Lorentz Invariants
We found above that for an event (
x′, y′, z′, t′
) for which
the coordinates
of the event (
x
,
y
,
z
,
t
) as measured in the other frame
S
satisfy
The
quantity
is said to be a
Lorentz invariant
: it doesn’t vary on going from one
frame to another.
A simple twodimensional analogy to this invariant is given by considering two sets of axes,
Oxy
and
Ox′ y′
having the same origin
O
, but the axis
Ox′
is at an angle to
Ox
, so one set of axes is
the same as the other set but rotated. The point
P
with coordinates (
x
,
y
) has coordinates (
x′
,
y′
)
measured on the
Ox′ y′
axes. The square of the distance of the point
P
from the common origin
O
is
x
2
+
y
2
and is also
x′
2
+
y′
2
, so for the transformation from coordinates (
x
,
y
) to (
x′
,
y′
),
x
2
+
y
2
is an invariant. Similarly, if a point
P
1
has coordinates (
x
1
,
y
1
) and (
x
1
′,
y
1
′) and another point
P
2
has coordinates (
x
2
,
y
2
) and (
x
2
′,
y
2
′) then clearly the two points are the same distance apart as
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 Fall '10
 DavidJudd
 Physics, Special Relativity

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