# New asymptotic Anti-de Sitter solution with a timelike extra dimension in 5D relativity

## TL;DR

This paper derives a new asymptotic Anti-de Sitter solution in 5D relativity with a timelike extra dimension, showing how such a geometry naturally emerges and discussing its holographic implications.

## Contribution

It presents the first explicit 5D asymptotic AdS solution with a timelike extra dimension, expanding the understanding of higher-dimensional geometries in Kaluza-Klein theory.

## Key findings

- Negative cosmological constant emerges with timelike extra dimension
- AdS space is naturally induced on a brane from 5D Kaluza-Klein manifold
- Holographic duality principles are applicable in this setup

## Abstract

In 5D relativity, the usual 4D cosmological constant is determined by the extra dimension. If the extra dimension is spacelike, one can get a positive cosmological constant $\Lambda$ and a 4D de Sitter (dS) space. In this paper we present that, if the extra dimension is timelike oppositely, the negative $\Lambda$ will be emerged and the induced 4D space will be an asymptotic Anti-de Sitter (AdS). Under the minimum assumption, we solve the Kaluza-Klein equation $R_{AB} = 0$ in a canonical system and obtain the AdS solution in a general case. The result shows that an AdS space is induced naturally from a Kaluza-Klein manifold on a hypersurface (brane). The Lagrangian of test particle indicates the equation of motion can be geodesics if the 4D metric is independent of extra dimension. The causality is well respected because it is appropriately defined by a null higher dimensional interval. In this 5D relativity, the holographic principle can be used safely because the brane is asymptotic Euclidean AdS in the bulk. We also explore some possible holographic duality implications about the field/operator correspondence and the two-points correlation functions.