Existence and stability of circular orbits in static and axisymmetric spacetimes

Abstract

The existence and stability of timelike and null circular orbits (COs) in the equatorial plane of general static and axisymmetric (SAS) spacetime are investigated in this work. Using the fixed point approach, we first obtained a necessary and sufficient condition for the non-existence of timelike COs. It is then proven that there will always exist timelike COs at large $\rho$ in an asymptotically flat SAS spacetime with a positive ADM mass and moreover, these timelike COs are stable. Some other sufficient conditions on the stability of timelike COs are also solved. We then found the necessary and sufficient condition on the existence of null COs. It is generally shown that the existence of timelike COs in SAS spacetime does not imply the existence of null COs, and vice-versa, regardless whether the spacetime is asymptotically flat or the ADM mass is positive or not. These results are then used to show the existence of timelike COs and their stability in an SAS Einstein-Yang-Mills-Dilaton spacetimes whose metric is not completely known. We also used the theorems to deduce the existence of timelike and null COs in some known SAS spacetimes.

Appendices

Appendix A: Sufficient conditions for (in-)stabilities of timelike COs

When Eq. (38) is replaced by

$$\begin{aligned} h(\rho )=\chi \rho ^2A^{\prime 3}~(\chi \ne 0), \end{aligned}$$

(A.1)

an implicit solution can be obtained

$$\begin{aligned} \frac{\sqrt{1+8\chi }A^{\frac{3}{2}+\frac{\sqrt{1+8\chi }}{2}}}{2\rho ^2} -c_2 A^{\sqrt{1+8\chi }}+c_1=0. \end{aligned}$$

(A.2)

In particular, when $\chi =1$ or $\chi =-2/25$, two explicit solution can be found

$$\begin{aligned} \chi= & {} 1, \quad A (\rho )=\left( {\frac{2c_1\rho ^2}{2c_2\rho ^2-3}}\right) ^{1/3} \end{aligned}$$

(A.3)

$$\begin{aligned} \chi= & {} -\frac{2}{25}, \quad A(\rho )=\left( \frac{1}{3} x+\frac{10}{3}x^{-1}\right) ^{5/3},\nonumber \\ x= & {} \left( 5\sqrt{81c_1^2\rho ^4-40c_2^3\rho ^6}-45c_1\rho ^2\right) ^{1/3}. \end{aligned}$$

(A.4)

The former will only have stable COs (if any) and the later will only have unstable COs (if any) and none of them will allow MSCOs.

When Eq. (38) is replaced by

$$\begin{aligned} h(\rho )=\chi A^2A^\prime ~(\chi \ne 0), \end{aligned}$$

(A.5)

an implicit solution can be obtained

$$\begin{aligned} ( 2-\chi )\left( c_1- c_2 A^{\sqrt{1+4\chi }}\right) \rho ^5 + \sqrt{1+4\chi }A^{\frac{3}{2}+\frac{\sqrt{1+4\chi }}{2}}\rho ^{\chi +3}=0. \end{aligned}$$

(A.6)

In particular, when $\chi =-4/25$, an explicit solution can be found

$$\begin{aligned}&A(\rho )=\left[ \left( y/5+6c_2\rho ^{\frac{4}{25}} y^{-1} \right) \rho \right] ^{5/3},\nonumber \\&y=\left( 15\sqrt{225c_1^2- 120c_2^3 \rho ^{\frac{54}{25}}}-225c_1 \right) ^{1/3} \rho ^{-\frac{7}{25}}. \end{aligned}$$

(A.7)

This solution will only have unstable COs (if any) and no MSCOs.

When Eq. (38) is replaced by

$$\begin{aligned} h(\rho )=\chi \rho AA^{\prime 2} ~(\chi \ne 0), \end{aligned}$$

(A.8)

an implicit solution can be obtained

$$\begin{aligned} c_1- c_2 A^{\sqrt{\chi ^2+6\chi +1}}+ \frac{\sqrt{\chi ^2+6\chi +1} A^{\frac{\chi }{2}+ \frac{3}{2}+\frac{\sqrt{\chi ^2+6\chi +1}}{2}}}{2\rho ^2}=0. \end{aligned}$$

(A.9)

In particular, when $\chi =-3+5\sqrt{3}/3$, an explicit solution can be found

$$\begin{aligned}&A(\rho )=\left( \frac{u}{3}+2\sqrt{3}c_2\rho ^2 u^{-1}\right) ^{\sqrt{3}},\nonumber \\&u=\left( 9\sqrt{27{c_1}^2\rho ^4-8\sqrt{3}c_2^3\rho ^6} -27\sqrt{3}c_1\rho ^2\right) ^{1/3}. \end{aligned}$$

(A.10)

This solution will only have unstable COs (if any) and no MSCOs.

One can also replace Eq. (38) by

$$\begin{aligned} h(\rho )=\chi \rho A^2A^{\prime \prime } ~(\chi \ne 0).\end{aligned}$$

(A.11)

Although the sign of $A^{\prime \prime }$ and consequently that of the right hand side cannot be fixed before solving $A(\rho )$, this transformation do allow the equation to be solvable and then a determination of $A^{\prime \prime }$’s sign. The implicit solution to this equation is given by

$$\begin{aligned} ( \chi +2 ) c_2 A^{-\frac{1+\chi }{\chi -1}}+ ( 1+\chi ) A^{-\frac{\chi +2}{\chi -1}}\rho ^{\frac{\chi +2}{\chi -1}} +c_1=0. \end{aligned}$$

(A.12)

In particular, when $\chi =-3$, two explicit solution can be found

$$\begin{aligned}&A(\rho )=\left[ \left( \rho ^{1/4}+\sqrt{c_1c_2+\sqrt{\rho }}\right) /c_1\right] ^4, \end{aligned}$$

(A.13)

$$\begin{aligned}&A(\rho )=\left[ \left( -\rho ^{1/4}+\sqrt{c_1c_2+\sqrt{\rho }}\right) /c_1\right] ^4 , \end{aligned}$$

(A.14)

where in both cases $c_1c_2>0$. For the former solution, since $A^{\prime \prime }<0$ it will only have unstable COs (if any) and no MSCOs. For the later one, $A^{\prime \prime }>0$ and it will only have stable COs (if any) and no MSCOs. When $\chi =-1/2$, or $\chi =-5/2$, Eq. (A.12) can also generate some explicit solutions of $A(\rho )$. However, since sign of their second derivatives cannot be determined, and we will not list them here.

Appendix B: Proof of Theorem F

To prove the theorem, we define a $\mu (\rho )$ first

$$\begin{aligned} \mu (\rho )=A(\rho )/\rho \end{aligned}$$

(B.1)

such that

$$\begin{aligned} \mu ^\prime (\rho )=\left[ A^\prime (\rho )\rho -A(\rho ) \right] /\rho ^2 .\end{aligned}$$

(B.2)

Then using Eq. (43) it is seen that the existence of a null CO is equivalent to the existence of a point such that $\mu ^\prime (\rho )=0$.

Now since $A(\rho =a)=0$, we have

$$\begin{aligned} \mu ^\prime (\rho =a)=A^\prime (\rho =a)/a^2. \end{aligned}$$

(B.3)

Since $A(\rho )>0$ for at least the neighborhood (a,  b) of $\rho$, we must have $A^\prime (\rho =a)\ge 0$. Therefore

$$\begin{aligned} \mu ^\prime (\rho =a)\ge 0 .\end{aligned}$$

(B.4)

If the equal sign is true, then the CO exists. If it is not true, we further show by contradiction that $\mu ^\prime (\rho )$ cannot be positive definite for $\rho \in [a,~\infty )$ and therefore due to the continuity of $\mu ^\prime (\rho )$ there must exist a point $\rho _*$ satisfying $\mu ^\prime (\rho _*)=0$, which is again a CO. Suppose $\mu ^\prime (\rho )>0$ for all $\rho \in [a,~\infty )$, then clearly $A(\rho )=\mu (\rho )\rho$ will diverge faster than $\rho ^1$, which contradicts the assumption.