# RELATIVISTIC KINEMATICS - Desy

Mechanics

Nov 13, 2013 (4 years and 4 months ago)

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RELATIVISTIC KINEMATICS
- four-vector:
),( rtr
i
r
=

c = 1

Minkowski metrics:
222
rtr
i
r
−=

- Lorentz transformation: x’ moves with velocity v w.r.t. x:

'
'
t
x
= γ

= γ

1
1
v
v

t
x

t
x

1
1
v
v

'
'
t
x
1²1/1 >−= vγ

- transformation t (x’=0) , x’(t=0) :

t

= γ t’
time dilatation: t’ = time in moving system
x’ = γ x
length contraction: x = length in rest system

- r
i
' r
i
' = r
i
r
i
square of 4-vectors Lorentz invariant
r
i
'

s
i
' = r
i
s
i
scalar product of ″ ″

- line element
ds =
22
rddt
r

= dt
2
1 v
r

= dt / γ

- four-velocity:
v
i
= dr
i
/ ds = γ dr
i
/ dt = γ (1,
v
r
) | ⋅ m

... invariant mass

- four-momentum:
p
i
= mv
i
= ( γ m, γ m
v
r
) = ( E,
p
r
)

m/E=
γ

Epv/
r
r
=

22
pEpp
i
i
r
−=

- rest system: ⇒
0p =
r
0v
=
r
, γ = 1
222
mEp
i
==
four-momentum
2
= invariant mass
2
- moving system:
0pmE
222
>+=
r

- non-relativistic: :
22
mp <<
r
kin
EmmpmpmE +=+≈+= 2/
222
r
r

- ultra-relativistic: :
22
mp >>
r
pE =