John Baez

2005-04-30 08:30:31 UTC

For a course on a classical mechanics I decided to have my

students work out the geodesics on a 5-dimensional manifold

M x U(1) with the metric h given by

h_{ij} = g_{ij}

h_{i5} = A_i

h_{55} = 1

where i,j = 1,2,3,4, g is a metric on M and A is a 1-form

on M describing the electromagnetic vector potential.

I was hoping to get the equation for the motion of

a charged particle in a electromagnetic field, namely

m (D^2q/Dt^2)^i = e F^i_j (dq/dt)^j

where:

dq/dt is the derivative of the path q(t),

D^2q/Dt^2 is its covariant 2nd derivative,

F_{ij} = d_i A_j - d_j A_i is the electromagnetic field,

m is the particle's mass, and

e is its charge.

And indeed, I get this assuming that the particle moves

around the circle U(1) at velocity e/m:

(dq/dt)^5 = e/m

HOWEVER, I don't find that the velocity of the particle around

the circle is constant!

(dq/dt)^5 seems to have nonzero derivative, which is annoying:

the particle's effective charge-to-mass ratio changes with time!

Am I making a mistake or what?

It's all the more annoying because the spacetime M x U(1) has

rotational symmetry in the U(1) coordinate, so by Noether's theorem,

the momentum in the 5 direction is conserved.

However, the velocity in the 5 direction appears not to be

conserved, basically because we obtain the velocity from the

momentum by raising an index:

(dq/dt)^5 = h^{5a} (dq/dt)_5

and h^{5a} is time-dependent.

I wish I were making some mistake here - am I?

students work out the geodesics on a 5-dimensional manifold

M x U(1) with the metric h given by

h_{ij} = g_{ij}

h_{i5} = A_i

h_{55} = 1

where i,j = 1,2,3,4, g is a metric on M and A is a 1-form

on M describing the electromagnetic vector potential.

I was hoping to get the equation for the motion of

a charged particle in a electromagnetic field, namely

m (D^2q/Dt^2)^i = e F^i_j (dq/dt)^j

where:

dq/dt is the derivative of the path q(t),

D^2q/Dt^2 is its covariant 2nd derivative,

F_{ij} = d_i A_j - d_j A_i is the electromagnetic field,

m is the particle's mass, and

e is its charge.

And indeed, I get this assuming that the particle moves

around the circle U(1) at velocity e/m:

(dq/dt)^5 = e/m

HOWEVER, I don't find that the velocity of the particle around

the circle is constant!

(dq/dt)^5 seems to have nonzero derivative, which is annoying:

the particle's effective charge-to-mass ratio changes with time!

Am I making a mistake or what?

It's all the more annoying because the spacetime M x U(1) has

rotational symmetry in the U(1) coordinate, so by Noether's theorem,

the momentum in the 5 direction is conserved.

However, the velocity in the 5 direction appears not to be

conserved, basically because we obtain the velocity from the

momentum by raising an index:

(dq/dt)^5 = h^{5a} (dq/dt)_5

and h^{5a} is time-dependent.

I wish I were making some mistake here - am I?