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On Euclidean space, every Killing vector field is a constant translation generator plus a constant infinitesimal rotation generator: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 areThe notation ${\partial }_x$ denotes a coordinate basis vector; $dx$ denotes a differential one-form. The corresponding flow of a vector field maps areFor translation, all arrows have the same direction and length. For rotation, arrows are tangent to circles, have length $\sqrt{x^2+y^2}$, 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).

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.

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 ${\Phi }_s$ is its local flow of a vector field, the equivalent statements areThe 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 $s$.

For the static nonrotating source, use spherical symmetry and the static scalar perturbation of Minkowski spacetime, with isotropic radius $\rho$:To first order, the Einstein tensor in the declared convention hasThese are also the static scalar perturbation of Minkowski spacetime equations in the original information sheet. The source has $T_{00}=M{\delta }^{(3)}(\mathbf{x})$ and zero spatial stress. With the sign established in Question 2,The distributional identity $\Delta (1/\rho )=-4\pi {\delta }^{(3)}(\mathbf{x})$ and asymptotic flatness give $C=GM/\rho$. For $F=A+C$, the spatial field equation has trace $-2\Delta F=0$ and therefore ${\partial }_i{\partial }_{j}F=0$. Its decaying spherical solution is $F=0$, so $A=-GM/\rho$. This determines both potentials; the spatial correction cannot be inferred from the Newtonian gravitational potential alone.

Now use the isotropic-to-areal weak-field radius change. The angular coefficient defines $r^2=(1+2GM/\rho ){\rho }^2$, hence $r=\rho +GM$ to first order. Thus $dr=d\rho$ to the same accuracy, and replacing $1/\rho$ by $1/r$ inside a perturbation changes only second-order terms. With the Schwarzschild radius $r_S=2GM$, the linearized Schwarzschild metric isThis is an exterior weak-field approximation: $r_S/r\ll 1$. The point source is used distributionally to fix the mass, not to claim a regular or weak field at the origin.

For a null geodesic choose its orbital plane as $\theta =\pi /2$, using rotational symmetry, and let a dot denote an affine parameter derivative. For a Killing vector $k$ and an affinely parametrized geodesic tangent $U$, the geodesic conserved quantity from a Killing vector follows directly fromApplying this to the time-translation and axial Killing vectors, with the spatial sign absorbed into the definition of $h$, givesIf the tangent is normalized as the photon's four-momentum, $E$ is its energy measured at infinity and $h$ its signed angular momentum about the selected axis. Rescaling an arbitrary affine parameter rescales both; the ratio $b=|h|/E$ is the invariant impact parameter. Future-directed orbits have $E>0$. Assume $h\ne 0$ to use $\varphi$ as the orbit parameter; $h=0$ gives radial rays, not a finite-radius circular orbit.

The null vector constraint givesExpand consistently to first order in the weak-field approximation correction: $(1-u^2{)}^{-1}=1+O(u^2)$ and $(1+u{)}^{-1}=1-u+O(u^2)$. Also $du/d\varphi =-r_S\dot{r}/h$. Therefore the first-order orbital equation isThe cubic term is the first correction relative to the flat inverse-radius term; the equation is not a controlled exact equation for large $u$.

A constant solution of a first integral is not by itself sufficient to establish a circular geodesic. For a circular trajectory, the radial Euler-Lagrange equation also requires $A^{\prime }(r){\dot{t}}^2/2=r{\dot{\varphi }}^2$, where now $A(r)=1-r_S/r$ is the metric's time coefficient. Combining this with the null vector constraint gives $r{A}^{\prime }=2A$, or $u=2/3$. Equivalently, the formal potential $V(u)=u^2(1-u)$ must have a stationary point. Apart from the infinite-radius endpoint $u=0$, its only stationary point is $u_c=2/3$, with $V(u_c)=4/27$. Thus the formal circular solution requiresThe reduced radial equation is $u^{\prime \prime }=-u+\frac{3}{2}{u}^2$, with prime denoting $d/d\varphi$. It follows by differentiating the first integral on moving segments and agrees with the regular Euler-Lagrange equation at turning points. Perturbing $u=u_c+ϵ$ givesThere is an exponentially growing linear instability mode; equivalently $V^{\prime \prime }(u_c)=-2<0$. The formal circular orbit is unstable.

The weak-field photon-orbit validity limitation is essential: $u_c=2/3$ is not small. There is therefore no finite-radius circular photon orbit established within the controlled weak-field region. The formal extrapolation happens to reproduce the radius and impact parameter of the exact Schwarzschild photon sphere, but that physical result needs the exact geometry. The calculation also cannot establish a Schwarzschild event horizon from a first-order expansion at $u\sim 1$.

For the full unit-radius two-dimensional de Sitter spacetime, the coordinate ranges areOne can use $0\le \chi <2\pi$ with endpoints identified. If the identification is omitted, the same line element describes the universal covering spacetime instead. The periodicity follows from the hyperboloid embeddingThe induced ambient Minkowski metric is precisely the given Lorentzian metric.

Using an affine parameter $s$, the geodesic Lagrangian is $\frac{1}{2}({\dot{t}}^2-{\cosh }^{2}t{\dot{\chi }}^2)$. The nonzero Christoffel symbols areThus the geodesic equations and their first integrals areHere $\epsilon =1$ for unit-speed timelike geodesics, $0$ for null geodesics and $-1$ for unit-speed spacelike geodesics. The conserved $J$ is the geodesic conserved quantity from a Killing vector for the angular symmetry.

All geodesics of two-dimensional de Sitter spacetime can be written especially simply in the embedding. If $P$ is the initial point and $V$ the initial tangent, with $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, the geodesic has normal ambient acceleration $X^{\prime \prime }=\epsilon X$. ThereforeThese curves stay in the two-plane spanned by $P$ and $V$, preserve both constraints and have no tangential acceleration. They consequently solve the geodesic equation and exhaust all initial directions.

An isometry can move the chosen point to $P=(0,1,0)$, namely $(t,\chi )=(0,0)$. For future-directed timelike geodesics take $V=(\cosh \alpha ,0,\sinh \alpha )$, obtainingThe continuous angular branch has $|\chi |<\pi /2$, and every real $\alpha$ gives a distinct timelike direction. The two null geodesics have $V=(1,0,\pm 1)$ andFor spacelike geodesics take $V=(\sinh \alpha ,0,\cosh \alpha )$, givingThey are closed with period $2\pi$ in affine arclength and $|t|\le |\alpha |$. In particular, $\alpha =0$ is the equatorial geodesic $t=0$. These three families, with reversals of parametrization allowed, are the requested complete set through the chosen point.

The conformal time substitution givesHenceMultiplying by ${\cos }^{2}T$ gives the regular conformal cylinder of two-dimensional de Sitter spacetime. Adding $T=\pm \pi /2$ completes the conformal compactification; both boundary circles are spacelike. Null geodesics are straight lines $\chi =\pm T$ through the origin. The other families have the useful diagram equationsThe image sketches representative directions from each family; the formulas allow all intermediate directions.

For the complete observer $\chi =0$, let $d(\chi ,0)\in [0,\pi ]$ be the shortest angular distance on the circle. Since light travels one unit of angular distance per unit conformal time, an event at $(T,\chi )$ can signal to the observer at some finite future time precisely whenEquality defines its future observer event horizon. Likewise an event can receive a signal emitted by the observer precisely whenand equality defines the past horizon. In the lift $-\pi \le \chi \le \pi$, their branches are $\chi =\pm (\pi /2-T)$ and $\chi =\pm (T+\pi /2)$. The common communication region is the diamond $|\chi |+|T|<\pi /2$, the observer's static patch of de Sitter spacetime. These are the observer horizons in two-dimensional de Sitter spacetime. Every complete timelike observer has analogous horizons after a de Sitter isometry; they depend on the observer, rather than enclosing an intrinsic black-hole region.

Despite these horizons, each constant-$T$ circle is a Cauchy hypersurface of the full spacetime: any inextendible causal curve crosses it exactly once. The global spacetime is therefore globally hyperbolic and has no Cauchy horizons. The conformal boundaries lie at infinite proper time for complete timelike geodesics and infinite affine distance for complete null geodesics, so their finite height in the diagram is not a physical incompleteness.

Use metric compatibility and the symmetric Levi-Civita connection. Expanding $k_a=g_{ac}{k}^c$ givesThe Christoffel formula makes the bracket ${\partial }_c{g}_{ab}$. Thus the Killing equation is equivalent to the displayed coordinate formula in the question, also the vanishing Lie derivative of the metric.

Along an affinely parametrized geodesic with tangent $V$,The first term vanishes by the geodesic equation and the second by the Killing equation, because $V^a{V}^b$ is symmetric. This proves the geodesic conserved quantity from a Killing vector.

For $d{s}^2=-d{u}^2+u^{2}d{v}^2$, the three independent coordinate Killing conditions areThe field $(0,1)$ satisfies all three immediately. For $e^{-v}(1,u^{-1})$, the second equation reads $u{e}^{-v}-u{e}^{-v}=0$, and the third reads $e^{-v}-e^{-v}=0$. For $e^v(-1,u^{-1})$, they read $-u{e}^v+u{e}^v=0$ and $e^v-e^v=0$. The first equation holds for both, verifying all three Killing vector fields.

Their conserved contractions with $(\dot{u},\dot{v})$ may be denotedAdding the last two expressions after multiplying by $e^v,e^{-v}$ givesTo identify these loci, set $T=u\cosh v$, $X=u\sinh v$. The metric becomes $-d{T}^2+d{X}^2$, and the equation becomes $(\alpha +\beta )T+(\alpha -\beta )X=2\gamma$, a straight line in Minkowski spacetime. Nontrivial such lines, parametrized affinely, are exactly the geodesics in this coordinate patch. The all-zero choice of constants is a vacuous equation rather than an individual geodesic.

Let $u$ be the tangent vector to a geodesic with affine parameter $\lambda$. Its geodesic equation is ${\nabla }_{u}u=0$. For a Killing vector field $\xi$, metric compatibility givesThusThis geodesic conserved quantity from a Killing vector is independent of whether the geodesic is timelike, spacelike or null. For a timelike geodesic one can use proper time; for a null geodesic choose an affine parameter. An arbitrary nonaffine tangent rescales the displayed quantity and need not make it constant.

Let $D_{abc}={\nabla }_a{\nabla }_b{K}_c$. Differentiating the Killing equation gives $D_{abc}=-D_{acb}$. For a covector, the Ricci identity isCall this difference $C_{abc}$. The differentiated Killing equation then givesTaking the first minus the second plus the third yieldsThe pair symmetries of the Riemann curvature tensor and the first Bianchi identity reduce the coefficient to $2R_{cbad}$. Therefore the second covariant derivative of a Killing vector is

For an affine parameter $\lambda$ on a geodesic, its tangent obeys $V^b{\nabla }_b{V}^a=0$. ThusThe second term vanishes by the geodesic equation; the first contracts a symmetric product $V^a{V}^b$ with the antisymmetric derivative from the Killing equation. This establishes the geodesic conserved quantity from a Killing vector:

With $u=t-z$, the quadratic geodesic Lagrangian isIts transverse Euler-Lagrange equations areThe longitudinal equations areafter using the difference of those same equations. In particular,This conserved quantity also follows from the Killing vector field ${\partial }_t+{\partial }_z$ and the geodesic conserved quantity from a Killing vector.

The angular Euler-Lagrange equations are homogeneous in $\dot{\theta }$ and $\dot{\varphi }$. Therefore initial data with both angular velocities zero give the unique solution with constant $\theta$ and $\varphi$, so a radial geodesic remains radial.

Time-translation symmetry supplies the geodesic conserved quantity from a Killing vectorMetric compatibility makes the squared tangent norm another constant,For a proper-time parametrized timelike geodesic $k=-1$, for a null geodesic $k=0$, and for a unit-speed spacelike geodesic $k=1$. The constant $e$ is the conserved energy per unit mass associated with the static Killing vector ${\partial }_t$.

Write $f(r)=1-r_{+}^2/r^2$ and let a dot denote differentiation with respect to an affine parameter $\tau$. The time-translation and $z$-translation Killing vector fields, together with rotational symmetry of the unit $S^3$, give the conserved quantitiesDefine $\epsilon =-g_{ab}{\dot{x}}^a{\dot{x}}^b$, so $\epsilon =1,0,-1$ for timelike, null and spacelike geodesics respectively. The normalization equation becomesIt therefore has the effective potential form

A Killing vector field $K$ obeys ${\mathcal{L}}_{K}g=0$, or equivalently ${\nabla }_{(a}{K}_{b)}=0$. Along an affinely parametrized geodesic with tangent $U$,The second term vanishes by the geodesic equation, while the first contracts the symmetric tensor $U^a{U}^b$ with the antisymmetric part selected by the Killing equation. Thus $g(K,U)$ is a geodesic conserved quantity from a Killing vector. In Minkowski spacetime, translational Killing fields give conserved energy-momentum and rotational or boost Killing fields give the corresponding angular-momentum and boost charges; the same construction works for any spacetime isometry.