RELATIVISTIC KINEMATICS - Desy

conjunctionfrictionΜηχανική

13 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

105 εμφανίσεις

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 =