What is the relationship between the Schwarzschild radius and Black hole Singularity?

Question

1. What is the relationship between the Schwarzschild radius and Black hole Singularity?

2. Can the Planck length be the length of singularity?

3. Or is the length of the Schwartzschild shorter than the Planck length?

Answer 1

The Schwarzschild metric is given by,

$$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}t^2 -\left( 1-\frac{2GM}{r}\right)^{-1}\mathrm{d}r^2 -\underbrace{r^2\mathrm{d}\theta^2 -r^2 \sin^2 \theta \mathrm{d}\phi^2}_{r^2\mathrm{d}\Omega^2}$$

in spherical coordinates. Notice the metric tensor is singular at the origin, $r=0$, and at the Schwarzschild radius $r=2GM$. The former is a true physical curvature singularity. However, as one can verify by computing various curvature scalars, the singularity $r=2GM$ is unphysical, and can be removed by a coordinate transformation. In particular, the metric in Kruskal coordinates is given by,

$$\mathrm{d}s^2 = \frac{32G^3M^3}{r}e^{-r/2GM} \left( \mathrm{d}U^2 -\mathrm{d}V^2\right) + r^2 \mathrm{d}\Omega^2$$

The event horizon $r=2GM$ (in natural units) corresponds to $V=\pm U$, and indeed the metric is not singular at the point, reflecting the fact the singularity arose simply because we chose an inappropriate coordinate system. Regarding the $r=0$ singularity, we cannot define a notion of length or associated any length to the singularity; it is a single point on the manifold.

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At the Planck scale, we cannot resort solely to general relativity, and quantum gravity effects become important. From a quantum field theory perspective, this is typical. Short distances correspond to a high energy scale where our theory does not provide a description. As Prof. Tong states,

The question of spacetime singularities is morally equivalent to that of high-energy scattering. Both probe the ultra-violet nature of gravity. A spacetime geometry is made of a coherent collection of gravitons. The short distance structure of spacetime is governed - after Fourier transform - by high momentum gravitons. Understanding spacetime singularities and high-energy scattering are different sides of the same coin.

Nevertheless, gravity is different from simply an effective field theory, such as Fermi's theory of the weak interaction. We may still make predictions regarding gravity for high energies; e.g. if we collide at energies above the Planck mass, we know we form a black hole.

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A final remark as other answers mentioned naked singularities. When quantum gravity effects become important, after we pass the event horizon, we cannot communicate them to others behind the horizon; this is an example of cosmic censorship. However, violations have been hypothesized, c.f. the Gregory-Laflamme instability of $p$-branes and black strings.

Answer 2

1. The Schwarzschild radius depends on the matter content within the black hole and is given by

   $$r=\frac{2GM}{c^2},$$where $G$ is the gravitational constant, $M$ the mass inside the black hole and $c$ the speed of light. The singularity is related to the horizon and the corresponding radius by the statement that there are no "naked singularities". This means that singularities always come with an event horizon, which is given by the Schwarzschild radius.

2. The singularity, at least within general relativity, is point-like. This means that the concept of length does not apply to it. However, quantum gravity might change the nature of the singularity within a black hole and turn it into something extended. But this problem, the solution to which might also concern your first question, is not settled as of today.
3. By the above formula, you can in principle construct black holes with a Schwarzschild radius below the Planck length, but for reasons related to quantum gravity, it is not clear whether such a concept would make sense or not. Furthermore, one might argue that a sufficiently small black hole cannot exist because it would instantly evaporate due to Hawking radiation.

Answer 3

We don't know the answers to these questions.

The Schwarzschild metric that describes a static black hole is a solution obtained from general relativity and this is a classical theory. The singularity is not an object and it doesn't have a size. The singularity is a place where the spacetime curvature becomes infinite, and consequently we can't say what happens there (because we can't do the arithmetic with infinity).

Few of us believe the curvature really goes to infinity because we expect quantum effects to become important as we approach the Planck length. However we have no theory of quantum gravity to tell us what happens as the centre of black holes. There are many speculative theories, the more extreme of which propose that spacetime simply ends at the event horizon and black holes have no interior. Generally speaking we'd expect that distances shorter than the Planck length can't be measured, so there will be some physics happening at the centre over a length scale of around the Planck length. But what goes on we simply don't know at the moment.