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Unformatted text preview: (θ ) =
− B ( θ ) A( θ )
A( θ ) 2 + B ( θ ) 2 = 1
A π /2) = 0
B π /2) = 1 A(0) = 1
B (0) = 0 2 Upon inspection, M (θ) reveals itself to be3
M (θ ) =
− sin (θ) sin (θ)
cos (θ) Transformations in Physics
Stuﬀ happens. Events happen. Events happen independent of any system we
may choose to describe them in. An event happens some-where and some-when
once we place it in a frame of reference. A valid description of an event in some
frame will consist of a temporal (time) coordinate4 and as many as three spacial
Frames of Reference:
S y S’ y’ O O’ v =β c
Lab Rocket Following the usual convention, the “Rocket Frame” (S ) moves through the
“Lab Frame” (S ) with a velocity = β c (measured in the lab frame). Measurev
ments in the rocket frame will generally be primed, measurements in the lab
frame will generally not be primed.
As was the case in geometric transformations, and for essentially the same reason, it is incredibly beneﬁcial to have the two frames share a common origin
(in space and time!) - this is satisﬁed if we require the two spacial coordinate
frames (x,y ,z and x ,y ,z ) coincide at t = t = 0.
Setting up the Math
Provided the frames share a common origin in space and time, it is relatively
straightforward to transform the description of events from one inertial frame
to another. . .
3 How does this diﬀer from the rotation operator you memorized in your introductory
calculus class? Why does it diﬀer?
4 The factor of c in the temporal element is there provisionally to give that element the
same dimensionality as the spacial elements. We will revisit this choice later. 3 r = L( β ) r ct
L 0 ,0 ( β ) x L 1 , 0 ( β ) = y L 2 , 0 ( β )
L 3 ,0 ( β ) L 0 ,1 ( β )
L 1 ,1 ( β )
L 2 ,1 ( β )
L 3 ,1 ( β ) L 0 ,2 ( β )
L 1 ,2 ( β )
L 2 ,2 ( β )
L 3 ,2 ( β ) L 0 ,3 ( β )
L 1 ,3 ( β ) L 2 ,3 ( β ) L 3 ,3 ( β ) ct x
y z Leading to. . .
ct = L0,0 (β ) ct + L0,1 (β ) x + L0,2 (β ) y + L0,3 (β...
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