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Black Holes

Field: General Relativity

A region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it.

Sequence of Expressions

Definition

Event Horizon

The Event Horizon H\mathcal{H} is the null hypersurface defined by the condition that the outgoing null geodesics are trapped, corresponding to the location where the metric component grrg_{rr} diverges or where the time component gttg_{tt} vanishes for the Schwarzschild metric: gtt(r)=0 or grr(r)→∞g_{tt}(r) = 0 \text{ or } g_{rr}(r) \to \infty
Definition

Singularity

For the interior of a black hole described by the Schwarzschild solution, the singularity is located at r=0r=0. It is characterized by the divergence of curvature invariants, such as the Kretschmann scalar KK: K=RμνρσRμνρσ→r→0∞K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} \xrightarrow{r \to 0} \infty
For a static, spherically symmetric spacetime described by the Schwarzschild metric ds2=−(1−2GMrc2)c2dt2+(1−2GMrc2)−1dr2+r2dΩ2ds^2 = -\left(1 - \frac{2GM}{r c^2}\right) c^2 dt^2 + \left(1 - \frac{2GM}{r c^2}\right)^{-1} dr^2 + r^2 d\Omega^2, the Schwarzschild radius RsR_s is defined by the location where the coefficient of dt2dt^2 vanishes, yielding: Rs=2GMc2R_s = \frac{2GM}{c^2}
The Einstein Field Equations relate the geometry of spacetime (represented by the Einstein tensor GμνG_{\mu\nu}) to the distribution of mass and energy (represented by the Stress-Energy Tensor TμνT_{\mu\nu}): Gμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} where Gμν=Rμν−12gμνRG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R, and RμνR_{\mu\nu} is the Ricci curvature tensor.
For a static observer at radial coordinate rr in the Schwarzschild spacetime, the relationship between the proper time dτd\tau and the coordinate time dtdt is given by the metric component g00g_{00}: \ndτdt=−g00=1−2GMrc2 \frac{d\tau}{dt} = \sqrt{-g_{00}} = \sqrt{1 - \frac{2GM}{rc^2}}
Consider the vacuum solution to the Einstein Field Equations (Rμν=0R_{\mu\nu} = 0) in the asymptotically flat limit. The theorem asserts that the metric tensor gμνg_{\mu\nu} is uniquely determined by the three conserved charges: the mass MM, the electric charge QQ, and the angular momentum JJ. The resulting metric is the Kerr-Newman solution, which is characterized by the parameters (M,Q,J)(M, Q, J) alone.
The path xμ(λ)x^\mu(\lambda) of a test particle in a curved spacetime, parameterized by λ\lambda, follows a geodesic and is governed by the equation:\nd2xμdλ2+Γνσμdxνdλdxσdλ=0 \frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\sigma} \frac{dx^\nu}{d\lambda} \frac{dx^\sigma}{d\lambda} = 0 \nwhere Γνσμ=12gμρ(∂νgρσ+∂σgρν−∂ρgνσ)\Gamma^\mu_{\nu\sigma} = \frac{1}{2} g^{\mu\rho} (\partial_\nu g_{\rho\sigma} + \partial_\sigma g_{\rho\nu} - \partial_\rho g_{\nu\sigma}) are the Christoffel symbols.
The Schwarzschild radius RsR_s, representing the radius at which the escape velocity equals the speed of light, is defined solely by the mass MM of the object: \nRs=2GMc2 R_s = \frac{2GM}{c^2} \nwhere GG is the gravitational constant, MM is the mass, and cc is the speed of light.
The principle of General Covariance requires that the physical laws, expressed by an action SS or field equations, remain invariant under arbitrary coordinate transformations xμ→x′μx^{\mu} \to x'^{\mu}. Mathematically, this means that the equations must be formulated using tensors T...μ...T^{\mu...}_{...} that transform according to the Jacobian of the transformation, ensuring that the action integral remains invariant: δS=∫L−gd4x  ⟹  δS=0\delta S = \int \mathcal{L} \sqrt{-g} d^4x \implies \delta S = 0
For a rotating mass described by the Kerr metric, the angular velocity ω\omega of the local inertial frames (the dragging effect) is determined by the off-diagonal metric component gtϕg_{t\phi}: \nω=−gtϕgϕϕ=2GJrc2r2+a2(r2+a2)−2GMr \omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2G J r}{c^2 r^2 + a^2 (r^2 + a^2) - 2G M r} \nwhere JJ is the angular momentum and a=J/(Mc)a = J/(Mc) is the spin parameter.