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>> Now we're going to continue with Poincare patch, and

Poincare coordinates in the de Sitter space, and

other possible solution of this equation that ABXBXB

equals 2H minus to the second power is falling.

We can write HX, well, this can be related to this.

+HXD squared

equal to

1-HXI+.

The reason for the notations will become clear in a moment.

So, we can define this to be equal to this, H tau+.

And HX1 squared,

+ HXD-1 squared

equal to HX plus i

squared to h tow plus.

So using this, we can write the following.

Well, this can be now solved

as HX0 = hyperbolic sine

of H Tau+ + HX+I squared

over 2 exponent of H of tau + HXi,

this, and will be sold like this.

HXi + e to the exponent of H tau+.

Where i, both here, here, and here,

and here is ranging from 1 to D-1, and

finally, HXD, which solves this,

together with this solve this equation,

is minus hyperbolic assigned,

of H tau + + HX + i squared divided by 2 exponent of H tau +.

So, if we plug now these expressions

into the ambient space Minkowskian metric,

then we get metric induce on this space of the form,

ds + squared = d tao + squared minus

exponent of 2H tao + dx + vector squared.

This guy is just this thing here.

Now, and so the problem was this metric and with this coordinate.

One can see immediately is that

-X0 + XD as falls from here, and

here is nothing but -1 over H exponent of HT+,

which is always less than zero,

which means that X0 is always greater than XD.

So these coordinates do not cover entire.

Unlike the previous coordinates, which we have been considering.

They are called global coordinates and they cover entire de Sitter space,

entire hyperboloid.

These coordinates cover only half of it, so if those coordinates were covering all

of this hyperboloid, these coordinates cover only upper

half of it, we have to cut it by a plane.

This and these coordinates cover only that half.