel axes:

• How do x’ and t’ axes look in the x and t axes?

• t’ axis:

• x’ axis:

7.1

0'x

zzyy

ctxxxctct

';'

)(');('

0 ctx

v

x

c

xt

x’

t’t

x0't

0 xct 2c

vx

c

xt

Page 21: Relativistic Classical Mechanics

Minkowski spacetime

• When

• How do x’ and t’ axes look in the x and t axes?

• t’ axis:

• x’ axis:

7.1

0'x 0 ctx

c

xt

t

x0't

0 xct

1cv

ctx

xct c

xt

Page 22: Relativistic Classical Mechanics

Minkowski spacetime

• Let us synchronize the clocks of the S and S’ frames at the origin

• Let us consider an event

• In the S frame, the event is to the right of the origin

• In the S‘ frame, the event is to the left of the origin

7.1

x’

t’t

x

Page 23: Relativistic Classical Mechanics

Minkowski spacetime

• Let us synchronize the clocks of the S and S’ frames at the origin

• Let us consider an event

• In the S frame, the event is after the synchronization

• In the S‘ frame, the event is before the synchronization

7.1

x’

t’t

x

Page 24: Relativistic Classical Mechanics

Minkowski spacetime7.1

0)( 2 s

0)( 2 s

Page 25: Relativistic Classical Mechanics

223

0

cmpp

Four-momentum

• For an object moving relative to a laboratory system, we define a contravariant vector of four-momentum:

• Magnitude of four-momentum

7.4

mup 00 mup cm

11 mup ;xvm zy vmpvmp 32 ;

z

y

x

vm

vm

vm

cm

3

0

pp

3

0

3

0

ppg 2222 )()()()( zyx vmvmvmcm

2

2

1c

vmc mc

Page 26: Relativistic Classical Mechanics

Four-momentum

• Rest-mass: mass measured in the system where the object is at rest

• For a moving object:

• The equation has units of energy squared

• If the object is at rest

223

0

cmpp

7.4

m

2222 )()()()( zyx vmvmvmcm

mm~

2223

0

)()~()~( mcvmcmpp

222 )~()()~( vmmccm

222222 )~()()~( cvmmccm

2222 )~

()( cpmc

2~cmE

22222 )~

()( cpmcE

0~p

2220 )(mcE

20 mcE

Page 27: Relativistic Classical Mechanics

Four-momentum7.4

20 mcE

z

y

x

vm

vm

vmc

E

~

~

~

Page 28: Relativistic Classical Mechanics

Four-momentum

• Rest-mass energy: energy of a free object at rest – an essentially relativistic result

• For slow objects:

• For free relativistic objects, we introduce therefore the kinetic energy as

7.4

20 mcE

22~ cmcmE 2/122 )1( mc

02/122 )1( mcE

21

22

mc

2

22

21

c

vmc

2

2

0

mvE

0

2

2EE

mv

0EET

2mcET 22 mcmc 22222 )~

()( mccpmc

Page 29: Relativistic Classical Mechanics

Non-covariant Lagrangian formulation of relativistic mechanics

• As a starting point, we will try to find a non-covariant Lagrangian formulation (the time variable is still separate)

• The equations of motion should look like

3,2,1

ix

V

dt

dpi

i

7.9

i

i

dx

dVmv

dt

d

21 iii

i

x

L

x

V

v

Lmv

;1 2

22

21

1

mcv

mvi

i

VmcL 22 1

VTL

Page 30: Relativistic Classical Mechanics

Non-covariant Lagrangian formulation of relativistic mechanics

• For an electromagnetic potential, the Lagrangian is similar

• The equations of motion should look like

• Recall our derivations in “Lagrangian Formalism”:

3,2,1

ix

L

dt

dpi

i

7.9

3

121 i

i

ii

ii

i

x

Av

xqqA

mv

dt

d

)()(1 2

BvEqAvqt

Aq

mv

dt

d i

3

1

22 1i

iivAqqmcL

ii

i qAmv

p

21

Page 31: Relativistic Classical Mechanics

Non-covariant Lagrangian formulation of relativistic mechanics

• Example: 1D relativistic motion in a linear potential

• The equations of motion:

• Acceleration is hyperbolic, not parabolic

7.9

maxc

xmc

dt

d

22

constaxmaxcmcL ;22

c

at

xc

x

22

22 )(

atc

catcx

22220 )( catc

a

cxx

Page 32: Relativistic Classical Mechanics

Useful results

21

1

2

2)(1

1

cv

2

22

)(1

1

cv

2222 ))(( cvc 2

222)(c

cv

22222 )( ccv 222 )( cvc

Page 33: Relativistic Classical Mechanics

Non-covariant Hamiltonian formulation of relativistic mechanics

• We start with a non-covariant Lagrangian:

• Applying a standard procedure

• Hamiltonian equals the total energy of the object

LvpHi

ii

3

1

7.98.4

VmcL 22 1

Vmcvvmi

ii

223

1

1 Vmc

vm

2

2)(

Vmcc

cm

2

2

22 Vcm 2

22 mcmcT

VmcT 2 VET 0

VETH 0

2

222)(c

cv

Page 34: Relativistic Classical Mechanics

Non-covariant Hamiltonian formulation of relativistic mechanics

• We have to express the Hamiltonian as a function of momenta and coordinates:

7.98.4

VcmH 2

222 )( cvc

Vcvmc 222 )(

Vcmvmc 4222 )(

z

y

x

vm

vm

vm

cm

p

VEpc 20

22 )(

VEpcH 20

22 )(

Page 35: Relativistic Classical Mechanics

More on symmetries

• Full time derivative of a Lagrangian:

• Form the Euler-Lagrange equations:

• If

dt

dL

M

m

M

mm

mm

m

qq

Lq

q

L

t

L

1 1

M

m

M

mm

mm

m

qq

Lq

q

L

dt

d

t

L

1 1

M

mm

m

qq

L

dt

d

t

L

1

Lqq

L

dt

d

t

L M

mm

m1

dt

dH

0t

L constLqq

LH

M

mm

m

1

Page 36: Relativistic Classical Mechanics

Non-covariant Hamiltonian formulation of relativistic mechanics

• Example: 1D relativistic harmonic oscillator

• The Lagrangian is not an explicit function of time

• The quadrature involves elliptic integrals

7.98.4

constEHt

Ltot

0

2

2

22

32 kx

xc

mcVcmH

totEkx

xc

mc

2

2

22

3

2

222 kx

xcmcL

2

2

322

2

2

kxE

mccx

tot

2322

2

0)2()2(

)2(

mckcxcE

dxkxEtt

tot

tot

Page 37: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan A

• So far, our canonical formulations were not Lorentz-invariant – all the relationships were derived in a specific inertial reference frame

• We have to incorporate the time variable as one of the coordinates of the spacetime

• We need to introduce an invariant parameter, describing the progress of the system in configuration space:

• Then

7.10

3,2,1,0;'

d

dxx

2

1

),',(

dxxI

Page 38: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan A

• Equations of motion

• We need to find Lagrangians producing equations of motion for the observable behavior

• First approach: use previously found Lagrangians and replace time and velocities according to the rule:

7.10

3,2,1

idt

d

d

dx

dt

dx ii

2

1

),',(

dxxI

xxd

d

'

c

xt

0

ddtddxi

cx

dd

x i

0

'

0'

'

x

xc

i

Page 39: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan A

• Then

• So, we can assume that

• Attention: regardless of the functional dependence, the new Lagrangian is a homogeneous function of the generalized velocities in the first degree:

7.10

3,2,1;'

'0 i

x

xc

dt

dx ii

2

1

2

1

0

0

0

'

',,),,(

c

xd

x

xc

c

xxLdtxtxLI

ii

t

t

ii

c

xt

0

2

1

0

0

0 1

'

',,

dd

dx

cx

xc

c

xxL

ii

2

1

0

00

'

',,

'

dx

xc

c

xxL

c

x ii

0

00

'

',,

')',(

x

xc

c

xxL

c

xxx

ii

)',()',( xxaaxx

Page 40: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan A

• From Euler’s theorem on homogeneous functions it follows that

• Let us consider the following sum

7.10

3

0 ''

xx

)',()',( xxaaxx

3

0

'

xx

3

0

3

0 '''

xx

xx

3

0, '''

xx

xx

3

0,

2

'''

xxxx 0

3

0,

23

0,

2

''"'

'''

xxxx

xxxx

Page 41: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan A

• If three out of four equations of motion are satisfied, the fourth one is satisfied automatically

7.10

3

0,

23

0,

23

0 '''"

''''

xxxx

xxxx

xx

3

0,

22

'''"

'''

xxxx

xxxx

3

0

3

0

"''

''

'

xxx

xxx

x

3

0 ''

xd

dx

0''

3

0

xxxd

d xxd

d

'

Page 42: Relativistic Classical Mechanics

Example: a free particle

• We start with a non-covariant Lagrangian

7.10

22 1 mcL222

2 1

c

z

c

y

c

xmc

0

00

'

',,

')',(

x

xc

c

xxL

c

xxx

ii

2

0

32

0

22

0

1

2 ''

''

''

1

cxx

c

cxx

c

cxx

cmc

3

1

2

02

'

'1

i

i

x

xmc

3,2,1'

'0

ix

xc

dt

dx ii

3

1

2

0

02

0

00

'

'1

'

'

',,

'

i

iii

x

x

c

xmc

x

xc

c

xxL

c

x

3

1

220 ''i

ixxmc

Page 43: Relativistic Classical Mechanics

Example: a free particle

• Equations of motion

7.10

3

1

2200

00

'''

',,

'

i

ii

i xxmcx

xc

c

xxL

c

x

3

0

''

xxmc

3

0

''

xxmc

xxd

d

'

0'''

3

0

xxmcxd

d0

''

'

2

13

0

xx

mcx

d

d

d

d

d

dx

d

dxx '

d

du

dx

ud

'

Page 44: Relativistic Classical Mechanics

Example: a free particle

• Equations of motion of a free relativistic particle

7.10

02

'3

0

dd

udd

u

dd

mcu

d

d

u

x0

3

0

uu

mcu

d

d

cuu

3

0

0

c

mcu

d

d

0

)(

d

mud0

d

dp

0)()()(

d

vmd

d

vmd

d

vmd zyx

0d

dE

Page 45: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan B

• Instead of an arbitrary invariant parameter, we can use proper time

• However

• Thus, components of the four-velocity are not independent: they belong to three-dimensional manifold (hypersphere) in a 4D space

• Therefore, such Lagrangian formulation has an inherent constraint

• We will impose this constraint only after obtaining the equations of motion

7.10

2

3

0

3

0

cd

dx

d

dxuu

Page 46: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan B

• In this case, the equations of motion will look like

• But now the Lagrangian does not have to be a homogeneous function to the first degree

• Thus, we obtain freedom of choosing Lagrangians from a much broader class of functions that produce Lorentz-invariant equations of motion

• E.g., for a free particle we could choose

7.10

xud

d

3

0 2

umu

0)(

d

mud

Page 47: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan B

• If the particle is not free, then interaction terms have to be added to the Lagrangian – these terms must generate Lorentz-invariant equations of motion

• In general, these additional terms will represent interaction of a particle with some external field

• The specific form of the interaction will depend on the covariant formulation of the field theory

• Such program has been carried out for the following fields: electromagnetic, strong/weak nuclear, and a weak gravitational

7.10

Page 48: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan B

• Example: 1D relativistic motion in a linear potential

• In a specific inertial frame, the non-covariant Lagrangian was earlier shown to be

• The covariant form of this problem is

• In a specific inertial frame, the interaction vector will be reduced to

7.10

constaxmaxcmcL ;22

3

0

3

0 2

xGumu

1 maG

Page 49: Relativistic Classical Mechanics

Example: relativistic particle in an electromagnetic field

• For an electromagnetic field, the covariant Lagrangian has the following form:

• The corresponding equations of motion:

7.107.6

3

0

32103

0

),,,(2

uxxxxqAumu

3

2

1

/

A

A

A

c

A

3

0

)()(

x

uAAq

d

mud

AB

t

AE

3

0

uFq

0

0

0

0

1

xyz

xzy

yzx

zyx

cBcBE

cBcBE

cBcBE

EEE

cF

3

01

1)(

uFq

d

mud

Page 50: Relativistic Classical Mechanics

Example: relativistic particle in an electromagnetic field

• Maxwell's equations follow from this covariant formulation (check with your E&M class)

7.107.6

0

0

0

0

1

xyz

xzy

yzx

zyx

cBcBE

cBcBE

cBcBE

EEE

cF

3

01

1)(

uFq

d

mud)(

)( zyyzxx

vcBvcBcEc

q

dt

vmd

)( zyyzxx

vBvBEc

q

dt

dp xBvE

c

q)(

Page 51: Relativistic Classical Mechanics

Covariant Lagrangian formulation of relativistic mechanics: plan B

• What if we have many interacting particles?

• Complication #1: How to find an invariant parameter describing the evolution? (If proper time, then of what object?)

• Complication #2: How to describe covariantly the interaction between the particles? (Information cannot propagate faster than a speed of light – action-at-a-distance is outlawed)

• Currently, those are the areas of vigorous research

7.10

Page 52: Relativistic Classical Mechanics

Covariant Hamiltonian formulation of relativistic mechanics: plan A

• In ‘Plan A’, Lagrangians are homogeneous functions of the generalized velocities in the first degree

• Let us try to construct the Hamiltonians using canonical approach (Legendre transformation)

• ‘Plan A’: a bad idea !!!

8.4

)',()',( xxaaxx

3

0 ''

xx

3

0

'

x

3

0

''

x

x 'x

0

Page 53: Relativistic Classical Mechanics

Covariant Hamiltonian formulation of relativistic mechanics: plan B

• In ‘Plan B’: instead of an arbitrary invariant parameter, we use proper time

• We have to express four-velocities in terms of conjugate momenta and substitute these expressions into the Hamiltonian to make it a function of four-coordinates and four-momenta

• Don’t forget about the constraint:

8.4

3

0

up

3

0

u

u up

23

0

cuu

Page 54: Relativistic Classical Mechanics

Covariant Hamiltonian formulation of relativistic mechanics: plan B

• For a free particle:

8.4

3

0

up

3

0 2

umu

up

3

0 2

umu

u

3

0 2

uumg

u

umg mu

mpu /

3

0 2

m

pp

3

0

3

0

3

0 22

m

pp

m

pp

m

pp

3

0 2

m

pp

Page 55: Relativistic Classical Mechanics

Covariant Hamiltonian formulation of relativistic mechanics: plan B

• For a particle in an electromagnetic field:

8.4

3

0

up

3

0

3

0 2

uqAumu

up

3

0

3

0 2

uqAumu

u qAmu

m

qApu

3

0

3

0

3

0

)(2

))(()(

qApqA

m

qApqAp

m

qApp

3

0 2

))((

m

qApqAp

Page 56: Relativistic Classical Mechanics

Relativistic angular momentum

• For a single particle, the relativistic angular momentum is defined as an antisymmetric tensor of rank 2 in Minkowski space:

• This tensor has 6 independent elements; 3 of them coincide with the components of a regular angular momentum vector in non-relativistic limit

7.8

pxpxm

0)()(/

)(0)(/

)()(0/

///0

zyzxz

yzyxy

xzxyx

zyx

yvzvmxvzvmmvctczE

zvyvmxvyvmmvctcyE

zvxvmyvxvmmvctcxE

czEmvctcyEmvctcxEmvct

m

Page 57: Relativistic Classical Mechanics

Relativistic angular momentum

• Evolution of the relativistic angular momentum is determined by:

• For open systems, we have to define generalized relativistic torques in a covariant form

7.8

)(

pxpx

d

d

d

dm

d

dpx

d

dxp

d

dpx

d

dxp

d

dpx

d

dpxumuumu

From the equations of motion

d

dpx

d

dpx

d

dm

N

d

dm

Page 58: Relativistic Classical Mechanics

Relativistic kinematics of collisions

• The subject of relativistic collisions is of considerable interest in experimental high-energy physics

• Let us assume that the colliding particles do not interact outside of the collision region, and are not affected by any external potentials and fields

• We choose to work in a certain inertial reference frame; in the absence of external fields, the four-momentum of the system is conserved

• Conservation of a four-momentum includes conservation of a linear momentum and conservation of energy

7.7

0

d

dp

Page 59: Relativistic Classical Mechanics

Relativistic kinematics of collisions

• Usually we know the four-momenta of the colliding particles and need to find the four-momenta of the collision products

• There is a neat trick to deal with such problems:

• 1) Rearrange the equation for the conservation of the four-momentum of the system so that the four-momentum for the particle we are not interested in stands alone on one side of the equation

• 2) Write the magnitude squared of each side of the equation using the result that the magnitude squared of a four-momentum is an invariant

7.7

Page 60: Relativistic Classical Mechanics

Relativistic kinematics of collisions

• Let us assume that we have two particles before the collision (A and B) and two particles after the collision (C and D)

• Conservation of the four-momentum of the system:

• 1) Rearrange the equation (supposed we are not interested in particle D)

• 2) Magnitude squared of each side of the equation:

7.7

)()()()( DCBA pppp

)()()()( CBAD pppp

3

0

)()()()()()(

CBACBA pppppp

3

0

)()(

DD pp

Page 61: Relativistic Classical Mechanics

Relativistic kinematics of collisions7.7

3

0

)()()()()()(

CBACBA pppppp

3

0

)()(

DD pp

3

0

2222

)()()()()()(2

)()()()(

CBCABA

CBAD

pppppp

cmcmcmcm

3

0

3

0,

3

0

)()()()()()(

jijiji pppgppp

3

0

3

0

3

0

)()(2)()()()(

jijiji pppppp

ji

23

0

)()()( cmpp iii

Page 62: Relativistic Classical Mechanics

Example: electron-positron pair annihilation

• Annihilation of an electron and a positron produces two photons

• Conservation of the four-momentum of the system:

• Let us assume that the positron is initially at rest:

• 1) Rearrange the equation

21 ee

)()()()(21

pppp

2;0 mcEp

)()()()(

12pppp

Page 63: Relativistic Classical Mechanics

Example: electron-positron pair annihilation

• 2) Magnitude squared of each side of the equation:

3

0

)()()()()()(11

pppppp

3

0

)()(22

pp

3

0

2222

)()()()()()(2

)()()()(

11

12

pppppp

cmmcmccm 021 mm

0)()()()()()()(3

0

2

11

ppppppmc

)()()()(

12pppp

Page 64: Relativistic Classical Mechanics

Example: electron-positron pair annihilation

0)()()()()()()(3

0

2

11

ppppppmc

3

0

)()(

pp

pp

c

E

c

E

0;2 pmcE

mE

3

0

)()(1

pp

1

1

ppc

E

c

E

mE1

3

0

)()(1

pp

1

1

ppc

E

c

E

12

cos1

1 pp

c

EE

Page 65: Relativistic Classical Mechanics

Example: electron-positron pair annihilation

• The photon energy will be at a maximum when emitted in the forward direction, and at a minimum when emitted in the backward direction

0cos)( 122

1

1

1

pp

c

EEmEmEmc

11 pcE

0cos)( 122 11

1

pc

E

c

EEmEmEmc

mEmcc

p

c

EmE

21

2)(

cos1

12

22

cos)(

)(1

pcEmc

EmcmcE