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)

What we obtain is the following picture

By using the negative of , i.e. , we can draw the lower end as follows

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

We obtain the following picture.

The only thing left is to draw the “compact set” of the spacetime. In order to make the whole smooth, I need to know the tangent line at the point of the function . I use

D(h)(3.4)

To draw the upper half compact core, I use an ellipse of the form for some .

Suppose and , let us consider the function

By solving , we find that

Since

we know that

Therefore, the condition gives

Using , we obtain .

In this case, we can also use a cylinder via

plot::Surface([sqrt(3.4)*cos(v), sqrt(3.4)*sin(v), u], u = -3 .. 3, v = 0 .. 2*PI)

Eventually, this is what we have

Note that in the above picture, I have used the point instead of the point .

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