Today, I will show how to draw an asymptotically flat spacetime with two ends using MuPad. To draw ends, we use hyperbolic functions.

First, we talk about the end. The function that I am going to use is . It is clear that blow up at and approaches at infinity. To use , we use the following

h := proc(t) begin (1-4*t)/(t-2)+12 end_proc

I have added to the function so that approaches at infinity. Then to draw the (upper) end, I use the following function

f := proc(x, y) begin if x^2 + y^2 > 3.4 then h(x^2+y^2) else end_if end_proc

I have used the number because I do not want my end is too tall, keep in mind that is the blow-up number. We can calculate

h(3.4)

to see that this number is nothing but which is close to as I need. We are now able to draw the first end using the following

plot( plot::Function3d(f, x = -6 .. 6, y = -6 .. 6, Submesh = [3, 3]), ViewingBox = [Automatic, Automatic, 3 .. 8], Scaling = Constrained)