system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a limit on how big/dense an object can be in the universe? Or how much gravity it produces?

Eventually would it tear the space time 'fabric'?
gpt
There exists a limit, called the Buchdahl limit, which describes the maximum amount of mass that can exist in a sphere before the sphere must undergo gravitational collapse to a black hole.

If a spherically symmetric star has mass *M* and radius *R*, then the limit is

> R > R*_B_* = 9GM/4c^(2)

If a star in equilibrium with mass M and radius R*_B_* is given a spherically symmetric push inward, the star has no choice but to collapse inwards and eventually become a black hole. The star never reaches a steady state again.

You will often read or hear that if an object is contained within its Schwarzschild radius R*_S_* = 2GM/c^(2), then it will become a black hole. Although that is a true statement, it is misleading. The inevitable collapse occurs once the object is within its Buchdahl radius, which is slightly larger than the Schwarzschild radius.

That answers how big an object can be. Your question about a maximum density is now easily answered. The average density of a spherical star is

> ρ = kM/R^(3)

where k = 3/(4π). The density itself must satisfy the inequality

> ρ < kM/R*_B_*^(3) = K/M^(2)

where K is some constant. Insofar that the mass *M* can be arbitrarily small, the average density can be arbitrarily large. So there is no limit.

>Or how much gravity it produces?

This question can be interpreted in a number of ways. If by "gravity" you mean the curvature (i.e., the Ricci scalar), then that can be arbitrarily large. At the center of a spherically symmetric star of constant density, the curvature behaves like 1/(M-4R/9). The Buchdahl limit is M>4R/9, so this curvature is always positive, but it can be arbitrarily large for stars close to the Buchdahl limit. Of course, for a Schwarzschild black hole, the curvature is infinite at the singularity.