# Noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole

## TL;DR

This paper calculates noncommutative corrections to minimal surface areas in pure AdS and Schwarzschild-AdS spacetimes, revealing specific logarithmic and mass-related modifications, and explores their implications for holographic entanglement entropy.

## Contribution

It introduces a novel perturbative approach to incorporate noncommutative effects into minimal surface calculations in AdS spacetimes, including a carefully constructed tetrad ensuring real corrections.

## Key findings

- Noncommutative effects introduce logarithmic corrections.
- Corrections include mass and noncommutative parameter terms.
- Holographic entanglement entropy obeys a thermodynamics-like relation.

## Abstract

Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincar\'e coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a logarithmic term, while for the Schwarzschild-AdS black hole, it contains a logarithmic contribution plus both a mass term and a noncommutative parameter related term. Furthermore, we show that the holographic entanglement entropy with noncommutativity obeys a relation which is similar to the first law of thermodynamics in the pure AdS spacetime.