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PHY2061
R. D. Field
Department of Physics
relativity_4a.doc
University of Florida
Analogy with Rotations
Consider two frames of reference
the
Oframe
(label points
according to x,y) and the
O'
frame
(label points according to
x',y').
Let the origins of the two
frames coincide and rotate the
O'
frame
about the zaxis by an angle
θ
.
The two frames are related by
the following transformation (
i.e.
by a rotation).
θ
cos
sin
sin
cos
y
x
y
y
x
x
′
+
′
=
′
−
′
=
cos
sin
sin
cos
y
x
y
y
x
x
+
−
=
′
+
=
′
Vector Notation:
′
′
−
=
y
x
y
x
cos
sin
sin
cos
r
R
r
′
=
r
r
−
=
cos
sin
sin
cos
R
=
y
x
r
r
′
′
=
′
y
x
r
r
Rotational Invariant:
2
2
2
2
2
2
r
r
r
y
x
y
x
r
r
r
′
=
′
⋅
′
=
′
+
′
=
+
=
⋅
=
r
r
r
r
Lorentz Transformation:
Let
cosh
θ
=
γ
and
sinh
θ
=
βγ
βγ
then
′
′
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 Spring '08
 FRY
 Physics

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