In search for anti-de Sitter space (I)
Let us start by considering the direct product of a 3d Minkowski space with a 4d Minkowski space. The coordinates for this space then go as
If we impose the condition we obtain the following relation:
We can notice that if we fix the value of the coordinate we obtain one of the conditions for five-dimensional anti-de Sitter space. This will not produce anti-de Sitter space since the surface is not embeded in . Nevertheless, I am not looking for anti-de Sitter space: I am looking for some space that will reduce to anti-de Sitter space in some limit. Defining we can write
with the understanding that the etas will represent flat metrics. Going back to the original, seven-dimensional metric we need
Now define the following tensors:
, and .
Then I claim the metric can be written as
Now we impose the condition that . With this condition we can use the five coordinates used in the Alday & Maldacena article as a starting point:
, and .
The next task is to write the metric in terms of these coordinates and hope that we get something that will look like anti-de Sitter space with some extra terms.