Transcript CHAPTER 2: Special Theory of Relativity

The Lorentz Velocity Transformations
defining velocities as: ux = dx/dt, uy = dy/dt,
u’x = dx’/dt’, etc. it is easily shown that:
With similar relations for uy and uz:
The Lorentz Velocity Transformations
In addition to the previous relations, the Lorentz
velocity transformations for u’x, u’y , and u’z can
be obtained by switching primed and unprimed and
changing v to –v:
The Lorentz Velocity Transformations: an
object moves with the speed of light
u’x = c (light or, if neutrinos are massless, they must travel at the speed of light)
v )
c
(1

u 'x  v
cv
c
ux 


c
vu x
vc
v
1

1

1
2
c
2
c
c
Spacetime diagrams
For two events
lightlines
( x1 , t1 ),( x2 , t2 ); x  x2  x1 ; t  t2  t1
Spacetime interval
s   x  ( c  t )
2
2
s  0
lightlike
s  0
spacelike
s  0
timelike
2
2
Causality: cause-and-effect relations
Invariant:
s  x  y  z  (ct )
2
2
2
2
2
2
2
Courtesy: Wikimedia Commons and John Walker
Doppler effect
source
observer
λ=(c-v)t/n=λ0(1-v/c)
f=1/T=1/(λ/c)=f0/(1-v/c)
Light wave proper period seen in the
source frame T0 (f0=1/ T0)
v
light, x=ct=n
source, x=vt
period in the observer’s frame T=T0
f0
1
1
1
1
f 


(1  v / c) T (1  v / c)  T0 (1  v / c)
1 (1  v 2 / c 2 ) 1 (1  v / c)
(1  v / c)



f0
T0 (1  v / c)
T0 (1  v / c)
(1  v / c)
x=(c-v)t=nλ’
Astronomy: redshift
(1  v / c )
f 
f0
(1  v / c )
f  f0
EXAMPLE: DOPPLER EFFECT IN FAST ION BEAM PRECISION LASER
SPECTROSCOPY IN COLLINEAR AND ANTI-COLLINEAR GEOMETRIES
REFERENCED TO A FREQUENCY COMB
Another example with Doppler effect: laser cooling of atoms
Exclusion of relativistic frequency shifts by combining
collinear and anticollinear measurements
Frequency of light perceived by a moving ion
' 
 [ 1   cos(  )]
1  2
Laser tuned to resonance; the perceived frequency equals the transition frequency
 '   tr
1 
 tr
1 
1 

 tr
1 
Thus, the resonance frequencies for collinear geometry (  0 )  c 
and for anticollinear geometry (
 180 )  ac
To obtain the transition frequency we take the product
and this is an exact relativistic formula!
 tr   c ac
2.6
#31
2.6
#32
Solution:
Twin Paradox
(Courtesy: Wikimedia Commons)
2.7: Experimental Verification
Time Dilation and Muon Decay
Figure 2.18: The number of muons detected with speeds near 0.98c is much different (a)
on top of a mountain than (b) at sea level, because of the muon’s decay. The
experimental result agrees with our time dilation equation.
v=0.98c
Atomic Clock Measurement
Figure 2.20: Two airplanes took off (at different times) from Washington, D.C., where the U.S. Naval
Observatory is located. The airplanes traveled east and west around Earth as it rotated. Atomic clocks on
the airplanes were compared with similar clocks kept at the observatory to show that the moving clocks
in the airplanes ran differently.
Flight time
(41.2 h)
(48.6 h)
The time is changing in the moving frame, but the calculations must also take
into account corrections due to general relativity (Einstein). Analysis shows that
the special theory of relativity is verified within the experimental uncertainties.
2.11: Relativistic Momentum
Because physicists believe that the conservation
of momentum is fundamental, we begin by
considering collisions where there do not exist
external forces and
dP/dt = Fext = 0
Relativistic Momentum
Frank (fixed or stationary system) is at rest in system K holding a ball of
mass m. Mary (moving system) holds a similar ball in system K that is
moving in the x direction with velocity v with respect to system K.
Relativistic Momentum

If we use the definition of momentum, the
momentum of the ball thrown by Frank is
entirely in the y direction:
pFy = mu0
The change of momentum as observed by
Frank is
ΔpF = ΔpFy = −2mu0
According to Mary (the Moving frame)

Mary measures the initial velocity of her own
ball to be u’Mx = 0 and u’My = −u0.
In order to determine the velocity of Mary’s
ball as measured by Frank we use the
velocity transformation equations:
(we used velocity summation formula
with ux=0)
Relativistic Momentum
Before the collision, the momentum of Mary’s ball as measured
by Frank (the Fixed frame) becomes
Before
Before
(2.42)
For a perfectly elastic collision, the momentum after the collision is
After
After
(2.43)
The change in momentum of Mary’s ball according to Frank is
(2.44)
Relativistic Momentum (con’t)
 The conservation of linear momentum requires the
total change in momentum of the collision, ΔpF + ΔpM,
to be zero. The addition of Equations (2.40) and (2.44)
clearly does not give zero.
ΔpF = ΔpFy = −2mu0
Linear momentum is not conserved if we use the
conventions for momentum from classical physics
even if we use the velocity transformation equations
from the special theory of relativity.
There is no problem with the x direction, but there is a
problem with the y direction along the direction the ball
is thrown in each system.
Relativistic Momentum


Rather than abandon the conservation of linear
momentum, let us look for a modification of the
definition of linear momentum that preserves both it
and Newton’s second law.
To do so requires reexamining mass to conclude that:
Relativistic momentum (2.48)
With modified (relativistic) momentum
pFy  
2mu0
1  u0 2 / c 2
Now ΔpF + ΔpM =0
and momentum conserved!
2m0u0 1  v 2 / c 2
pMy 
1  [v  u0 (1  v ) / c ] / c
2
2
2
2m0u0 1  v 2 / c 2
(1  v / c )(1  u0 / c )
2
2
2
2

2
2

2m0u0
(1  u0 2 / c 2 )
Relativistic Momentum: two points of view
physicists
like to refer to the mass in Equation (2.48) as
the rest mass m0 and call the term m = γm0 the relativistic
mass. In this manner the classical form of momentum,
p=mu, is retained. The mass is then imagined to increase
at high speeds.
other
physicists prefer to keep the concept of mass as an
invariant, intrinsic property of an object. We adopt this latter
approach and will use the term mass exclusively to mean
rest mass.
Behavior of relativistic momentum and classical momentum for v/c->1
2.11
#60
(a)
1
(0.58312) 2
2  
2
v
v


1  
 
c
 
c
Homework (will be not graded):
Problems
2.5. #20,21,23,27
Homework
2.6 #31, #32
2.7 #37
2.11 #60
Thank you for your attention!