Environmental Engineering Reference
In-Depth Information
Before we press ahead and begin to present some specific examples of symmetry
in action it is perhaps worth re-emphasising that symmetries play a very fundamen-
tal role in modern physics. Time translational invariance is the symmetry which
says that the laws of physics do not change over time and it embodies the idea that
an experiment performed today should yield the same result as the same experi-
ment performed tomorrow, all other things being equal. Remarkably, the law of
conservation of energy arises as a direct consequence of this symmetry.
2
Similarly,
the law of conservation of momentum can be derived if we insist that the laws of
physics should be invariant under translations in space (loosely speaking we might
say that it does not matter where an experiment is performed) and the law of con-
servation of angular momentum can be derived by insisting on invariance under
rotations in space (i.e. it does not matter what the orientation of an experiment
is). These three symmetries are very intuitive symmetries of space and time. They
embody the idea that there is no fundamentally special place, time or direction in
the Universe. We have seen in Part II that Einstein added a new symmetry of space
and time to this list, namely Lorentz invariance, and it is the purpose of this part
of the topic to emphasise the central role of that symmetry in his theory.
11.1.1 Rotations and translations
Vectors and scalars: a recap
Laws of physics, such as Coulomb's Law or Newton's laws, are built using only
scalar (such as mass and electric charge), vector (such as force and acceleration)
and occasionally tensor (such as the moment of inertia) quantities. By their very
definition, these objects do not change if we decide to use a different system of
co-ordinates. Of course the components of a vector (or tensor) do depend upon the
choice of co-ordinates but the vector is still the same old vector. Objects which do
depend upon the details of our co-ordinate system are not of interest to physicists,
since the way we choose to parameterise points in space and time should not be
important.
In this and the next subsection we explore this idea in a little more detail. Let
us begin by considering two frames of reference
T
and
T
, which differ in some
way that does not depend upon time. A general vector
does not care about the
change of co-ordinates, although its components do generally change:
V
V
i
e
i
.
V
=
V
i
e
i
=
(11.1)
i
i
The summation is over the three spatial components and the
e
i
are the unit basis
vectors in
T
whilst the
e
i
are the unit basis vectors in
T
. So although the com-
ponents of the vector do change (
V
i
V
i
) the vector remains the same. We say
that the two frames of reference lead to different representations of the same vec-
tor. Scalars are even simpler, for their numerical value is independent of reference
frame.
=
2
It is outside of our remit to provide the proof of the link between symmetry and conservation laws in
this topic.
