# AdS Space Horizon - Black Holes & Research

• PDEagle
In summary, a good source for information on the global structure of Schwarzschild-dS and Schwarzschild-AdS solutions is available at http://www.arxiv.org/abs/gr-qc/9507019. These solutions can be written in various coordinate charts and parameterizations, with some authors using \Lambda as a parameter. The event and cosmological horizons can be located by solving a polynomial, with the help of mathematical techniques such as the discriminant, Sturm's method, and multivariable Taylor series. Additionally, students of general relativity can study the orbits of test particles in these solutions using effective potentials. The second question about the relationship between AdS space and black hole is too vague to answer without context and
PDEagle
Where is the horizon of the AdS space?
What's the relationship between AdS space and black hole?
Is there any related articles?
thanks.

Last edited:

PDEagle said:
Where is the horizon of the AdS space?
What's the relationship between AdS space and black hole?
Is there any related articles?
thanks.

A good source for basic information about the global structure of the Schwarzschild-dS and Schwarzschild-AdS solutions (and their electrically charged generalizations) is http://www.arxiv.org/abs/gr-qc/9507019

Note that there are many distinct coordinate charts in which these solutions may be written, which are useful in different contexts, and in addition the metric functions can be parameterized in different ways. In particular, some authors prefer to write them in terms of $$\Lambda$$ in which case the same line element can be used for either dS or AdS. Others prefer to write a form in which the approximation location of the cosmological horizon is given by one parameter.

An interesting pedagogical point here is that we locate the event and cosmological horizons by solving a polynomial, which in this case happens to be a cubic, e.g. in Schwarschild-dS it can be written $$1-2m/r-r^2/a^2$$. This is a good opportunity to put a solid mathematical education to use by employing the discriminant to ensure that we have three real roots (we only care about the two positive real roots, of course), then using Sturm's method to analyze the disposition of the roots, and finally using multivariable Taylor series to approximation the roots. In the Schwarzschild-dS case, we find $$r \approx 2m, \; r \approx a-m$$ for the location of the event horizon and cosmological horizon, where $$a^2 > 27 m^2$$ and where $$\Lambda=3/a^2$$.

A very good exercise for gtr students is to follow the model of the analysis via effective potentials of test particle orbits for the Schwarzschild vacuum to study the orbits of test particles in the Schwarzschild-dS or AdS lambdavacuums. In the dS case, for some values of orbital angular momentum (of the test particle), we have two unstable and one stable circular orbits. More interestingly, test particles with zero orbital angular momentum can "hover" at just the right radius to balance the gravitational attraction of the massive object at the "center" with the effect of positive $$\Lambda$$ (This configuration is unstable.)

Your second question is a bit too vague for me to attempt to answer. Can you provide some context? What level of mathematical sophistication did you desire in a reply?

Chris Hillman

The horizon of the AdS space is located at the boundary of the space, which is infinitely far away from any observer within the space. This boundary is often referred to as the "boundary at infinity" and is where the AdS space meets the flat spacetime outside of it.

The relationship between AdS space and black holes is that AdS space is often used as a theoretical model for studying the properties of black holes. This is because AdS space has a negative curvature, which is similar to the curvature near a black hole's event horizon. Additionally, the AdS/CFT correspondence, a theoretical framework that relates AdS space to a conformal field theory, has been used to gain insights into the behavior of black holes.

There have been numerous articles and research studies exploring the connection between AdS space and black holes. Some recent articles include "Black Holes in AdS Space and the Gauge-Gravity Duality" by Andrew Strominger, "AdS/CFT Correspondence and Black Holes" by Roberto Emparan, and "Holographic Black Holes" by Gary Horowitz. These articles delve into the mathematical and theoretical aspects of the relationship between AdS space and black holes, providing valuable insights into these complex phenomena.

## 1. What is AdS Space Horizon?

AdS (Anti-de Sitter) Space Horizon is a theoretical boundary that surrounds a black hole in which the escape velocity exceeds the speed of light. It marks the point of no return, beyond which nothing, including light, can escape the gravitational pull of the black hole.

## 2. How are black holes and AdS Space Horizon related?

Black holes are objects with such strong gravitational pull that they can trap even light within their event horizon. AdS Space Horizon is a concept that applies specifically to black holes in Anti-de Sitter space, a type of spacetime described by Einstein's theory of general relativity. In this space, the event horizon is the AdS Space Horizon.

## 3. What is the significance of studying AdS Space Horizon in black hole research?

Studying AdS Space Horizon allows us to better understand the properties and behavior of black holes, which are still not fully understood. It also has implications for our understanding of gravity and the laws of physics in extreme environments.

## 4. How do scientists study AdS Space Horizon in black holes?

Scientists use mathematical models and simulations to study AdS Space Horizon. They also use observations and data from black hole mergers and other phenomena to test and refine their theories.

## 5. What are some potential applications of research on AdS Space Horizon and black holes?

Research on AdS Space Horizon and black holes can have implications for a variety of fields, including astrophysics, cosmology, and even technology. It can also help us better understand the fundamental laws of the universe and potentially lead to new discoveries and advancements in our understanding of the cosmos.

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