Environmental Engineering Reference
In-Depth Information
Similarly we can determine the y -component of the velocity:
d y
γ( d t +
d y / d t
d y
d t =
v y =
u d x /c 2 ) =
γ( 1
+
u( d x / d t )c 2 )
v y
=
uv x /c 2 ) .
(6.34)
+
γ( 1
Notice that for uv x
c 2 and u
c these results reduce to the expectation based on
classical thinking. Eqs. (6.33) and (6.34) are known as the velocity transformation
equations and their use is pretty straightforward. Perhaps the only place where
there is room for error is when it comes to figuring out the signs. For example, if
S were moving in the negative x -direction then we should replace u
u in the
equations. We can quickly check to see that the velocity transformation equations
satisfy the 2nd postulate, i.e. if v x =
→−
c and v y
=
0wehave
+
c
u
=
=
v x
c,
(6.35)
uc/c 2
1
+
which is as it should be.
6.3.2 Stellar Aberration Revisited
It is at this point that we can confirm that although Einstein has abolished
the ether his new theory is still capable of explaining the phenomenon of stellar
aberration. To understand this, let us consider the particular situation illustrated in
Figure 6.10. We imagine that the Sun, Earth and star all lie in the same plane and
that the Sun is at rest in S . Suppose that light emitted from the star arrives at an
angle angle α to the vertical in S . We shall take the relative speed between the
EarthandSuntobe u and α is the angle at which the starlight arrives on Earth.
Using the velocity addition formulae with v x =−
c sin α and v y =−
c cos α we
have that
c sin α
u
v x =
,
(6.36)
u
c
1
sin α
y
S
a
Earth
x
S
y
u
a′
Sun
x
Figure 6.10
Incident starlight in the Earth and Sun rest frames.
Search WWH ::




Custom Search