Environmental Engineering Reference
In-Depth Information
Fortunately, this does not prevent Eq. (11.25) from remaining true since the depen-
dence upon
R
always cancels in the right hand side, regardless of the dependence of
R
upon the time, whilst it also disappears from the left hand side after differentating
twice with respect to time. Thus the law of gravity is invariant under Galilean trans-
formations. Notice that it would not be invariant if
R
were to depend upon some
higher power of
t
.
11.2 LORENTZ SYMMETRY
At the end of the last section we discussed invariance under Galilean transfor-
mations. However we know from Part II that although Galilean transformations
are a good approximation at low velocities they ought really to be replaced by the
Lorentz transformations if physics is to accord with both of Einstein's postulates.
Since it is our intention that all laws of physics should be consistent with Einstein's
theory it would be to our advantage to find a way of representing physical objects
such that Lorentz invariance is explicit from the outset, in much the same way
that the use of vectors makes explicit invariance under co-ordinate transformations
(rotations and translations).
Let us state our intention. We would like to build all of the equations in physics
using only mathematical objects which do not change as one alters the inertial frame
of reference. Ordinary vectors and scalars provide the paradigm since equations
built out of them do not change under a change of co-ordinates. Ordinary scalars
and vectors will not suffice however, since transformations between inertial frames
mix up the spatial and temporal co-ordinates of an event. Now physics is concerned
entirely with the relationships between
events
in space and time. For every event
we can represent it, in any given inertial frame, by a list of four numbers (
t,x,y,z
).
Now these numbers may change as we move from inertial frame to inertial frame
but the event remains the same. The invariant idea of 'an event' suggests immedi-
ately that we might try to represent events by vectors in a four-dimensional space.
At this stage in our development this is little more than an idea but it is an idea
that will soon gain in stature.
Let us begin by recapping the Lorentz transformations. As in Part II, when it is
useful to focus on two particular inertial frames we shall always pick the frames
S
and
S
related in the usual way, i.e. the axes are aligned, the origins coincide
0and
S
moves in the positive
x
direction with speed
u
. Accordingly
we can write the Lorentz transformations written in Eq. (6.28) in a particularly
suggestive manner:
t
=
at
t
=
ct
=
ct
cosh
θ
−
x
sinh
θ,
(11.27a)
x
=−
ct
sinh
θ
+
x
cosh
θ,
(11.27b)
y
=
y,
(11.27c)
z
=
z,
(11.27d)
