PC1672 Advanced dynamics

1.1 Symmetry in physics

Physical laws should not depend on our choice of coordinates.

If the form of a physical law is invariant (unchanged) under a change of coordinates, then this gives us a symmetry principle. Such principles have often provided clues about how to construct better physical theories. Examples include

Einstein's special relativity (which we shall be looking at in more detail)
general relativity (invariance under general coordinate transformations)
the standard model which describes the strong, electromagnetic and weak interactions in particle physics (invariance under SU(3)xSU(2)xU(1) gauge transformations).

For example, our description of the universe should not depend on our choice of coordinate axes; it should be invariant under rotations. This means that we need to formulate our physical laws in terms of mathematical quantities with geometrical meanings that are independent of our choice of axes. For example the position vector of a particle ${\bf x}=(x_1,x_2,x_3)$ has a definite magnitude and a definite direction in space, whatever axes we use to define its components (notation). Under a rotation of the axes, the three numbers we use to specify the position will change, but its meaning remains invariant. Any other three numbers which transform in the same way under rotations form the components of a vector whose geometrical meaning does not change. As well as vectors, we have scalars: single numbers which do not depend on our choice of axes.

Scalars and vectors are not the only quantities with geometrical meanings which are invariant under rotations. Later in the course we shall meet tensors. A tensor acts linearly on a vector to generate a new vector with a different magnitude pointing in a different direction. Its components can be written in the form of a 3-by-3 matrix. The numbers in this matrix do depend on our choice of axes; what the tensor does to a vector does not.

Newton's laws involve the accelerations of bodies and the forces between them, all of which are vector quantities. The forms of these laws are invariant under

translations of the origin
rotations of the coordinate axes
Galilean boosts to frames moving with constant relative velocity.

These correspond to the facts that (as far as we can tell) there are no special places in the universe, no special directions and no special inertial frame.

The speed of light is not invariant under Galilean boosts, contrary to what we observe. Hence Newton's dynamics cannot be the whole story. Hidden in it is the assumption that time is universal. In special relativity we must give up this idea: time is not universal. Instead we need to start thinking of time as just another "coordinate"!

Notation: In this course the three coordinate axes will be denoted 1, 2, and 3 rather than x, y, and z. The basis vectors and components of vectors will be labelled with subscripts 1, 2 and 3.

Textbook references

K&K 14.2, 11.4
M 1, 10.2
M&T 1, 14.2
Feynman lectures, volume I, 11

Mike Birse
17th May 2000