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nLab > Latest Changes: anti de Sitter spacetime

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 19th 2011
- PermaLink
Author: Urs
Format: MarkdownItexearlier today I had created a stub for [[anti de Sitter spacetime]]

earlier today I had created a stub for anti de Sitter spacetime
- CommentRowNumber2.
- CommentAuthorDavid_Corfield
- CommentTimeDec 13th 2016
- PermaLink
Author: David_Corfield
Format: MarkdownItexPresumably [[super anti de Sitter spacetime]] >A supergeometric analog of de Sitter spacetime. is missing an 'anti'. Can you have plain 'super de Sitter'? If so, I guess that wants an entry. By the way, aren't I supposed to get the impression from [[anti de Sitter spacetime]] of an $(n-1, 2)$ signature, as at [[anti de Sitter group]]? >Up to [[isometry]], the **anti de Sitter spacetime** of [[dimension]] $d + 1$ is the [[pseudo-Riemannian manifold]] whose underlying [[manifold]] is the submanifold of the [[Cartesian space]] $\mathbb{R}^{d+2}$ that solves the equation > $$ \sum_{i = 1}^{d+1} (x^i)^2 - (x^{d+2})^2 = 0$$

Presumably super anti de Sitter spacetime

A supergeometric analog of de Sitter spacetime.

is missing an ’anti’.

Can you have plain ’super de Sitter’? If so, I guess that wants an entry.

By the way, aren’t I supposed to get the impression from anti de Sitter spacetime of an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1, 2)</annotation></semantics></math>$ signature, as at anti de Sitter group?

Up to isometry, the anti de Sitter spacetime of dimension $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d + 1</annotation></semantics></math>$ is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Cartesian space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d+2}</annotation></semantics></math>$ that solves the equation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \sum_{i = 1}^{d+1} (x^i)^2 - (x^{d+2})^2 = 0</annotation></semantics></math>$
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeDec 13th 2016
- PermaLink
Author: Urs
Format: MarkdownItexWoops. All fixed now. Regarding super de Sitter: not in the usual sense, but see [arXiv:1610.01566](https://arxiv.org/abs/1610.01566)

Woops. All fixed now.

Regarding super de Sitter: not in the usual sense, but see arXiv:1610.01566
- CommentRowNumber4.
- CommentAuthorDavid_Corfield
- CommentTimeDec 15th 2016
- (edited Dec 15th 2016)
- PermaLink
Author: David_Corfield
Format: MarkdownItexIs this right yet? Don't we need two coordinates of negative sign in the signature? From [here](http://www.ift.unesp.br/users/nastase/AdS-CFTnotes.pdf) (2.30) >$-x_0^2 + \sum_{i = 1}^{d-1} x^2_i - x^2_{d+1} = -R^2$ (why no $d$-coordinate?) From [here](http://www.aei.mpg.de/~gielen/ads.pdf) > an $n + 1$ dimensional flat spacetime with signature $(n - 1, 2)$, i.e. the set of points $(X_1, X_2, \ldots , X_{n+1})$ satisfying $(X_1)^2 + (X_2)^2 + \ldots + (X_{n-1})^2 - (X_n)^2 - (X_{n+1})^2 = -1$.

Is this right yet? Don’t we need two coordinates of negative sign in the signature?

From here (2.30)

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">−</mo><msubsup><mi>x</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>+</mo><msubsup><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msubsup><mi>x</mi> <mi>i</mi> <mn>2</mn></msubsup><mo>−</mo><msubsup><mi>x</mi> <mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow> <mn>2</mn></msubsup><mo>=</mo><mo lspace="0.11111em" rspace="0em">−</mo><msup><mi>R</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">-x_0^2 + \sum_{i = 1}^{d-1} x^2_i - x^2_{d+1} = -R^2</annotation></semantics></math>$

(why no $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>$ -coordinate?)

From here

an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math>$ dimensional flat spacetime with signature $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n - 1, 2)</annotation></semantics></math>$ , i.e. the set of points $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1, X_2, \ldots , X_{n+1})</annotation></semantics></math>$ satisfying $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mi>…</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(X_1)^2 + (X_2)^2 + \ldots + (X_{n-1})^2 - (X_n)^2 - (X_{n+1})^2 = -1</annotation></semantics></math>$ .
- CommentRowNumber5.
- CommentAuthorDavid_Corfield
- CommentTimeDec 16th 2016
- PermaLink
Author: David_Corfield
Format: MarkdownItexI gave what I think is the right definition, and mentioned the more general case of $(p, q)$ signature.

I gave what I think is the right definition, and mentioned the more general case of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p, q)</annotation></semantics></math>$ signature.
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeNov 15th 2018
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer: *** Discussion of thermal [[Wick rotation]] on global [[anti-de Sitter spacetime]] (which is already periodic in _real_ time) to [[Euclidean field theory]] with periodic _imaginary_ time is in * B. Allen, A. Folacci, [[Gary Gibbons]], _Anti-de Sitter space at finite temperature_, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (<a href="https://doi.org/10.1016/0370-2693(87)91437-7">doi:10.1016/0370-2693(87)91437-7</a>) <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/23">v23</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>
added pointer:

Discussion of thermal Wick rotation on global anti-de Sitter spacetime (which is already periodic in real time) to Euclidean field theory with periodic imaginary time is in
- B. Allen, A. Folacci, Gary Gibbons, Anti-de Sitter space at finite temperature, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (doi:10.1016/0370-2693(87)91437-7)
diff, v23, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeNov 20th 2019
- PermaLink
Author: Urs
Format: MarkdownItexadded a cartoon of AdS sliced by hyperbolic slices <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/27">v27</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>

added a cartoon of AdS sliced by hyperbolic slices

diff, v27, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeMay 6th 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded this pointer: * Jiri Podolsky, Ondrej Hruska, _Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes_, Phys. Rev. D 95, 124052 (2017) ([arXiv:1703.01367](https://arxiv.org/abs/1703.01367)) <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/34">v34</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>
added this pointer:
- Jiri Podolsky, Ondrej Hruska, Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes, Phys. Rev. D 95, 124052 (2017) (arXiv:1703.01367)
diff, v34, current
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeDec 22nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to the recent * Anjan A. Sen, Shahnawaz A. Adil, Somasri Sen, *Do cosmological observations allow a negative $\Lambda$?* ([arXiv:2112.10641](https://arxiv.org/abs/2112.10641)) <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/37">v37</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>
added pointer to the recent
- Anjan A. Sen, Shahnawaz A. Adil, Somasri Sen, Do cosmological observations allow a negative $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math>$ ? (arXiv:2112.10641)
diff, v37, current
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeJun 28th 2024
- (edited Jun 28th 2024)
- PermaLink
Author: Urs
Format: MarkdownItexhave re-written and expanded a little the section on coordinate charts ([here](https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime#CoordinateCharts)) (it's still minimalistic, though, not even mentioning global charts yet) <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/41">v41</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>

have re-written and expanded a little the section on coordinate charts (here)

(it’s still minimalistic, though, not even mentioning global charts yet)

diff, v41, current
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeJun 28th 2024
- PermaLink
Author: Urs
Format: MarkdownItexhave spelled out ([here](https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime#CartanGeometry)) Cartan connection and curvature tensors for AdS spacetime in Poincaré coordinates <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/41">v41</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>

have spelled out (here) Cartan connection and curvature tensors for AdS spacetime in Poincaré coordinates

diff, v41, current
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJul 1st 2024
- (edited Jul 1st 2024)
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * S. J. Avis, [[Chris J. Isham]], D. Storey: *Quantum field theory in anti-de Sitter space-time*, Phys. Rev. D **18** 3565 (1978) [[doi:10.1103/PhysRevD.18.3565](https://doi.org/10.1103/PhysRevD.18.3565)] <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/42">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/42">v42</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>
added pointer to:
- S. J. Avis, Chris J. Isham, D. Storey: Quantum field theory in anti-de Sitter space-time, Phys. Rev. D 18 3565 (1978) [doi:10.1103/PhysRevD.18.3565]
diff, v42, current
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeJul 1st 2024
- PermaLink
Author: Urs
Format: MarkdownItexand to: * Peter Breitenlohner, [[Daniel Z. Freedman]], *Positive energy in anti-de Sitter backgrounds and gauged extended supergravity*, Physics Letters B **115** 3 (1982) 197-201 [<a href="https://doi.org/10.1016/0370-2693(82)90643-8">doi:10.1016/0370-2693(82)90643-8</a>] <a href="https://ncatlab.org/nlab/revision/diff/anti+de+Sitter+spacetime/42">diff</a>, <a href="https://ncatlab.org/nlab/revision/anti+de+Sitter+spacetime/42">v42</a>, <a href="https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime">current</a>
and to:
- Peter Breitenlohner, Daniel Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Physics Letters B 115 3 (1982) 197-201 [doi:10.1016/0370-2693(82)90643-8]
diff, v42, current

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nLab > Latest Changes: anti de Sitter spacetime