Anti-de-sitter room

The anti-de-sitter space ( AdS ) is a space-time with a negative cosmological constant . The name was given as a counterpart ( positive cosmological constant) to the De Sitter space , named after the Dutch astronomer Willem de Sitter . The form of energy in the anti-de-sitter room is attractive in contrast to the de-sitter room . This corresponds to the usual effect of gravity.

The anti-de-sitter space is similar to the hyperbolic space plus a temporal dimension. In contrast to our universe , an anti-de-sitter space can neither expand nor contract - it looks the same at all times.

Despite this difference, anti-de-sitter space is proving to be very useful in finding quantum theories of spacetime and gravity . Thus the universe in the Randall-Sundrum models is a five- dimensional anti-de-sitter space.

definition

2-dimensional anti-de-sitter room. The time increases in azimuthal direction.

The -dimensional anti-de-sitter space is definable as the hyperboloid ${\displaystyle d}$

${\displaystyle u^{2}+v^{2}=\left(a^{2}+\mathbf {x} ^{2}\right)/c^{2},}$

with line element . Here is a -dimensional isotropic spatial vector, is a constant, is the speed of light . For constant the anti-de-sitter space consists of a circle parallel to the plane, i.e. H. time runs in a circle around the hyperboloid. ${\displaystyle {d\tau }^{2}=du^{2}+dv^{2}-d\mathbf {x} ^{2}/c^{2}}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle d-1}$ ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle u-v}$ ${\displaystyle S^{1}}$

The topology of the AdS room is . The fact that time is cyclical is not a problem, as one can also use its universal overlay instead of the hyperboloid . ${\displaystyle S^{1}\times \mathbb {R} ^{d-1}}$

The intrinsic structure of the AdS space is better expressed by so-called global coordinates , i.e. H. ' For the sake of simplicity is set here . The time variable is initially -periodic. The line element becomes ${\displaystyle (t=a\arctan \left(v/u\right),r=\mathbf {\left|x\right|} ,{\hat {\mathbf {\Omega } }}_{d-2}=\mathbf {x} /r)}$ ${\displaystyle u={\sqrt {a^{2}+r^{2}}}\cos \left(t/a\right)}$ ${\displaystyle v={\sqrt {a^{2}+r^{2}}}\sin \left(t/a\right).}$ ${\displaystyle c=1}$ ${\displaystyle t}$ ${\displaystyle 2\pi a}$

${\displaystyle d\tau ^{2}=\left(1+r^{2}/a^{2}\right)dt^{2}-\left(1+r^{2}/a^{2}\right)^{-1}dr^{2}-r^{2}d\Omega _{d-2}^{2},}$

where the line element is the -dimensional unit sphere . ${\displaystyle d\Omega _{d-2}^{2}}$ ${\displaystyle d-2}$

Penrose diagram of the anti-de-sitter space. The radial direction is conformally compacted to a finite interval. All geodesics are periodic in time. Massless particles move in a diagonal direction and reach the edge twice in one period.

The causal relationship of the AdS space can be seen from his Penrose diagram . This is created from the global coordinates with the help of . The coordinate compacts the radial direction on the interval , the line element is given a Minkowski shape except for a conforming factor. For applications, it is important that the edge of AdS is time-like, and thus can be used as a stage for a -dimensional physical model. Exchange of information with the infinitely distant AdS edge is possible in a finite time, the edge is therefore also physically important. ${\displaystyle r=\tan(\rho )}$ ${\displaystyle \rho }$ ${\displaystyle |\rho |\leq \pi /2}$ ${\displaystyle d-1}$

Poincaré coordinates

This type of intrinsic coordinates describes only half of the AdS space, but has a particularly simple shape for the line element. Be it . The embedding of half of the -dimensional AdS space in the -dimensional space is then given by ${\displaystyle \left\{X^{1},...,X^{d}\right\}}$ ${\displaystyle \mathbf {X} _{\left(d-2\right)}=\left\{X^{1},...,X^{d-2}\right\}}$ ${\displaystyle d}$ ${\displaystyle d+1}$ ${\displaystyle X^{d-1}>0}$

${\displaystyle {\begin{aligned}\mathbf {x} &={\frac {1}{X^{d-1}}}\left\{a\mathbf {X} _{\left(d-2\right)},\;{\frac {1}{2}}\left(\mathbf {X} _{\left(d-2\right)}^{2}+\left(X^{d-1}\right)^{2}-\left(X^{d}\right)^{2}-a^{2}\right)\right\},\\u&=aX^{d}/X^{d-1},\\v&={\frac {1}{2X^{d-1}}}\left(\mathbf {X} _{\left(d-2\right)}^{2}+\left(X^{d-1}\right)^{2}-\left(X^{d}\right)^{2}+a^{2}\right).\end{aligned}}}$

The line element

${\displaystyle d\tau ^{2}={\frac {a^{2}}{\left(X^{d-1}\right)^{2}}}\left(\left(dX^{d}\right)^{2}-d\mathbf {X} _{\left(d-1\right)}^{2}\right)}$

is conformally equivalent to the Minkowski line element. For constant time variables it is a hyperbolic space , described by Poincaré half-space coordinates . ${\displaystyle X^{d}}$

The special physical properties

If you are floating somewhere freely in such a space, you have the impression that you are at the bottom of a gravitational potential : every object that you hurl away returns like a boomerang . It is even more surprising that the time to return does not depend on the force of the throw: the object will move further away on its round trip the more momentum you give it, but the return time always remains the same. If you send out a flash of light that consists of photons with the maximum possible speed, it moves away infinitely far and still returns in a finite time. The reason for this strange phenomenon is a kind of time contraction that increases with distance from the observer.

Geometric properties

The -dimensional anti-de-sitter space is a Lorentz manifold of constant negative section curvature . ${\displaystyle d}$

Its isometric group is . ${\displaystyle PO(d-1,2)}$

Its edge at infinity can be identified with for . ${\displaystyle \left\{\left[x\right]\in \mathbb {R} P^{d}\colon \langle x,x\rangle =0\right\}}$ ${\displaystyle \langle x,x\rangle =x_{1}^{2}+\ldots +x_{d-1}^{2}-x_{d}^{2}-x_{d+1}^{2}}$

The anti-de-sitter room and the holographic theory

If we approximate hyperbolic space by stacked disks, then the anti-de-sitter spacetime is like a stack of these disks that form a cylinder . Time passes along the cylinder axis.

The easiest way is to initially imagine the disks as two-dimensional and their edge as a circular line. However, a hyperbolic space can have more than two dimensions. The anti-de-sitter space, which most closely resembles our spacetime with its three spatial dimensions, creates a three-dimensional projection of these "disks" as a cross-section of the four-dimensional cylinder.

In the four-dimensional anti-de-sitter space, the boundary of space - in relation to the universe - is a very large spherical surface at any point in time. The hologram of the holographic theory lies on this limit . This corresponds to the idea that a quantum gravity theory inside such a space is equivalent to an ordinary quantum field theory of point particles that is valid on the edge. If this is the case, one can use a relatively manageable “quantum particle theory” to define a hypothetical quantum gravity theory about which we know practically nothing.

See also

• De-sitter room
• De-Sitter model

literature

• Ugo Moschella: The de Sitter and anti-de Sitter Sightseeing Tour. In: Thibault Damour (ed.): Einstein, 1905–2005 - Poincaré Seminar 2005. Birkhäuser, Basel 2005, ISBN 978-3-7643-7435-8 , pp. 120-134.
• Birgit Jovanović: Masses of anti-de sitter spacetimes. Dipl.-Arb., Techn. Univ. Vienna, 2008 ( PDF , accessed February 17, 2009; 477 kB)
• Carlos Barceló, Matt Visser: Living on the edge: cosmology on the boundary of anti-de Sitter space. In: Physics Letters B . Vol. 482, Issue 1-3, June 1, 2000, pp. 183-194, doi: 10.1016 / S0370-2693 (00) 00520-7 , arxiv : hep-th / 0004056 .

Individual proof

1. ↑ Juan Maldacena: Gravity - An Illusion . In: Spectrum of Science . March 2006, p. 40. Retrieved January 2017.