m theory function
Many thanks for going through these notes. This is a good point. The reason why m does not have a time dependence is given in chapter seven of Carroll’s online notes for "Spacetime and Geometry: an Introduction to General Relativity" which is cited often in the UFT series as you know. The reason is that all spherically symmetric vacuum metrics produce a time like Killing vector and are stationary metrics. As in previous UFT papers the most general spherically symmetric vector is :
ds squared = – exp (2 alpha(r)) dt squared + exp (beta(r)) dr squared + c squared d cap omega squared
where r is defined not to depend on t in the metric. In the Minkowski metric, which is spherically symmetric, alpha(r) = beta(r) = 0. In the m theory metric:
exp(2 alpha(r)) := m(r), exp( beta(r)) := 1 / m(r)
In the orbit, on the other hand, r is a function of t for the following reason. In the spherically symmetric Minkowski space the Cartesian metric is diag (1,-1, -1, -1) and does not depend on t, but the hamiltonian and lagrangian defined in this metric depend on t. In the spherically symmetric m space the metric, similarly, does not depend on t but the hamiltonian and lagrangian depend on t. In both cases the hamiltonian and lagrangian are dynamic quantities defined in a stationary metric. So I followed this received wisdom in setting up the m theory.