# Geodesics of AdS3: Find a Reference

• A
• ShayanJ
In summary, the conversation is about finding the geodesics of AdS3 spacetime and requesting a reference for their calculation. The first response provides a link to a paper on the topic, which leads to another link with plots and images. However, these are not what the person is looking for and they mention needing a rigorous derivation of the geodesics. The next response suggests using the embedding of AdS3 in ##\mathbb{R}^{(n-1,2)}## and explains how the geodesic equation can be solved using two points on the quadric surface. They also provide a link to a post on Stack Exchange that further explains how the answer is "obvious".
ShayanJ
Gold Member
I need to find the geodesics of AdS3 spacetime. But my searches have given me nothing. Can anyone give a reference where they're calculated?
Thanks

Can't tell if this is what is needed,

From, https://arxiv.org/pdf/1604.02687v2.pdf

Which is from, https://inspirehep.net/record/1445018/plots

Which is from, https://www.google.com/search?q=geo...tvvRAhXF5oMKHZVQDBQQ_AUICCgB&biw=1024&bih=490

Interesting, but not what I needed. Thanks anyway.

P.S.
I can try to find a parametrization for curves in those pictures but I need a rigorous derivation of the geodesics.

You can easily get geodesics in ##AdS_n## by considering the embedding in ##\mathbb{R}^{(n-1,2)}##,

$$X_{-1}^2 + X_0^2 - X_1^2 - X_2^2 - \ldots - X_{n}^2 = r^2.$$
Since ##AdS_n## is a maximally-symmetric space (its isometry group ##SO(n-1,2)## has the maximal number of generators ##\frac12 n(n+1)## ), it follows that the geodesic equation is also ##SO(n-1,2)##-invariant. Generically, a solution of the geodesic equation (i.e., a geodesic) is defined by giving two points: the start and end points (caveat: not every two points on ##AdS_n## can be joined by a geodesic!). Given two points on the quadric surface above, together with the origin ##(0,0,0,0,\ldots,0)##, one has a 2-plane, and you should be able to show that the intersection of this 2-plane with the quadric surface is in fact a geodesic. (In the same sense, the geodesics on a 2-sphere are given by the intersections of 2-planes through the origin of 3-space with that 2-sphere).

I can't quite come up with an argument right now that makes this fact "obvious", but it should be straightforward enough for you to write out the geodesic equation and find its first integrals. There should be enough first integrals (i.e. integration constants) to define the orientation of a 2-plane in ##(n+1)##-dimensional space, and you should be able to derive an algebraic condition which is precisely finding the intersection of that 2-plane with the quadric surface that defines the embedding.

This post on Stack Exchange does a bit more work to show how the answer is obvious ;)

http://physics.stackexchange.com/questions/116813/ads-space-boundary-and-geodesics

Spinnor

## 1. What is the AdS3 space?

The AdS3 space, also known as Anti-de Sitter space, is a three-dimensional space with constant negative curvature. It is a solution to Einstein's equations of general relativity and is used in various areas of theoretical physics, including string theory.

## 2. What are geodesics in AdS3?

Geodesics in AdS3 are the shortest paths between two points in the AdS3 space. They are curves that follow the natural path of least resistance in the space, similar to a straight line in flat space.

## 3. How do I find a reference for geodesics in AdS3?

There are several resources available for finding references on geodesics in AdS3. These include research papers, textbooks, and online resources such as lecture notes and videos. It is recommended to consult with a mentor or colleague in the field for guidance on the most relevant and reliable references.

## 4. What are the applications of studying geodesics in AdS3?

Geodesics in AdS3 have numerous applications in theoretical physics, including in the study of black holes, quantum gravity, and holography. They also have implications for understanding the geometry and topology of the AdS3 space itself.

## 5. Are there any challenges in understanding geodesics in AdS3?

Yes, there are several challenges in understanding geodesics in AdS3. One of the main challenges is the highly non-linear nature of the equations involved, which makes it difficult to find exact solutions. Additionally, the concept of geodesics in curved spaces can be challenging to grasp for those unfamiliar with differential geometry.

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