## 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:

and

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:

with

and

Inserting this into Mathematica gives:

which means

Notice that the case is interesting: for the constraints we have used the solution is an exponential function of a quadratic polynomial.

Index Guysaid, on September 2, 2008 at 6:35 pmThis should be more correct if one takes to be a constant and keeps all the dependence explicit. In that case the PDE looks interesting, but hard to solve.

Index Guysaid, on September 10, 2008 at 7:54 pmThis is all non-sense.