Environmental Engineering Reference
In-Depth Information
14
Acceleration and General
Relativity
14.1
ACCELERATION IN SPECIAL RELATIVITY
There is nothing to stop us from describing accelerated motion in Special Rela-
tivity. Perhaps the most natural question to ask is: what are the components of an
acceleration in
S
given the corresponding components in
S
(where
S
and
S
are the
usual two inertial frames)? Starting from the velocity addition formula, Eq. (6.33),
we have that
d
v
x
uv
x
/c
2
)
2
.
v
x
)u/c
2
1
(u
+
d
v
x
=
uv
x
/c
2
−
(14.1)
1
+
(
1
+
uv
x
/c
2
)
d
t
In conjunction with d
t
=
γ(u)(
1
+
this equation implies that
a
x
1
1
γ(u)
u
2
/c
2
−
1
a
x
=
uv
x
/c
2
,
(
1
+
uv
x
/c
2
)
2
1
+
3
1
a
x
,
i.e.
a
x
=
(14.2)
γ(u)(
1
+
uv
x
/c
2
)
d
v
x
/
d
t
are the accelerations in the
x
-direction in
S
and
S
. Simlarly we can use Eq. (6.34) to establish that
d
v
x
/
d
t
and
a
x
=
where
a
x
=
2
a
y
−
.
uv
y
1
a
x
a
y
=
(14.3)
uv
x
/c
2
)
c
2
uv
x
γ(u)(
1
+
+
