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The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.

Definitions of the Riemann curvature tensor can differ by an overall sign, independent of the metric signature. Reversing that definition reverses the Ricci tensor, Ricci scalar and Einstein tensor, as well as the curvature side of the geodesic deviation identity and the Palatini identity. A derivation must choose the contraction and derivative ordering explicitly before combining formulas from different conventions. Changing a metric signature alone is a separate operation.

For $D>2$, rescale the string-frame metric by $g^{(E)}=e^{-4\phi /(D-2)}{g}^{(s)}$. The Weyl transformation of the Ricci scalar makes the coefficient of the Einstein-frame Ricci term independent of the dilaton. Indeed $g^{(s)}=e^{4\phi /(D-2)}{g}^{(E)}$ contributes factors $e^{2D\phi /(D-2)}$ from the volume, $e^{-2\phi }$ from the original coupling and $e^{-4\phi /(D-2)}$ from the Ricci term; their product is one. Derivative terms supply the transformed dilaton kinetic term. The two-dimensional case is excluded from this particular rescaling.

The Einstein-Hilbert action is the integral of the Ricci scalar against the metric volume form. Varying it with respect to the metric gives the vacuum Einstein field equations, up to a boundary term.

Every four-dimensional Einstein metric with $R_{ab}=\Lambda g_{ab}$ has constant Ricci scalar $R=4\Lambda$. The vacuum metric f(R) gravity tensor reduces to $(\Lambda f^{\prime }(4\Lambda )-f(4\Lambda )/2)g_{ab}$, giving the displayed necessary and sufficient condition for a fixed cosmological constant. Both function and derivative may vanish, so division by $f^{\prime }$ is unnecessary. Requiring this for all real $\Lambda$ and a smooth $f$ instead yields $f(R)=C{R}^2$.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, negative extrinsic curvature, and $K=-3H+\kappa$, the Hamiltonian constraint gives the displayed scalar perturbation equation. Indeed $K^2=9H^2-6H\kappa$ and $K^i{}_j{K}^j{}_i=3H^2-2H\kappa$ to first order. The spatial Ricci scalar is $4\Delta \Phi$; subtract the background equation $6H^2=16\pi G\bar{\rho }$ and divide by four.

For $h_{ab}=\delta g_{ab}$, the Ricci scalar varies asThe last two terms form a covariant divergence and hence contribute only a boundary term to a compactly supported variation of the Einstein-Hilbert action.

An FLRW metric with $p=-{\varrho }_0$ and constant negative energy density ${\varrho }_0$ has ${\alpha }^2=-8\pi G{\varrho }_0/3>0$ and obeys ${\dot{a}}^2+k=-{\alpha }^2{a}^2$. Positive solutions exist only for $k=-1$. They are $a={\alpha }^{-1}\sin [\alpha (t-t_B)]$ on $0<\alpha (t-t_B)<\pi$. This is an open slicing of Anti-de Sitter spacetime, whose Ricci scalar is $-12{\alpha }^2$ and Kretschmann scalar is $24{\alpha }^4$. The zeros mark boundaries of this slicing rather than curvature singularities.

A pressureless relativistic fluid has stress-energy tensor $T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }$, with four-velocity normalized by $u^{\mu }{u}_{\mu }=-c^2$. In its rest frame, $u^i=0$, so $\boxed{T^0{}_0=-\rho c^2,T^i{}_j=0}$. Here $\rho$ is rest-frame mass density, or energy density divided by $c^2$.

For the linearized Ricci tensor and scalar, discard the quadratic connection terms. The contracted connection is ${\Gamma }^{\mu }{}_{\mu \nu }={\partial }_{\nu }(\Phi -3\Psi )/c^2$. Differentiation then givesThe Ricci scalar uses the zeroth-order inverse metric, $R=-R_{00}+{\delta }^{ij}{R}_{ij}$, because multiplying first-order curvature by another metric perturbation would again be second order. These are the linearized curvature of a two-potential weak-field metric in the paper's Riemann curvature tensor sign convention.

A semicolon denotes the covariant derivative for the Levi-Civita connection. For example, the four lower indices of the Riemann curvature tensor contribute four connection terms:Here ${\partial }_{\eta }=\partial /\partial x^{\eta }$ is a partial derivative, ${\Gamma }_{b\eta }^a$ are Christoffel symbols, and repeated indices are summed from $0$ to $3$. Write $g_{\mu \nu }$ for the metric tensor, $g^{\mu \nu }$ for its inverse metric, $R_{\mu \nu }=g^{\lambda {\nu }^{\prime }}{R}_{\lambda \mu {\nu }^{\prime }\nu }$ for the Ricci tensor, and $R=g^{\mu \nu }{R}_{\mu \nu }$ for the Ricci scalar. This tensor contraction agrees with the paper's $R_{\mu \nu }=-R^a{}_{\mu \nu a}$.

By metric compatibility, the inverse metric may pass through each covariant derivative. Contracting the second Bianchi identity with $g^{\lambda \nu }$ giveswhere ${\nabla }_{\eta }$ is the same covariant derivative as the semicolon. Next contract with $g^{\mu \kappa }$. Antisymmetry of the first pair gives $g^{\mu \kappa }{R}_{\lambda \mu \kappa \eta }=-R_{\lambda \eta }$, soThus the contracted Bianchi identity is ${\nabla }^{\mu }{R}_{\mu \eta }=\frac{1}{2}{\partial }_{\eta }R$. Raising the remaining index and using metric compatibility provesThe Einstein tensor $G^{\mu \nu }$ therefore has identically zero covariant divergence, without imposing any field equation. This makes the Einstein field equations compatible with stress-energy conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, where $T^{\mu \nu }$ is the stress-energy tensor. The connection terms describe how energy and momentum are compared at neighboring points; ordinary componentwise partial derivatives would not give a coordinate-independent conservation law in curved spacetime.

For the equation of state $p=-\varrho$, the cosmological continuity equation gives $a\dot{\varrho }=0$. Hence $\varrho ={\varrho }_0$ is constant. First take the physically intended positive energy density and defineThe Friedmann equation and the original Friedmann acceleration equation becomeSolve the second equation as $a=A{e}^{H_{\Lambda }t}+B{e}^{-H_{\Lambda }t}$. The first equation then requires $4H_{\Lambda }^{2}AB=k$. Choosing the time origin separately in each solution gives all positive branches:For $k=-1$, there is also the time reverse $a=H_{\Lambda }^{-1}\sinh [H_{\Lambda }(t_B-t)]$ on $t<t_B$, which contracts to a future zero. For $k=+1$, the cosmological bounce joins contraction to expansion at the positive minimum $a(t_b)=H_{\Lambda }^{-1}$. Retaining $\ddot{a}=H_{\Lambda }^{2}a$ at the minimum excludes a spurious permanently static solution that would satisfy the first-order constraint alone.

These are vacuum-energy FLRW solutions: the expanding flat branch, closed slicing of de Sitter spacetime, and open slicing of de Sitter spacetime. To compare them at the same energy density, the figure sets $T=H_{\Lambda }(t-t_c)$, takes $A_0=H_{\Lambda }^{-1}$, and aligns the open boundary and closed minimum at $T=0$. Thus the three plotted functions are $H_{\Lambda }a=e^T$, $\cosh T$, and $\sinh T$, the last only for $T>0$.

Only the expanding $k=-1$ branch has a finite past zero. The expanding $k=0$ branch tends to zero only as $t\to -\infty$, and the $k=+1$ branch is bounded below. All three describe portions of de Sitter spacetime, with Ricci scalar $R=12H_{\Lambda }^2$ and Kretschmann scalar $24H_{\Lambda }^4$. In particular the $k=-1$ zero is a boundary of that slicing, not a curvature singularity. The inequality in part (d) does not apply, because $\varrho +3p=-2{\varrho }_0<0$.

For zero energy density, the remaining solutions are $a=A_0>0$ for $k=0$, and the expanding or contracting Milne model $a=\pm (t-t_B)>0$ for $k=-1$; no positive real solution exists for $k=+1$. If negative energy density is admitted mathematically, write ${\alpha }^2=-8\pi G{\varrho }_0/3>0$. Only $k=-1$ is possible, with $a={\alpha }^{-1}\sin [\alpha (t-t_B)]$ for $0<\alpha (t-t_B)<\pi$, an open slicing of anti-de Sitter spacetime. These limiting and negative-density cases do not alter the positive-density comparison above.

In the order $(u,v,x,y)$ the metric tensor and inverse metric areOnly $g_{uu}$ has nonzero partial derivatives. Substitution into the formula for the Levi-Civita connection,therefore givesEvery other Christoffel symbol vanishes. For example $g^{ud}$ selects $d=v$, and the metric tensor is independent of $v$ with constant $g_{av}$; hence ${\Gamma }^u{}_{bc}=0$.

Use the paper's Riemann curvature tensor conventionFor the independent components,The quadratic Christoffel symbol terms vanish here: any potentially contributing intermediate $u$ index gives a factor ${\Gamma }^u{}_{ab}=0$, and the remaining possible products have a lower $v$ index and vanish. Since the symmetric matrix $M$ gives $F_{,ij}=2M_{ij}$, lowering the first index givesAll remaining nonzero components follow from the Riemann curvature tensor symmetriesThere are no other independent components: terms involving a $v$ index vanish, while nonzero terms require one $u$ in each antisymmetric pair.

The Ricci tensor is consequentlyThe Ricci scalar vanishes even when the trace is nonzero, since $g^{uu}=0$. With zero cosmological constant, the vacuum Einstein field equations reduce to $R_{ab}=0$. Therefore the necessary and sufficient condition isThis is the Ricci tensor of a Brinkmann plane wave: a nonzero, trace-free profile can have nonzero Riemann curvature tensor despite vanishing Ricci tensor.

The Brans-Dicke theory couples a scalar field $\Phi$ to the Ricci scalar. Assume $\Phi \ne 0$, constant $\omega$, matter independent of $\Phi$, and variations of compact support so boundary terms may be discarded. Use the paper's curvature sign convention and signature $(+---)$. Set $X=g^{ab}{\partial }_a\Phi {\partial }_b\Phi$ and $\square \Phi ={\nabla }_a{\nabla }^a\Phi$. Define the stress-energy tensor consistently byThis convention fixes the sign of the matter term.

Vary with respect to $g^{ab}$. The variation of metric volume density contributes $-g_{ab}/2$ times each scalar Lagrangian, while $\delta R=R_{ab}\delta g^{ab}+g^{ab}\delta R_{ab}$. Applying the Palatini identity followed by two rounds of integration by parts gives the metric variation of a scalar-curvature coupling,The differentiated $\Phi$ terms remain because the curvature multiplier is not constant. Varying the kinetic term at fixed $\Phi$ givesCombining these terms with the matter variation, the Euler-Lagrange equation isRaising both free indices reproduces the contravariant form, because ${\nabla }_a{g}_{bc}=0$.

The independent scalar variation givesor $R+\omega {\Phi }^{-2}X-2\omega {\Phi }^{-1}\square \Phi =0$. The trace of the metric tensor equation in four dimensions isMultiplying the scalar equation by $\Phi$ eliminates the same combination $\Phi R+\omega {\Phi }^{-1}X$. Thus the scalar equation of Brans-Dicke theory isAt the degenerate coupling of Brans-Dicke theory, $\omega =-3/2$, the combined equations impose $T=0$ instead; division by $3+2\omega$ is unavailable. If matter depended directly on $\Phi$, its scalar variation would add a source and the displayed scalar equation would change.

The following lettered sections justify the supplied variation identities; they are supporting identities in the PDF, rather than three independent field-equation problems.

The local metric tensors do not determine the global coordinate identifications. The usual complete two-dimensional de Sitter spacetime is the hyperboloid $(X^0{)}^2-(X^1{)}^2-(X^2{)}^2=-1$ in ambient signature $(+--)$, withPulling back the ambient metric tensor gives $d{t}^2-{\cosh }^{2}td{\chi }^2$, withUnwrapping $\chi$ gives the universal cover of two-dimensional de Sitter spacetime instead. The usual embedded two-dimensional anti-de Sitter spacetime has $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in signature $(++-)$ andHere $r\in \mathbb{R}$ and $t$ is initially periodic modulo $2\pi$. The time circles are closed timelike curves. Its physically standard universal cover removes that identification:The two signs of $r$ describe two spatial ends; imposing $r\ge 0$ would retain only half of this complete model.

Both spacetimes have constant curvature and are geodesically complete. In the paper's convention their Ricci scalars are respectively $+2$ and $-2$. Their embeddings show that geodesics are intersections with planes through the ambient origin. They nevertheless have very different causal structures.

For the geodesics of two-dimensional de Sitter spacetime, use an affine parameter $\lambda$, normalization $\kappa ={\dot{t}}^2-{\cosh }^{2}t{\dot{\chi }}^2\in \{1,0,-1\}$ and conserved momentum $L={\cosh }^{2}t\dot{\chi }$. ThenTimelike geodesics extend to both infinite proper times; $\chi =\text{constant}$ is a simple example. Null geodesics obey $\sinh t=\pm |L|(\lambda -{\lambda }_0)$, so their affine parameter is also infinite at either temporal end. Spacelike geodesics have $|L|\ge 1$, obey $\sinh t=\sqrt{L^2-1}\sin (\lambda -{\lambda }_0)$ and are closed curves on the embedded cylinder; $t=0$ is its simplest spacelike geodesic circle. Unwrapping the spatial circle removes closure but not completeness.

For the geodesics of two-dimensional anti-de Sitter spacetime, the timelike Killing vector ${\partial }_t$ gives $E={\cosh }^{2}r\dot{t}$. The normalization is $\kappa ={\cosh }^{2}r{\dot{t}}^2-{\dot{r}}^2$, henceChoose a future-directed timelike geodesic, so $E\ge 1$ andFor $E>1$, it oscillates radially with proper period $2\pi$; $E=1$ gives the central geodesic $r=0$. In both cases, global time advances by $2\pi$ over a proper time interval $2\pi$. The curve is closed on the original hyperboloid, while its lift on the universal cover is nonclosed and future-directed. Null geodesics have $\sinh r=\pm E(\lambda -{\lambda }_0)$ and reach either spatial end only at infinite affine parameter. Spacelike geodesics satisfy $\sinh r=\sqrt{E^2+1}\sinh (\lambda -{\lambda }_0)$ and also have infinite proper length toward either end. Thus the finite coordinate time discussed below does not imply physical geodesic incompleteness.

The conformal cylinder of two-dimensional de Sitter spacetime follows by setting $\eta =\arctan (\sinh t)$:Its past and future conformal boundaries are spacelike circles. Null geodesics have $d\chi /d\eta =\pm 1$ and can travel only a finite angular distance over the entire infinite proper-time history. For the complete observer $\chi =0$, an event can send a signal to that observer precisely when its shortest angular distance $d(\chi ,0)$ is less than $\pi /2-\eta$. The observer horizons in two-dimensional de Sitter spacetime are therefore the null curves $d(\chi ,0)=\pi /2-\eta$; its past signal horizon has $d(\chi ,0)=\eta +\pi /2$. These are observer horizons, without curvature singularities. Their intersection bounds a static patch with metric tensorThe horizons at $\rho =\pm 1$ are regular Killing horizons beyond which this static chart fails, while the global coordinates remain smooth.

For the conformal strip of two-dimensional anti-de Sitter spacetime, set $\psi =\arctan (\sinh r)$. On its universal cover,The two conformal boundaries are timelike. A null ray from the center approaches a boundary after coordinate time $\pi /2$, though its affine parameter diverges. Every interior event can signal to the complete central static observer in finite global time, so that observer has no event horizon. A Poincaré horizon concerns a restricted coordinate patch, and accelerated observers can have different causal horizons.

Finally, the global de Sitter cylinder and its spatial cover are globally hyperbolic: constant-time slices are Cauchy hypersurfaces. The anti-de Sitter universal cover is not globally hyperbolic, because timelike infinity admits incoming signals within finite global time. A field's evolution therefore requires boundary conditions at its two conformal boundaries in addition to initial data. The opposite placement of the conformal boundaries explains why de Sitter observers have cosmological horizons despite global hyperbolicity, while the central anti-de Sitter observer has no event horizon despite the failure of global hyperbolicity.

The starting point is a physical massless closed-string stateIts symmetric traceless part is a spacetime spin-two field, the graviton; the antisymmetric part is the Kalb–Ramond field, and the scalar field part gives the dilaton. The physical-state conditions impose transverse polarizations, and longitudinal null states identifyfor the symmetric component. In position space this is the linearized diffeomorphism $\delta h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$. Thus the massless spin-two state has the gauge redundancy required of a gravitational field, rather than an arbitrary massive tensor interaction.

Closed-string amplitudes factorize on this graviton pole. Its leading soft coupling is universal and couples to the energy-momentum tensor of the other string states. Decoupling longitudinal polarizations enforces the gravitational Ward identities. The two-derivative cubic graviton vertex extracted from the low-momentum three-point amplitude is the cubic vertex of the Einstein-Hilbert action; continuing these Ward identities requires the nonlinear diffeomorphism-invariant completion. A complementary way to derive the same dynamics is to demand quantum consistency of propagation in a curved background.

Replace the flat target metric in the worldsheet action by $G_{\mu \nu }(X)$, and allow the two-form and dilaton backgrounds. After Euclidean continuation, their couplings have the schematic normalizationThese backgrounds are couplings of a string nonlinear sigma model. Quantum Weyl invariance requires their Weyl anomaly coefficients to vanish. To first order in ${\alpha }^{\prime }$, the metric equation isThe dilaton and two-form equations accompany it. In a critical-dimensional background with $H=0$ and constant $\Phi$, the leading metric equation is $R_{\mu \nu }=0$, precisely the vacuum Einstein field equations. Matter and other massless string fields supply the corresponding stress-energy when retained. This argument also explains why arbitrary curved target metrics are not consistent string backgrounds.

The equations arise from the leading string-frame massless effective actionA constant-dilaton expansion of its curvature term directly gives the Einstein-Hilbert kinetic and self-interaction terms. For a varying dilaton, introduce the Einstein-frame metric $G_{\mu \nu }^E=e^{-4\Phi /(D-2)}{G}_{\mu \nu }$. The Weyl rescaling of the metric and the corresponding transformation of the Ricci scalar and an integration by parts giveThis displays ordinary Einstein gravity coupled to the dilaton and two-form. Gravity need not be the only low-energy field: those fields, compactification moduli and gauge fields have to be retained or stabilized as appropriate. Four-dimensional gravity additionally requires compactification from the critical target dimension, with its Newton constant depending on the compactification volume.

The approximation is controlled when external energies obey $E\sqrt{{\alpha }^{\prime }}\ll 1$ and curvatures obey ${\alpha }^{\prime }|R_{\mu \nu \rho \sigma }|\ll 1$. Massive string modes then produce suppressed higher-derivative interactions, such as higher-curvature corrections. Small string coupling $g_s=e^{{\Phi }_0}$ also suppresses higher-genus quantum corrections; dimensionally $G_D$ scales as $g_s^2{\alpha }^{\prime (D-2)/2}$. Thus Einstein gravity is the universal leading two-derivative dynamics of the massless closed-string spin-two field, with additional massless matter, string-length corrections and quantum corrections systematically controlled.

For the bosonic theory this is the massless-sector effective description in 26 dimensions; the tachyon means that it is not by itself a proof of a stable vacuum. Tachyon-free supersymmetric string theories have the same massless graviton mechanism in ten dimensions before compactification. Removing the tachyon and obtaining a realistic four-dimensional model require extra physical input beyond the emergence of the Einstein action.

Take both curvature radii to be one. Coordinate ranges depend on the global identification, which a line element alone cannot fix. For the ordinary two-dimensional de Sitter spacetime,This is the unit hyperboloid $X_0^2-X_1^2-X_2^2=-1$ in ambient signature $(+--)$, parametrized by $X_0=\sinh t$, $X_1=\cosh t\cos \chi$, $X_2=\cosh t\sin \chi$. Unwrapping $\chi$ instead gives its universal cover.

For two-dimensional anti-de Sitter spacetime, the embedding $X_0^2+X_1^2-X_2^2=1$ in ambient signature $(++-)$ is parametrized by $X_0=\cosh r\cos t$, $X_1=\cosh r\sin t$, $X_2=\sinh r$. On this hyperboloid $r\in \mathbb{R}$ and $t$ is periodic modulo $2\pi$, creating closed timelike circles. The physically usual universal cover removes that periodicity:Both signs of $r$ are needed for the full two-dimensional spatial line. Taking $r\ge 0$ without an additional construction would omit one of its two ends. The following causal comparison uses the AdS universal cover.

Both models are maximally symmetric Lorentzian manifolds, with constant sectional curvature and no curvature singularities. In the paper's convention their Ricci scalars are respectively $+2$ and $-2$. The de Sitter spatial scale factor $a(t)=\cosh t$ contracts to a nonzero minimum and then expands; $t=0$ is not a big-bang singularity. The AdS metric is static, and ${\partial }_t$ is timelike everywhere because $g_{tt}={\cosh }^{2}r>0$.

The embeddings give a concise classification of all affinely parametrized geodesics. Geodesic acceleration in the ambient space is normal to the hyperboloid, hence proportional to $X$. Differentiating the constraint and setting $X^{\prime 2}=\kappa$, with $\kappa =1,0,-1$ for timelike, null and spacelike curves, givesEvery solution stays in the two-plane spanned by its initial point $P$ and tangent $V$, so every geodesic is a plane section through the ambient origin. For de Sitter, these are $P\cosh s+V\sinh s$ for timelike curves, $P+sV$ for null curves, and $P\cos s+V\sin s$ for spacelike curves. Thus timelike geodesics and null geodesics are complete and nonclosed, while spacelike geodesics are closed circles on the ordinary hyperboloid. These are the geodesics of two-dimensional de Sitter spacetime.

For AdS the roles of the circular and hyperbolic solutions are reversed: timelike curves are $P\cos s+V\sin s$, null curves are $P+sV$, and spacelike curves are $P\cosh s+V\sinh s$. Timelike curves close after proper time $2\pi$ on the hyperboloid, but their lifts on the universal cover continue to ever later time rather than forming closed curves. All three kinds extend for arbitrary affine parameter. This is geodesics of two-dimensional anti-de Sitter spacetime, not an assertion that geodesics reach the conformal boundary in finite physical affine length.

The coordinate first integrals make the behavior concrete. In de Sitter, spatial rotational symmetry gives a conserved $P_{\chi }={\cosh }^{2}td\chi /ds$, and normalization givesConstant-$\chi$ observers are timelike geodesics. Nontrivial null geodesics have affine parameter proportional to $\sinh t$, which diverges at both $t\to \pm \infty$. In AdS the conserved static energy is $E={\cosh }^{2}rdt/ds$, withFor timelike curves, $E\ge 1$ andThey oscillate through the center and never reach spatial infinity; $r=0$ is the $E=1$ geodesic. For null curves, $\lambda =\pm \sinh r/E+\text{constant}$, so both ends are at infinite affine parameter. Spacelike AdS geodesics likewise extend to the two spatial ends.

For the conformal structure, introduce de Sitter conformal timeMultiplying by ${\cos }^2\eta$ gives the conformal cylinder of two-dimensional de Sitter spacetime. The past and future conformal infinities are spacelike circles at $\eta =\mp \pi /2$. Null curves are $\chi =\pm \eta +\text{constant}$ modulo $2\pi$. Although the conformal-time interval is finite, their physical affine parameters and timelike observers' proper times are infinite at its ends. Constant-$\eta$ circles are Cauchy hypersurfaces; global de Sitter is globally hyperbolic.

For AdS putThe conformal strip of two-dimensional anti-de Sitter spacetime has unbounded time and timelike conformal boundaries at $\psi =\pm \pi /2$. Null curves have $t\pm \psi =\text{constant}$. A signal from any finite $r$ reaches the center in coordinate time $|\psi |<\pi /2$, and a null curve crosses the entire conformal strip in time $\pi$. This finite coordinate travel time coexists with infinite physical affine length to the boundary. The covering spacetime has no closed timelike curves, but is not globally hyperbolic: causal curves can arrive from timelike infinity without meeting a proposed initial slice. Field evolution consequently requires boundary conditions as well as initial data. Unwrapping time removes closed timelike curves, not the timelike boundary.

The contrast in observer event horizons follows directly from these null curves. For the complete de Sitter observer $\chi =0$, let $d(\chi ,0)\in [0,\pi ]$ be shortest angular distance. An event can send a signal to this observer before its future endpoint precisely whenEquality is its future observer event horizon. It can receive a signal emitted by the observer after its past endpoint precisely when $d(\chi ,0)<\eta +\pi /2$, whose equality is the past horizon. In the observer's fundamental domain these horizons are null linesTheir intersection encloses the observer's static patch of de Sitter spacetime, the diamond $d(\chi ,0)<\pi /2-|\eta |$. They are observer-dependent cosmological horizons, not curvature singularities or Cauchy horizons. Every inertial observer has the corresponding horizons by de Sitter symmetry.

For the complete AdS static observer at $r=0$, every event at finite $r$ can send a signal to the observer, and can receive one, because time is unbounded and the coordinate distance $|\psi |$ is finite. Thus there is no corresponding global static-observer event horizon. The lapse never vanishes in these coordinates. Restricted accelerated-observer patches can have observer horizons, but those are distinct from the global static model. On the original periodic-time AdS hyperboloid, the more serious issue is closed timelike curves.

The figure identifies the de Sitter spatial edges and displays only a finite time window of the AdS covering strip, whose time continues indefinitely. Solid red curves are null geodesics; the oscillating AdS curve is a timelike geodesic. The shaded de Sitter diamond is the static observer's two-way communication region, not the entire spacetime.

The concise comparison is de Sitter has spacelike conformal infinities and cosmological observer horizons; covering AdS has timelike conformal boundaries and no horizon for an eternal global static observer. Both are constant-curvature and geodesically complete, but their geodesic recurrence, global topology, causal boundaries and initial-value properties differ.

Work on a regular patch with a smooth nonzero conformal factor. Its sign is constant there, so replace it by $|\Omega |$ if necessary and assume $\Omega >0$. Put $u=\log \Omega$, and let all partial derivatives, index raising on $u_i,{\Omega }_i$, and the operators $\Delta ,|\nabla \cdot {|}^2$ in the following formulas use the flat Cartesian metric $\delta$. The orthonormal coframe $E^i=\Omega d{x}^i$ hasThe proposed connection 1-forms reduce toLowering the first frame index with $\delta$ makes these antisymmetric, which establishes metric compatibility. Moreover,Thus Cartan's first structure equation has zero torsion. Uniqueness of the metric-compatible torsion-free connection identifies these forms as the Levi-Civita connection; this also explains why merely guessing a skew connection would not have sufficed.

To calculate the Riemann curvature tensor, use Cartan's second structure equation. Direct differentiation and multiplication giveIntroduce the Schouten tensor in Cartesian coordinate components,The preceding calculation groups into the curvature 2-formsKeep the convention $R_{ijkl}=g({\partial }_i,R({\partial }_k,{\partial }_l){\partial }_j)$, with $R(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}$, as in the first solution. Converting frame indices to coordinate indices gives the curvature of a conformally Euclidean metric:These are coordinate components. Components in the dual orthonormal frame are ${\Omega }^{-4}{R}_{ijkl}$, hence the same expression with the overall factor ${\Omega }^{-2}$. This distinction is important when contracting the Riemann curvature tensor.

Contract with $g^{ik}={\Omega }^{-2}{\delta }^{ik}$. In four dimensions the Ricci tensor becomes $R_{jl}=2S_{jl}+{\delta }_{jl}{\sum }_k{S}_{kk}$, and thereforeTaking the remaining metric trace yields the Ricci scalarThe cancellation of the gradient-squared terms in the last expression is special to dimension four.

For the Tolman wormhole, compute on $r>0$:Substitution into the Ricci tensor and Ricci scalar givesFor example, the coordinate radial eigenvalue is $-12\lambda /(r^2+\lambda {)}^2$, while each of the three tangential eigenvalues is $4\lambda /(r^2+\lambda {)}^2$. Their sum vanishes, but none vanishes when $\lambda \ne 0$. The Vacuum Einstein equations with zero cosmological constant require $R_{ij}-\frac{1}{2}R{g}_{ij}=0$ and hence $R_{ij}=0$. The metric is scalar-flat but not a vacuum solution for any nonzero $\lambda$; only $\lambda =0$ gives flat vacuum space. A nonzero cosmological constant does not rescue it: its trace would force $R=4\Lambda$, so $R=0$ would first force $\Lambda =0$.

The smooth wormhole interpretation uses $\lambda >0$; $r=0$ is another asymptotically flat end, not an included origin, as inversion $\rho =\lambda /r$ shows. For $\lambda <0$, the conformal factor vanishes at $r=\sqrt{-\lambda }$, where the metric is degenerate. The local non-vacuum calculation remains valid on each regular region away from that sphere, but such a metric is not a smooth wormhole across it.

The gravitational variable and the fields held fixed in a variation must be specified. For bosonic matter a second-order metric tensor formulation is sufficient. For spinorial matter a vierbein and a spin connection are needed to define the local Lorentz representation and its covariant derivative. “Second order” refers to a connection already expressed through derivatives of the vierbein; “first order” treats that connection as an independent variable.

For a metric tensor, a scalar field and an Abelian gauge field, consider the Einstein-Hilbert action with matter,Here $e=\sqrt{-g}$ and the connection in $R$ is Levi-Civita. The metric tensor variation uses $\delta e=-e{g}_{\mu \nu }\delta g^{\mu \nu }/2$ andMetric compatibility makes the last two terms a divergence. ThusStationarity for arbitrary compactly supported variations givesExplicitly,Scalar field and gauge-potential variations give ${\nabla }^2\varphi -V^{\prime }=0$ and ${\nabla }_{\mu }{F}^{\mu \nu }=0$. Their equations imply stress-energy conservation, consistently with the contracted Bianchi identity. If the boundary metric tensor is fixed instead of using compactly supported variations, add the Gibbons–Hawking–York boundary term, ${\kappa }^{-2}{\int }_{\partial M}{\epsilon }_{\partial M}\sqrt{|h|}K$, with the outward-normal orientation. Fixing the metric tensor alone does not otherwise remove the normal derivatives of its variation.