Black Holes

The Ultimate Enigma: Regions of spacetime where gravity is so strong that nothing, not even light, can escape.

Singularities: Coordinate vs. Curvature

General relativity predicts the existence of singularitiesā€”points where spacetime curvature becomes infinite. We distinguish coordinate singularities (removable by coordinate transformations) from true physical singularities, and explore the Penrose-Hawking singularity theorems.

1. Curvature Invariants

To detect true singularities, we examine scalar quantities constructed from the curvature tensor. These are coordinate-independentā€”if they diverge, the singularity is physical.

Ricci Scalar

The Ricci scalar R = g^Ī¼Ī½ R_Ī¼Ī½ measures the trace of the Ricci curvature. For the Schwarzschild solution:

$R = 0$

This is zero everywhere (vacuum solution), so it doesn't reveal the singularity. We need a stronger invariant.

Kretschmann Scalar

The Kretschmann scalar is the contraction of the Riemann tensor with itself:

$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$

For the Schwarzschild spacetime, explicit calculation gives:

$K = \frac{48 G^2 M^2}{c^4 r^6} = \frac{12 r_s^2}{r^6}$

Behavior:

  • As r ā†’ āˆž: K ā†’ 0 (flat spacetime)
  • At r = r_s: K = 12/r_sā“ ā‰ˆ 0.75/r_sā“ (large but finite)
  • As r ā†’ 0: K ā†’ āˆž (true singularity!)

Conclusion: The Kretschmann scalar diverges at r = 0, proving this is a genuine curvature singularity. At r = r_s, K is finite, confirming the horizon is only a coordinate singularity.

2. Tidal Forces Near Singularities

Tidal forces arise from the inhomogeneity of the gravitational field and are described by the Riemann curvature tensor via the geodesic deviation equation.

For a radially infalling observer, the tidal force stretching them radially while compressing them tangentially is given by components of the Riemann tensor:

$R_{r t r t} \sim \frac{GM}{r^3}$

This radial-radial component diverges as r ā†’ 0, meaning infinite tidal stretching (spaghettification).

Geodesic Deviation

If two neighboring geodesics have separation vector Ī¾^Ī¼, the relative acceleration is:

$\frac{D^2 \xi^\mu}{D\tau^2} = -R^\mu_{\;\nu\rho\sigma} u^\nu u^\rho \xi^\sigma$

where u^Ī¼ is the four-velocity. Near r = 0, this becomes arbitrarily large, tearing apart any physical object.

3. Coordinate Singularities

Not all apparent singularities are physical. A coordinate singularity occurs when the coordinate system becomes ill-defined, but the spacetime geometry itself remains smooth.

Example: The Horizon

In Schwarzschild coordinates, the metric components diverge at r = r_s:

$g_{tt} = -\left(1 - \frac{r_s}{r}\right) \to 0, \quad g_{rr} = \left(1 - \frac{r_s}{r}\right)^{-1} \to \infty$

However, we can introduce new coordinates that are regular at the horizon. For example, in Eddington-Finkelstein coordinates with v = t + r*:

$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dv^2 + 2c\,dr\,dv + r^2 d\Omega^2$

This metric has no singularity at r = r_s! All components remain finite, and infalling observers cross smoothly.

How to Identify Coordinate Singularities

  • Check curvature invariants (Ricci scalar, Kretschmann scalar): if finite, likely coordinate singularity
  • Attempt coordinate transformation to remove the singularity
  • Examine geodesic completeness: can freely falling observers cross?

4. Penrose-Hawking Singularity Theorems

In the 1960s-70s, Roger Penrose and Stephen Hawking proved that singularities are inevitable in general relativity under fairly general conditions. These are among the most important results in GR.

Penrose Theorem (1965)

Statement: If a non-compact region of spacetime contains a trapped surface and satisfies the null energy condition, then spacetime is null geodesically incompleteā€”it contains a singularity.

Key concepts:

  • Trapped surface: A closed 2-surface where both outgoing and ingoing null geodesics converge (area decreases)
  • Null energy condition: R_Ī¼Ī½ k^Ī¼ k^Ī½ ā‰„ 0 for all null vectors k^Ī¼ (holds for normal matter)
  • Geodesic incompleteness: Geodesics cannot be extended to arbitrary affine parameter

For gravitational collapse beyond the Schwarzschild radius, a trapped surface forms. The theorem then guarantees a singularity forms inside.

Hawking Theorem (1967)

Statement: Under similar energy conditions plus assumptions about cosmology (expanding universe), the Big Bang singularity is inevitable.

The Penrose-Hawking theorems show that singularities are not artifacts of symmetry assumptions (like spherical symmetry in Schwarzschild), but generic features of general relativity. They arise from the focusing effect of gravity on geodesics.

5. Cosmic Censorship Conjecture

The singularity theorems guarantee singularities form, but do they always hide behind event horizons? Or can "naked singularities" exist, visible to distant observers?

Weak Cosmic Censorship

Conjecture (Penrose, 1969): Singularities arising from gravitational collapse of realistic matter are always hidden behind event horizonsā€”no naked singularities form from generic initial data.

This remains unproven but is widely believed. If violated, predictability of GR would break down, as singularities could influence distant regions unpredictably.

Known Counterexamples

Certain fine-tuned scenarios do produce naked singularities:

  • Extremal Reissner-Nordstrƶm black hole: QĀ² = MĀ² produces naked singularity
  • Extremal Kerr black hole: a = M also produces naked singularity (ring singularity)
  • Special solutions like the "shell-focusing" singularity

However, these require exact conditions (measure zero in initial data space), suggesting cosmic censorship holds generically.

6. Physical Implications

Breakdown of Classical GR

At singularities, curvature diverges and classical general relativity breaks down. We expect quantum gravitational effects to become important at the Planck scale:

$\ell_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6 \times 10^{-35} \text{ m}$

Information Paradox

If matter falls into a black hole and eventually reaches the singularity, what happens to the quantum information it carried? Hawking radiation is thermal and carries no information. This leads to the black hole information paradox, one of the deepest puzzles in theoretical physics.

Observational Status

We cannot observe singularities directly (cosmic censorship). However, we infer their existence from:

  • Event horizons detected via X-ray binaries and gravitational waves
  • Supermassive black holes at galactic centers (e.g., Sgr A*)
  • EHT imaging of M87* black hole shadow

Key Takeaways

  • Curvature invariants (Kretschmann scalar) distinguish coordinate singularities from physical ones
  • The Schwarzschild singularity at r = 0 is a true curvature singularity with K ā†’ āˆž
  • The horizon at r = r_s is merely a coordinate singularity (K finite)
  • Penrose-Hawking theorems prove singularities are generic in GR, not artifacts of symmetry
  • Cosmic censorship conjectures that singularities are always hidden behind horizons
  • Singularities signal the breakdown of classical GRā€”quantum gravity needed
ā† Previous: Event HorizonNext: Geodesics Near Black Holes ā†’