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Calling the new metric h, you can do

h = M.metric('h')
h[Y.frame(),:,Y] = g[Y.frame(),:,Y]

This will initialize the components of h w.r.t. to chart Y to be identical to those of g in the same chart.

Calling the new metric h, you can do

h = M.metric('h')
M.lorentzian_metric('h')
h[Y.frame(),:,Y] = g[Y.frame(),:,Y]

This will initialize the components of h w.r.t. to chart Y to be identical to those of g in the same chart. chart.