SLIDE 7 An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same direction as the initial particle; the second photon has energy 4 MeV and travels in the direction
- pposite to that of the first. Write the relativistic equations for conservation of momentum and
energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable particle. before after photon 1 photon 2
Problem
An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same direction as the initial particle; the second photon has energy 4 MeV and travels in the direction
- pposite to that of the first. Write the relativistic equations for conservation of momentum and
energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable particle.
( )
1 2 2
1 /
ph ph
E E mv c c v c = − −
before after momentum conservation
( )
2 1 2 2
1 /
ph ph
mc E E v c = + −
energy conservation
( )
1 2 2
8 4 4 1 /
ph ph
mvc E E MeV MeV MeV v c = − = − = −
( )
2 1 2 2
8 4 12 1 /
ph ph
mc E E MeV MeV MeV v c = + = + = − ( ) 4 1 ( ) 12 3 v a c b = = =
(a) (b)
( )
( ) ( )
2 2 1 2
1 / 12 1 1/ 9 11.3
ph ph
mc E E v c MeV MeV = + − = − ≈
photon 1 photon 2
Problem Problem
A moving electron collides with a stationary electron and an electron-positron pair comes into being as a result. When all four particles have the same velocity after the collision, the kinetic energy required for this process is a minimum. Use a relativistic calculation to show that Kmin=6mc2, where m is the electron mass. before after energy conservation
2 1 2
4 E mc E + =
1 2
4 p p =
momentum conservation
( )
( )
2 2 2 2 1 1
E mc p c = +
( )
( )
2 2 2 2 2 2
E mc p c = +
2 1 2
4 E mc E + =
1 2
4 p p =
( )
( )
( )
( )
( )
2 2 2 2 2 2 2 2 1 1 2 1
1 2 16 16 16 E E mc mc E mc p c ! " + + = = + # $ % &
( ) ( )
( ) ( )
2 2 2 2 2 2 2 1 1 1
2 16 E p c E mc mc mc − + + =
( )
2 2
mc
( )
2 2 2 2 1
14 / 2 7 E mc mc mc = =
1
p
2
4p
2 2 1 1
6 K E mc mc = − =
In the center-of-mass RF: before after
1 '
p
1 '
p
2 1
2 ' 4 E mc =
( )
2 2 1'
' 2 ' 2 E mc mc γ γ = = → =
( )
2 2
' 1 3 1 ' 4 2 v v c c − = → =
relative speed
' 2 v V =
( )
2
' 3 4 3 48 1 3/ 4 7 49 1 '/ v V v vv c + = = = = + +
( )
2 2 2 1
6 1 48/ 49 mc K mc mc = − = −
- General relativity is the geometric theory of gravitation published by
Albert Einstein in 1916.
- It is the current description of gravitation in modern physics.
- It unifies special relativity and Newton's law of universal gravitation,
and describes gravity as a geometric property of space and time.
- In particular, the curvature of space-time is directly related to the
four-momentum (mass-energy and momentum).
- The relation is specified by the Einstein’s field equations,
a system of partial differential equations. (graduate level course)
General Relativity
