Solving equations of motions in some gravity background
I would like to consider the gravity background:
with the case that We saw previously that the equations of motion were given by:
Now we take the warp factor to have the form . Then we have
We will introduce a new symbol,
with a function of the worldsheet coordinates.
The differential equation now looks like:
We now assume that can be factored into functions of each coordinate, . Then we can solve this PDE when
with a constant.
Separation of variables gives the following differential equation:
Inserting this into Mathematica gives:
Notice that the case is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.