maybe spacetime under gravity?

I found this nice relationship between the unit circle and a distribution I name in the picture...

In words it is a distribution that is tangent to the circle at 0*, and has the same area as the circle from -infinity to infinity. So it's pretty interesting on it's own but I found it thinking about how relativity is always explained as how mass warps spacetime like a ball on a rubber sheet. I had the idea that maybe a unit mass of a unit area displaces a unit area of spacetime from minus infinity to infinity. I don't know enough differential geometry to say whether this fits with the very complex relativity formulas or not but it was just a thought.

IMPORTANT NOTE:
The area of the formula for the green region from minus infinity to infinity is pi, but subtracting the bottom half of the circle's area of pi/2 gives pi/2. As I'm considering it as a displacement it makes sense this way.

* Dr. Rose helped me with this part I originally though it stayed tangent to the circle over an interval...