Physics 123

1
Special relativityenergy, momentum and mass

• Physics 123

2
Outline

• Lorentz transformations
• 4-dimentional
• energy-momentum
• Mass is energy
• Doppler shift

3
Lorentz transformations
System (x,yz,t) is moving with respect to
system (x,y,z,t) with velocity v

• Lorentz
• xg(xvt)
• yy
• zz
• tg(tvx/c2)

• Galileo
• xxvt
• yy
• zz
• tt

4
Time dilation

• Clocks moving relative to an observer are
  measured by the observer to run more slowly ( as
  compared to clocks at rest)
• Dt measured in v0 frame, Dt0- measured in
  moving frame

Hendrik Antoon Lorentz Derived time and space
transformations before Einstein
5
Twin paradox

• Two twins Joe and Jane. Joe stays on Earth and
  Jane goes to Pluto at vltc
• Joe observes that Jane's on-board clocks
  (including her biological one), which run at
  Jane's proper time, run slowly on both outbound
  and return leg. He therefore concludes that she
  will be younger than he will be when she returns.
  
• On the outward leg Jane observes Joe's clock to
  run slowly, and she observes that it ticks slowly
  on the return run. So will Jane conclude that Joe
  will have aged less? And if she does, who is
  correct?

6
Length contraction

• No change in directions perpendicular to velocity

• The length of an object is measured to be shorter
  when it is moving relative to the observer than
  when it is at rest

7
4-dimensional space time

• Add time to space metric x1x, x2y, x3z,
  x4ict
• 4- dimensional lengthinterval - Lorentz
  invariant
• AB real space-like interval, there exists a
  frame of reference where the two events happen at
  the same time (t1t2 ), but at different places
  (r12?0)
• AB imaginary time-like interval, there exists
  a frame of reference where the two events happen
  at the same place (r120), but at different times
  (t1?t2)

8
Energy, mass and momentum

• m0 mass at rest
• Relativistic energy

• Energy at rest
• Em0c2
• Kinetic energy

• Relativistic momentum

• 4-dimensional Energy momentum vector
• (pxc, pyc, pzc, iE)
• Lorentz invariant interval

9
Conservation laws

• Both energy and momentum are conserved in the
  relativistic case
• Mass must be considered as an integral component
  of energy Egmc2

10
Conservation laws

• Energy could be used to create mass
• To conserve momentum electron and positron must
  collide head on. Then Z-boson is produced at
  rest.

11
Conservation laws

• Mass could be destroyed and converted into energy
• To conserve momentum (zero initially) the photons
  must be flying in the opposite direction with the
  same absolute values of momenta

12
Mass and energy

• Mass and energy are interchangeable
• Energy can be used to create mass (matter)
• Mass can be destroyed and energy released

13
Doppler shift

• Light emitted at f0,l0
• In the sources r.f.
• the distance between crests is l0
• The time between crests is t01/f0 l0/c
• Where are crests in the r.f. moving with speed v
  wrt sources r.f. (chasing the wave)
• lcDt-vDt(c-v)Dt
• DtgDt0g l0/c
• l(c-v) g l0/c

14
Doppler shift

• When the source and the observer move towards
  each other the wavelength decrease (red?violet)
• When the source and the observer move away from
  each other the wavelength increase (violet ? red)
  Redshift used to measure galaxies velocities
  ?universe expansion (Hubble)