Lecture 38: Relativistic mechanics (2 Dec 09)
Project:
2 homeworklike problems from FetterWalecka or Goldstein on
“underrepresented topics” in the homework, for instance HamiltonJacobi
theory, continuum mechanics, hydrodynamics, or relativity.
A. Review: 4vector formalism
1. The Lorentz transformation along ˆ
x
is
ct
*
=
γ
(
ct

βx
)
x
*
=
γ
(
x

βct
)
y
*
=
y
;
z
*
=
z
;
β
=
v/c
;
γ
= 1
/
q
1

β
2
2. Consider this as a special case of transformations on 4vector
x
0
(=
ct
)
, x, y, z
preserving the length
r
2

x
2
0
.
(
x
μ
, x
μ
)
≡
X
μ
g
μ
x
2
μ
;
g
0
=

1
, g
1
=
g
2
=
g
3
= 1
3. In matrix notation (this transform is the defining property of a 4
vector):
x
*
μ
=
X
ν
a
μν
x
ν
→
X
μ
g
μ
a
μν
a
μλ
=
g
ν
δ
νλ
and the inverse matrix is
a

1
νμ
=
g
ν
g
μ
a
μν
→
x
ν
=
X
μ
g
ν
g
μ
a
μν
x
*
μ
4. Form a 4vector from the gradient (and partial with respect to
t
):
2
μ
=
g
μ
∂
∂x
μ
;
g
μ
∂
∂x
*
μ
=
X
ν
a
μν
g
ν
∂
∂x
ν
and the wave operator is a 4scalar:
(
2
μ
,
2
μ
) =
X
μ
g
μ
2
μ
2
μ
=
∇
2

1
c
2
∂
2
∂t
2
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
B. Mechanics
1.
r
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 BRUCH
 Energy, Kinetic Energy, Momentum, Work, Special Relativity, p2 c2

Click to edit the document details