PHYS 230
Fall 2007
Assignment 7
1. In class, as in the textbook, we “proved” that
(
c
Δ
t
)
2

(Δ
x
)
2
= (
c
Δ
τ
)
2
= (
c
Δ
t
±
)
2

(Δ
x
±
)
2
using a geometric construction. Prove it from the Lorentz transformation equations
using the following steps:
(a) Show that
(
ct
)
2

(
x
)
2
= (
ct
±
)
2

(
x
±
)
2
Use the Lorentz transformation equations:
ct
=
γ
(
ct
±
+
βx
±
)
and
x
=
γ
(
x
±
+
βct
±
)
and substitute into the LHS.
We obtain
γ
2
[(
ct
±
+
βx
±
)
2

(
x
±
+
βct
±
)
2
]
or
γ
2
(1

β
2
)[(
ct
±
)
2

(
x
±
)
2
]
since the cross terms cancel.
But,
γ
2
(1

β
2
)
is just
1
, so the identity is proved.
(b) Diﬀerentiate, use an appropriate expansion, and show therefore that
(
c
Δ
t
)
2

(Δ
x
)
2
= (
c
Δ
τ
)
2
= (
c
Δ
t
±
)
2

(Δ
x
±
)
2
The simplest way is to diﬀerentiate the Lorentz transformation equations directly,
with respect to
τ
, the proper time, to obtain
dx
±
=
γ
(
dx

βcdt
)
and
cdt
±
=
γ
(
cdt

βdx
)
Thus
(
dx
±
)
2

(
cdt
±
)
2
=
γ
2
[(
dx

βcdt
)
2

(
cdt

βdx
)
2
]
or
(
dx
±
)
2

(
cdt
±
)
2
=
γ
2
(1

β
2
)[(
dx
)
2

(
cdt
)
2
] +
γ
2
[

2
βdxdt
+ 2
βdxdt
]
Since, as before,
γ
2
(1

β
2
) = 1
, this gives
(
dx
±
)
2

(
cdt
±
)
2
= (
dx
)
2

(
cdt
)
2
1
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An alternative way is to replace
x
by
x
+ Δ
x
, and
ct
by
ct
+
c
Δ
t
on the LHS, to
multiply out, and then show that, order by order, the expression is equal to the
RHS.
Thus, show that
(
x
+ Δ
x
)
2

(
ct
+
c
Δ
t
)
2
= (
x
±
+ Δ
x
±
)
2

(
ct
±
+
c
Δ
t
±
)
2
Multiplying out, the zero order terms are equal from part (a). The ﬁrst order
terms give
2
x
Δ
x

2
c
2
t
Δ
t
= 2
x
±
Δ
x
±

2
c
2
t
±
Δ
t
±
and the second order terms give what is required.
So, we must show that the ﬁrst order terms
are
equivalent. Use the Lorentz trans
formations again, and substitute on the LHS to give
γ
2
(Δ
x
±
+
βc
Δ
t
±
)(
x
±
+
βct
±
)

γ
2
(
c
Δ
t
±
+
β
Δ
x
±
)(
ct
±
+
βx
±
)
The cross terms cancel, and, once again,
γ
2
(1

β
2
) = 1
. QED.
2. French 7.1.
A
K
meson traveling through the laboratory breaks up into two
π
mesons. One of
the two
π
mesons is left at rest. What was the kinetic energy of the
K
? What is the
kinetic energy of the remaining
π
meson? (Note: Mass in MeV is just a measure of
Mc
2
, momentum in MeV is a measure of
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 Fall '07
 Harris
 Mass, Momentum, Special Relativity, Fundamental physics concepts, Centre of mass, rest mass

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