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Relativistic Momentum
Classical physics:
Definition of momentum:
p = mv
Conservation of momentum: p1 + p2 = p3 + p4
Coordinate transformation (Galilei; velocity of object: u; frame: v):
p' = m u' = m(u + v) = p + mv
 p1' + p2' = p1 + p2 + mv + Mv = p3 + p4 + mv + Mv = p3' + p4'
p4
p3
m
p1
p2
M
Relativistic physics:
Same definition of momentum does not work due to complicated velocity transformation
 Use four-velociy (has simpler transformation law!): define four-momentum: P = m U
Assume four-momentum conservation in one frame: P1 + P2 = P3 + P4
Coordinate transformation (Lorentz):
P' = m U' = m Lv-1U = Lv-1 m U = Lv-1 P
 P1' + P2' = Lv-1 P1 + Lv-1 P2 = Lv-1(P1 + P2) = Lv-1(P3 + P4 ) = P3' + P4'
c
()
c
()
U = u u  P = m u u
Space component (velocity of object: v): p =  mv
This is the relativistic momentum (conserved in all inertial frames!)
Define dynamic mass: md = m  p = md v (equivalent to classical case)
To distinguish between the masses we call m the rest mass
Relativistic Momentum and Energy
energy
mc
1 v2 = –
1 (mc2 + –1 mv2)
Time component :  mc = –––––––
=
–––––––
≈
mc
(1
+
–
––
)
2 2
2
c
1 – v /c
2 c
c
2
Time component of four-momentum conservation (same approximation, v << c ):
(mc2 + ½ mv12) + (Mc2 + ½ Mv22 ) = (mc2 + ½ mv32) + (Mc2 + ½ Mv42 )
E1 +
E2
=
E3
+
E4
Conservation of Energy!
Total energy:
E = mc2 = md c2
Rest energy:
E0 = mc2 (Energy for v = 0)
Kinetic energy:
Ekin = E – E0 =  mc2 – mc2 = ( – 1) mc2
Some important relations
Analog to the spacetime interval we calculate the ‘magnitude’ of the four-momentum:
P2 = (E/c)2 – p2 = ( mc)2 – ( mv)2 = 2m2c2 (1 – v2/c2) = m2c2  E2 = p2c2 + m2c4
For m  0 (p2c2 >> m2c4): E = pc
Velocity in terms of E and p : v = p/(m) = pc2/E ; for m  0 : v = c
Coordinate transformation:
P = m U = m Lv U' = Lv P'  E' =  (E – v px); px' =  (px – v/c2 E); py' = py ; pz' = pz
Relativistic Momentum and Energy
Energy transformation and Doppler shift
Energy transformation for massless objects ( p = E/c; assume p = px)
1 – v/c = E
E' =  (E – v px) =  (E – v E/c) = E –––––––
2 2

1––––
– v/c
1 + v/c
If px = 0: E' =  E
Identical to Doppler shift formulas  Energy proportional to frequency (for m = 0)
p
 Can be used to easily deduce general formula for Doppler shift:
px = p cos q
E' =  (E – v px) =  (E – v p cos q) = E  (1 – v/c cos q) (for m = 0)
q
 fE = fR (1 – v/c cos q)
px
v
1 – v /c
Relativistic Momentum and Energy
Energy transformation and Doppler shift
Energy transformation for massless objects ( p = E/c; assume p = px)
1 – v/c = E
E' =  (E – v px) =  (E – v E/c) = E –––––––
2 2

1––––
– v/c
1 + v/c
If px = 0: E' =  E
Identical to Doppler shift formulas  Energy proportional to frequency (for m = 0)
p
 Can be used to easily deduce general formula for Doppler shift:
px = p cos q
E' =  (E – v px) =  (E – v p cos q) = E  (1 – v/c cos q) (for m = 0)
q
 fE = fR (1 – v/c cos q)
px
v
1 – v /c
Equivalence of Mass and Energy: Einstein’s box
M M
L L
Dx
Centre of mass cannot shift in isolated system
 light pulse has a mass equivalent:
Conservation of momentum: p = E/c = Mv
Total time of process: t = L/c
CM shift of box:
MDx = Mvt = p L/c = E/c2 L
CM shift of light:
mlightDxlight = mlight L
 mlight = E/c2
Headlight Effect
Laboratory frame
E-M Radiation emitted isotropically
Rest frame of emitter
v = 0.9 c
v = 0.9 c
Galilean
transf.
v = 0.99 c
Relativistic Wire
at rest in S
at rest in S'
S'
S
charged
wire
uncharged wire
I
S
v
–Q
B = µI/2d
FB = – Qv×B
Lorentz force
(attractive)
I
S'
–Q
B = µI/2d
FB = – Qv×B = 0
Electrostatic force
(attractive)
General Relativity: Spacetime curvature
General Relativity: Spacetime curvature
General Relativity: Gravitational Lensing
General Relativity: Spacetime Diagram, Light Cones
ct
ct'
General Relativity
Future
y
Elsewhere
x'
x
Past
Special Relativity
Object ….
falling
into
Black hole