PHYSICAL REVIEW D 97, 044005 (2018)
Massive graviton geons
† ‡ Katsuki Aoki,1,* Kei-ichi Maeda,1, Yosuke Misonoh,1, and Hirotada Okawa2,1,§ 1Department of Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan 2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
(Received 24 October 2017; published 5 February 2018)
We find vacuum solutions such that massive gravitons are confined in a local spacetime region by their gravitational energy in asymptotically flat spacetimes in the context of the bigravity theory. We call such self-gravitating objects massive graviton geons. The basic equations can be reduced to the Schrödinger- Poisson equations with the tensor “wave function” in the Newtonian limit. We obtain a nonspherically symmetric solution with j ¼ 2, l ¼ 0 as well as a spherically symmetric solution with j ¼ 0, l ¼ 2 in this system where j is the total angular momentum quantum number and l is the orbital angular momentum quantum number, respectively. The energy eigenvalue of the Schrödinger equation in the nonspherical solution is smaller than that in the spherical solution. We then study the perturbative stability of the spherical solution and find that there is an unstable mode in the quadrupole mode perturbations which may be interpreted as the transition mode to the nonspherical solution. The results suggest that the nonspherically symmetric solution is the ground state of the massive graviton geon. The massive graviton geons may decay in time due to emissions of gravitational waves but this timescale can be quite long when the massive gravitons are nonrelativistic and then the geons can be long-lived. We also argue possible prospects of the massive graviton geons: applications to the ultralight dark matter scenario, nonlinear (in) stability of the Minkowski spacetime, and a quantum transition of the spacetime.
DOI: 10.1103/PhysRevD.97.044005
I. INTRODUCTION in a box. Gravitational geons in asymptotically AdS spacetimes are constructed in [16–18]. Recent observations found gravitational waves from In the present paper, we consider asymptotically flat black hole mergers in which a few percent of the energy spacetimes and construct a gravitational geon of massive of the system is radiated by the gravitational waves [1–4]. gravitons. The extra ingredient in the present analysis is a In this way, it is well known that the gravitational waves mass of a graviton. Although a graviton in general relativity have their energy and change the background geometry. (GR) is massless, theories with a massive graviton have Therefore, it seems possible that the gravitational waves are received much attention. For instance, models with extra gravitationally bounded in a local region of the spacetime dimensions predict the existence of massive gravitons as by their own gravitational energy. This time-dependent well as a massless graviton as Kaluza-Klein modes. In other self-gravitating object is called a gravitational geon, short examples, the massive graviton has been discussed to give for gravitational-electromagnetic entity, which was intro- rise to the cosmic accelerating expansion (see [19,20] for duced by Wheeler [5]. Although a gravitational geon can be reviews and references therein) and recently discussed as a constructed in asymptotically flat spacetimes [6,7], the – – candidate of dark matter [21 32]. geon is not exactly periodic in time [8 15], and it has a Massive bosonic fields can form self-gravitating objects finite lifetime. However, the instability of geons could be which are called oscillatons in a real massive scalar field – remedied in asymptotically anti de Sitter (AdS) spacetimes [33], boson stars in a complex massive scalar field [34,35], since the AdS boundary can confine particles and waves as and Proca stars in a complex massive vector field [36] (see also [37,38]). Hence, we can naturally expect that the *katsuki-a12@gravity.phys.waseda.ac.jp massive graviton also yields self-gravitating solutions † [email protected] which we call massive graviton geons. Indeed, the paper ‡ [email protected] [27] showed the coherent oscillations of the massive §[email protected] graviton lead to the Jeans instability and then the massive gravitons could be self-gravitated. However, the analysis Published by the American Physical Society under the terms of [27] is based on the linear perturbation theory around the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to homogeneous background. It has not been cleared whether the author(s) and the published article’s title, journal citation, or not the massive gravitons indeed yield a nonperturbative and DOI. localized object.
2470-0010=2018=97(4)=044005(17) 044005-1 Published by the American Physical Society AOKI, MAEDA, MISONOH, and OKAWA PHYS. REV. D 97, 044005 (2018)
κ2 κ2 κ2 We study nontrivial vacuum solutions to the bigravity defined by ¼ g þ f. Just for simplicity, to admit the theory. The bigravity theory contains a massless graviton Minkowski spacetime as a vacuum solution, we restrict the and a massive graviton [39]. The energy-momentum tensor potential U as the form [41,42] of the massive graviton was derived in [24,27] in the similar way to general relativity [40]. We find that the basic X4 equations in the Newtonian limit can be reduced to the U ¼ ciUiðKÞ; ð2:2Þ Schrödinger-Poisson equations with the tensor “wave i¼2 function.” Due to the tensorial feature, the solutions in 1 αβρσ μ ν this system have different properties from the scalar case. U2ðKÞ¼− ϵμνρσϵ K αK β; We construct two types of the massive graviton geons: the 4 1 monopole geon and the quadrupole geon. The monopole αβγσ μ ν ρ U3ðKÞ¼− ϵμνρσϵ K αK βK γ; geon is the spherically symmetric configuration of the 3! massive graviton which corresponds to the eigenstate of 1 U K − ϵ ϵαβγδKμ Kν Kρ Kσ the zero total angular momentum. On the other hand, the 4ð Þ¼ 4! μνρσ α β γ δ; ð2:3Þ quadrupole geon is not spherically symmetric, instead, it is given by the quadrupole modes of the spherical harmonics. with The quadrupole geon is the eigenstate of the zero orbital qffiffiffiffiffiffiffiffiffiffi μ angular momentum. Although the field configuration of the μ μ −1 K ν ¼ δ ν − g f ; ð2:4Þ quadrupole geon is not spherical, it yields the spherically ν symmetric energy density and then the gravitational poten- pffiffiffiffiffiffiffiffiffiffi −1 μ tial of the quadrupole geon is spherically symmetric. where ð g fÞ ν is defined by the relation We also discuss the stability of the geons. The monopole qffiffiffiffiffiffiffiffiffiffi μ qffiffiffiffiffiffiffiffiffiffi ρ geon is unstable against quadrupole mode perturbations. −1 −1 μρ g f g f ¼ g fρν: ð2:5Þ We then expect a transition from the monopole geon to the ρ ν quadrupole one. The quadrupole geon would be a ground state of the massive graviton geons. We can set c2 ¼ −1 by using the normalization of the The paper is organized as follows. In Sec. II,we parameter m . Then gμν ¼ fμν ¼ ημν is a vacuum solution introduce the bigravity theory and take the Newtonian G of the bigravity, and the parameter mG describes the mass limit with the self-gravity. We numerically construct the of the massive graviton propagating on the Minkowski massive graviton geons in Sec. III. We find that not only the background. spherically symmetric configuration of the massive grav- We assume that the curvature scales of the spacetimes are iton but also a nonspherically symmetric configuration smaller than the graviton mass yields spherically symmetric energy density distributions. The perturbative stability of the monopole geon is studied 2 2 2 2 j∂ gμνj ≪ m ; j∂ fμνj ≪ m : ð2:6Þ in Sec. IV. We give a summary and discuss some physical G G aspects of the massive graviton geons in Sec. V.In 1 Appendix A, we summarize multipole expansions of the This means that we consider only weak gravity limit. tensor field and the definitions of the angular momenta. We perform the perturbations up to second orders around We also find an octupole configuration of the massive Minkowski spacetime and discuss a bound state of per- graviton geon in Appendix B. turbed gravitational field. The perturbed gravitational fields (δgμν and δfμν) are rewritten into two modes, that is, II. NEWTONIAN LIMIT OF BIGRAVITY 1 massless mode∶ ðδgμν þ αδfμνÞ; ð2:7Þ We assume the existence of a massive graviton as well 1 þ α as a massless graviton. The gravitational action is given α by [39] massive mode∶ ðδgμν − δfμνÞ: ð2:8Þ 1 þ α Z Z pffiffiffiffiffiffi 1 4 pffiffiffiffiffiffi 1 4 − − R 2 2 S ¼ 2 d x gRðgÞþ 2 d x f ðfÞ α ≔ κ κ 2κ 2κ where g= f. Those two modes are split into four g f Z components by those frequencies: the low-frequency m2 pffiffiffiffiffiffi − G d4x −gU g; f ; : κ2 ð Þ ð2 1Þ 1The inequalities (2.6) also lead to that we do not need to take into account the Vainshtein mechanism [43]. For instance, the where gμν and fμν are two dynamical metrics, and RðgÞ and exterior region of a localized object with a mass M yields R κ2 κ2 2 3 ðfÞ are their Ricci scalars. The parameters g and f j∂ gμνj ∼ GM=r and then (2.6) suggests the region outside the κ ≔ 2 1=3 are the corresponding gravitational constants, while is Vainshtein radius rV ðGM=mGÞ .
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ð0Þ δ ð0Þ ð0Þα massless mode g μν, the low-frequency massive mode ∇ ∇ − 2 φ 0 O φ2 ð α mGÞ μν ¼ þ ð μνÞ; ð2:15Þ Mμν, the high-frequency massless mode hμν, and the high- φ frequency massive mode μν, i.e., with the constraints
ð0Þ ð0Þ ∶ δ μν 2 μ 2 massless mode g μν þ hμν=Mpl ð2:9Þ ∇μφ ¼ 0 þ OðφμνÞ; φ μ ¼ 0 þ OðφμνÞ; ð2:16Þ
μν massive mode∶ Mμν þ φ =MG; ð2:10Þ where we have ignored the higher order corrections of φμν to the Einstein-Klein-Gordon equations. The Einstein where ð0Þ tensor Gμν is constructed by the low-frequency background κ κ ð0Þ M ≔ ;M≔ : 2:11 μν pl κ κ G 2 ð Þ g μν while TG is the energy-momentum tensor of the high- g f κg frequency perturbations φμν. The energy-momentum tensor Here the high frequency means that the frequency is same of massive graviton is defined by or larger than the graviton mass mG, while the low 0 μα 0 νβ 1 0 μν 0 αβ ð2Þ μν − ð Þ ð Þ − ð Þ ð Þ δ φ frequency is that much smaller than mG. Intuitively, TG ¼ g g 2 g g R αβ½ 0 δð Þ g μν and Mμν represent the Newtonian potential and the 2 0 μν − mG 4φμαφν − ð Þ φαβφ O φ3 Yukawa potential, respectively, while the high-frequency 8 ð α g αβÞþ ð μνÞ; ð2:17Þ modes hμν and φμν are the propagating massless and massive spin-2 fields (see [27] for more details). with We shall focus on the scale beyond the Compton 2 −1 ð Þ wavelength of the massive graviton (mG ). In this scale, δR μν½φ Mμν can be ignored due to the Yukawa suppression. Since 1 ð0Þ ð0Þ ð0Þα ð0Þ we are interested in a localized object composed of the ∇ φαβ ∇ φ ∇ φβ ∇ φ ¼ 4 μ ν αβ þ ν ½α β μ massive graviton φμν, we assume the massless graviton hμν 0 0 0 0 0 0 does not appear. As a result, the metrics gμν and fμν are 1 ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ φαβ ∇ ∇ φ ∇ ∇ φ − 2∇ ∇ φ approximated by þ 2 ð ν μ αβ þ α β μν α ðμ νÞβÞ 1 ð0Þβ ð0Þ ð0Þ 1 ð0Þ 0 φ α αβ ð Þ μν þ ∇ φ α − ∇αφ ∇ μφν β − ∇βφμν : gμν ≃ g μν þ ; ð2:12Þ 2 ð Þ 2 MG ð2:18Þ ð0Þ φμν fμν ≃ g μν − ; ð2:13Þ The time average of this energy-momentum tensor gives αMG ð0Þ the source of the low-frequency background g μν.Wehave ð0Þ ð0Þ ð0Þ φ with jφμνj=MG ≪ j g μνj where g μν ¼ ημν þ δ g μν. used the notations such that the suffixes on μν are raised 0 ð0Þ We then consider the perturbations up to second orders. ð Þ ∇ Just the similar to the GR case [40], the high frequency and lowered by g μν and μ is the covariant derivative with φ ð0Þ modes μν give the source for the low frequency gravita- respect to g μν. tional potential, when we take the Isaacson average. The We then take the Newtonian limit: the nonrelativistic low-frequency projection (the Isaacson average) is usually limit of the massive graviton as well as the weak chosen as the spatial average (or the spacetime average) gravitational field approximation as for the background. because of massless fields. However, here we consider the The background metric is given by massive graviton, where momentum is much smaller 0 ð Þ μ ν 2 i j compared with the rest mass energy mG. Hence we perform g μνdx dx ¼ −ð1 þ 2ΦÞdt þð1 þ 2ΨÞγijdx dx ; the time average over the time interval T ¼ 2π=mG and we do not take any spatial average. ð2:19Þ Under this setting, the Einstein equations in bigravity whereas the massive graviton is expressed by without matter fluids are reduced into the Einstein-Klein- ψ 00 ψ 0 1 Gordon equations i −im t φμν ¼ pffiffiffi e G þ c:c:; ð2:20Þ ψtr 3 γij þ ψ ij 2 ð0Þμν 1 μν Φ Ψ ψ ψ ψ ψ G ¼ 2 hTG ilow; ð2:14Þ where f ; ; 00; 0i; tr; ijg are slowly varying func- M i 2 pl tions of ðt; x Þ, e.g. ∂t ψ ij ≪ mG∂tψ ij, and ψ ij is traceless,
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i ψ i ¼ 0. The three-dimensional Euclidean metric is the massive graviton. The polarization eigenstates are denoted by γij and ∂i is the derivative with respect to explicitly shown in Appendix A. Δ γij. We shall retain the three-dimensional covariance Note that the Laplace operator is connected with the l thus γij and ∂i are not necessary to be δij and the partial orbital angular momentum whereas the symmetry of the derivatives, respectively. The indices i, j are raised and field configuration is determined by j and jz. For instance, the spherically symmetric configuration is given by j ¼ lowered by γij. The constraint equations (2.16) yield jz ¼ 0 but this mode does not correspond to the zero orbital i i ψ ψ ψ ∂iψ ψ ∂jψ angular momentum mode as shown just below. tr ¼ 00; 00 ¼ i0; 0i ¼ ij; mG mG In what follows, we shall find two types of the massive graviton geons: the monopole geon and the quadrupole ð2:21Þ geon which correspond to the zero total angular momentum 0 which indicate the inequalities mode j ¼ and the zero orbital angular momentum mode l ¼ 0, respectively. We discuss them in order. We will ψ ψ ≪ ψ ≪ ψ j 00j; j trj j 0ij j ijj: ð2:22Þ summarize their properties in Table V.
Using the time average h ilow, we obtain A. Monopole geons 2 μν m We first consider spherically symmetric bound states of ≃ G ψ ψ ij 0 0 0 hTG ilow diag 4 ij ; ; ; ; ð2:23Þ Eqs. (2.24) and (2.25) which we call the monopole geon. The spherical symmetry leads to pffiffiffiffiffiffiffiffi where denotes the complex conjugate. The equations are i j −iEt −2 i j ψ dx dx ¼ 16πψ 0ðrÞe ðT0 0Þ dx dx reduced to the Poisson-Schrödinger equations ij rffiffiffi ; ij 2 −iEt 2 2 2 2 ¼ ψ 0ðrÞe ð2dr − r dΩ Þ; ð3:1Þ mG 3 ΔΦ ¼ ψ ψ ij; ð2:24Þ 8M2 ij pl where E is the “energy eigenvalue” of the bound state. ∂ Δ Since we are interested in the bound state, E is a real ψ − Φ ψ constant and ψ 0 is a real function of r. The Poisson- i ∂ ij ¼ 2 þ mG ij; ð2:25Þ t mG Schrödinger equations are then given by i where Δ ¼ ∂i∂ . The spatial component of the Einstein 1 d 2 dΦ 2 2 equation leads to Ψ ¼ −Φ. Note that these equations (2.24) r ¼ 4πGm ψ 0; ð3:2Þ r2 dr dr G and (2.25) are invariant under the rescaling
Φ → λ2Φ ψ → λ2ψ i → λ−1 i and ; ij ij; jx j jx j; −2 1 ψ 3 t → λ t; 2:26 d 2 d 0 2 ð Þ r ¼ 2 þ m Φ ψ 0 − 2m Eψ 0; ð3:3Þ r2 dr dr r2 G G thus, the system is scale invariant. We also note that the 1 8π 2 “wave function” of the Schrödinger equation is a tensor where G ¼ = Mpl. The mass of the geon is defined by ψ Z field ij which yields differences from the scalar wave ∞ function case as we will see in the following sections. ≔ 4π 2 2 ψ 2 M drr mG 0: ð3:4Þ 0
III. SELF-GRAVITATING MASSIVE GRAVITONS The mass M is rescaled under the scaling (2.26) as In this section, we study the periodical solutions with the → λ localized massive gravitons, i.e., the bound states of the M M: ð3:5Þ system (2.24) and (2.25). To find the bound states, we focus The asymptotic forms of the solutions are analytically ψ on some eigenstates of the angular momentum of ij. found as As detailed in Appendix A, the angular dependence of each eigenstate is given by the so-called pure-orbital spherical GM Φ → − ; ð3:6Þ harmonics ðTs Þ , where j and j inside the parentheses r j;jz ij z pffiffiffiffiffiffiffiffiffi are the total angular momentum quantum number and the −1 ~ ψ 0 → Cr W ~ −1=2 5 2 −2Er~ ð−2EÞ ;2 total angular momentum projection quantum number. pffiffiffiffiffiffiffi ~ The total angular momentum j and the orbital angular e− −2Er~ momentum l are related by j ¼ l þ s with s ¼ 0; 1; 2. → C0 pffiffiffiffiffiffiffi ; ð3:7Þ ~ We note that the label s does not refer to polarizations of r1−1= −2E
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TABLE I. Energy eigenvalues and 95% mass radii of monopole geon. ~ ~ Node E R95% n ¼ 0 −0.02708 37.72 n ¼ 1 −0.01140 89.85 n ¼ 2 −0.006302 161.9 n ¼ 3 −0.003995 254.3 n ¼ 4 −0.002757 367.4 n ¼ 5 −0.002016 500.0
n increases as shown in Table I. In Table I, we summarize the scale-invariant energy eigenvalues E~ and the 95% ~ mass radii R95%, below which 95% of the mass energy is included. Since the 95% mass radius increases in the higher-node state, the lowest-node (the lowest-energy) state gives the most compact object. The orbital angular momentum l does not vanish in the j ¼ 0 mode since the zero total angular momentum in the tensor field is realized only when l ¼þ2 and s ¼ −2.Asa 2 result, the orbital angular momentum term 3=mGr exists in Eq. (3.3). The regularity at the center of the monopole geon leads to that ψ 0 vanishes at r ¼ 0. As a result, the monopole geon has a shell-like field configuration as shown in Fig. 1. On the other hand, the Newtonian potential Φ is a scalar field, which has no orbital angular momentum for a spherical configuration, and then Φ is finite at the center. FIG. 1. The solutions to the Poisson-Schrödinger equations of the spherically symmetric bound state j ¼ 0. The red curves B. Quadrupole geons show the configuration of the massive graviton ψ~ 0 and the Next we consider a nonspherically symmetric configu- blue dashed curves correspond to the gravitational potential Φ~ . ration of the massive graviton and find its bound state. The solutions are specified by the number of nodes of ψ~ 0. We assume a quadrupole configuration of ψ ij, which is given by with pffiffiffiffiffiffiffiffi X2 −iEt þ2 E ψ ij ¼ 16π ψ 2;j ðrÞe ðT2 Þ ; ð3:10Þ E~ ¼ ; r~ ¼ GMm2 r; ð3:8Þ z ;jz ij 2 3 G jz¼−2 ðGMÞ mG þ2 0 where ðT2;j Þij is given by the spherical harmonics of the where Wκ;μðzÞ is the Whittaker function, and C and C are z integration constants. We notice that the dimensionless quadrupole mode. We call this bound state (3.10) the ~ quadrupole geon. As shown in Appendix A, ðTþ2 Þ has variables E and r~ are invariant under the scaling (2.26). 2;jz ij Some numerical solutions to the Poisson-Schrödinger special properties which lead to equations are shown in Fig. 1 where we have used the dimensionless scale-invariant variables: X2 ψ ψ ij ¼ 4 ψ 2 ; ð3:11Þ ij 2;jz jz¼−2 Φ ψ 0=M Φ~ ψ~ pl ¼ 2 ; 0 ¼ 2 : ð3:9Þ ffiffiffiffiffiffiffiffi ðGMmGÞ ðGMmGÞ X2 p 16π d dψ 2 Δψ −iEt 2 ;jz þ2 ij ¼ e 2 r ðT2 Þ : ð3:12Þ r dr dr ;jz ij The boundary condition is assumed to be dΦ=dr, jz¼−2 dψ 0=dr → 0 as r → 0, and Φ and ψ 0 vanish at infinity. The solutions are parametrized by the number of nodes. As a result, the energy density of ψ ij turns to be spherically We call the solution with n nodes the n-th eigenstate of the symmetric. We then assume Φ ¼ ΦðrÞ. The basic equa- monopole geon because the energy eigenvalue increases as tions are now given by
044005-5 AOKI, MAEDA, MISONOH, and OKAWA PHYS. REV. D 97, 044005 (2018) X2 Whereas the radial dependence of the quadrupole geon is 1 d 2 dΦ 2 2 r ¼ 4πGm ψ 2 ; ð3:13Þ ψ 2 G ;jz uniquely determined by 2E for each E, the angular r dr dr −2 jz¼ dependence is not unique because the expansion coeffi- þ2 cients aj of five tensor spherical harmonics ðT2;j Þij are 1 d dψ 2 z z 2 ;jz 2 Φ − ψ 2 r ¼ mGðmG EÞ 2;jz : ð3:14Þ arbitrary although there is one constraint (3.16). Each r dr dr energy eigenstate of the quadrupole geon is degenerated. Needless to say, the scalar bosonic field with j ¼ 0 has no The orbital angular momentum l vanishes in the quadru- degeneracy. Although the equations (3.17) and (3.18) are pole geon although that is not the case in the monopole the same, there are different features between the gravitons geon. Note that the orbital angular momentum vanishes and the scalar bosons. only when j ¼ 2 and s ¼þ2. Another difference from the bosonic case is that the field The variables ψ 2 are the solutions to the Sturm- ;jz configuration is not spherically symmetric. The quadrupole Liouville equations with the same eigenvalue E under geons have anisotropic pressures although they can be the boundary conditions dψ 2 =dr → 0 at the center and ;jz ignored to discuss the bound states. As a result, the ψ 2 → 0 at infinity. Hence, the functional forms of ψ 2 ;jz ;jz quadrupole geons may emit gravitational waves. We will ψ are uniquely determined by some function 2EðrÞ for each return some consequences of the gravitational wave emis- E ψ 2 and then ;jz are expressed by sion in Sec. V. Before ending this section, we comment on the polar- ψ ψ 2;jz ¼ ajz 2E; ð3:15Þ izations of the geons. The massive graviton has spin-0, spin-1, and spin-2 polarization modes. The monopole geon where ajz are constants which are normalized as is an eigenstate of the spin-0 polarization while the X quadrupole geon is not an eigenstate of the polarizations. 2 1 The relation between the polarization eigenstates and the jajz j ¼ : ð3:16Þ jz angular momentum eigenstates is given by Eq. (A26) in Appendix A which explicitly shows that ðTþ2 Þ is not the 2;jz ij Then, the equations are reduced to polarizations eigenstate. 1 d dΦ 2 4π 2 ψ 2 IV. LINEAR FLUCTUATIONS AROUND 2 r ¼ GmG 2E; ð3:17Þ r dr dr THE MONOPOLE GEON 1 d 2 dψ 2E In the previous section, we find two types of geon r ¼ 2m ðm Φ − EÞψ 2 ; ð3:18Þ r2 dr dr G G E solutions; monopole and quadrupole. Since the binding energy (−E~ ) of the quadrupole geon is much larger than which are exactly the same equations of the Poisson- that of the monopole one, the monopole geon is expected to Schrödinger system for the scalar field with the j ¼ 0 mode. be unstable. In order to confirm it, we study the perturba- The solutions to Eqs. (3.17) and (3.18) have been already tions around the monopole geon. As long as the perturba- investigated, e.g. in [44–49]. The energy eigenvalues and the tions do not spoil the Newtonian approximation, we can use the equations (2.24) and (2.25). Hence, we consider the explicit forms of ψ 2E are shown in Table. II and Fig. 2. Compared to those of the monopole geon, the lowest energy configurations eigenvalue of the quadrupole geon (E~ ¼ −0.1628)ismuch Φ ¼ Φ0 þ δΦ; ψ ¼ ψ 0 þ δψ ; ð4:1Þ lower than that of the monopole geon (E~ ¼ −0.0278). Since ij ;ij ij the lower energy state should be more stable than the higher where Φ0 and ψ 0 are the solutions of the n-th state of the state, the monopole geon may transit to the quadrupole geon. ;ij monopole geon. The perturbations δψ can be decomposed In the next section, we will indeed find that the monopole ij into the odd parity perturbations and the even parity geon is unstable against quadrupole mode perturbations. perturbations,
TABLE II. Energy eigenvalues and 95% mass radii of quadru- δψ δψ ðevenÞ δψ ðoddÞ ij ¼ ij þ ij ; ð4:2Þ pole geon. ~ ~ and the different parity modes are not coupled with Node E R95% each other. n ¼ 0 −0.1628 7.830 There is no odd parity mode of the gravitational n ¼ 1 −0.03080 36.00 δψ ðoddÞ 2 −0 01250 potential. The odd parity perturbations ij are obtained n ¼ . 85.04 by only solving the Schrödinger equation with the potential n ¼ 3 −0.006747 154.4 Φ ¼ Φ0 and then the problem can be reduced to the
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background spherical symmetry, we just omit the suffixes j and jz hereafter. The perturbed Poisson equation yields
1 ∂2 1 jðj þ Þ ðrδϕÞ − δϕ ¼ 4πGψ 0ðδξ þ δξ Þ; ð4:5Þ r ∂r2 r2 while the Schrödinger equation with j ≥ 2 gives 1 ∂2 jðj þ 1Þþ6 − ðrδξÞþ þ m U δξ 2m r ∂r2 2m r2 G G pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG 3 1 ∂ ψ δϕ jðj þ Þ δχ δξ þ mG 0 þ 2 ¼ i ∂ ; ð4:6Þ mGr t 1 ∂2 jðj þ 1Þþ4 − ðrδχÞþ þ m U δχ 2m r ∂r2 2m r2 G pG ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG 3 1 − 1 2 ∂ jðj þ Þ δξ ðj Þðj þ Þ δσ δχ þ 2 þ 2 ¼ i ∂ ; mGr mGr t FIG. 2. The configurations of ψ 2 (red curves) and Φ (blue dashed curves) in the case of the quadrupole mode. ð4:7Þ 1 ∂2 ðj − 1Þðj þ 2Þ eigenvalue problem of a self-adjoint operator. Hence, the − rδσ m U δσ 2 ∂ 2 ð Þþ 2 2 þ G δψ ðoddÞ mGr r mGr eigenvalues, i.e., the frequencies of ij , are real and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 2 ∂ then there is no growing mode (and no decaying mode). ðj Þðj þ Þ δχ δσ þ 2 ¼ i ∂ ; ð4:8Þ The monopole geons are linearly stable against the odd mGr t parity perturbations. We then study the even parity perturbations. We intro- where U ¼ Φ0 − E=m. We note j ¼ 0 and j ¼ 1 of the duce a different basis of the multipole expansion form perturbations are exceptional modes. The variables δχ and ðTs Þ which is called the pure-spin spherical harmonics δσ are undefined for the j ¼ 0 mode in which (4.7) and j;jz ij A 2 1 δσ denoted by ðY Þ with A ¼ S0;E1;E2;B1;B2. The label (4.8) are not obtained. For the j ¼ mode, the variable j;jz ij A characterizes the polarization states of the massive spin-2 is undefined and the equation (4.8) does not exist. However, 0 field and the parity (see Appendix A). Then, the even parity Eq. (4.6) [and Eq. (4.7)] is correct even for the j ¼ mode 1 perturbations can be expressed by (and the j ¼ mode) since the coefficients in front of the undefined variable vanish. pffiffiffiffiffiffiX X δΦ 4π δϕ We find solutions of the form ¼ j;jz ðt; rÞYj;jz ; ð4:3Þ ≥0 ≤ j jjzj j W − ω W ω ffiffiffiffiffiffiffiffi X X δϕ ¼ e i t þ ei t; ð4:9Þ p r r δψ ðevenÞ ¼ 16πe−iEt δξ ðt; rÞðYS0 Þ ij j;jz j;jz ij ≥0 j jjzj≤j 1 1 ffiffiffiffiffiffiffiffi X X δξ ξ ξ −iωt ξ − ξ iω t p ¼ ð A þ BÞe þ ð A BÞe ; ð4:10Þ þ 16πe−iEt δχ ðt; rÞðYE1 Þ r r j;jz j:jz ij j≥1 ≤ jjzj j 1 1 pffiffiffiffiffiffiffiffi X X −iωt iω t − E2 δχ χ χ χ − χ þ 16πe iEt δσ ðt; rÞðY Þ ; ð4:4Þ ¼ ð A þ BÞe þ ð A BÞe ; ð4:11Þ j;jz j;jz ij r r ≥2 j jjzj≤j 1 1 δσ σ σ −iωt σ − σ iω t ¼ ð A þ BÞe þ ð A BÞe ; ð4:12Þ where Yj;jz is the scalar spherical harmonics. Since the r r different modes of j and jz are not coupled due to the where fW;ξA;B; χA;B; σA;Bg are complex functions of r and ω is a complex constant. The regularity conditions 2We have used two types of the orthonormal sets of the tensor s A at the center lead to the boundary condition fW;ξA;B; harmonics ðTj;j Þij and ðYj;j Þij which are related by the z z s orthogonal transformation (A26). Although ðT Þ are useful χA;B; σA;Bg → 0. At infinity we assume the boundary j;jz ij to discuss the bound state solutions, the perturbation equations condition δξ; δψ; δχ; δσ → 0. Except for the j ¼ 0 mode, are given by a simpler form by using ðYA Þ . δϕ δξ δχ δσ −1 j;jz ij all variables f ; ; ; g decay faster than r . Hence,
044005-7 AOKI, MAEDA, MISONOH, and OKAWA PHYS. REV. D 97, 044005 (2018) we can assume the boundary condition fW;ξA;B; χA;B; the equations for j ¼ 1 are σA;Bg → 0 at infinity. However, δϕ of the j ¼ 0 mode −1 decays as r , i.e., W → W0 (constant). We cannot assume → 0 0 W at infinity in the case of j ¼ ; instead, we should pffiffi 0 6 1 assume dW=dr → 0. 0 Oψ 0 2 0 1 mGr ξ The Poisson equation yields that W is formally given by B pffiffi C A B −1 6 CB C B Oψ − 8πGmGψ 0Oϕ ψ 0 0 2 0 C ξ B mGr CB B C −8π O−1ψ ξ B pffiffi CB C W ¼ G ϕ 0 A; ð4:13Þ B 6 C@ χ A 0 2 0 Oχ A @ mGr A −1 pffiffi χ where Oϕ is the inverse operator of Oϕ, which is 6 0 O 0 B m r2 χ defined by 0 1 G ξA 2 1 B C d jðj þ Þ B ξB C Oϕ ¼ − þ : ð4:14Þ ωB C dr2 r2 ¼ @ A; ð4:16Þ χA
The perturbed equations for the j ¼ 0 mode are given by χB 0 Oψ ξ ξ A ω A −1 ¼ ; Oψ − 8πGm ψ 0O ψ 0 0 ξ ξ G ϕ B B and the equations for the general modes j ≥ 2 are ð4:15Þ written as
0 1 pffiffiffiffiffiffiffiffiffiffiffiffi 3jðjþ1Þ B 0 Oψ 0 2 00C B mGr C0 1 0 1 B pffiffiffiffiffiffiffiffiffiffiffiffi C 3j j 1 B −1 ð þ Þ C ξA ξA B Oψ − 8πGmGψ 0Oϕ ψ 0 0 2 000CB C B C B mGr CB C B C B pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CB ξB C B ξB C B 3jðjþ1Þ ðj−1Þðjþ2Þ CB C B C B 0 2 0 Oχ 0 2 CB χ C B χ C B mGr mGr CB A C B A C B pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CB C ¼ ωB C; ð4:17Þ 3 1 −1 2 B jðjþ Þ ðj Þðjþ Þ CB χB C B χB C B 2 0 Oχ 0 2 0 CB C B C B mGr mGr CB σ C B σ C B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C@ A A @ A A B ðj−1Þðjþ2Þ C B 0002 0 Oσ C σ σ B mGr C B B @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ðj−1Þðjþ2Þ 002 0 Oσ 0 mGr where easily constructed from the definitions of the variables (4.9)–(4.12). 1 d2 jðj þ 1Þþ6 We use the spectral method and then the problem is Oψ ¼ − þ þ m U; ð4:18Þ 2m dr2 2m r2 G reduced to the eigenvalue problem of the matrix. Since the G G infinity cannot be treated in numerical calculations, we choose a range ð0; 10R95%Þ for r for which we have 1 2 1 4 O − d jðj þ Þþ confirmed the numerical results are sufficiently converged. χ ¼ 2 2 þ 2 þ mGU; ð4:19Þ mG dr 2mGr When all eigenvalues are real, the solution is stable; while if there exists an imaginary part of ω, there are always 1 d2 ðj − 1Þðj þ 2Þ exponentially growing modes because fω; −ω; ω ; −ω g Oσ ¼ − þ þ m U: ð4:20Þ 2m dr2 2m r2 G are the eigenvalues of the perturbations. G G In Tables III and IV, we show some eigenvalues of ω ≥ 0 ω ≥ 0 ω ω Reð Þ ,Imð Þ , where the rescaled is defined by Once we find one solution with the eigenvalue , we can ω −ω ω ω~ easily obtain the solutions with the eigenvalues ; ¼ 2 3 : ð4:21Þ ðGMÞ m and −ω . Supposing fW;ξA; ξB; g with ω is a solution, G the solution with −ω is given by fW;ξA; −ξB; g. We note that the j ¼ 0 mode perturbations have a trivial The solutions with eigenvalues ω and −ω are also solution such that ω ¼ 0 and
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~ ~ ~ TABLE III. The seven lowest eigenvalues ωi of the j ¼ 0 mode perturbations with ReðωiÞ,ImðωiÞ ≥ 0 and jω~ 0j < jω~ 1j < jω~ 2j < .
Zeroth state First state Second state
ω~ 0 00 0 ω~ 1 0.005127 0.001389 0.0005677 ω~ 2 0.01326 0.003794 þ 0.0007111i 0.001571 þ 0.0003633i ω~ 3 0.01699 0.005369 0.002495 ω~ 4 0.01936 0.006537 0.002879 þ 0.0001624i ω~ 5 0.02095 0.007374 0.003445 ω~ 6 0.02197 0.008009 0.003839
j 0 TABLE IV. The seven lowest eigenvalues ω~ i of perturbations We first discuss the ¼ mode perturbations. We find around the zeroth state of the monopole geon. the n-th state has the n different unstable modes as shown in Table III. The unstable modes could be interpreted as the 1 2 3 j ¼ j ¼ j ¼ transition modes to the lower energy eigenstate from the ω~ 0 0.00000 0.00000 0.004440 higher one. Only the zeroth state of the monopole geon is ω~ 1 0.004674 0.0005155i 0.004918 linearly stable against the j ¼ 0 mode perturbation. ω~ 2 0.00622 0.008190 0.005600 Henceforth, we shall only show the results of the ω~ 3 0.01078 0.008469 0.01133 perturbations around the zeroth state. Our numerical ω~ 4 0.01132 0.008660 0.01189 ω~ calculations show that the zeroth state is linearly stable 5 0.01551 0.01070 0.01346 against the j ≠ 2 mode perturbations. However, we find ω~ 6 0.01581 0.01358 0.01559 there exists an unstable mode in the j ¼ 2 mode perturba- tions as shown in Table IV. The explicit forms of the unstable mode are depicted in Fig. 3. ξ ∝ ψ 0=r; ξ 0; δϕ 0; 4:22 B A ¼ ¼ ð Þ As a result, the zeroth state of the monopole geon is unstable against the j ¼ 2 mode perturbation which is which corresponds to a phase shift of the background consistent with the energy argument. However, it is not so iϵ solution, i.e., ψ 0;ij þ δψij ¼ e ψ 0;ij with a small constant manifest whether the unstable mode indeed represents the ϵ. We also find almost zero eigenvalues in the j ¼ 1 and transition to the zeroth state of the quadrupole geon from j ¼ 2 modes perturbations which are not trivial solutions. the zeroth state of the monopole geon. The quadrupole geon is purely given by the j ¼ 2 mode and the gravita- tional potential is exactly spherically symmetric. We expect that the j ≠ 2 modes of ψ ij and the nonspherical part of Φ decay in time and then the configurations of Φ and ψ ij will converge towards the zeroth state of the quadrupole geon. However, we need to fully solve (2.24) and (2.25) for the monopole geon with perturbations in order to discuss this process, which we leave for a future work. Furthermore, the stability of the quadrupole geon is an open question. Since the field configuration is not spheri- cally symmetric in this case, different modes of the multi- pole expansion should be coupled and the stability analysis becomes very difficult. We hope that the quadrupole geon is linearly stable since it is the only mode with the zero orbital angular momentum and thus the lowest energy eigenstate of (2.24) and (2.25).
V. SUMMARY AND DISCUSSIONS We find vacuum solutions such that the massive grav- FIG. 3. The solution to the j ¼ 2 mode perturbation equations itons are localized by their gravitational energy in the with ω~ ¼ 0.00052i (the unstable perturbation around the mo- context of the bigravity theory, which we have called the nopole geon). The amplitudes of the solutions are scale free massive graviton geon by following Wheeler’s idea [5].In because the equations are linear. the Newtonian limit, the vacuum equations of the bigravity
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2 ≲ −1 TABLE V. Properties of the massive graviton geons. unstable when GM mG , in which case we may expect a hairy black hole surrounded by time-dependent massive Monopole geon Quadrupole geon gravitons is formed. Angular momentum j ¼ 0; l ¼ 2 j ¼ 2; l ¼ 0 The quadrupole geons may emit gravitational waves. We μν Configuration Angular spherical quadrupole note, however, that the low-frequency part of TG has no Radial shell (Fig. 1) ball (Fig. 2) oscillation due to the phase cancellation up to the second Lowest energy E~ ¼ −0.027 E~ ¼ −0.16 order. Hence, we study the backreaction of the massive Degeneracy 1 5 graviton φμν on the high-frequency massless gravitons hμν. Polarization scalar not polarized 00 We find TG is static and spherically symmetric, while the spatial components are given by 3 theory can be reduced to the Poisson-Schrödinger equa- ij 2 ik j ij kl −2im t T ≃ m −ψ ψ þ γ ψ ψ e G þ c:c:; ð5:4Þ tions with the tensor “wave function.” We numerically G G k 4 kl construct two types of the massive graviton geons: the spherically symmetric configuration of the massive grav- which clearly has the oscillating anisotropic stress and then iton (the monopole geon) and the configuration is repre- the quadrupole geons emit the gravitational waves hμν. The sented by the quadrupole modes of the spherical harmonics gravitational waves observed sufficiently far from the geon (the quadrupole geon). We summarize their properties in are given by Table V. We find the energy density distribution is spheri- Z 4G cally symmetric not only for the monopole geon but also hTT ¼ eik·xP d3x0Tklðt; x0Þe−ik·x0 þ c:c:; ð5:5Þ for the quadrupole geon. The lower energy eigenstate ij r ij;kl G should be more stable than the higher eigenstate. Indeed, the perturbation analysis around the monopole geon reveals where k is the wave vector of the gravitational waves with 2 2 that there exists an unstable mode in the quadrupole mode k ¼ð2mGÞ and Pij;kl is the transverse-traceless projec- perturbations which is consistent with the energy argument. tion operator. Note that the pure-orbital spherical harmon- Since the system is scale invariant, we can construct a ics with j ¼ 2 and s ¼þ2 are just constant matrices in massive graviton geon with any size of R95% as long as the Cartesian coordinates (see Appendix A). Hence, the ≫ −1 R95% mG . The most stable and compact massive grav- integration in (5.5) is iton geon would be given by the zeroth state of the Z Z quadrupole geon whose mass and typical amplitude are 3x0 kl −ik·x0 ∝ 3x0ψ 2 0 −ik·x0 d TG e d 2Eðr Þe Z 7.8 2π ≃ 0 0 2 0 0 GM 2 ; ð5:1Þ ¼ dr r ψ 2 ðr Þ sin½2m r : ð5:6Þ m R95% E G G mG
1=2 2 jφμνj=MG ≃ 0.3 × α ðGMmGÞ : ð5:2Þ As a result, the gravitational wave emission is suppressed (probably exponentially) for the nonrelativistic geons, i.e., If the size of the geon becomes smaller such as ∂rψ 2E ≪ mGψ 2E. The nonrelativistic quadrupole geons are ∼ −1 R95% mG , we cannot ignore the relativistic effects and long-lived objects even if they have the oscillating part of the present approximations are no longer valid. Nevertheless, the anisotropic stress. we may evaluate the maximum mass of the massive graviton The gravitational potential of the quadrupole geon has ≲ −1 the same structure as the case of a massive scalar field of the geon from the expression (5.1) as follows: if R95% mG , −1 0 M ≳ ðGm Þ , then the Schwarzschild radius 2GM j ¼ mode because Eqs. (3.17) and (3.18) are the same G – becomes larger than the size of the massive graviton geon ones as those in the scalar case (see e.g., [44 49]). Hence, R95% the quadrupole geon can represent a central part of a dark , which may become a black hole in a relativistic −22 situation. Then we expect that the massive graviton geon matter halo if the mass of the graviton is about 10 eV ≲ just as the case with the ultralight scalar dark matter exists only when M Mmax, where (see [49] for example and references therein). −10 Because the matter fields are coupled with gμν given by −1 10 eV M ∼ ðGm Þ ∼ 1 M⊙ : ð5:3Þ φ max G m (2.12), the massive graviton μν can be observed as G localized “gravitational waves” with the frequency and This mass bound of the geon is consistent with the stability amplitude of analysis of the Schwarzschild black hole in the bigravity. m The papers [50,51] showed that the Schwarzschild black f ≃ m =2π ∼ 10−8 G Hz; ð5:7Þ 2 ≳ −1 G 10−22 eV hole solution is stable when GM mG while it turns to be
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2 eigenvalue is negative. When all fluctuation energy is jφμνj M ∼ 10−7α1=2 : ð5:8Þ converted into the geons, the production of geons will stop. M 109 G M⊙ This is just like a structure formation by Jeans instability. Nonetheless, extended analysis may be required to con- In the case α M =M 2 ∼ 1, i.e., M ∼ M , this ¼ð pl GÞ pl G clude the nonlinear (in)stability of the Minkowski spacetime oscillation has a large amplitude; thus, such a possibility with massive gravitons. We have also to study a quantum is already excluded by the pulsar timing array (PTA) mechanical (in)stability against quantum fluctuations. observations [52]. One possibility to be a consistent Since the ground state of the massive graviton geon scenario is introducing the hierarchy of the mass scales would be the quadrupole geon, the produced geons should ≳ 108 α1=2 ≲ 10−8 MG Mpl ( ). Another possibility is intro- eventually transit to the quadrupole geon. Although the ducing the Z2 symmetry of the massive graviton [27]; the resultant quadrupole geon will emit the gravitational waves, theory is symmetric under φμν → −φμν. The symmetry the gravitational wave emission is drastically suppressed if leads to MG ¼ Mpl and the matter fields are coupled with the produced geon is nonrelativistic. As a result, we may the metric [53] expect that the geon may not decay soon and the localized qffiffiffiffiffiffiffiffiffiffi structure would remain. h α i 1 −1 Although we have only discussed the classical features g μν ¼ gμν þ 2gμα g f þ fμν eff 4 ν of the Poisson-Schrödinger equations (2.24) and (2.25), 0 ψ ð Þ 1 α one may quantize the massive graviton regarding ij as ¼ g μν − φμαφ ν þ : ð5:9Þ 4M2 indeed the wave function of the massive graviton. The pl “Hamiltonian operator” could be defined by In this case, the amplitude of the oscillation is given by Δ α 2 −15 ˆ jφμαφ ν=4M j ∼ 10 , which can be detectable by the H ≔− þ m Φ: ð5:10Þ pl 2m G future observations of PTA. We note that the ultralight G scalar dark matter predicts oscillations of the gravitational The massive graviton geon may be interpreted as a Bose- potential because the coherent oscillation of dark matter Einstein condensate of the massive gravitons just by the leads to the oscillation of the pressure [54]. In the tensor analogy to the massive scalar field and then it can be seen as a case, we also expect the oscillation of the gravitational macroscopic object. If the massive graviton geon can transit potential [see (5.4)]. quantum mechanically from one state to another, this To discuss the dark matter scenario, we have to take into describes a quantum transition of spacetime since the account other observational constraints, such as the spacetime metric is given by (2.12) [or (5.9)]. On the other Yukawa force constraints and the gravitational wave con- hand, introducing a matter field and supposing the gravita- straints, on the graviton mass. Supposing the linear theory tional potential is dominated by the matter with a mass M⋆, is a good approximation in the observational scales, the i.e., Φ ¼ −GM⋆=r, the Schrödinger equation is exactly 10−23 ≲ ≲ 10−4 mass range eV mG eV is excluded when solved in a way similar to the quantization of the ∼ – Mpl MG (see [55 58] for examples). However, if Hydrogen atom. The nonrelativistic massive graviton may ≫ MG Mpl, the interaction of the massive graviton be quantized by the method of the canonical quantization. becomes weak and then the constraints are relaxed (For Finally, we notice that interesting phenomena must exist instance, the lunar laser ranging experiment can give a in relativistic extensions of our study. For instance, we need 2 ≳ 10−11 graviton mass constraint only if ðMpl=MGÞ [59]). to discuss the relativistic system to obtain the precise value Furthermore, the observational constraints cannot be of the maximum mass of the geon. In the Newtonian limit, applied to the Z2 symmetric model since the Z2 symmetry the quadrupole geon is degenerated because the projection −1φ μν angular momentum j does not contribute to the equations prohibits the Yukawa interaction MG μνT . As a result, z the graviton mass constraints cannot be applied to the of the geons. However, in the relativistic system, it is no longer true. We may expect that the degeneracy is resolved. graviton dark matter models with the hierarchy MG ≳ 8 Furthermore, the Vainshtein mechanism is irrelevant to the 10 M or with the Z2 symmetry. pl nonrelativistic geons while it should be important for the The gravitational geons are recently discussed in the relativistic geons. All these investigations require treat- context of the nonlinear instability of the AdS spacetime ments beyond the Newtonian approximation. [16–18,60]. The energy eigenvalue of the massive graviton geon is negative which suggests that the massive graviton ACKNOWLEDGMENTS geons can be produced from fluctuations around the Minkowski vacuum. Does it imply the instability of K. A. would like to thank the Institute of Cosmology and Minkowski spacetime if a graviton is massive? We expect Gravitation, University of Portsmouth for their hospitality that is not the case, because the mass energy of the during his visit. K. A. also acknowledges Kazuya Koyama produced geon is not negative although the energy and Jiro Soda for useful discussions. K. M. would like to
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i thank Gary Gibbons for the information about the geons ξy ¼ð0; cos ϕ; − cot θ sin ϕÞ; ðA7Þ and the periodic solutions. This work was supported in part by JSPS KAKENHI Grant No. JP15J05540 (K. A.), i ξz ¼ð0; 0; 1Þ; ðA8Þ No. JP16K05362 (K. M.), No. JP17H06359 (K. M.), and No. JP17K18792 (H. O.). in the spherical coordinates, i; j; k ¼ðr; θ; ϕÞ. The spin ˆ angular momentum operators SI are defined so that ˆ ˆ ˆ APPENDIX A: PURE-SPIN AND PURE-ORBITAL JI ¼ LI þ SI. The magnitude squared of the operators SPHERICAL HARMONICS are defined by the sum of the squared operators, e.g., As is well known, the angular dependence of a quantity Lˆ 2 Lˆ 2 Lˆ 2 Lˆ 2 (which can be either a scaler, a vector, or a tensor) can be ¼ x þ y þ z : ðA9Þ expanded in terms of the spherical harmonics. Here, we focus on a symmetric and traceless three-dimensional We note tensor t and consider a multipole expansion of it. We ij Sˆ 2 2 2 1 introduce two types of the orthonormal bases of the tij ¼ × ð þ Þtij; ðA10Þ multipole expansion which are called the pure-spin spherical harmonics ðYA Þ and the pure-orbital spherical which explicitly shows that the symmetric and traceless j;jz ij 3 tensor is a spin-2 field. harmonics ðTs Þ , respectively (see [61,62] for more j;jz ij We assume that some tensor spherical harmonics ðY Þ details). Note that the suffix j inside the parentheses j;jz ij are the eigenstates of the operators Jˆ 2 and Jˆ : represents the total angular momentum while the indices z outside the parentheses are the spatial indices of the tensor. 2 Jˆ ðY Þ ¼ jðj þ 1ÞðY Þ ; ðA11Þ We first introduce the orbital angular momentum oper- j;jz ij j;jz ij ators Lˆ and the total angular momentum operators Jˆ I I Jˆ which satisfy the commutator relations zðYj;jz Þij ¼ jzðYj;jz Þij; ðA12Þ X ˆ ˆ ˆ where j 0; 1; 2; and j 0; 1; …; j. However, ½LI; LJ ¼i ϵIJKLK; ðA1Þ ¼ z ¼ K these conditions do not determine the tensor spherical X harmonics completely. We can further assume a property ˆ ˆ ˆ ½JI; JJ ¼i ϵIJKJK; ðA2Þ of them. K The pure-spin spherical harmonics ðYA Þ are under- j;jz ij X stood as polarization eigenstates of a spherical wave of a Jˆ Lˆ ϵ Lˆ ½ I; J ¼i IJK K; ðA3Þ spin-2 field where the polarization states are labeled by K A ¼ S0;E1;E2;B1;B2. The propagating direction of the spherical wave is given by the unit radial vector ni ¼ ri=jrj. where I, J, K are the indices of the Cartesian coordinates, The spin-2 polarization modes E2, B2 are transverse to the I;J;K ¼ðx; y; zÞ. The operators are explicitly given by propagating direction, Lˆ ≔−iðr × ∂=∂rÞ ; I I i E2 i B2 0 n ðYj;j Þij ¼ n ðYj;j Þij ¼ ; ðA13Þ − ξi ∂ z z ¼ i I i; ðA4Þ the spin-1 polarization modes E1, B1 are mixed longi- Jˆ ≔−L I i ξI ; ðA5Þ tudinal and transverse
ξi i E1 i B1 where I is the rotational Killing vectors around I ¼ x, y, z n ðY Þ ≠ 0;nðY Þ ≠ 0; L j;jz ij j;jz ij axes and ξI are the Lie derivatives with respect to them. ninjðYE1 Þ ¼ ninjðYB1 Þ ¼ 0; ðA14Þ We recall that the indices i, j, k are not necessary to be j;jz ij j;jz ij those of the Cartesian coordinates. For instance, the Killing vectors are written by and the spin-0 polarization mode S0 is the longitudinal mode i ξx ¼ð0; − sin ϕ; − cot θ cos ϕÞ; ðA6Þ niðYS0 Þ ≠ 0;ninjðYS0 Þ ≠ 0: ðA15Þ j;jz ij j;jz ij
The modes E1 and E2 are called the electric-type parity, i.e., 3 θ ϕ → π − θ π ϕ If tij pisffiffiffi not traceless, the trace part can be expanded in terms of the even parity according to ð ; Þ ð ; þ Þ while γ 3 Yj;jz ij= which corresponds to the spin-zero state, i.e., B1 and B2 are the magnetic-type parity, i.e., the odd parity. Sˆ 2 γ 0 ðYj;jz ijÞ¼ . The pure-spin spherical harmonics are orthonormal
044005-12 MASSIVE GRAVITON GEONS PHYS. REV. D 97, 044005 (2018) Z A A0 ij TABLE VI. Defined harmonics in each total angular momentum. dΩðY Þ ðY 0 0 Þ ¼ δ 0 δ 0 δ 0 ; ðA16Þ j;jz ij j ;jz AA jj jzjz Pure-orbital Pure-spin (orbital angular where the complex conjugate is given by ðYA Þ ¼ (helicity) momentum) j;jz ij −1 jz A 0 −2 ð Þ ðYj;−j Þij. In the spherical coordinates, they are j ¼ A ¼ S0 s ¼ z 0 l 2 explicitly given by (h ¼ ) ð ¼ Þ j ¼ 1 A ¼ S0;E1;B1 s ¼ 0; −1; −2 0 1 l 1 2 3 ðh ¼ ; Þð¼ ; ; Þ 1 2Y 0 j ≥ 2 All modes are defined S0 j;jz ðY Þ ¼ pffiffiffi ; ðh ¼ 0; 1; 2Þðl ¼ 0; 1; 2; …Þ j;jz ij 6 −r2Y γˆ j;jz ab 0 E1 rðYj;jz Þb 0 S0 ðY Þ ¼ ; ðYj;j Þij ¼ðYj;j Þij; j;jz ij 0 z z 1 þ1 ffiffiffi E1 B1 00 ðY Þij ¼ p ½ðY Þij þ iðY Þij ; E2 j;jz 2 j;jz j;jz ðY Þ ¼ 2 ; j;jz ij r Y ð j;jz Þab 1 −1 ffiffiffi E1 − B1 c ðYj;j Þij ¼ p ½ðYj;j Þij iðYj;j Þij ; 0 rϵb ðYj;j Þ z 2 z z ðYB1 Þ ¼ z c ; j;jz ij 0 1 ðYþ2 Þ ¼ pffiffiffi ½ðYE2 Þ þ iðYB2 Þ ; 00 j;jz ij 2 j;jz ij j;jz ij ðYB2 Þ ¼ ; ðA17Þ j;jz ij 2 c 1 r ϵ ðY Þ −2 E2 B2 ða j;jz bÞc ðY Þ ¼ pffiffiffi ½ðY Þ − iðY Þ ; ðA21Þ j;jz ij 2 j;jz ij j;jz ij with which also indicates that A ¼ S0, A ¼ E1;B1, and A ¼ E2;B2 represent the polarization states of the spin-2 field, 0 1 respectively. In the j ¼ mode, there exists only the − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ helicity-zero mode and there are the h ¼ 0; 1 modes in ðYj;jz Þa ¼ DaYj;jz ; ðA18Þ 2jðj þ 1Þ the j ¼ 1 modes. The helicity-two modes appear when j ≥ 2. In a massless spin-2 field, the helicity-zero mode and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the helicity-one modes are not dynamical degrees of 2 freedom whereas all modes are dynamical in the massive ðYj;j Þ ¼ case. z ab ðj − 1Þjðj þ 1Þðj þ 2Þ On the other hand, the pure-orbital spherical harmonics 1 s ˆ ˆ ˆ 2 ðTj;j Þij are defined by the eigenstates of the orbital angular × D D − D γˆ Y ; ðA19Þ z a b 2 ab j;jz momentum operator
2 Lˆ ðTs Þ ¼ lðl þ 1ÞðTs Þ ; ðA22Þ j;jz ij j;jz ij and the spherical harmonics Yj;jz where a, b, c are the 2 a b 2 indices in the unit two-sphere d Ω ¼ γˆabdx dx ¼ d θ þ with the orthonormality 2 2 ˆ Z sin θdϕ and Da is the covariant derivative with respect to γˆ s s0 ij ab. The two-dimensional Levi-Civita tensor is denoted by dΩðT Þ ðT 0 0 Þ ¼ δ 0 δ 0 δ 0 : ðA23Þ j;jz ij j ;jz ss jj jzjz ϵab. Clearly from the explicit forms, the modes E1, B1 are ≥ 1 defined only for j and the modes E2, B2 are defined The possible total angular momenta of the spin-2 field are ≥ 2 only for j . In Table VI, we summarize which tensor given by j ¼ l; l 1; l 2 which are labeled by spherical harmonics are defined in each total angular s ¼ 0; 1; 2, i.e., j ¼ l þ s. We note that the label s momentum. does not represent the spin states since the pure-orbital The helicity eigenstates ðYh Þ are defined by j;jz ij spherical harmonics are no longer the eigenstates of the polarizations of the spin-2 field. The pure-orbital spherical ˆ n · SðYh Þ ¼ hðYh Þ ; ðA20Þ harmonics are characterized by their angular momenta. We j;jz ij j;jz ij note that
n ˆ 2 2 i j i where is the unit radial vector in the Cartesian coor- L ¼ −r Δ þ r r ∂i∂j þ 2r ∂i; ðA24Þ dinates and h ¼ 0; 1; 2. The helicity eigenstates are given by linear combinations of the pure-spin spherical thus, the pure-orbital spherical harmonics are also eigen- harmonics as states of the Laplace operator. As a result, we obtain
044005-13 AOKI, MAEDA, MISONOH, and OKAWA PHYS. REV. D 97, 044005 (2018) 1 d d l l 1 function as well as the (scalar) spherical harmonics (see Δ s 2 − ð þ Þ s ½fðrÞðT Þ ¼ 2 r f 2 f ðT Þ ; j;jz ij r dr dr r j;jz ij [61,62]). Here, we only show the relations between the pure-spin spherical harmonics and the pure-orbital ðA25Þ spherical harmonics by which one can explicitly obtain the pure-orbital spherical harmonics from Eq. (A17). The pure- where f is a function of r. The pure-orbital spherical orbital spherical harmonics are obtained by the orthogonal harmonics are explicitly constructed by the so-called spin transformation as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðj þ 1Þðj þ 2Þ 2jðj þ 2Þ jðj − 1Þ ðT−2 Þ ¼ ðYS0 Þ þ ðYE1 Þ þ ðYE2 Þ ; j;jz ij 2ð2j þ 1Þð2j þ 3Þ j;jz ij ð2j þ 1Þð2j þ 3Þ j;jz ij 2ð2j þ 1Þð2j þ 3Þ j;jz ij sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðj þ 1Þ 3 3ðj − 1Þðj þ 2Þ ðT0 Þ ¼ − ðYS0 Þ þ ðYE1 Þ þ ðYE2 Þ ; j;jz ij ð2j − 1Þð2j þ 3Þ j;jz ij ð2j − 1Þð2j þ 3Þ j;jz ij ð2j − 1Þð2j þ 3Þ j;jz ij sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3jðj − 1Þ 2ðj − 1Þðj þ 1Þ ðj þ 1Þðj þ 2Þ ðTþ2 Þ ¼ ðYS0 Þ − ðYE1 Þ þ ðYE2 Þ ; j;jz ij 2ð2j þ 1Þð2j − 1Þ j;jz ij ð2j þ 1Þð2j − 1Þ j;jz ij 2ð2j þ 1Þð2j − 1Þ j;jz ij sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi j þ 2 j − 1 ðT−1 Þ ¼ ðYB1 Þ þ ðYB2 Þ ; j;jz ij 2j þ 1 j;jz ij 2j þ 1 j;jz ij sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi j − 1 j þ 2 ðTþ1 Þ ¼ ðYB1 Þ − ðYB2 Þ : ðA26Þ j;jz ij 2j þ 1 j;jz ij 2j þ 1 j;jz ij
We note that ðT0 Þ and ðT−1 Þ are defined only for j ≥ 1, modes of the pure-orbital spherical harmonics which are j;jz j;jz whereas ðTþ2 Þ and ðTþ1 Þ are defined only for j ≥ 2 (see not spherically symmetric. j;jz j;jz 2 Table VI). The relations (A26) can be used even for j ¼ 0, We find that the j ¼ s ¼ pure-orbital spherical 1 since the coefficients in front of the undefined variables in harmonics further have a special property: orthonormal the right-hand side vanish when j ¼ 0,1. without integration The total angular momentum operators are given by the 1 Lie derivatives. The symmetry of the field configuration is þ2 þ2 ij δ ðT2;j ÞijðY2 0 Þ ¼ j j0 ; ðA27Þ determined by the total angular momentum. For instance, z ;jz 4π z z the spherically symmetric mode is j ¼ jz ¼ 0 and the 0 which hold only for the j ¼ 2 modes of ðTþ2 Þ . The axisymmetric modes are jz ¼ . On the other hand, the j;jz ij orbital angular momentum vanishes only for the j ¼ s ¼ 2 explicit forms in the spherical coordinates are given by
0 pffiffi 1 1ffiffi 6 p ð1 þ 3 cos 2θÞ − 2 r sin 2θ 0 B 6 C 1 B 2 C þ2 pr ffiffi 1 − 3 2θ 0 ðT2 0Þ ¼ pffiffiffiffiffiffiffiffi B ð cos Þ C; ðA28Þ ; ij 16π @ 6 qffiffi A 2 2 2 − 3r sin θ
0 1 − sin 2θ −r cos 2θ −ir sin θ cos θ eiϕ B C ðTþ2Þ ¼ pffiffiffiffiffiffiffiffi @ r2 sin 2θ ir2 sin2 θ A; ðA29Þ 2;1 ij 16π 0
0 1 sin2 θ r sin θ cos θ ir sin2 θ 2iϕ þ2 e B 2 2 2 C ðT2 2Þ ¼ pffiffiffiffiffiffiffiffi @ r cos θ ir sin θ cos θ A; ðA30Þ ; ij 16π −r2 sin2 θ
044005-14 MASSIVE GRAVITON GEONS PHYS. REV. D 97, 044005 (2018)
0 þ2 and jz < are obtained by the relation ðT2;−j Þij ¼ TABLE VII. Energy eigenvalues and 95% mass radii of octu- 2 z ð−1Þjz ðT Þ . On the other hand, in the Cartesian pole geon. 2;jz ij coordinates, they are just constant matrices ~ ~ Node E R95% 0 1 n ¼ 0 −0.05411 20.30 −100 n ¼ 1 −0.01760 60.23 1 B C þ2 ffiffiffiffiffiffiffiffi @ −10A n ¼ 2 −0.008631 120.6 ðT2;0Þij ¼ p ; ðA31Þ 24π n ¼ 3 −0.005109 201.4 2
0 1 APPENDIX B: OCTUPOLE GEONS 00 1 2 1 B C An octupole configuration also gives a spherically ðTþ Þ ¼∓ pffiffiffiffiffiffiffiffi @ 0 i A; ðA32Þ 2; 1 ij 16π symmetric energy density distribution. We find 0 1 ðTþ2Þ ðTþ2Þij ¼ðTþ2 Þ ðTþ2 Þij ¼ ; ðB1Þ 0 1 3;2 ij 3;2 3;−2 ij 3;−2 4π 1 i 0 2 2 2 1 B C Tþ Tþ ij 0; B2 ðTþ Þ ¼ pffiffiffiffiffiffiffiffi @ −10A: ðA33Þ ð 3;2Þijð 3;−2Þ ¼ ð Þ 2; 2 ij 16π 0 where
0 1 3 cos θsin2θ rð1 þ 3 cos 2θÞ sin θ=22ir cos θsin2θ 2iϕ þ2 e B 2 2 C ðT3 2Þ ¼ pffiffiffiffiffiffiffiffi @ r ðcos θ þ 3 cos 3θÞ=4 ir cos 2θ sin θ A: ðB3Þ ; ij 16π −r2 cos θsin2θ
Hence, if the configuration of the massive graviton is A few lower energy eigenvalues and the 95% mass radii given by are summarized in Table VII. Since the orbital angular pffiffiffiffiffiffiffiffi momentum is smaller than the monopole geon ðl ¼ 2Þ but ψ 16π −iEtψ þ2 þ2 ij ¼ e 3ðrÞ½aþðT3;2Þij þ a−ðT3;−2Þij ; ðB4Þ larger than the quadrupole geon ðl ¼ 0Þ, the energy eigenvalues are intermediate between them. Φ Φ we obtain ¼ ðrÞ, where aþ and a− are constants with Although the energy eigenvalue of the octupole geon is 2 2 1 aþ þ a− ¼ . The orbital angular momentum of this mode smaller than that of the monopole geon, we could not find is l ¼ 1. the unstable mode in the linear stability analysis of the The solutions can be found under the boundary conditions j ¼ 3 mode perturbations in Sec. IV. This is not surprising ψ 3 → 0 both at the center and at infinity. We note dψ 3=dr because the linear stability does not conclude the back- does not vanish at the center because the j ¼ 3 modes are not ground is indeed stable. No unstable mode in the j ¼ 3 symmetric about the equatorial plane. On the other hand, mode linear perturbations implies that the monopole geon the boundary conditions of the gravitational potential are does not immediately transit to the octupole geon even if dΦ=dr → 0 at the center and Φ → 0 at infinity. the transition is possible due to the nonlinear effects.
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