PHY190 Lecture #15
November 23, 2007
R9.4 Properties of Four Momentum
•
On a spacetime diagram
d
⇒
R
dτ
is tangent to the worldline as an object moves
•
time coordinate is treated on equal footing with the three space coordinates
•
By defining fourvelocity in terms of a fourvector (
d
⇒
R
) and an invariant (
dτ
) we can calculate
d
⇒
R
dτ
in other frames (once we
know it in one frame).
◦
Do this with the Lorentz transformations that we already know
•
Similarly
⇒
P
is also the ratio of a fourvector (
d
⇒
R
and two invariants (
m
and
dτ
)
P
t
=
mdt /dτ
=
m
γ
(
dt

vdx
)
dτ
=
γm
dt
dτ

dx
dτ
=
γ
(
P
t

vP
x
)
•
In general,
P
t
=
γ
(
P
t

vP
x
)
P
x
=
γ
(
P
x

vP
t
)
P
y
=
P
y
P
z
=
P
z
•
These are the same transformations as
(
t
;
x
)
→
(
t
;
x
)
•
In general if you have any fourvector:
⇒
A
= (
A
t
;
A
x
, A
y
, A
z
)
•
Can transform them to another frame with:
⇒
A
=
γ

γv
0
0

γv
γ
0
0
0
0
1
0
0
0
0
1
⇒
A
A
t
A
x
A
y
A
z
=
γ

γv
0
0

γv
γ
0
0
0
0
1
0
0
0
0
1
A
t
A
x
A
y
A
z
•
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 Fall '08
 BERTOLDI
 Momentum, Special Relativity, Lorentz

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