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A black hole is a region from whose events no future-directed causal curve can reach the distant exterior. In an asymptotically flat spacetime with future null infinity ${\mathcal{I}}^{+}$, the standard definition isHere $J^{-}$ denotes the causal past and ${\mathcal{H}}^{+}$ is the future event horizon. This is a statement about the complete causal geometry, rather than a material surface or a local force threshold.

For a Schwarzschild black hole, ingoing Eddington-Finkelstein coordinates give outgoing radial null geodesics the slope $dr/dv=(1-2M/r)/2$ in units $G=c=1$. Inside $r=2M$ even this outgoing family moves toward smaller areal radius; the other future radial null family does too. A regular horizon can therefore prevent escape without being a curvature singularity.^[ref]

Rotating examples are described by a Kerr black hole. A dynamically forming event horizon need not coincide with a slice's apparent horizon; locating the former requires global escape information.

A conformally flat metric is locally $g={\Omega }^2\eta$, where $\Omega >0$ is a smooth conformal factor and $\eta$ is a flat metric tensor. Its domain is a conformally flat manifold. The conformal factor preserves null cones and unparametrized null geodesics, but can change curvature.^[ref] For example, $g=a(\eta {)}^2(-d{\eta }^2+d{\mathbf{x}}^2)$ has Ricci scalar $R=6a^{\prime \prime }/a^3$ in the usual general relativity curvature convention, so conformal flatness need not mean flatness.

A covariant derivative differentiates a tensor field while accounting for the changing local basis. For a vector field, an affine connection gives ${\nabla }_{\mu }{V}^{\nu }={\partial }_{\mu }{V}^{\nu }+{\Gamma }^{\nu }{}_{\mu \rho }{V}^{\rho }$. Each additional contravariant index contributes a plus connection term, and each covariant index contributes a minus term.

The correction makes the result a tensor, even under nonlinear changes of coordinates. For example, a constant Cartesian vector on a flat plane has varying components in polar coordinates; the connection terms cancel that coordinate variation. In general relativity, the Levi-Civita connection supplies the metric-compatible derivative. Parallel transport sets the derivative along a curve to zero, while the geodesic equation sets a freely falling body's tangent derivative to zero.^[ref]

For a Kerr black hole, let $K={\partial }_t$ be the stationary Killing vector field normalized at infinity. The exterior Kerr ergoregion is where $g(K,K)>0$, so the tangent to motion following $K$ alone would be a spacelike vector. A physical observer there cannot remain at fixed angular coordinates relative to infinity.

For nonzero spin its outer stationary-limit surface isIt lies outside the outer event horizon except where they meet on the rotation axis. A causal curve can escape this exterior Kerr ergoregion; forced co-rotation does not make it the black-hole interior. The conserved Killing energy $E=-p\cdot K$ can be negative there, enabling the Penrose process, while locally measured future-particle energy remains positive.^[ref][ref]

A tidal force is a difference of gravitational acceleration across an extended region. In Newtonian gravity, $g_i=-{\partial }_i\Phi$. Two neighboring freely falling particles separated by ${\xi }^j$ have, to first order,For a point mass, $\Phi =-GM/r$. In a local orthonormal radial/transverse frame,A small freely falling cloud stretches radially and compresses transversely. Outside the source the three eigenvalues sum to zero, consistently with ${\nabla }^2\Phi =0$; within matter the trace is $-4\pi G\rho$. These are accelerations of separations, not the much larger common acceleration of the cloud's centre.

In general relativity, freely falling test particles follow timelike geodesics. A freely falling frame can eliminate the connection at one event, but not the Riemann curvature tensor throughout a finite laboratory. Let $u$ be the geodesic tangent and $\xi$ a connecting vector for neighboring geodesics with $[u,\xi ]=0$. In the explicitly chosen curvature conventionthe geodesic deviation equation isIt follows by commuting ${\nabla }_u{\nabla }_{\xi }u$, using ${\nabla }_{u}u=0$ and zero torsion. In a parallel-propagated orthonormal frame, its spatial components directly describe measured relative acceleration. The weak-field, slow-motion limit recovers the Newtonian Hessian above. Reversing the curvature sign convention reverses the displayed curvature side, without changing the physical stretching. The equivalence principle removes locally uniform gravity, while geodesic deviation measures the residual tidal field. The connection between parallel transport, curvature and neighboring geodesics is developed systematically in the standard geometric construction.^[ref]

Ordinary component differences at neighboring points depend on how the local bases vary. The covariant derivative corrects for that variation:Each upper tensor index adds a connection term and each lower index subtracts one. A scalar has ${\nabla }_{a}f={\partial }_{a}f$, while the derivative of a tensor transforms as a tensor with one additional covariant index. The Christoffel symbols themselves are not tensor components; their inhomogeneous transformation cancels that of the ordinary derivative.

For the Levi-Civita connection,These express metric compatibility and zero torsion. Along $x^a(\lambda )$, parallel transport is the initial value problemIt preserves inner products for the Levi-Civita connection. Parallel transport of the tangent itself gives the affinely parametrized geodesic equation, ${\ddot{x}}^a+{\Gamma }_{bc}^a{\dot{x}}^b{\dot{x}}^c=0$.

Transport generally depends on the path. On a round sphere, transport a tangent vector around the triangle bounded by two meridians $90$ degrees apart and the intervening equatorial arc. All three sides are geodesics; the enclosed spherical area is $\pi a^2/2$. The returning vector is rotated by $90$ degrees, with sign set by traversal orientation. Locally this holonomy is controlled by curvature, even though the connection can be made zero at a single point. A vector can therefore have changing coordinate components while remaining parallel, and constant components do not generally mean parallel transport.

This construction compares directions at separate points through a chosen connection, rather than assuming that a curved manifold supplies a path-independent comparison.^[ref]

Use Cartesian coordinates $x^a=(t,x,y,z)$ with $c=1$ and ${\eta }_{ab}=\mathrm{diag}(1,-1,-1,-1)$. Spatial dot and cross products below use the Euclidean three-metric. The alternating tensor is given the orientation ${ϵ}^{0123}=+1$.

Vector and covector transformation. The vector $A=A^b{\partial }_b$ is independent of coordinates, and the chain rule gives ${\partial }_b=(\partial x^{\prime a}/\partial x^b){\partial }_a^{\prime }$. Likewise the one-form $A_{b}d{x}^b$ is independent of coordinates. Comparing coefficients provesTheir contraction $A_a{B}^a$ is unchanged because the forward and inverse Jacobians cancel. Here lowering an index uses the metric, not an identification of the numerical upper and lower components.

Why antisymmetrization matters. Write $J^c{}_a=\partial x^c/\partial x^{\prime a}$. Differentiating the covector law givesThe last term obstructs the general tensor transformation law. It is symmetric in $a,b$, so it cancels from ${\partial }_a^{\prime }{A}_b^{\prime }-{\partial }_b^{\prime }{A}_a^{\prime }$. ThereforeEquivalently $F=dA$ is the exterior derivative of the electromagnetic four-potential one-form. The statement that ${\partial }_a{A}_b$ is not a tensor needs a qualification: under affine Cartesian Lorentz transformations it does transform tensorially, because the second derivatives vanish. The failure concerns arbitrary nonlinear coordinate changes, including curvilinear coordinates in flat spacetime. This distinction is precisely the difference between general covariance and covariance under the restricted inertial-frame transformations.^[ref]

Electric and magnetic components. Choose $A^a=(\Phi ,\mathbf{A})$, so $A_a=(\Phi ,-\mathbf{A})$, $\mathbf{E}=-\nabla \Phi -{\partial }_t\mathbf{A}$ and $\mathbf{B}=\nabla \times \mathbf{A}$. ThenThus the lower-index field tensor isThis is a derived component representation of the field, with the source's sign convention.

Raised indices and the first invariant. Raising both indices changes the time-space signs and leaves the spatial block unchanged:In the scalar contraction, each independent antisymmetric pair is counted twice, givingUnder a Lorentz transformation each inverse transformation from a lower index cancels its matching transformation from an upper index, leaving the same scalar. In particular a plane electromagnetic wave has $E=B$ in these units and makes this invariant vanish, while a pure electric field makes it negative.

The printed dual convention. The PDF explicitly defines ${}^{\ast }{F}^{ab}={ϵ}^{abcd}{F}_{cd}$, with no $1/2$. Each independent pair therefore contributes twice: for example ${}^{\ast }{F}^{01}=F_{23}-F_{32}=-2B_x$. The complete result isThe time-space pairs contribute $-4\mathbf{E}\cdot \mathbf{B}$ and the spatial pairs contribute another $-4\mathbf{E}\cdot \mathbf{B}$. The conventional Hodge dual includes $1/2$ and would instead give the contraction $-4\mathbf{E}\cdot \mathbf{B}$; it must not silently replace this paper's definition. With fixed orientation the latter contraction is a pseudoscalar: invariant under proper Lorentz transformations but odd under orientation reversal if compared using a fixed alternating symbol.

Energy density and flux. Substitution into the given mixed stress-energy tensor yieldsFor the first formula, $F^{0c}{F}_{0c}=-E^2$ and $F_{cd}{F}^{cd}/4=(B^2-E^2)/2$. For the second, $F^{0j}{F}_{ij}=(\mathbf{E}\times \mathbf{B}{)}_i$. The minus sign on $T^0{}_i$ arises from its lowered spatial index; it does not reverse the physical energy flow. Raising that index gives $T^{0i}=S_i$, and symmetry gives $T^{i0}=S_i$, with $\mathbf{S}=\mathbf{E}\times \mathbf{B}/{\mu }_0$ the Poynting vector in $c=1$ units. Thus $T^{00}$ is energy density and $T^{i0}$ represents energy flux. In SI coordinates $x^0=ct$, restore $T^{0i}=S_i/c$ and $T^{00}=({ϵ}_0{E}_{\mathrm{SI}}^2+B^2/{\mu }_0)/2$, with ${ϵ}_0{\mu }_0{c}^2=1$. This keeps the mixed-index signs and the units distinct.

The Riemann curvature tensor measures the local path dependence of parallel transport. For a torsion-free connection in a coordinate basis, the conventiondefines its action through the failure of covariant derivatives to commute. Parallel transport around a small loop can consequently change a vector even when its starting and ending points coincide.

In general relativity, geodesic deviation connects curvature with the relative acceleration of neighbouring freely falling bodies: tidal stretching is a local physical effect. Nonzero connection coefficients alone do not establish curvature, since even a flat plane has nonzero coefficients in polar coordinates. Conversely, choosing coordinates that remove the connection at one point does not remove a nonzero curvature tensor. The overall curvature-sign convention must be preserved when comparing component formulas.^[ref]

Sean M. Carroll develops and teaches the geometric account of gravity in general relativity. His 1997 lecture notes connect differential geometry to Einstein field equations and applications in gravitational radiation, black holes and cosmology.

A central example is the covariant derivative: differentiating the coordinate components of a vector field alone does not produce another tensor. An additional affine connection term accounts for the changing coordinate basis. This construction allows physical equations to retain their meaning when coordinates change, supplying the mathematical foundation for the later gravitational applications.^[ref]

Use metric signature $(-,+,+,+)$ and an arbitrary increasing parameter $\lambda$ on a massive particle's timelike curve. Write $v^a=d{x}^a/d\lambda$ without yet imposing proper-time normalization. The free relativistic particle action isIts derivative with respect to $v^a$ is $p_a=mc{g}_{ab}{v}^b/\sqrt{-v^2}$. The square root makes the action independent of the chosen increasing parameter. Varying it first preserves the nontrivial velocity dependence that produces the geodesic equation.^[ref]

After variation, choose $\lambda =\tau$, so $v^a=U^a$ is the four-velocity, $U^2=-c^2$ and $p_a=m{U}_a$. Substituting $\sqrt{-U^2}=c$ into the integrand before an unconstrained variation would instead make it look constant and lose the momentum equation. The choice of proper time must therefore follow the variation, or be handled by an equivalent constrained action.