Anti-de Sitter spacetime

Anti-de Sitter spacetime is the maximally symmetric Lorentzian spacetime of constant negative curvature.

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.

Embed radius-$L$ Anti-de Sitter spacetime as $X\cdot X=-L^2$ in signature with two timelike directions. Initial tangent $V_P$ satisfies $X_P\cdot V_P=0$ and $V_P\cdot V_P=-1$. The displayed curve remains on the hyperboloid and has unit timelike tangent; its ambient acceleration is normal, proving it is a geodesic. Every such curve reaches $-X_P$ after proper time $\pi L$. On the universal cover, this is a later antipodal event, not a closed causal loop. Global time advances by $2\pi$ over a complete ambient period.

For ambient quadratic form $Q=-x_0^2-x_4^2+x_1^2+x_2^2$, associate $M=\left(\begin{array}{cc}{x}_4+x_1& x_2+x_0\\ x_2-x_0& x_4-x_1\end{array}\right)$. Its determinant is $-Q$, so the unit Anti-de Sitter spacetime hyperboloid is the special linear group of real two-by-two matrices. On its tangent space at the identity, the induced bilinear form is $\frac{1}{2}\mathrm{tr}(UV)$. The Killing form is $4\mathrm{tr}(UV)$, and left and right matrix multiplication preserve the ambient form. Consequently the induced bi-invariant pseudo-Riemannian metric is $B/8$. This model has periodic time; its causally unwrapped version is the universal covering Lie group.

The action $(A,B):M\mapsto AM{B}^{-1}$ preserves the quadratic form $-\det M$ on real two-by-two matrices. Its kernel consists of the simultaneous scalar pairs $(I,I)$ and $(-I,-I)$. The differential $(a,b):M\mapsto aM-Mb$ is injective on the two traceless Lie algebras, and both source and target have dimension six. Its image is therefore open and, since the source is connected, equals the identity component of a Lie group $S{O}_0(2,2)$. This gives a two-sheeted covering map onto the identity component, not onto the disconnected full special orthogonal group $SO(2,2)$.

For Lorentzian boundary metric $\eta$ and $z>0$, this Anti-de Sitter spacetime patch has unit curvature radius. Its orthonormal coframe has connection ${\omega }^a{}_3=-{\theta }^a$ and curvature ${\Omega }^a{}_b=-{\theta }^a\wedge {\theta }_b$, giving sectional curvature $-1$. The full local isometry algebra is $\mathfrak{so}(d-1,2)$ in spacetime dimension $d$. A patch need not support every local Killing flow for all parameter values.

For unit-radius metric $d{y}^2+e^{-2y}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$, translations conserve $p^{\mu }=e^{-2y}{\dot{x}}^{\mu }$. Timelike normalization gives $K^2=-{\eta }_{\mu \nu }{p}^{\mu }{p}^{\nu }>0$ and ${\dot{y}}^2=K^2{e}^{2y}-1$. Setting $w=e^{-y}$ yields ${\dot{w}}^2+w^2=K^2$, hence the displayed proper-time solution where the cosine is positive. Its two patch-horizon crossings are separated by proper time $\pi$, although the globally extended geodesic is complete.

The boundary of a chosen Poincare coordinates on anti-de Sitter spacetime patch inside the full spacetime is a null coordinate horizon. It is not the timelike conformal boundary, and not a curvature singularity. Timelike geodesics can cross it in finite proper time while the patch time diverges. In a signed-diameter conformal strip, the patch containing $T=\rho =0$ satisfies $|T|<\pi /2+\rho$.

The unit hyperboloid $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in ambient signature $(++-)$ has $X^0=\cosh r\cos t$, $X^1=\cosh r\sin t$, $X^2=\sinh r$. It gives the displayed metric with $r\in \mathbb{R}$ and periodic $t$ modulo $2\pi$. Its time circles are timelike, so the universal cover, with $t\in \mathbb{R}$, removes these closed timelike curves. The metric alone does not prescribe the global identification. Both versions have constant negative curvature and two spatial ends.

A geodesic of the unit embedded hyperboloid lies in the plane spanned by its initial point and tangent. The constraint $X^2=1$ forces the displayed ambient acceleration. Timelike geodesics are $X=P\cos s+V\sin s$, null ones are $X=P+sV$, and spacelike ones are $X=P\cosh s+V\sinh s$. Timelike geodesics are closed on the hyperboloid but lift to complete nonclosed curves on its universal cover. In static coordinates, conserved $E={\cosh }^{2}rdt/ds$ gives $(dr/ds{)}^2=E^2/{\cosh }^{2}r-1$ and $\sinh r=\sqrt{E^2-1}\sin (s-s_0)$. Null affine parameters obey $\lambda =\pm \sinh r/E+\text{constant}$, proving completeness at both spatial ends.

On the universal cover, $-\pi /2<\psi <\pi /2$ and $t\in \mathbb{R}$. The two conformal boundaries are timelike. Null geodesics are $t\pm \psi =\text{constant}$ and can travel from either boundary to the center in finite coordinate time, despite taking infinite physical affine parameter to reach the boundary. There is no event horizon for the complete central static observer. The spacetime is not globally hyperbolic, unlike the global conformal cylinder of two-dimensional de Sitter spacetime; well-posed field evolution requires boundary conditions as well as initial data.

Associate $(u,v,x,y)$ with $X=\left(\begin{array}{cc}u+x& y+v\\ y-v& u-x\end{array}\right)$. The ambient quadratic form is $-\det X$; hence the radius-$\ell$ anti-de Sitter quadric is $X=\ell g$ with $g\in SL(2,\mathbb{R})$. The action $X\mapsto AX{B}^{-1}$ gives $S{O}_0(2,2)\cong (SL(2,\mathbb{R})\times SL(2,\mathbb{R}))/\{(I,I),(-I,-I)\}$. Its induced bi-invariant pseudo-Riemannian metric is $g={\ell }^{2}K/8$, giving $\mathrm{Ric}=-2g/{\ell }^2$ and cosmological constant $-1/{\ell }^2$. The antipodal quotient is $PSL(2,\mathbb{R})$; unwrapping global time gives the universal covering group instead. The connected component is essential in the displayed orthogonal-group isomorphism.

In four dimensions, global static coordinates have

In static coordinates on anti-de Sitter spacetime with curvature radius $L$ and $c=1$, put $f=1+r^2/L^2$. The conserved energy per unit rest mass is $E=fdt/d\tau$. Timelike normalization for a radial geodesic gives $(dr/d\tau {)}^2=E^2-1-r^2/L^2$, hence $d^{2}r/d{\tau }^2=-r/L^2$. An outward launch from the origin at proper time ${\tau }_0$ gives the displayed harmonic solution until its first return, after proper time $\pi L$, independently of $E>1$. Beyond that return, use a signed diameter coordinate or change the radial direction to keep the areal radius nonnegative. This is a coordinate realization of Anti-de Sitter timelike geodesic refocusing.

In global static coordinates, an outward radial null geodesic takes infinite affine parameter to approach the anti-de Sitter conformal boundary but only finite static coordinate time $\pi a/2$.

In static coordinates on anti-de Sitter spacetime, write $f=1+r^2/L^2$, conserved energy per unit rest mass $E=f\dot{t}$, and conserved specific angular momentum $h=r^2\dot{\varphi }$. Timelike normalization gives ${\dot{r}}^2=E^2-V_{\mathrm{eff}}$ with $V_{\mathrm{eff}}=f(1+h^2/r^2)$. A circular geodesic at $r_0>0$ obeys $V_{\mathrm{eff}}^{\prime }=0$, so $h^2=r_0^4/L^2$ and $E=f(r_0)$. Thus $\dot{t}=1$ and $\dot{\varphi }=\pm 1/L$. Both its static-coordinate period and its own proper time period are $2\pi L$, independent of radius. Stability follows from $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8/L^2>0$. A static clock at the orbital radius measures the different period $2\pi L\sqrt{f(r_0)}$, which does depend on radius.