relativistic mechanics - department of physics and …physics.gmu.edu/~joe/phys428/topic4.pdf ·...

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Relativistic Mechanics SR as a meta-theory:  laws of physics must be Lorentz-invariant. Laws of classical mechanics are not:  they are invariant under Galilean transformations, not Lorentz transformations. =>  we need a revised, relativistic, mechanics! Starting point:  Assume each particle has an invariant proper mass                           (or rest mass)  m 0  Define 4-momentum  m 0  U Basic axiom: conservation of in particle collisions:  * P n  =  0 (in the sum, P of each particle going into the collision is counted   positively while of each particle coming out is counted negatively) Since a sum of 4-vectors is a 4-vector, this equation is Lorentz-invariant. 1

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Page 1: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Relativistic Mechanics

SR as a meta­theory:  laws of physics must be Lorentz­invariant.

Laws of classical mechanics are not:  they are invariant under Galilean transformations, not Lorentz transformations.

=>  we need a revised, relativistic, mechanics! 

Starting point:  Assume each particle has an invariant proper mass                           (or rest mass)   m

Define 4­momentum  P = m0 U

Basic axiom:  conservation of P in particle collisions:   * Pn =  0

(in the sum, P of each particle going into the collision is counted  positively while P of each particle coming out is counted negatively)

Since a sum of 4­vectors is a 4­vector, this equation is Lorentz­invariant.

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Page 2: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

P = m0 U = m

0 (u) (u, c)  ≡ (p, mc)

relativistic mass:             m = (u) m0  

relativistic momentum:   p = m u

* p = 0        * m = 0Conservation of 4­momentum yields both:

These are Newtonian conservation laws when  c  ∞

What about conservation of energy?  

When v/c ≪ 1 :

Einstein proposed the equivalence of mass and energy:  E = mc2

=>  P = (p, E / c)

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Page 3: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Relativistic energies 

rest energy:           m0c2  

kinetic energy:      T = mc2 – m0c2 = ( ­ 1) m

0c2

Note:  Potential energy associated with position of particle in an            external electromagnetic or gravitational field does not contribute           to the relativistic mass of a particle.  When particle's PE changes,           it's actually the energy of the field that's changing.  

H atom:  energy of electron = ­13.6 eV   (binding energy)

=>  matom

 c2 = mp c2 + m

e c2 – 13.6 eV

=>  atom has less mass than its constituents!

1 eV = 1.602 × 10­12 ergs = 1.602 × 10­19 J

mp = 1.67 × 10­24 g   ;   m

e = 9.11 × 10­28 g

=>  m / m = 13.6 eV / (mp + m

e ) c2 ≈ 1.8 × 10­8

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Page 4: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Nuclear physicists use the atomic mass unit  u, defined so that the massof the 12C atom is exactly 12 u.

1 u = 931.502 MeV / c2

mp = 1.00727647 u   ;   m

n = 1.00866501 u   ;   m

e = 5.485803 × 10­4 u

=>  m / m ≈ 0.8%   for the 12C  nucleus

Binding energy ≈ 90 MeV    (≈ 7.5 MeV per nucleon)

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Page 5: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

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Page 6: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

P  =  m0 (u) (u, c)  =  (p, mc)  =  (p, E / c)

=>  P2 = E2/c2 – p2

In particle's rest frame:  P = (0, m0c)  =>  P2 = m

02 c2

=>  E2/c2 – p2 = m0

2 c2   =>   E2 = m0

2 c4 + p2c2

For a photon:  m0 = 0   =>   E = pc   ,   P = E/c (n, 1)

E = h = hc/    ,    n = a unit vector in dir of propagation

Since P is a 4­vector, E and p transform as follows under a standard LT:

px' = (p

x ­ E/c)

py' = p

y

pz' = p

z

E' = (E ­ pxc)

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Page 7: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Applications

1.  Compton scattering:  photon strikes stationary electron

Before:E e­

E'

After:

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Page 8: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

h / mec  is called the 

“Compton wavelength”of the electron.

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SP 4.1

Page 9: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

2.  Inverse Compton scattering:  photon scatters off a highly energetic                                                    charged particle

BeforeE

m0

After

E2

E1

As before:  

=>  

Assume E ≫ m0c2 ≫ E

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Page 10: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

(E ≫ m0c2)

(from previous slide)

=>

Maximum energy is transferred to photon when cos  = 1:  

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Page 11: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Cosmic Microwave Background  (CMB):  3 K  blackbody spectrum

=>  typical photon energy ~ kT = 3 × 10­4 eV

Very high energy cosmic ray proton:  1020 eV ;   mp c2 = 938 MeV

=>  CMB photon can be upscattered to ~ 1019 eV   (source of gamma rays)

3.  Particle accelerators:  “available energy” for creating new particles                                        is the energy in the CM frame

Consider 2 arrangments:  

a)  a 30 GeV proton collides with a proton at rest

b)  two 15 GeV protons collide head­on

(lab and CM frames are identical  =>  30 GeV of                                                              available energy)

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Page 12: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

a)   lab frame:  Pi = [ m

pc, (+1) m

pc ]         CM frame:  P = (0, E/c)

To reach an available energy of 30 GeV in arrangement (a), we would need a 480 GeV proton!

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Page 13: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

4.  Cosmic ray cutoff:  high­energy nucleons can undergo the                                      reaction     + N    N + 

Reactions with CMB photons (E ~ 3 × 10­4 eV) probably produces a cutoff in the cosmic ray spectrum:  no CRs with energy above the threshold energy for the reaction.

Assuming a head­on collision, what's the threshold energy?

We need E = (mN + m

) c2  in CM frame  =>  P

f = (0, m

Nc + m

c)

Lab frame, before collision:

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Page 14: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

=>  EN ≫ m

Nc2   =>   LHS  2 E

N     =>   E

N ≈ 3 × 1014 MeV

(mNc2 ≈ 940 MeV ,  m

c2 ≈ 140 MeV)  

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SP 4.2­8

Page 15: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

What about forces?

Define 4­force  F = dP/d = d(m0U) / d = m

0A + (dm

0/d) U

Limited version of Newton's 3rd Law for contact collisions between 2 particles:   during contact,  is the same for both particles.  

F1 + F

2 = d(P

1 + P

2) / d = 0    =>

    F

2 = ­ F

1

Relativistic 3­force  f = dp/dt = d(mu)/dt  

f becomes Newtonian force when   0

4th component of F is proportional to the power absorbed by the particle.

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Page 16: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

=>  F⋅U = proper rate at which particle's internal energy increases

Also:  

Equating above 2 expressions for F⋅U  =>  rest mass­preserving forces are characterized by:  

F⋅U = 0

f ⋅u = dE/dt     =>       f ⋅dr = dE    F = (u) (f, f ⋅u /c)

(as in Newtonian theory; the added energy  is entirely kinetic)

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Page 17: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

4­force transformation:           (Q ≡ dE/dt)

Note:  for rest mass­preserving force, f is invariant among IFs with            relative velocities parallel to force.

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SP 4.9

Page 18: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

For a rest mass­preserving force:  

a is not generally parallel to f

A moving particle offers different inertial resistances to the same force, depending on whether it is applied longitudinally or transverse.

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Page 19: Relativistic Mechanics - Department of Physics and …physics.gmu.edu/~joe/PHYS428/Topic4.pdf · Relativistic Mechanics SR as a metatheory: laws of physics must be Lorentzinvariant

Also for a rest mass­preserving force:

F  =  m0A + dm

0/d U  =  m

0 d2R/d2  =  (u) (f, c­1 dE/dt)

=>   (u) f = m0 d2r/d2

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