General relativity

A weak gravitational field is a regime in which the metric differs little from a suitable flat reference metric. In a slow-motion, approximately stationary situation,

g_{tt} ≃ 1 + 2 ϕ / c^{2}

for signature

(+ - - -)

, with Newtonian potential

ϕ

. Proper-clock rates then differ fractionally by

Δ ϕ / c^{2}

. Small metric perturbations do not by themselves imply every velocity is nonrelativistic; slow motion is a separate approximation when deriving Newtonian dynamics.

Shapiro time delay

Extra propagation time of a light or gravitational-wave signal through a gravitational field compared with an appropriate flat-space reference. In a weak static field with Newtonian potential of a point mass

Φ_{N} < 0

, the leading delay along an approximately straight ray is

Δ t = - 2 \int Φ_{N} d l / c^{3} > 0

. The local speed of light remains

c

; the integrated geometry changes the relation between emission and reception times.

Spin four-vector ( $s^{μ}$ )

The spin of a small torque-free test gyroscope is represented by a spacelike vector orthogonal to its four-velocity,

s^{μ} u_{μ} = 0

. Its temporal component vanishes in its own rest frame. Along a geodesic,

D s^{μ} / d τ = 0

: parallel transport preserves the orthogonality and the invariant spin magnitude. For an accelerated observer the corresponding nonrotating transport is Fermi-Walker transport.

Laboratory spin in rectilinear Fermi-Walker transport ( $s_{∥} = const, γ s_{⊥} = const$ )

For the orthogonal parametrization

s^{μ} = γ (s \cdot v / c, s)

, torque-free acceleration along a fixed direction keeps the longitudinal laboratory component constant and the transverse component proportional to

1/ γ

. The invariant proper-spin length remains constant; these changing three-components are not a torque-induced loss of spin.

General covariance

The ability to express a theory consistently under arbitrary changes of spacetime coordinates. A tensor equation has this property when all its fields, including any background structures, transform appropriately. It does not by itself say that every active transformation is a dynamical symmetry, or that no preferred background structure exists. General relativity has the additional substantive feature that its metric tensor is dynamical.

Covariant equation with a preferred timelike vector ( $(η^{μν} - α n^{μ} n^{ν}) \partial_{μ} \partial_{ν} ϕ = 0$ )

In units

c = 1

and metric signature

(-, +, +, +)

, take a fixed unit timelike vector field

n

and

α > 0

. The displayed equation for a scalar field is tensorial; in arbitrary coordinates use the flat Levi-Civita connection in place of ordinary second derivatives. In the rest inertial frame of

n

it becomes

- (1 + α) ϕ_{tt} + \nabla^{2} ϕ = 0

, with characteristic speed

1/ 1 + α

. Its fixed

n

selects a preferred frame. A boost that does not preserve

n

is not a symmetry with this background held fixed, although transforming the components of every quantity gives an equivalent coordinate description. This separates general covariance from a substantive relativity principle.

Bertotti-Robinson spacetime ( $AdS_{2} (m) \times S^{2} (m)$ )

The simply connected Bertotti-Robinson geometry is the product of the universal cover of two-dimensional anti-de Sitter spacetime and a two-sphere with equal radius

m > 0

. In metric signature

(-, +, +, +)

g = m^{2} sec^{2} X (- d T^{2} + d X^{2}) + m^{2} d Ω_{2}^{2}, T \in R, ∣ X ∣ < π /2.

Its radial Penrose diagram is an infinite conformal strip of two-dimensional anti-de Sitter spacetime. The metric tensor

- λ^{2} d t^{2} + m^{2} λ^{- 2} d λ^{2} + m^{2} d Ω_{2}^{2}

is a patch of it, with

λ = m (sin T - sin X) / cos X

. Its Killing vector field

\partial_{t}

extends across alternating degenerate Killing horizons

λ = 0

. Both factors are geodesically complete, giving a maximal analytic extension of this patch without periodic time identification.

Parametrized post-Newtonian formalism (PPN)

The parametrized post-Newtonian formalism expresses a weak-field metric tensor using phenomenological parameters so that different gravitational theories can be compared with the same observations. Its spatial-curvature parameter

γ_{PPN}

is one in general relativity; it is distinct from the kinematic Lorentz factor.

Parametrized post-Newtonian light deflection ( $α = 2 (1 + γ_{PPN}) GM / (b c^{2})$ )

The leading Schwarzschild light deflection of a distant ray passing an isolated spherical mass is

α = 2 (1 + γ_{PPN}) GM / (b c^{2})

. Thus measurements of angular displacement can constrain

γ_{PPN}

rather than merely detect a nonzero gravitational field. In general relativity this becomes the Schwarzschild light deflection

4 GM / (b c^{2})

Synchronous coordinates ( $d s^{2} = c^{2} d t^{2} - h_{ij} (t, x) d x^{i} d x^{j}$ )

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and

t

is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor

c^{2}

. Indeed

g_{tt} = c^{2}

and

g_{t i} = 0

give

Γ^{μ}_{tt} = 0

. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.

Synchronous coordinates for irrotational dust ( $u^{♭} = - d t, g = - d t^{2} + g_{ij} d x^{i} d x^{j}$ )

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity

u

has hypersurface orthogonality, then

u^{♭} \land d u^{♭} = 0

and

ι_{u} d u^{♭} = 0

. The interior product of the first identity gives

d u^{♭} = 0

. The Poincaré lemma supplies

u^{♭} = - d t

locally; spatial coordinate functions constant along the smooth flow of

u

give

u = \partial_{t}

g_{tt} = - 1

and

g_{t i} = 0

. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.

Gravitational singularity theorem

A theorem giving conditions under which some geodesic is incomplete. The Penrose singularity theorem is one example.

Israel junction conditions ( $[h_{ab}] = 0, [K_{ab}] - h_{ab} [K] = - κ_{D}^{2} S_{ab}$ )

Across a thin timelike matter layer, the induced metric is continuous while the jump in its extrinsic curvature is determined by the surface stress-energy tensor. Choose one common spacelike normal and

K_{ab} = \frac{1}{2} \partial_{y} h_{ab}

in Gaussian normal coordinates. The singular tangential Einstein tensor is

- ([K_{ab}] - h_{ab} [K]) δ (y)

; equating it to

κ_{D}^{2} S_{ab} δ (y)

gives the displayed conditions. Reversing the normal or the sign convention for extrinsic curvature changes the corresponding jump signs consistently. These conditions determine the cosmological expansion law of a thin gravitating brane.

Trapping horizon

A trapping horizon is a hypersurface foliated by marginal two-surfaces with one null expansion zero and the other nonzero. A future outer trapping horizon has

θ_{+} = 0

θ_{-} < 0

and

L_{n} θ_{+} < 0

for a future ingoing null normal

n

. Unlike an event horizon, this is a quasi-local definition. Under the null energy condition, a regular future outer horizon is spacelike or null. The ingoing derivative condition distinguishes its outer type from an inner trapping horizon.

Congruence (general relativity)

A congruence in general relativity is a family of curves through an open spacetime region such that each point lies on exactly one member. Its tangent field describes the family locally. A geodesic congruence is the special case in which every member is a geodesic.

Causal structure

The causal structure of a spacetime classifies separations and tangent directions as timelike, null or spacelike through its light cones. It determines which events can communicate by causal curves. A null curve is one member of this broader geometric structure.

Causal reachability on a spatial circle ( $J^{\pm} (0) = {(t, x) : \pm c t \geq min_{n} ∣ x + 2 π R n ∣}$ )

For flat spacetime with

x

periodic of circumference

2 π R

, lift endpoints of causal curves to the covering Minkowski spacetime. A signal can choose any winding, so the causal future and causal past use the shortest circular distance rather than a fixed coordinate separation. After a duration

π R / c

, every point on the circle is reachable. A constant tangential electric field has a nonzero loop electromotive force; maintaining it while a charge gains energy each lap requires an energy reservoir, not a single-valued electrostatic potential.

Flat-background scalar-force theory of gravity ( $d p^{i} = \partial^{i} Φ p_{j} d x^{j} - (p^{j} \partial_{j} Φ) d x^{i}$ )

This covariant scalar-force model uses the Minkowski metric and the displayed momentum law, with

□ Φ = - 4 π G T^{i}_{i}

. For a massive test particle, write

p^{i} = m u^{i}

u^{i} u_{i} = 1

. Contracting the force law with

p_{i}

gives constant rest mass; the acceleration is

d u^{i} / d τ = \partial^{i} Φ - u^{i} (u^{j} \partial_{j} Φ)

, independent of the mass. A static weak field obeys the Poisson equation and yields Newtonian acceleration

- \nabla Φ

. For a photon,

p \cdot d x = 0

and

d x

is parallel to

p

, so

d p^{i} = - p^{i} d Φ

: photon energies change but their unparametrized paths remain straight. The motion law has the same unparametrized curves as the conformal metric

e^{2Φ} η

, but the field equation must not be silently identified with a different scalar-gravity theory.

Scalar-force gravitational redshift ( $ν_{r} / ν_{e} = e^{Φ_{e} - Φ_{r}}$ )

For a photon in the flat-background scalar-force theory of gravity,

e^{Φ} p^{i}

is constant along its path. In a static field, energies measured by stationary laboratories therefore satisfy the displayed ratio. At first order,

Δ ν / ν = - ΔΦ

in units

c = 1

, matching the weak-field Pound-Rebka experiment. This agreement does not imply light bending: a momentum change parallel to the original null tangent changes its scale, not its direction.

Taub–NUT space

The single-centre Euclidean Taub–NUT geometry is a gravitational instanton with a circle fibre over three-dimensional space. Its fibre shrinks smoothly at the centre when its period is chosen appropriately. Adding flat worldvolume directions gives the local geometry of a Kaluza-Klein monopole.

// Target: algebra.bigb

Near-horizon limit

A limit retaining the geometry near a horizon while discarding terms describing its distant asymptotic region. For a regular extremal brane throat, removing the constant from its harmonic function can yield an Anti-de Sitter spacetime factor. Local symmetries and supersymmetries of the limiting geometry need not extend to the full original spacetime.

Rindler limit of a Schwarzschild horizon

Outside a Schwarzschild metric horizon

r_{s}

, write

r - r_{s} = ρ^{2} / (4 r_{s})

. To leading order the radial metric is

d s^{2} ≃ (\frac{c ρ}{2 r _{s}})^{2} d t^{2} - d ρ^{2} .

It has Rindler coordinates with

α = c / (2 r_{s})

. The proper-distance-like coordinate

ρ

has null paths exponential in

\pm c t / (2 r_{s})

; the Schwarzschild coordinate difference

r - r_{s} \propto ρ^{2}

is exponential in

\pm c t / r_{s}

Inside, writing

r_{s} - r = ρ^{2} / (4 r_{s})

instead gives

d s^{2} ≃ d ρ^{2} - [c ρ / (2 r_{s})]^{2} d t^{2}

, where the radial coordinate is timelike. The negative-

x

Rindler wedge is therefore not the black-hole interior: its time coordinate retains the same causal type as in the positive-

x

wedge. The flat radial limit does not remove the genuine curvature of the full Schwarzschild spacetime.

Einstein-Maxwell theory

The Einstein-Maxwell system couples the metric to an electromagnetic field. In geometrized units its source-free equations are

G_{ab} = 8 π T_{ab}^{EM}

\nabla_{a} F^{ab} = 0

and

d F = 0

, with

T_{ab}^{EM} = (F_{a c} F_{b}^{c} - \frac{1}{4} g_{ab} F_{c d} F^{c d}) / (4 π)

. Electrovacuum need not have vanishing Ricci tensor, so a Komar integral need not be unchanged between different enclosing surfaces.

Majumdar–Papapetrou solution ( $d s^{2} = - H^{- 2} d t^{2} + H^{2} d x^{2}$ )

The four-dimensional Majumdar–Papapetrou solution is a static multi-black-hole Einstein-Maxwell solution with

H = 1 + \sum_{i} M_{i} /∣ x - x_{i} ∣

and positive geometrized masses

M_{i}

. The electromagnetic potential can be written

A = \pm (H^{- 1} - 1) d t

, with a common sign of charge. Its mass at infinity is

\sum_{i} M_{i}

. Each distinct centre is an extremal horizon of zero surface gravity rather than a pointlike naked metric singularity.

Analytic extension of a Majumdar–Papapetrou horizon

Near a distinct centre of mass

M

, write

H = M / r + h + r F (r, θ, ϕ)

, where

h

is the regular part evaluated at that centre and

F

is analytic in local angular charts. Set

A = M + h r

Q = rH = A + r^{2} F

and

d v = d t + (h + M / r)^{2} d r

. The metric becomes

d s^{2} = - \frac{r ^{2}}{Q ^{2}} d v^{2} + 2 \frac{A ^{2}}{Q ^{2}} d v d r + F \frac{( Q + A ) ( Q ^{2} + A ^{2} )}{Q ^{2}} d r^{2} + Q^{2} d Ω_{2}^{2} .

All coefficients extend analytically across

r = 0

, with

Q (0) = M > 0

and two-dimensional determinant

- 1

. The surface is null, its generator is

\partial_{v}

, and the double zero of its squared norm gives zero surface gravity. The construction uses a constant

h

so that the coordinate differential is integrable despite the nonspherical regular part of

H

Twistor theory

Twistor theory encodes the conformal geometry of four-dimensional spacetime using incidence geometry in a complex vector space. In the flat correspondence, a spacetime point is a two-plane in a four-dimensional twistor space, and hence a projective line in projective twistor space. Null separation becomes intersection of these lines. Choosing a spacetime signature is an additional reality condition on the complex geometry.

Twistor correspondence ( $Gr (2, C^{4}) \leftarrow F_{1, 2} (C^{4}) \to CP^{3}$ )

The flag manifold of a line

ℓ

contained in a two-plane

W

has projections

(ℓ, W) \mapsto W

and

(ℓ, W) \mapsto ℓ

. Their fibers are respectively

CP^{1}

and

CP^{2}

. This is the flat twistor correspondence: complexified compactified Minkowski space is the Grassmannian of two-planes in twistor space, while projective twistor space is its projective space of lines.

Penrose-Ward correspondence

A holomorphic vector bundle on an appropriate twistor domain, trivial on its space-time incidence lines, determines a complex anti-self-dual gauge gauge connection. Split its patching matrix on each line and apply the tangent vector fields

π^{A^{'}} \partial_{A A^{'}}

. They annihilate the unsplit twistor patching matrix, so the two logarithmic derivatives of the factors agree and define a global degree-one function of

π

. It is consequently

π^{A^{'}} A_{A A^{'}}

. The inverse splitting bundle frame is parallel along each alpha-plane; commuting these parallel equations proves that the primed gauge curvature two-component spinor vanishes. Reality conditions recover unitary Euclidean gauge fields; Euclidean instanton bundles need triviality on the real twistor lines, not on every possible complex line.

Abelian Ward splitting by Laurent series ( $f_{x} (ζ) = \sum_{n \in Z} f_{n} (x) ζ^{n} = h_{1} - h_{0}$ )

On a twistor line use

π_{0^{'}} = ζ, π_{1^{'}} = 1

and

ω^{A} = x^{A 0^{'}} ζ + x^{A 1^{'}}

. For a degree-zero holomorphic patching logarithm, its Laurent series splits as

h_{0} = - \sum_{n \geq 0} f_{n} ζ^{n}

h_{1} = \sum_{n < 0} f_{n} ζ^{n}

. The twistor incidence relation implies

\partial_{A 0^{'}} f_{n} = \partial_{A 1^{'}} f_{n - 1}

. Hence

(- \partial_{A 0^{'}} + ζ \partial_{A 1^{'}}) h_{α} = \partial_{A 1^{'}} f_{- 1}

on both patches. The gauge potential is

A_{A 0^{'}} = - \partial_{A 1^{'}} f_{- 1}

A_{A 1^{'}} = 0

. Exponentiating the two additive factors produces the linewise holomorphic splitting of

e^{f}

Contour potential for an anti-self-dual Maxwell field ( $A_{B B^{'}} = \frac{1}{2 πi} \oint_{Γ} \frac{ι _{B^{'}}}{ι _{C^{'}} π ^{C^{'}}} f_{, B} π_{D^{'}} d π^{D^{'}}$ )

The Abelian Ward splitting by Laurent series gives this gauge potential through the Cauchy integral formula, with

f_{, B} = \partial f / \partial ω^{B}

evaluated on the twistor line. The integrand has total projective degree zero. Holding the contour fixed locally and differentiating gives

F_{A A^{'} B B^{'}} = ϵ_{A^{'} B^{'}} (2 πi)^{- 1} \oint_{Γ} f_{, A B} π_{D^{'}} d π^{D^{'}}

, using

π_{A^{'}} ι_{B^{'}} - π_{B^{'}} ι_{A^{'}} = ϵ_{A^{'} B^{'}} (ι_{C^{'}} π^{C^{'}})

. Its unprimed two-component spinor is symmetric, so the primed gauge curvature vanishes and the field obeys the anti-self-dual Maxwell equations. These signs use

ϵ_{0^{'} 1^{'}} = 1

ϵ^{0^{'} 1^{'}} = - 1

and raising by left multiplication; other consistent raising conventions can reverse both displayed overall signs.

Line-trivial bundles in the Ward correspondence ( $E ∣_{L_{x}} ≅ O^{\oplus r}$ )

On a twistor domain containing the lines

L_{x}

for

x

in a space-time domain

U

, the Ward construction uses a holomorphic vector bundle

E

with

E ∣_{L_{x}} ≅ O^{\oplus r}

for every

x \in U

. A degree-zero restriction is not sufficient:

O (1) \oplus O (- 1)

has degree zero but is not holomorphically trivial. A trivial restriction has

r

independent global holomorphic sections; their evaluations give a linewise bundle frame and hence a factorization of its patching matrix. Nearby factors can be chosen holomorphically because

H^{1} (CP^{1}, O)^{r^{2}} = 0

. Lines on which the splitting type is nontrivial are called jumping lines and obstruct this regular reconstruction there. For a globally regular field on affine complexified Minkowski space the condition is needed on all its incidence lines; on a smaller domain only those lines are required.

Holomorphic splitting of a twistor patching matrix ( $F = H H^{- 1}$ )

A bundle trivial on a twistor line has a global holomorphic bundle frame there. Comparing it with the given two patch bundle frames produces invertible matrices

H, H

, holomorphic on their respective affine patches, with

F = H H^{- 1}

. The obstruction to varying such a bundle frame locally with space-time is

H^{1} (CP^{1}, O)^{n^{2}} = 0

, so the splitting factors can be chosen holomorphically in a neighbourhood. If two splittings differ by

H \mapsto H g

and

H \mapsto H g

, then

g

is holomorphic on the compact complex projective line and hence independent of its projective coordinate. It is the space-time gauge freedom.

Alpha-plane ( $A_{ℓ} = {W : ℓ \subset W} ≅ CP^{2}$ )

An alpha-plane in the compact twistor correspondence is the set of two-planes containing a fixed line

ℓ

in twistor space. Its affine portion has tangent vectors

u^{A} π^{A^{'}}

for fixed primed spinor

π

, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.

Twistor line ( $L_{x} = P (W_{x}) ≅ CP^{1}$ )

The projective line associated to a two-plane

W_{x}

representing a point of complexified compactified Minkowski space. Two distinct twistor lines intersect precisely when their spacetime points are null separated. On the Lorentzian real slice, the line lies entirely in projective null twistor space and parametrizes the null geodesics through the point.

Complexified compactified Minkowski space ( $M_{C} = Gr (2, C^{4})$ )

The four-complex-dimensional Grassmannian of two-planes in twistor space. Its Plücker embedding is the Klein quadric. An affine chart consists of graphs

W_{x} = {(i x π, π)}

. A Lorentzian reality structure on twistor space or an Euclidean reality structure on twistor space selects different four-real-dimensional compact spacetimes inside this same complex manifold.

Twistor space ( $T ≅ C^{4}$ )

In flat twistor theory, a twistor is a pair

Z = (ω^{A}, π_{A^{'}})

of two-component spinors. The twistor incidence relation with a spacetime point is

ω^{A} = i x^{A A^{'}} π_{A^{'}}

. The nonzero vector

Z

and its projective class play different roles: the latter belongs to projective twistor space.

Euclidean reality structure on twistor space ( $J^{2} = - 1, J (W) = W$ )

Choose an antilinear quaternionic structure

J

C^{4}

. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a

J

-invariant complex two-plane, equivalently a quaternionic line in

H^{2}

. These form

HP^{1} ≅ S^{4}

, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line

P span_{C} (Z, J Z)

, giving a fibration

CP^{3} \to S^{4}

with fiber

CP^{1}

Twistor Hermitian form ( $h (Z, W) = ω^{A} \overline{ρ}_{A} + π_{A^{'}} \overline{η}^{A^{'}}$ )

For

Z = (ω, π)

and

W = (η, ρ)

the displayed Hermitian form has signature

(2, 2)

. It is zero on every pair of vectors in

W_{x} = {(i x π, π)}

when

x

is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with

x

Hermitian. This gives the Lorentzian reality structure on twistor space.

Lorentzian reality structure on twistor space

Real compactified Minkowski points are the two-dimensional totally isotropic subspaces for the signature-

(2, 2)

twistor Hermitian form. In a graph chart this is the usual Hermitian reality condition on

x^{A A^{'}}

. Their twistor lines lie in projective null twistor space. On the Klein quadric the same real form is the projectivized null cone of a real quadratic form of signature

(2, 4)

Null twistor ( $h (Z, Z) = 0$ )

A nonzero twistor null for the twistor Hermitian form. Its projective class determines a real null geodesic in compactified Minkowski space. For nonzero

π

, its affine real solutions of the twistor incidence relation have the form

x = x_{0} + r \overline{π}^{A} π^{A^{'}}

r \in R

. Twistors with

π = 0

encode generators at infinity rather than finite affine solutions.

Projective null twistor space ( $PN = {[Z] \in CP^{3} : h (Z, Z) = 0}$ )

The five-real-dimensional space of projective null twistors. A projective line lies entirely in it precisely when its corresponding two-plane is totally isotropic for the twistor Hermitian form. Such lines represent real points of Minkowski conformal compactification. Two independent null twistors lie on one such line exactly when

h (X, Y) = 0

Twistor incidence relation ( $ω^{A} = i x^{A A^{'}} π_{A^{'}}$ )

For a fixed affine spacetime point

x

, this linear relation cuts out a two-dimensional subspace

W_{x} = {(i x π, π)}

of twistor space. Projectivizing gives its twistor line. For a fixed projective twistor with

π \neq = 0

, its solutions are

x = x_{0} + u^{A} π^{A^{'}}

, a totally null affine alpha-plane. The compact description replaces affine solutions by the two-planes containing the fixed twistor line in the vector space.

Projective twistor space ( $PT ≅ CP^{3}$ )

The projective space of one-dimensional complex subspaces of twistor space. A point corresponds to an alpha-plane in complexified compactified Minkowski space; a projective line corresponds to a spacetime point.

Affine projective twistor space ( $P T_{aff} = CP^{3} ∖ CP_{\infty}^{1}$ )

For affine complexified Minkowski space, the relevant twistor domain is

P T_{aff} = {[ω^{A}, π_{A^{'}}] \in CP^{3} : π \neq = 0} = CP^{3} ∖ CP_{\infty}^{1} ≅ Tot (O (1) \oplus O (1)) .

Incidence can be written

ω^{A} = x^{A A^{'}} π_{A^{'}}

, absorbing the conventional factor

i

into the definition of

x

. Each space-time point gives a twistor line

L_{x} ≅ CP^{1}

. On charts

π_{0^{'}} \neq = 0

and

π_{1^{'}} \neq = 0

, the fibre coordinates transform by

ω / π_{1^{'}} = (ω / π_{0^{'}}) / λ

, with

λ = π_{1^{'}} / π_{0^{'}}

, giving the two

O (1)

summands. The full projective twistor space corresponds to complexified compactified Minkowski space.

Analytic extension of a spacetime

An analytic extension embeds a spacetime isometrically as a proper open region in a larger analytic Lorentzian manifold with a nondegenerate metric tensor. A divergence of coordinate components does not obstruct extension if a new chart makes all metric coefficients analytic and the determinant nonzero. For a static spherical metric with an analytic lapse

f

and a positive simple horizon radius, ingoing coordinates replace

- f d t^{2} + d r^{2} / f

- f d v^{2} + 2 d v d r

. This removes the coordinate singularity at

f = 0

; transverse metric coefficients must also remain regular. An extension across such a horizon is distinct from extending through a curvature singularity.

Two-sheeted extension of superextremal Kerr spacetime ( $ρ^{2} / (r^{2} + a^{2}) + z^{2} / r^{2} = 1$ )

For

0 < M < a

Δ = (r - M)^{2} + a^{2} - M^{2} > 0

, so no horizon blocks need adjoining. The oblate Cartesian coordinates of the Kerr metric satisfy the displayed relation, where

ρ^{2} = x^{2} + y^{2}

. Keeping only the positive root

r

leaves a cut along the disk

z = 0

ρ < a

. Continue

r

through zero away from its boundary, the Kerr ring singularity, onto a negative-radius sheet. Equivalently cut two Cartesian sheets along the disk and join opposite banks, so

r

changes sign continuously. The metric is analytic at the disk interior in signed oblate coordinates since

Σ = a^{2} cos^{2} θ \neq = 0

there. The extension has asymptotically flat ends at

r \to + \infty

and

r \to - \infty

; only the singular ring remains inextendible by an analytic Lorentz metric.

Gravitational redshift ( $ν_{r} / ν_{e} = N_{e} / N_{r}$ )

In a static spacetime with

g_{tt} = - N^{2}

, a photon has conserved Killing energy

E = - p_{t}

. A static observer with four-velocity

N^{- 1} \partial_{t}

measures frequency proportional to

E / N

. Hence light emitted at one lapse and received at another obeys the displayed ratio. For

N = 1 + Φ/ c^{2} + O (Φ^{2} / c^{4})

it gives the weak-field shift

(ν_{r} - ν_{e}) / ν_{e} = (Φ_{e} - Φ_{r}) / c^{2}

. This local clock effect does not require light-ray bending.

An upward photon has lower received frequency when compared with the receiver's local clock:

Φ_{r} > Φ_{e}

makes the shift negative. The corresponding upper clock runs faster in a comparison using the same static time coordinate. These are reciprocal descriptions of gravitational redshift and gravitational time dilation, not opposite predictions. A locally measured speed of light remains

c

Static-emitter redshift in a de Sitter patch ( $1 + z = α_{O} / α_{E}$ )

For stationary emitter and observer in the same static patch, repeated rays between fixed radii have the same coordinate travel delay. Since a stationary clock obeys

d τ = α d t

, the received-to-emitted proper-period ratio is

α_{O} / α_{E}

. At a central observer and emitter radius

r_{E}

, this is

(1 - r_{E}^{2} / r_{S}^{2})^{- 1/2}

. Holding an emitter stationary requires unbounded proper acceleration as it approaches the horizon; the limiting redshift formula does not describe a timelike static worldline on the horizon itself.

Static-surface Schwarzschild redshift ( $1 + z = (1 - 2 GM / (R c^{2}))^{- 1/2}$ )

A static emitter at radius

R > 2 GM / c^{2}

and a static receiver at infinity compare photon wavelengths with

1 + z = (1 - 2 GM / (R c^{2}))^{- 1/2}

. Conservation of the time-translation Killing energy and the static clock rate

d τ = 1 - 2 GM / (r c^{2}) d t

prove the relation. Motion of the emitter adds a Doppler factor; a rotating stellar exterior may require a different metric.

Gravitational time dilation ( $d τ = N d t, Δ τ / τ ≃ ΔΦ/ c^{2}$ )

In a static spacetime, a clock held at a fixed position records proper time

d τ = N d t

for metric time component

- N^{2} c^{2} d t^{2}

. Two clocks compared over the same coordinate interval have rate ratio

N_{upper} / N_{lower}

. Each clock ticks normally in its own laboratory; the difference appears when their readings or transmitted frequencies are compared. For a weak Newtonian gravitational potential,

N ≃ 1 + Φ/ c^{2}

, so the fractional excess of the upper rate is

δ ≃ (Φ_{upper} - Φ_{lower}) / c^{2}

Take two clocks maintained at rest above Earth, separated vertically by

h = 1000 m

. Approximate the local gravitational acceleration as constant,

g = 9.81 m s^{- 2}

, neglect rotation, and use the exact speed of light

c = 299792458 m s^{- 1}

from NIST's constants database. Then

ΔΦ ≃ g h = 9810 m^{2} s^{- 2}

and

δ ≃ \frac{g h}{c ^{2}} = 1.09151 \times 1 0^{- 13}, Δ τ ≃ δ (86400 s) = 9.43064 \times 1 0^{- 9} s .

The upper clock gains about

9.4

nanoseconds during one

86400

-second interval on the lower clock. This is a leading weak-field estimate with

h ≪ R_{\oplus}

, not a precision prediction for a particular mountain: Earth's rotation and actual gravitational potential matter in accurate comparisons. The effect is experimentally accessible. Chou, Hume, Rosenband and Wineland raised an optical clock by

0.33 m

and measured a fractional rate change

(4.1 \pm 1.6) \times 1 0^{- 17}

; the same formula with

g = 9.80 m s^{- 2}

predicts

3.60 \times 1 0^{- 17}

. See their ^[ref]. Such comparisons make clocks probes of gravitational potential as well as instruments for timekeeping.

For the ideal spherical, nonrotating Schwarzschild spacetime, the exact static-clock factor relative to time at infinity is

N (r) = 1 - r_{s} / r

, where

r_{s} = 2 GM / c^{2}

is the Schwarzschild radius. Its weak-field expansion is

1 - r_{s} / (2 r)

. The figure compares these factors only in a vacuum exterior with

r > r_{s}

; for a material body, the radius must also exceed its surface radius. It does not extend the exterior formula through the body's interior. See Tong's lecture notes, section 6.1 on Schwarzschild exterior geometry.

Figure 1.
Clock gain at 1 km above Earth and exact versus weak-field Schwarzschild exterior clock factors
. **Details**.

Geostationary gravitational clock gain ( $Δ τ = tGM (R^{- 1} - r_{g}^{- 1}) / c^{2}$ )

For spherical weak-field gravity, a stationary clock at radius

r_{g} > R

gains the displayed proper time relative to a surface clock over coordinate interval

t

. A geostationary orbit has

r_{g} = (GM / Ω_{\oplus}^{2})^{1/3}

. This is the potential contribution only. Combining it with orbital and surface rotational special-relativistic terms requires a consistent reference frame and must not be silently included in a gravitational-only comparison.

Outgoing pulse redshift from a radial Schwarzschild infaller ( $ν_{\infty} / ν_{em} = f / (E + E^{2} - f)$ )

A radial infalling geodesic in Schwarzschild spacetime has

d t / d s = E / f

and

d r / d s = - E^{2} - f

, where

f = 1 - 2 M / r

. An outward radial null geodesic has fixed retarded time

u = t - r_{*}

with

d r_{*} / d r = 1/ f

. Consequently

d u / d s = (E + E^{2} - f) / f

. This factor stretches a short emitted pulse's duration, and its reciprocal multiplies the received frequency at infinity. Near horizon crossing

s_{H}

, it becomes

d u / d s \sim 4 M / (s_{H} - s)

, including both gravitational redshift and the emitter's Doppler shift. The local short-pulse approximation requires its emitted duration to be much smaller than

s_{H} - s

// Target: general-relativity.bigb

Pound-Rebka experiment

The experiment compares gamma-ray emission and resonant absorption at different heights in Earth's field, using the Mössbauer effect to resolve the small gravitational redshift. Its weak-field result is the frequency shift

Δ ν / ν = - g H / c^{2}

for an upward ray over height

H

. Any metric theory with the corresponding static lapse predicts this leading effect; the test alone does not distinguish general relativity from the weak-field prediction of Nordstrom theory of gravitation.

Planar warped spacetime ( $d s^{2} = A (z)^{2} (- d t^{2} + d x^{2} + d y^{2}) + d z^{2}$ )

For positive smooth

A

, this Lorentzian manifold has flat

(t, x, y)

slices scaled by

A (z)^{2}

. Translation in each slice coordinate gives a Killing vector field. An orthonormal coframe is

e^{0} = A d t

e^{1} = A d x

e^{2} = A d y

e^{3} = d z

. Its Levi-Civita connection and Ricci tensor are controlled by

a = A^{'} / A

and

q = A^{''} / A = a^{'} + a^{2}

Cartan curvature of a planar warped spacetime ( $a = A^{'} / A, q = A^{''} / A$ )

For the orthonormal coframe of a planar warped spacetime, Cartan's first structure equation gives

ω^{i}_{3} = a e^{i}

and

ω^{3}_{i} = - s_{i} a e^{i}

, where

i = 0, 1, 2

and

(s_{0}, s_{1}, s_{2}) = (- 1, 1, 1)

. All other connection 1-forms vanish. Cartan's second structure equation gives

Θ^{i}_{3} = - q e^{i} \land e^{3}

and

Θ^{i}_{j} = - s_{j} a^{2} e^{i} \land e^{j}

for

i \neq = j

. Lowering the first index makes both the connection 1-forms and the curvature 2-forms antisymmetric. Contraction yields

R_{00} = q + 2 a^{2}

R_{11} = R_{22} = - q - 2 a^{2}

, and

R_{33} = - 3 q

Diffeomorphism invariance of general relativity

Transforming a metric tensor and all matter fields together by the pullback of a diffeomorphism preserves the Einstein field equations and covariant matter equations. Passive coordinate changes only change the description of the fields; active pullbacks produce another representative of the geometry. The treatment of transformations acting on a boundary depends on the boundary conditions and possible charges. A diffeomorphism is not necessarily an isometry of a fixed metric.

Timelike geodesic ( $\nabla_{u} u = 0$ )

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives

g (\overset{γ}{˙}, \overset{γ}{˙}) = - 1

Physically, an ideal massive test particle in free fall follows a timelike geodesic when nongravitational forces and appreciable self-gravity can be neglected. The normalization above uses

c = 1

; in SI units it is

g (u, u) = - c^{2}

. Zero covariant acceleration does not mean constant coordinate velocity: a dropped object accelerates relative to the ground. Conversely, a person standing on the ground is pushed away from free fall by the supporting surface. A photon follows a null geodesic, whose parameter cannot be proper time. See Tong's geodesics in spacetime.

Timelike geodesics of a quadratic-time conformal metric ( $d s^{2} = (t^{2} / α) (d x^{2} - c^{2} d t^{2})$ )

For

α > 0

and

t \neq = 0

, the conformal factor

t^{2} / α

makes spatial translation a Killing vector field. Using proper time, its conserved momentum is

P = (t^{2} / α) d x / d τ

. With

v = d x / d t

and

d s^{2} = - c^{2} d τ^{2}

this becomes

P = \frac{∣ t ∣}{α} \frac{v}{1 - v ^{2} / c ^{2}} .

Thus

v / 1 - v^{2} / c^{2}

alone is not constant. Set

b = α P / c

. Every unparametrized timelike geodesic in a connected nonzero-

t

patch has the form

x (t) = x_{c} + c b arsinh (t /∣ b ∣) (b \neq = 0), x (t) = x_{c} (b = 0) .

The nonvertical paths have

v = c b / t^{2} + b^{2}

and proper-time parametrization

τ - τ_{c} = \frac{t t ^{2} + b ^{2} - b ^{2} arsinh ( t /∣ b ∣ )}{2 α} .

(3)

The chart excludes

t = 0

, where its metric is degenerate; a vertical family must not be lost by dividing by the conserved momentum.

Odd radial perturbations preserve timelike-geodesic periods ( $T_{τ} = T_{t} = 2 π$ )

For

d s^{2} = (1 + f (x))^{2} d x^{2} / (1 + x^{2}) - (1 + x^{2}) d t^{2}

, with smooth odd

f

and

∣ f ∣ < 1

, the conserved energy gives

d t / d τ = E / (1 + x^{2})

and

(d x / d τ)^{2} = (E^{2} - 1 - x^{2}) / (1 + f)^{2}

. Both half-cycle integrals over

[- E^{2} - 1, E^{2} - 1]

contain an even weight times

1 + f

. The odd contribution vanishes, leaving the unperturbed half-period

π

in each clock. The waveforms need not be sinusoidal, but both periods are unchanged for

E > 1

Timelike geodesics of an inverse-time conformal metric ( $d s^{2} = t^{- 2} (d x^{2} - d t^{2})$ )

In a positive-time patch, proper-time first integrals give

\overset{x}{˙} = p t^{2}

and

\dot{t}^{2} = t^{2} + p^{2} t^{4}

. If

p \neq = 0

, the unparametrized timelike geodesics are

t^{2} = (x - x_{c})^{2} - p^{- 2}

on an appropriate branch; in the expanded quadratic form their discriminant is positive. If

p = 0

, there is the separate family

x = x_{c}

t = t_{*} e^{\pm τ}

. A description using only the nonvertical quadratic family omits genuine timelike geodesics.

Coordinate singularity

A coordinate singularity is a failure of a coordinate description which disappears in a regular chart. The horizon singularity of the Schwarzschild metric in

(t, r)

coordinates is removable; its

r = 0

curvature singularity is not.

Areal radius ( $r$ )

In a spherically symmetric spacetime, the areal radius

r

is defined by the area

A = 4 π r^{2}

of each symmetry sphere. It is independent of which radial coordinate chart is used.

Geometrized units

Geometrized units set

G = c = 1

, allowing mass, length, and time to be expressed in common units. Restore

GM / c^{2}

for a mass parameter in a radial metric tensor.

Geometrized luminosity ( $L_{geom} = G L_{SI} / c^{5}$ )

A geometrized luminosity is dimensionless because in geometrized units an energy and a time have the same dimension. Starting from SI luminosity, multiply by

G / c^{5}

. The corresponding SI power unit is

c^{5} / G

. Radiating a fraction

f

of a mass

M

at constant SI luminosity

L

takes

f M c^{2} / L

, or

f (GM / c^{3}) / L_{geom}

in seconds.

Vaidya metric ( $d s^{2} = - (1 - 2 M (v) / r) d v^{2} + 2 d v d r + r^{2} d Ω^{2}$ )

The ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time

v

and areal radius

r

. In geometrized units it is

d s^{2} = - (1 - 2 M (v) / r) d v^{2} + 2 d v d r + r^{2} d Ω^{2}

. A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.

Teleological response of a Vaidya event horizon ( $ρ (v) = \frac{1}{2 M _{0}} \int_{v}^{\infty} e^{- (s - v) / (4 M_{0})} μ (s) d s$ )

For a small bounded mass perturbation

m (v) = M_{0} + μ (v)

that becomes constant in the future, write the event horizon radius as

r_{H} (v) = 2 M_{0} + ρ (v)

. The outgoing radial null geodesics of the Vaidya metric give

ρ^{'} - ρ / (4 M_{0}) = - μ / (2 M_{0})

to first order. Future stationarity selects the displayed solution and removes the exponentially growing homogeneous solution. If

μ^{'} \geq 0

, integration by parts gives

ρ - 2 μ = 2 \int_{v}^{\infty} e^{- (s - v) / (4 M_{0})} μ^{'} (s) d s \geq 0

. Thus the event horizon can grow before local matter arrives, lying outside the instantaneous spherical apparent horizon

r = 2 m (v)

. This is a consequence of the event horizon's global definition, not propagation of a local signal into the past.

Radial null geodesics of the Vaidya metric ( $v = const, d r / d v = (1 - 2 M / r) /2$ )

The two radial null geodesic families are

v = constant

and

d r / d v = (1 - 2 M (v) / r) /2

. The first has affine radius, while the second has tangent

k = \partial_{v} + (f /2) \partial_{r}

obeying

\nabla_{k} k = (M / r^{2}) k

, so advanced time is generally a nonaffine parameter. An affine parameter

λ

for the outgoing family satisfies

d lo g (d v / d λ) / d v = - M / r^{2}

Reconstructing Vaidya mass from a radial null geodesic ( $M (v) = \frac{1}{2} r_{0} (v) (1 - 2 r_{0}^{'} (v))$ )

An outgoing radial null geodesic of the Vaidya metric prescribed as

r = r_{0} (v)

determines the mass along that curve by

M (v) = r_{0} (v) (1 - 2 r_{0}^{'} (v)) /2

. This inverse relation comes from the radial null condition. It determines a function of advanced time wherever the prescribed curve is differentiable and has positive radius.

Christoffel symbols of the Vaidya metric ( $Γ^{μ}_{ν ρ}$ )

Writing

f = 1 - 2 M (v) / r

, the radial Christoffel symbols are

Γ^{v}_{vv} = M / r^{2}

Γ^{r}_{vv} = f M / r^{2} + M^{'} / r

and

Γ^{r}_{v r} = - M / r^{2}

. The angular couplings are

Γ^{v}_{θθ} = - r

Γ^{r}_{θθ} = - f r

, with

sin^{2} θ

factors for the corresponding

ϕϕ

terms,

Γ^{θ}_{r θ} = Γ^{ϕ}_{r ϕ} = 1/ r

Γ^{θ}_{ϕϕ} = - sin θ cos θ

and

Γ^{ϕ}_{θϕ} = cot θ

. The lower indices are symmetric. The plus sign of

M^{'} / r

follows from the ingoing

+ 2 d v d r

cross term.

Equivalence principle

The equivalence principle states that gravity couples universally to energy-momentum and that a freely falling local frame removes uniform gravitational acceleration at a point. Tidal curvature remains measurable over a finite region.

Imagine releasing a ball inside a small falling elevator. Relative to the cabin it initially floats, because cabin and ball share the same free fall. The weak equivalence principle says this trajectory is independent of the test particle's composition for identical initial conditions. The Einstein equivalence principle further identifies local nongravitational experiments with those of special relativity. It is a statement about experiments in a local inertial frame, not just equality of two masses.

The word local is essential. Coordinates can make the metric tensor equal to the Minkowski metric and its first derivatives vanish at one event. They generally cannot erase the Riemann curvature tensor, which involves second derivatives. Two separated balls can therefore approach or recede even inside the falling cabin. A sufficiently extended experiment detects these tidal forces; a long observation can accumulate them even when the cabin is small. The equivalence principle does not identify an entire planetary gravitational field with acceleration in flat spacetime. See Tong's discussion of normal coordinates and local equivalence.

Comma-to-semicolon prescription ( $\partial_{a} ⟼ \nabla_{a}$ )

The prescription replaces partial derivatives in flat-spacetime matter equations by the Levi-Civita connection and the Minkowski metric by the curved metric tensor. It implements minimal gravitational coupling and reproduces flat-spacetime physics at a point in a local inertial frame. It is not a unique rule for rearranged higher-derivative equations: commuting partial derivatives before replacement can conceal a Riemann curvature tensor term afterwards. The derivative-order ambiguity in covariant Maxwell potential equations is an explicit example. The equivalence principle removes uniform gravitational acceleration locally but does not erase tidal curvature over a finite experiment.

Einstein equivalence principle (EEP)

The Einstein equivalence principle combines the weak equivalence principle with local Lorentz invariance and local position invariance of nongravitational experiments. In a sufficiently small freely falling laboratory, their results agree with special relativity independently of its location and velocity. The strong equivalence principle extends this claim to local gravitational experiments.

Equivalence principle does not uniquely fix field equations

Universal metric coupling and local special-relativistic nongravitational physics do not select a unique dynamical metric action. Additional assumptions are needed to obtain the Einstein field equations. A local freely falling coordinate choice can remove connection coefficients at a point but cannot remove curvature or tidal effects over a finite region. Tests of redshift, free fall and self-gravitating dynamics therefore address related but distinct claims.

Local position invariance

Local position invariance requires identical local experiments to have the same outcome independently of where and when they are performed. For nongravitational experiments it is part of the Einstein equivalence principle. Comparisons of clocks through gravitational redshift test this principle once the gravitational potential difference is independently specified; they do not by themselves test the whole strong equivalence principle.

Local Lorentz invariance

Local Lorentz invariance requires the outcome of a local experiment, expressed in its freely falling apparatus, to be independent of the apparatus's velocity and orientation. In the Einstein equivalence principle it applies to nongravitational experiments; the strong equivalence principle also includes local gravitational experiments. It is distinct from local position invariance.

Weak equivalence principle

Freely falling test particles with identical initial position and velocity follow the same trajectory independently of their material composition. The principle neglects appreciable self-gravity and nongravitational forces. The Einstein equivalence principle extends the local statement to nongravitational experiments; the strong equivalence principle also includes gravitational experiments and self-gravitating bodies.

MICROSCOPE (satellite)

A satellite experiment testing the universality of free fall. Electrostatic control keeps test masses made from titanium and platinum alloys on matched orbital motion; the required differential forces constrain composition dependence. A reference sensor uses masses of the same composition. The experiment tests the weak equivalence principle, rather than the full strong equivalence principle.^[ref]

Eötvös experiment

A torsion-balance comparison of the gravitational acceleration of bodies of different composition, testing equality of inertial and passive gravitational mass up to a universal proportionality factor. A universal acceleration law independent of test-body mass and composition agrees with this weak equivalence principle test. It does not by itself establish the strong equivalence principle for self-gravitating bodies.

Strong equivalence principle (SEP)

The strong equivalence principle extends universality of free fall to bodies with appreciable gravitational self-energy and requires local experiments, including gravitational ones, in a freely falling laboratory to be independent of the laboratory's location and velocity. This statement neglects external tidal variation across the laboratory; it does not remove Riemann curvature tensor. In general relativity, universal metric coupling, fixed local gravitational coupling and the absence of extra preferred-frame gravitational fields implement this principle.

Nordtvedt effect ( $m_{G} / m_{I} = 1 + η_{N} E_{G} / (m c^{2})$ )

A violation of the strong equivalence principle can make the gravitational-to-inertial mass ratio depend on a body's negative gravitational self-energy

E_{G}

. Bodies with different self-binding fractions would then fall differently in the same external field, even after composition-dependent weak-equivalence effects are removed. For the Earth and Moon this produces a polarization of the lunar orbit toward the Sun, measurable by lunar laser ranging. General relativity predicts

η_{N} = 0

Absence of a local gravitational stress-energy tensor

The equivalence principle allows Riemann normal coordinates in which the first derivatives of the metric tensor vanish at a chosen point. A purported local gravitational stress-energy tensor built from those first derivatives would vanish there in that chart and therefore, if tensorial, in all charts. Such an expression cannot encode the gravitational energy assigned by a coordinate-dependent field description. This rules out an ordinary universal local covariant gravitational energy density, not the construction of local curvature tensors or boundary and averaged notions of gravitational energy. The Riemann curvature tensor still describes invariant tidal gravity.

Spacetime dimension

The spacetime dimension is the dimension of the smooth manifold used to model spacetime. Ordinary general relativity uses one temporal and three spatial dimensions.

Killing vector field ( $K$ )

A Killing vector field describes a direction in which geometry has a continuous symmetry: moving points together along its flow of a vector field preserves the metric tensor, including its pairing of tangent vectors. Such a symmetry gives a geodesic conserved quantity from a Killing vector, often a component of momentum or angular momentum. The direction may vary with position: rotation provides the basic example.

On a Riemannian manifold or Lorentzian manifold with metric tensor

g

, a smooth vector field

K

is a Killing vector field when its Lie derivative of

g

vanishes. If

Φ_{s}

is its local flow of a vector field, the equivalent statements are

L_{K} g = 0 ⟺ Φ_{s}^{*} g = g ⟺ \nabla_{(μ} K_{ν)} = 0.

The middle equality uses the pullback of a covariant tensor: the local flow maps are isometries on their domains. The last equality is the Killing equation, using the Levi-Civita connection. The infinitesimal condition holds throughout an open domain and integrates to exact local isometries; it is stronger than making the first derivatives of

g

vanish at a single point. See Tong, section 4.3.1, equation (4.29).

For example, on the interval

0 < x < 1

with Euclidean metric

d x^{2}

K = \partial_{x}

translates points by

x \mapsto x + s

. This is a local isometry, but a point reaches the end of the interval in finite flow parameter. A local flow of a vector field can be incomplete; see Killing fields on incomplete manifolds need not generate global isometries. The full-plane translation and rotation examples under Euclidean Killing vector field instead have flows for every real

s

Two-dimensional Killing algebra with a periodic generator is abelian ( $[T, Φ] = 0$ )

Suppose the full Lie algebra of Killing fields has basis

T, Φ

and the flow of

Φ

has a nonzero period

P

. Write

[T, Φ] = a T + b Φ

with constant coefficients. The Adjoint representation of the periodic flow satisfies

e^{P ad Φ} = 1

. Its real eigenvalues are

- a

and zero, so

a = 0

. The remaining adjoint operator is nilpotent and its exponential is

1 + P ad Φ

, forcing

b = 0

. This proves commutation from genuine periodic axial symmetry, without differentiating mere asymptotic component limits.

Bochner identity for Killing vector fields ( $\int_{M} ∣\nabla X ∣^{2} d V = \int_{M} Ric (X, X) d V$ )

On a closed Riemannian manifold, a Killing vector field

X

satisfies

\nabla^{*} \nabla X = Ric^{♯} X

for the nonnegative rough Laplacian. This follows by differentiating

\nabla_{i} X_{j} + \nabla_{j} X_{i} = 0

, contracting and commuting covariant derivatives; the contraction also gives

div X = 0

. Taking the inner product with

X

and using integration by parts gives the displayed identity. If Ricci curvature is nonpositive, the nonnegative left side must vanish, so

X

is a parallel vector field. In particular every Killing vector field on a closed flat Riemannian manifold is a parallel vector field.

Killing fields on incomplete manifolds need not generate global isometries

On a geodesically complete Riemannian manifold, every Killing vector field is complete and integrates to a global one-parameter subgroup of its isometry group. On the incomplete interval

(0, 1)

with its Euclidean metric,

\partial_{x}

satisfies the Killing equation, but translations leave the interval in finite time. The full isometry group is only the identity and reflection. Local solutions of the Killing equation therefore need not equal the Lie algebra of global isometries without completeness.

Killing transport ( $\nabla_{V} ξ_{a} = L_{ab} V^{b}, \nabla_{V} L_{ab} = - R_{ab c}^{d} V^{c} ξ_{d}$ )

The Killing equation and the second covariant derivative of a Killing vector transport a Killing field and its antisymmetric derivative along any curve. Here

L_{ab} = \nabla_{b} ξ_{a}

and the curvature convention is the one in curvature-sign convention in Killing derivative identities. In a parallel frame the equations form a homogeneous linear system of ordinary differential equations. Initial value zero has unique zero continuation. Thus on a connected pseudo-Riemannian manifold, not just on a Riemannian manifold, the field is determined by its value and first derivative at one point.

Dimension bound for Killing vector fields ( $dim kill (M, g) \leq n (n + 1) /2$ )

On a connected

n

-dimensional pseudo-Riemannian manifold, evaluation of a Killing vector field and its first derivative is injective by Killing transport. The value has

n

components and the Killing equation makes the derivative an antisymmetric two-tensor with

n (n - 1) /2

components. Their sum gives the bound. Flat metrics attain it through

ξ_{a} = A_{a} + B_{ab} x^{b}

with

B_{ab} = - B_{ba}

. Connectedness is essential for a uniform global bound in terms of

n

: independent fields can be supported on different connected components.

Killing field is determined by its value and first covariant derivative ( $K (p), (\nabla K)_{p}$ )

On a connected Riemannian manifold, these data determine a Killing field uniquely, without a completeness assumption. The second covariant derivative of a Killing vector makes its restriction to every geodesic a Jacobi field. The difference of two fields with identical data has zero Jacobi initial values and vanishes in a convex normal neighborhood. The set on which both its value and covariant derivative vanish is closed by continuity and open by the same local argument, so connectedness makes it the whole manifold.

Killing-flow four-acceleration ( $a_{a} = - \nabla_{a} lo g K^{b} K_{b}$ )

With metric signature

(+ - - -)

, normalize a timelike Killing vector field by

U^{a} = K^{a} / N

N^{2} = K^{a} K_{a}

. The Killing equation gives

K^{b} \nabla_{b} K_{a} = - \frac{1}{2} \nabla_{a} N^{2}

, while

K^{b} \nabla_{b} N = 0

. Thus the four-acceleration of its normalized flow is the displayed gradient. With metric signature

(- + + +)

the corresponding lower-index formula has the opposite sign.

Papapetrou electromagnetic field ( $F = d K^{♭}$ )

For a Killing vector field

K

, its metric-dual differential one-form produces the electromagnetic field

F_{ab} = 2 \nabla_{[a} K_{b]} = 2 \nabla_{a} K_{b}

. The exterior derivative gives

d F = 0

, while the second covariant derivative of a Killing vector gives

\nabla^{a} F_{ab} = - 2 R_{b d} K^{d}

. Thus this construction solves the source-free Maxwell equations in a vacuum spacetime with zero cosmological constant. The field is a test field: adding its stress-energy tensor would generally change the original metric tensor.

Kaluza-Klein decomposition along a Killing field ( $h = V (d x + A)^{2} + γ$ )

Where a Killing vector field is

\partial_{x}

and its squared norm

V = h_{xx}

is nonzero, an invariant metric tensor decomposes as

h = V (d x + A)^{2} + γ

. The quotient one-form has

A_{i} = h_{x i} / V

and the quotient metric has

γ_{ij} = h_{ij} - h_{x i} h_{x j} / V

. All coefficients are independent of

x

. Changing the fibre coordinate to

x^{'} = x + λ

sends

A

A - d λ

. A null Killing direction requires a different decomposition.

Kaluza-Klein geodesic Lorentz force ( $\frac{D x ˙ ^{i}}{d s} = k F^{i}_{j} \overset{x}{˙}^{j}$ )

For

d s_{5}^{2} = γ_{ij} d x^{i} d x^{j} + (d z + A_{i} d x^{i})^{2}

with

z

-independent coefficients, the geodesic Lagrangian makes

k = \overset{z}{˙} + A_{i} \overset{x}{˙}^{i}

constant. Its remaining Euler-Lagrange equations give the displayed Lorentz force, where

F = d A

and

D

uses the Levi-Civita connection of

γ

. For a particle of five-dimensional mass

M

with proper-time parameter

s

, the conserved momentum is

p_{z} = M k

and the four-dimensional rest mass is

m = M 1 + k^{2}

. Reparametrizing by four-dimensional proper time gives

D u^{i} / d τ = (p_{z} / m) F^{i}_{j} u^{j}

Sign of the projected Kaluza-Klein Lorentz force

Take

L = [η_{ab} \overset{x}{˙}^{a} \overset{x}{˙}^{b} + W^{2}] /2

with

W = \overset{y}{˙} + A_{a} \overset{x}{˙}^{a} = ξ

constant. The Euler-Lagrange equation for

x^{a}

η_{ab} \overset{x}{¨}^{b} + ξ (\partial_{c} A_{a} - \partial_{a} A_{c}) \overset{x}{˙}^{c} = 0.

Thus, with

F_{b c} = \partial_{b} A_{c} - \partial_{c} A_{b}

\overset{x}{¨}^{a} = ξ η^{ab} (\partial_{b} A_{c} - \partial_{c} A_{b}) \overset{x}{˙}^{c} = ξ F^{a}_{c} \overset{x}{˙}^{c} .

Reversing the explicit derivative order while retaining this metric and the same definition of

ξ

changes the equation rather than merely changing a field-tensor naming convention. More generally, a fibre term

σ W^{2}

σ = \pm 1

, gives

σ ξ F^{a}_{b} \overset{x}{˙}^{b}

; its conserved canonical fibre momentum is

p_{y} = σ ξ

In the normalized positive-fibre case with

∣ ξ ∣ < 1

, four-dimensional proper time gives

d u^{a} / d τ = c ξ F^{a}_{b} u^{b} / 1 - ξ^{2}

. This has the Lorentz force form. The scale relating the geometrical

A

to a physical electromagnetic four-potential fixes the conversion to an SI charge-to-mass ratio.

Affine normalization in a Kaluza-Klein geodesic reduction

For

d s_{5}^{2} = η_{ab} d x^{a} d x^{b} + (d y + A_{a} d x^{a})^{2}

, with

A

independent of

y

, an affine parameter

λ

makes

ξ = d y / d λ + A_{a} d x^{a} / d λ

and the tangent norm

N = g_{A B} \overset{x}{˙}^{A} \overset{x}{˙}^{B}

constant. The four-dimensional interval satisfies

d u^{2} = (N - ξ^{2}) d λ^{2} .

Consequently the formula

d λ^{2} = d u^{2} / (1 - ξ^{2})

assumes the additional normalization

N = 1

; affine parametrization alone does not set this value. With base metric signature

+ - - -

and

u = c τ

, the projected curve is timelike only for

N > ξ^{2}

. The normalized case gives

d τ / d λ = 1 - ξ^{2} / c

. The positive fibre term and the

+ - - -

base together have two positive directions; the same sign convention must be retained when interpreting the reduction.

Heisenberg metric Kaluza-Klein reduction ( $V = α + χ y^{2}$ )

For

h = α d x^{2} + β d y^{2} + χ (d z - y d x)^{2}

, the Kaluza-Klein decomposition along a Killing field

\partial_{x}

has

V = α + χ y^{2}

A = - χy d z / V

and

γ = β d y^{2} + α χ d z^{2} / V

. For a Riemannian metric, positive constants make the fibre norm and quotient metric positive everywhere.

Euclidean Killing vector field ( $K (x) = A + B x, B^{T} = - B$ )

On Euclidean space, every Killing vector field is a constant translation generator plus a constant infinitesimal rotation generator:

K (x) = A + B x, B^{T} = - B .

Here

A

is a constant vector and

B

is an antisymmetric matrix. The second covariant derivative of a Killing vector vanishes for the Euclidean metric, so the components are affine functions of Cartesian coordinates; the Killing equation makes the linear coefficient antisymmetric.

In the Euclidean plane with metric tensor

g = d x^{2} + d y^{2}

, two useful examples are

K_{tr} = \partial_{x}, K_{rot} = - y \partial_{x} + x \partial_{y} .

The notation

\partial_{x}

denotes a coordinate basis vector;

d x

denotes a differential one-form. The corresponding flow of a vector field maps are

Φ_{s}^{tr} (x, y) = (x + s, y), Φ_{s}^{rot} (x, y) = (x cos s - y sin s, x sin s + y cos s) .

(3)

For translation, all arrows have the same direction and length. For rotation, arrows are tangent to circles, have length

x^{2} + y^{2}

, and vanish at the origin. The rotation flow keeps

x^{2} + y^{2}

fixed, and its nonstationary integral curves of a vector field are circles. Both flows preserve the Euclidean metric. See Carroll's original notes, section 5, equations (5.45)-(5.46).

Figure 1.
Euclidean Killing vector fields: translation arrows and straight flow orbits, rotation arrows and circular flow orbits
. **Details**.

The circular integral curves of a vector field are symmetry orbits, whereas geodesics of the Euclidean plane are straight lines. A Killing vector field does not require its own orbits to be geodesics. The geodesic conserved quantity from a Killing vector pairs the field with the velocity of a separate geodesic; it does not require the velocity to equal the Killing vector field.

Helical orbit of a Euclidean Killing vector

An infinitesimal rotation in one coordinate plane plus a constant translation has integral curves of a vector field that rotate about a shifted centre in that plane and move uniformly in the remaining directions. Nonzero transverse drift gives helices; zero drift gives circles, with zero radius allowing straight or stationary curves.

Killing equation ( $\nabla_{(μ} K_{ν)} = 0$ )

The Killing equation tests whether a candidate vector field generates a local isometry, without needing to solve for its flow of a vector field. For the Levi-Civita connection of a metric tensor

g

, it is

\nabla_{(μ} K_{ν)} = \frac{1}{2} (\nabla_{μ} K_{ν} + \nabla_{ν} K_{μ}) = 0, K_{ν} = g_{ν ρ} K^{ρ} .

Parentheses mean averaging over the two index orders. The equation says the symmetric part of the covariant derivative vanishes, not that the entire covariant derivative vanishes. The Lie derivative of the metric tensor gives the coordinate form

(L_{K} g)_{μν} = K^{ρ} \partial_{ρ} g_{μν} + g_{ρ ν} \partial_{μ} K^{ρ} + g_{μ ρ} \partial_{ν} K^{ρ} = \nabla_{μ} K_{ν} + \nabla_{ν} K_{μ} .

The last two terms account for how tangent vectors move with the field; checking only

K^{ρ} \partial_{ρ} g_{μν}

misses this. See Carroll, section 5, equations (5.33) and (5.41)-(5.42).

For the Euclidean Killing vector field

K = - y \partial_{x} + x \partial_{y}

, the Euclidean metric in Cartesian coordinates makes covariant derivatives ordinary partial derivatives. Its diagonal equations give

\partial_{x} K_{x} = \partial_{y} K_{y} = 0

, and its mixed equation gives

\partial_{x} K_{y} + \partial_{y} K_{x} = 1 - 1 = 0

. Thus a varying vector field can preserve a constant metric tensor. In contrast,

K = x \partial_{x} + y \partial_{y}

gives

(L_{K} g)_{xx} = 2

and fails the Killing equation, despite the constant metric components.

K = \partial_{z}

is a coordinate basis vector, its components are constant and the coordinate form reduces to

\partial_{z} g_{μν} = 0

. Hence independence of the metric components from

z

supplies a Killing vector field. This shortcut applies to that coordinate generator, not to an arbitrary varying vector field.

Metric compatibility in Killing identities ( $\nabla g = 0$ )

For a torsion-free connection, the Lie derivative of a metric tensor obeys

(L_{K} g)_{ab} = 2 g_{c (a} \nabla_{b)} K^{c} + K^{c} \nabla_{c} g_{ab} = 2 \nabla_{(a} K_{b)} + K^{c} \nabla_{c} g_{ab} - 2 (\nabla_{(a} g_{b) c}) K^{c} .