# Geodesic Structure of Generalized Vaidya Spacetime through the K-Essence

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## Abstract

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## 1. Introduction

## 2. Summary of the Relation between K-Essence and Generalized Vaidya Spacetime

## 3. Geodesics for the Generalized K-Essence Vaidya Spacetime

#### 3.1. Case-I: $\mathcal{M}(t,r)=M+\frac{r}{2}{\varphi}_{t}^{2}$

#### 3.1.1. Time-like Geodesics for Case-I

#### 3.1.2. Null Geodesics for Case-I

#### 3.2. Case-II: $\mathcal{M}(t,r)=\mu t+\frac{r}{2}{\varphi}_{t}^{2}$

#### 3.2.1. Time-like Geodesics for Case-II

#### 3.2.2. Null Geodesics for Case-II

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Radius of the Dynamical Horizon

#### Detail Derivation

## References

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**Figure 4.**Time-like geodesics for $D=2.9$, $L=1$, ${t}_{0}=0.1$, and ${c}_{3}=0.1$. Blue-line for $r-{t}_{0}E(t)-At-{c}_{3}=0$ and red-line for $r-{t}_{0}E(t)-Bt-{c}_{3}=0$.

**Figure 5.**Time-like geodesics for $D=2.9$, $L=1$, ${t}_{0}=0.1$, and ${c}_{3}=10$. Blue-line for $r-{t}_{0}E(t)-At-{c}_{3}=0$ and red-line for $r-{t}_{0}E(t)-Bt-{c}_{3}=0$.

**Figure 6.**Circular time-like geodesics for $D=2.9$, $L=1$, and $M=1.45$. Blue-line for ${r}_{+}=\frac{M+\sqrt{{M}^{2}-2{L}^{2}}}{2}$ and red-line for ${r}_{-}=\frac{M-\sqrt{{M}^{2}-2{L}^{2}}}{2}$.

**Figure 10.**Null geodesics for $D=2.1$, $L=1$, ${t}_{0}=0.1$, and ${c}_{6}=0.1$. Blue-line for $r-{t}_{0}E(t)-At-{c}_{6}=0$ and red-line for $r-{t}_{0}E(t)-Bt-{c}_{6}=0$.

**Figure 11.**Null geodesics for $D=2.1$, $L=1$, ${t}_{0}=0.1$, and ${c}_{6}=10$. Blue-line for $r-{t}_{0}E(t)-At-{c}_{6}=0$ and red-line for $r-{t}_{0}E(t)-Bt-{c}_{6}=0$.

**Figure 12.**Circular null geodesics for $D=2.1$, $L=1$, and $M=1.05$. Blue-line for ${r}_{+}=M+\sqrt{{M}^{2}-{L}^{2}}$ and red-line for ${r}_{-}=M-\sqrt{{M}^{2}-{L}^{2}}$.

**Figure 13.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{8}=0.1$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 14.**Time-like geodesics for $r-{t}_{0}E(t)-\overline{A}-{c}_{8}=0$ when $\mu =0.05$, $L=1$, and ${t}_{0}=0.1$. Blue-line for ${c}_{8}=0.1$ and red-line for ${c}_{8}=30$.

**Figure 15.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{8}=30$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 16.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{8}=0.1$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 17.**Time-like geodesics for $r-{t}_{0}E(t)-\overline{B}-{c}_{8}=0$ when $\mu =0.05$, $L=1$, and ${t}_{0}=0.1$. Blue-line for ${c}_{8}=0.1$ and red-line for ${c}_{8}=30$.

**Figure 18.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{8}=30$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 19.**Time-like geodesics for ${r}_{+}=\frac{\mu t+\sqrt{{\mu}^{2}{t}^{2}-2{L}^{2}}}{2}$ for $\mu =0.05$ and $L=1$.

**Figure 20.**Time-like geodesics for ${r}_{-}=\frac{\mu t-\sqrt{{\mu}^{2}{t}^{2}-2{L}^{2}}}{2}$ for $\mu =0.05$ and $L=1$.

**Figure 21.**$\mu =0.01$, $L=3$, ${t}_{0}=0.1$, and ${c}_{9}=0.1$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 22.**$\mu =0.01$, $L=3$, ${t}_{0}=0.1$, and ${c}_{9}=20$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 23.**Null geodesics for $r-{t}_{0}E(t)-\overline{A}-{c}_{9}=0$ when $\mu =0.01$, $L=3$, and ${t}_{0}=0.1$. Blue-line for ${c}_{9}=0.1$ and red-line for ${c}_{9}=20$.

**Figure 24.**Null geodesics for $r-{t}_{0}E(t)-\overline{A}-{c}_{9}=0$ when $\mu =0.01$, $L=3$, and ${t}_{0}=0.1$. Blue-line for ${c}_{9}=0$.

**Figure 25.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{9}=0.1$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 26.**$\mu =0.05$, $L=1$, ${t}_{0}=0.1$, and ${c}_{9}=20$. Blue-line for $f({t}_{s})$ and red-line for $g({t}_{s})$.

**Figure 27.**Null geodesics for $r-{t}_{0}E(t)-\overline{B}-{c}_{9}=0$ when $\mu =0.05$, $L=1$, ${t}_{0}=0.1$. Blue-line for ${c}_{9}=0.1$ and red-line for ${c}_{9}=20$.

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Majumder, B.; Khlopov, M.; Ray, S.; Manna, G.
Geodesic Structure of Generalized Vaidya Spacetime through the K-Essence. *Universe* **2023**, *9*, 510.
https://doi.org/10.3390/universe9120510

**AMA Style**

Majumder B, Khlopov M, Ray S, Manna G.
Geodesic Structure of Generalized Vaidya Spacetime through the K-Essence. *Universe*. 2023; 9(12):510.
https://doi.org/10.3390/universe9120510

**Chicago/Turabian Style**

Majumder, Bivash, Maxim Khlopov, Saibal Ray, and Goutam Manna.
2023. "Geodesic Structure of Generalized Vaidya Spacetime through the K-Essence" *Universe* 9, no. 12: 510.
https://doi.org/10.3390/universe9120510