Problem Set 2

Problem 2

Show that the Lagrangian is invariant under the transform specified.  The transform depends only on position.  Thus the second term is invariant (you change the name of the position, but you do not change the value at that position).  The first term may have some position dependence because of the deriviatives.

First define the metric:

(g  = { {1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 
              0, -1}}) // MatrixForm

Now define the rotation matrix R_mu^nu (as given in the problem):

(r  =  {{0, b1, b2, b3}, {b1, 0, t3, -t2}, {b2, -t3, 0, 
              t1}, {b3, t2, -t1, 0}}) // MatrixForm

The contravariant derivative in the prime system is given by:

dphi/dx_mu = d phi/d xp_nu  d xp_nu/d x_mu = dphi/d xp_nu (delta_nu^mu+R_nu^mu)

To get the covariant derivative, we need:

xp^mu = g^munu xp_nu = x^mu + g^mu alpha R_alpha^beta g_beta nu x^nu

The transformed R matrix is:

MatrixForm[gr  = g . r . g]

Thus d phi*/d xp^mu = dphi*/dx^nu ( delta^nu_mu + R^nu_mu)

To first order in the R matrix,

dphi*/dxp^mu dphi/dxp_mu = dphi*/dx^mu dphi/dx_mu + 
  dphi*/dx^mu (R^nu_mu + R_mu^nu)dphi/dx_nu 
  
R^nu_mu+R_mu^nu = R^nu_mu + Transpose[R^nu_mu]=
R^nu_mu + Transpose[ g.R.g] = 0 as seen below:

MatrixForm[r + Transpose[gr]]

Thus, the first term in the Lagrangian is invariant also.