Yet another candidate
Today I found yet another space that looks promising. This one looks even more nicer than the previous cases. To obtain the metric one can start with the product of 3-d Minkowski and d-dimensional Minkowski space. The coordinates can be taken as
with a two-dimensional Minkowski index and a d-dimensional Minkowski index. Let us write the metric as:
Now one imposes the condition:
and introduces “Poincaré coordinates” that satisfy this constrain. These can be defined as:
Then writing the metric in terms of these coordinates one obtains:
One can see that when we set we obtain exact d-dimensional anti-de Sitter space. This metric is very atractive. It contains more symmetry than the other cases that I have consider previously. Besides the first term being invariant under Poincaré transformations, one also has an invariance under the rescalling of the coordinates. I shall explore this space further.