[Paper Review] Circular time-like geodesics around a charged spherically symmetric dilaton black hole

C. Blaga|arXiv (Cornell University)|Jun 28, 2014

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates circular time-like geodesics around a charged spherically symmetric dilaton black hole using the effective potential method derived from the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) metric. It determines the innermost stable circular orbit (ISCO) and shortest circular orbit (SCO) radii as functions of the charge-to-mass ratio $ b = Q^2/M^2 $, showing ISCO decreases from $ 6M $ (Schwarzschild) to $ 2M $ (extremal GMGHS), and SCO from $ 3M $ to $ 2M $, confirming known results for extreme cases.

ABSTRACT

In this note we examine the circular time-like geodesics near a spherically symmetric dilaton black hole, described using the exact solution for a static charged black hole found by Gibbons and Maeda and, independently, by Garfinkle, Horowitz and Strominger. The existence and stability of the circular orbits are analysed using the effective potential of a free material test particle moving on time-like geodesic near this black hole. We determine the radius of the innermost stable circular orbit, the radius of the shortest circular orbit and compare our results with those obtained by other authors for specific values of the parameters involved in our analysis.

Motivation & Objective

• To analyze the existence and stability of circular time-like geodesics around a charged spherically symmetric dilaton black hole described by the GMGHS solution.
• To determine the radius of the innermost stable circular orbit (ISCO) and the shortest circular orbit (SCO) for varying charge-to-mass ratios.
• To compare the results with known solutions for the Schwarzschild black hole ($ b = 0 $) and extremal GMGHS black hole ($ b = 2 $).
• To examine the behavior of the effective potential and its first derivative to identify critical radii for circular orbits.

Proposed method

• Derives the effective potential $ V_{\text{eff}} $ for time-like geodesics using the GMGHS metric, incorporating mass $ M $, charge $ Q $, and angular momentum $ L $.
• Introduces dimensionless variables $ u = r/M $, $ a = L^2/M^2 $, and $ b = Q^2/M^2 $ to simplify analysis of the effective potential.
• Analyzes the first derivative of $ V_{\text{eff}} $ with respect to $ u $, reducing it to a cubic equation whose roots determine circular orbit radii.
• Uses discriminant analysis of the cubic equation to classify the number of real roots and determine regions of orbital existence and stability.
• Applies stability criteria: minima of $ V_{\text{eff}} $ correspond to stable orbits, maxima to unstable ones.
• Plots the ISCO and SCO radii as functions of $ b $, using the critical value $ \bar{a}(b) $ where the number of real roots changes.

Experimental results

Research questions

• RQ1 What is the radius of the innermost stable circular orbit (ISCO) for a charged spherically symmetric dilaton black hole as a function of the charge-to-mass ratio $ b = Q^2/M^2 $?
• RQ2 What is the radius of the shortest circular orbit (SCO), and how does it vary with $ b $?
• RQ3 How do the number and nature (stable/unstable) of circular geodesics depend on the angular momentum parameter $ a = L^2/M^2 $ and charge parameter $ b $?
• RQ4 How do the ISCO and SCO radii behave in the limits $ b = 0 $ (Schwarzschild) and $ b = 2 $ (extremal GMGHS), and do they match known results?
• RQ5 What is the role of the discriminant of the cubic equation derived from $ dV_{\text{eff}}/dr = 0 $ in determining the existence and multiplicity of circular orbits?

Key findings

• The radius of the innermost stable circular orbit (ISCO) decreases monotonically from $ 6M $ at $ b = 0 $ (Schwarzschild) to $ 2M $ at $ b = 2 $ (extremal GMGHS).
• The radius of the shortest circular orbit (SCO) decreases from $ 3M $ at $ b = 0 $ to $ 2M $ at $ b = 2 $, matching known values for the Schwarzschild case.
• For $ a > \bar{a}(b) $, three real roots exist: one unstable orbit inside the ISCO and two orbits outside, with the larger radius being stable.
• The ISCO radius corresponds to the minimum of the effective potential and is stable for all $ b \in (0,2] $, with the stable orbit radius always $ \geq r_{\text{ISCO}} $.
• The critical value $ \bar{a}(b) $, where the number of real roots of $ dV_{\text{eff}}/dr = 0 $ changes, decreases from 12 to 2 as $ b $ increases from 0 to 2.
• The results for ISCO and SCO at $ b = 0 $ and $ b = 2 $ are consistent with Chandrasekhar’s work on Schwarzschild and Pradhan’s on extremal GMGHS black holes.