A point of infinite space time curvature, or infinite density

We are bounded in a nutshell of Infinite space: Blog Post #8, Worksheet # 3.1, Problem #5: A point of infinite space time curvature, or infinite density.

5. \(M( < r)\) is related to the mass density \(\rho (r)\) by the integral: \[M( < r) = \int^r_0 4 \pi r^{2\prime} \rho (r^\prime) dr^\prime\] (Recall that the \( 4\pi r^\prime \) comes from the surface area of each spherical shell, and the \(dr^\prime\) is the thickness of each thin shell; talk to a TF if this is not clear.) The fundamental theorem of calculus then implies that \(4\pi r^2 \rho(r) = d M( < r) / dr\). For the case in question 4, what is \(\rho(r) \)? Is the density finite as \(r \to 0\) in the case of a flat rotation curve?

Using the several definitions offered for \(M( < r)\), we can begin to unravel the formula that would yield the relationship of stellar density in the galaxy to the total mass. Using the previous definition of \(M( < r)\), we have: \[ M( < r) = \frac{V_c^2 r}{G}.\] Yet we also have the equation for  \( d M( < r) / dr\),

\[4\pi r^2 \rho(r) = d M( < r) / dr ,\] and if the derivative of the first equation was done, then these formulas could be equated. Thus, \[ dM( < r )/dr  = \frac {d\frac{V_c^2 r}{G}}  { dr} , \]   \[ M( < r)^\prime = \frac{V_c^2}{G}, \] since the velocity of the galaxy is an established value as well as the gravitational constant, they are treated as constants when deriving and thus only since the velocity of the galaxy is an established value as well as the gravitational constant, they are treated as constants when deriving and thus only r is differentiated. The equations seen now equated through \( M( < r)^\prime \) would be:

\[4\pi r^2 \rho(r) = = \frac{V_c^2}{G}, \] which can be re written for \( \rho (r)\) : \[ \rho (r) =\frac { V_c^2} {4\pi r^2  G } ,\] which can now serve to inform us on the pattern of \( \rho (r)\) when \(r \to 0\). We can see that as r approaches 0, the value of the denominator becomes increasingly small, and thus the total value of the expression grows substantially, resulting in the conclusion that \( \rho (r)\) tends towards \(\infty\) as \(r \to 0\).