Floating Orbits and Ergoregion Stability of an Exotic star-Black hole system. Marshall Scott1 and Daniel Kennefick2, 12870 Hwy 7 South, Camden, AR 71701, [email protected], 2University of Arkansas, Fay- etteville, AR 72701.
Introduction: A floating orbit is a metastable or- system emits gravitational radiation. This radiation bit around dense rotating astrophysical bodies wherein travels at the speed of light, possesses both transla- the orbiting body avoids orbital decay by stealing rota- tional and angular momentum, and is quadrupole in tional energy from the central body. Normally, the or- nature. The emission of this radiation decreases the biting body would emit gravitational waves as it orbits energy of the system and the orbital radius, and even- the central body, with the waves being directed toward tually the orbiting body inspirals. the central body and outer space. The emission of In the case of floating orbits, the inspiral due to these waves decreases the energy of the system and gravitational wave emissions can be prevented by lessens the orbital radius. In the case of a floating or- SRS. In SRS the waves lost to space are balanced by bit, the waves that are directed toward the central body the waves that are emitted from the exotic star toward interact with the central body's ergoregion, which is a the orbiting body, in a process is called the Penrose corotating region of spacetime around a dense, rotat- process. ing central body. In the Penrose process, an object enters the ergo- The waves take energy from this ergoregion, travel sphere of a dense astrophysical body moving in the back toward the orbiting body, and impart their en- prograde direction, and the object breaks into two ergy to the body. This process is called superradiant pieces, with one piece being directed toward the cent- scattering (SRS), for the returning gravitational waves ral body and the other piece directed outside of the er- possess more energy they had when they left the orbit- gosphere. Within this ergosphere, the spacetime and ing body. These impinging waves balance the total en- all matter is being dragged around at a velocity great- ergy loss by the orbiting body, thereby keeping the or- er than light. Consequently, the object while in the er- biting body's radius stable while decreasing the re- gosphere possesses a velocity greater than the local volution of the central body. spacetime due to the addition of its previous velocity These orbits are important for gravitational wave to that of the ergosphere. research and the further confirmation of Einstein's The inward going piece possesses a velocity less theory of General Relativity. Einstein's equations than the local spacetime and causes the central body to show that the gravitational interaction of two bodies decrease its rotational energy in order to accelerate it should emit gravitational waves. Unfortunately, these to the velocity of the spacetime. While the outward go- waves are of low amplitude and couple to matter ing piece has a velocity that is greater than the initial weakly, therefore they have been extremely difficult to object and escapes the ergosphere with more energy detect. However, if a floating orbit can be shown to than the initial body had. In our case the initial object exist theoretically and if one is discovered in nature, would be a gravitational wave and after this process it then the system would emit gravitational waves at a would impart its kinetic energy on the orbiting body regular frequency. With a constant source of waves in- thereby balancing the energy the body lost to space. teracting with the detectors, random noise and other Wronskian Calculation: Richartz et. al. argued anomalies can be eliminated and the general trend of that the Wronskian of the solutions of the Klein Gor- the radiation would be easier to detect and study. don equation can determine whether a system pos- The focus of my project was to investigate whether sesses SRS. For a system to have SRS, |R|, the reflec- by altering the previous solution employed by Ken- tion coefficient of the wave solution must be greater nefick and Glampedakis[1], which they applied to the than one. From their equation the following condi- case of a central black hole, to represent a central tions of SRS can be found: ∞ exotic star and employ a test developed Richartz et.al. ∗∣ − ∫ ∣ ∣2 iW f , f 2 f d 0 . Therefore, either [2] to see if our star would permit SRS. Furthermore, 0 0 iW0 and 0 over the interval, or iW0 and 0 by taking the Hartle process into consideration, we in- over the interval. Moreover, if iW 0 and 0 then tend to show that a floating orbit around an exotic star SRS is impossible and here is where we inserted our is theoretically possible. general solution for f . Superradiant Scattering and the Penrose Pro- The form of f that we used was a modification of cess: Einstein showed in his general relativistic field the wave solutions employed by Kennefick and Glam- equations that a non-spherically symmetric dynamical pedakis. Their solutions were came out of the radial part of the Teukolsky equation, which is a equation rose process. This self-perpetuating Penrose process derived from Einstein's equation describing a Kerr can radiate energy away from the exotic star far faster black hole. The altered equations are as follows: than in the black hole case. The faster the rotation the ∗ ∗ i n 2 −ikr −i r Rlm e A R r e for r r greater the ergoregion instability and consequently, ∗ − − ∗ − ∗ r 3 Bout ei r r 1 Bi n e i r A R re i r for r ∞ the longer a floating orbit can be sustained[3]. ∗ ∗ up out ikr 2 i n −ikr As outlined lined by Cardoso et. al., the timescale Rlm C e C e for r r ∗ 3 i r ∞ of ergoregion instability can be from a few seconds to r e for r . The Wron- − weeks depending on the mass and rotation of the cent- = n skian of these two solutions, for R r r , for any ral star. By equating their equation of the time scale ∈ℝ n , are: ergoregion instability, which details how long the Pen- i n up ∣ = out 4− 4 W R , R 2i C a M for any n lm lm r r rose process will last, and another equation of inspiral, i n up ∣ = i n 2 = ∞ W Rlm , Rlm r 4 i B A r for n 1 boundary conditions on the masses of the bodies can 2i Bi n r 2 for n 1 be established: × 22 = . If we go by Richartz et. al. then we will only experi- m 7.581 10 kg for M M ☉ ence SRS if the Wronskian contains a negative ima- = 6 which illustrates 3.783 kg for M 10 M ☉ ginary part. Therefore we would expect to see SRS that a massive central star is needed for a floating or- when evaluating at r , if M a , which is the case bit. However our exotic star is rotating nearly 300 in general, and if when evaluating at infinity, A is times that of the star in the Cardoso et. al. paper, i n negative and greater in magnitude than B for the which suggests that it would last long enough to sus- = n 1 case and we will never see any SRS if n 1 . tain a floating orbit. It should be noted that the sign of the Wronskian Lastly, the Hartle approach could also permit a changes depending on the ordering of our independent floating orbit. In this process, the orbiting body raises solutions, which is an ambiguity in Richartz et. al.'s a tide on the ergosphere, but due to the rotation of the condition for SRS. Fortunately, Kennefick and Glam- central body superseding that of the orbiting body the pedakis provide solutions for which they have proven tide is displaced an angle in the prograde direction. the existence of SRS utilizing their code that calcu- The tide pulls on the orbiting body and accelerates it, lates the full Teukolsky solution for a particle orbiting while decreasing the rotation of the central body. If a massive black hole. Since we know that SRS exits this increase in orbital energy balances the energy lost for their functions Ri n and Rup , we conclude that cor- to gravitational waves then a floating orbit is possible. rect ordering for the application of Richartz et. al.'s Conclusion: In summation, we have argued that up i n test is one in which we have the order W Rlm , Rlm . our solutions to the Wronskian, despite being positive, We find that, even with our more general solution, de- should display SRS for exotic stars due to the results signed to include the situation where the central black from Kennefick and Glampedakis paper, and the a hole becomes a material body without an event hori- floating orbit should be theoretically possible for our zon, the sign of the Wronskian remains unchanged, system. And showed that our exotic star should radiate thereby showing the possibility for SRS, unless A0 more gravitational energy than an similarly massed and ∣A∣∣Bi n∣ in the n=1 case and SRS is possible, in black hole do to ergoregion instability. Future research general, for any n1 . However, SRS is not possible will involve actually running the code from the Ken- for the solution evaluated at r . nefick and Glampedakis paper to see if SRS can be By placing our solutions into the equation for wave shown to exist for this system. If the calculations emission, we do find that SRS is possible for the solu- prove fruitful then the energy reflected back at the or- tion evaluated at infinity so long as ABi n in the n=1 biting body will be checked against the energy lost to case and is possible for any n1 : gravitational waves. So long as the gained energy su- ∣R∣2 = 1 2BinA r2 for n=1 persedes the lost energy, a floating orbit should exist. . 1 Bi n r 2 for n1, r ∞ Lastly, the physical nature of the exotic star would be Ergoregion Instability: The wave emission from checked against known star structures and if shown to an exotic star is potentially more potent due to ergore- be in accord with known stars, then a floating orbit gion instability. In this process, an exotic star under- could be shown to be possible in nature. goes the Penrose process and the inward wave propag- References: [1] K.G. and D. K. (2002) Physical ates back toward the center of the star. When this Review D 66, 1-33. [2] Richartz et. al. (2009) wave hits the center it continues in its trajectory since arXiv:2317v2 [gr-qc], 1-4. [3] Cardoso et al.,(2008) there is no horizon to absorb it. As this wave makes its arXiv:0709.05322v2 [gr-qc], 1-14. way back toward the ergosphere it undergoes the Pen-