Trapped Gravitational Waves in Jackiw-Teitelboim Gravity

We discuss the possibility that gravitational waves are trapped in space by gravitational interactions in 2-dimensional Jackiw-Teitelboim gravity. In the standard geon (gravitational electromagnetic entity) approach, the active region is introduced to confine gravitational waves spatially. In our approach, however, spacetime dependent traceless metric perturbations, i.e."gravitational waves"are trapped by the vacuum geometry and can be stable against the backreaction due to the metric fluctuations. We expect that our approach may shed light on finding similar self-trapping solutions in 4-dimensional gravity.


I. INTRODUCTION
In 1916, Einstein predicted that gravitational sources could produce waves of spacetime from his theory of general relativity [1]. In 2016, the LIGO (Laser Interferometer Gravitational-Wave Observatory) and VIRGO teams finally detected gravitational waves generated from a pair of merging black holes [2]. Since then, additional events have been observed, including the merger of two neutron stars [3]). Gravitational waves (GW) are no longer a possible mathematical solution of the theory, but a true physical object. The era of gravitational waves astronomy has begun.
In 1955, Wheeler introduced a particle-like object, geon (gravitational electromagnetic entity), where gravitational waves confined in space by electromagnetic interaction [4]. He hoped to construct the geon as an elementary particle but that did not seem fruitful. Brill and Hartle elaborated this idea by considering GW trapped by gravitational interactions [5], i.e., that GW are somewhat localized in space by their self-interaction. Given the dispersive nature of radiation, it seems such objects are metastable at best. Analyses in general relativity have devoted much effort to the discussion whether such a solution is selfconsistent and metastable [6][7][8][9][10]. These analyses assumed an empty asymptotic Minkowski background. We would like to consider the more realistic scenario of an FLRW or at least asymptotically dS such that the background is also nonstatic. Significant works have been done on asymptotic AdS in [11] and references therein.
In this age of gravitational waves detection, self-confining GW are therefore interesting and attractive as their very existence could be detected. Self-confining gravitational waves are obtained by splitting Einstein's tensor into a back-ground gravitational field γ µν , namely G 0 [g µν ] and the disturbance G[γ µν , h µν ] and taking the average over high frequencies and angular momentum. In this paper, we study fluctuations of the gravitational field ("gravitational waves") trapped in space by the vacuum geometry in 2-dimensional gravity in the framework of Jackiw-Teitelboim (JT) gravity [12,13]. Even though it is two-dimensional gravity, the existence of a lagrange multiplier makes it non-trivial as the (vacuum) equation of motion now becomes R = Λ. Using perturbation theory, the zeroth order gives the background solution, the first order gives the "wave" equation, and the averaged second order, the backreaction on the background geometry. We prefer to use "trapped gravitational waves" instead of geon because in the classical geon solution the effective energy-momentum that corrects the unperturbed solution is entirely deposited in a thin shell enclosing the geon (active region). Our motivation is to circumvent the need of an active region, which makes our solution more physically plausible. Clearly, the reason for choosing 2-dimensional (2D) gravitational theory is that calculations are tremendously simplified and the solution could shed light on the more complicated situation of 4D gravity. We assume a non-vanishing cosmological constant -intuitively the attractive self-gravity and the space-time expansion yields a potential in which the gravitational waves could be trapped. Our analysis yields trapped GW in some region of space. Furthermore, in JT gravity in two dimensions, we obtain the exact solution of the gravitational field equations in the synchronous gauge, the conformal gauge and in a spatially flat gauge. We discuss its connection to the self-trapping solution. As such, our perturbative analysis can, in principle, be described as an approximation to an exact solution, given a proper transformation. Nevertheless, our method is a step towards performing a similar analysis in 3D (where gravity-dilaton waves exist) or 4D, where true GW are known to exist.
The paper is organized as follows. In Section II, we apply the [9] method of finding gravitational geons to JT gravity. In Section III, we try to find analytic conditions for having gravitational waves trapped stably in space and present numerical results on the conditions. In Section IV, we display the exact solution in the synchronous and conformal and spatially flat frame of references. In Section V, we summarize our results and discuss future works.

II. HOW TO FIND A GEON IN JACKIW-TEITELBOIM GRAVITY
Our starting point is 1+1 gravity introduced by Jackiw and Teitelboim [12,13], where the Einstein equation is given by where R is the curvature scalar, Λ is the cosmological constant, and T is the energymomentum. As in [14,15], we put the metric ansatz to be where γ µν is the unperturbed metric with the signature of (−, +) and h µν is traceless and represnts the perturbations standing for a toy model of gravitational waves.
If we consider no matter, i.e., T = 0, Eq. (1) becomes Following [5,9], we expand it perturbatively as where (0), (1), (2), . . . imply the orders in h. We then solve this equation in the following three steps. First, the background geometry for the vacuum state comes from Second, the first order perturbation equation in h is a wave-type equation. A second order linear partial differential equation for h. Hence, the gravitational waves h trapped in space are determined by (6). Third, we test the stability of the solution h by considering the backreaction of gravitational waves to the metric through where the original metric γ changes intoγ by the backreaction and · · · means the time average.

A. Background Geometry for the Vacuum Solution
Consider the unperturbed metric The equation of motion Eq. (1), is then and the solution of Eq. (5) is given by where A and B are constants. This is similar to the dS solution in static coordinates in 4D if we suppress the angular part.

B. Gravitational Waves as Perturbations
Let us now consider the perturbed metric as in [14].
A geon would have the form of where the time part would be T (t) ∝ e −iωt and the spatial part R(r) should be confined in From Eq. (6), we can derive where prime denotes a derivative with respect to r. We expect that the possibility of trapped gravitational waves would be checked by exploring the form of asymptotic behavior of Eq. (13) with given p(r). Now we can find two asymptotic behaviors as follows.
[AB1]: The first asymptotic behavior is that the gravitational waves can be trapped in the region where p → 0. For p → 0, Eq. (13) becomes We may put and Eq. (14) bcomes Without loss of generality, we can consider the asymptotic behavior of the solutions around r = α. The solution is which means the solution R(r) becomes zero as p → 0 for r → α 1 .
[AB2]: The second asymptotic behavior is that the gravitational waves cannot be trapped in the region where p → ±∞. For p → ±∞, Eq. (13) becomes In this case, when p(r) = A + Br − Λ 2 r 2 , the solution is given by In Eq. (19), C 1 term is definitely divergent. The C 2 term diverges or goes to zero as r → ∞ on very particular cases such as a single degenerate root. Nevertheless, waves extending from some root of p(r) to infinity cannot be considered as finite and localized.
These two conditions seem simple but predict where the GW can be confined in space.
If R(r) has some finite support, then we get trapping. If not, then we cannot say that the GW are confined. We will turn back to this point in Section III D.

C. Backreaction of Gravitational Waves
The backreaction of gravitational waves to the vacuum metric is calculated by Eq. (7) which reduces into where p(r) is modified intop(r) and a dot dentoes a derivative with respect to time. After considering e iωt 2 = 1 2 , Eq. (21) becomes 1 An interesting situation occurs if there is a single root, i.e. α = β in region AB1. In such case, the lowest order approximation becomes a Bessel-type equation: R − p 2p R + ω 2 R = 0, with the solution In such a case, the limit r → α can actually be finite with It is difficult to find analytic solutions for Eq. (22) and we will try to solve it by numerical simulations. However, we can mention two main features that would be reflected in the numerical results: First, when R 1 and R 1, Eq. (22) givesp + Λ 0 reproducing Eq. (9). Hence, the background geometry will not change and p is nearly the same asp.
Second, when ω 1, ω terms get important in the right hand side of Eq. (22). One may expect that the mode of large ω cause substantial backreaction to the background metric.

D. Numerical Results
In JT gravity, the metric component p(r) is presented by quadratic curves, as given in 1. If there exists no zero of p(r) for r ≥ 0, GW cannot be trapped. 2. If there exists one zero r 1 of p(r) for r ≥ 0, GW can be trapped in the region where p(r) is finite or does not go to the infinity (i.e., 0 < r < r 1 ).
3. If there exist two zeros of p(r) for r ≥ 0, GW can be trapped in two regions, between the origin and the smaller zero (0 < r < r 1 ), and between the zeros (r 1 < r < r 2 ).
These conditions apply to all the cases regardless of the values of Λ as demonstrated in When p(r) has two distinct zeros, r 1 and r 2 (r 2 > r 1 > 0), we have chances to trap GW in 0 < r < r 1 or in r 1 < r < r 2 that is clear from the asymptotic behaviors AB1 and AB2.
The reasoning of FIG 4, FIG 5 and FIG 6 applies to the trapping region 0 < r < r 1 , and hence we focus on r 1 < r < r 2 which seems to be more interesting. In FIG 7, we consider p(r) = (r − 5) 2 − 4 with Λ < 0 whose two zeros are r = 3, 7. GWs are confined in the region  Fortunately, the two dimensional case is amenable to an exact analytical solution. Counting degrees of freedom, any spacetime in any dimensions can always be put into the synchronous form, and in the case of 1+1D into a conformally flat form. In particular, in such forms, we do not limit ourselves to a static ansatz plus a time-dependent perturbation.  Considering the synchronous gauge where F (r, τ ) can be any function. The solution of the JT equation of motion (EOM) is given by where f (r) , g(r) are arbitrary functions.
Similarly, considering the conformal gauge, The EOM can be solved by where c 1 , c 2 , c 3 are determined by the boundary conditions. Finally, considering a "spatially flat" gauge where yields a solution that clearly exhibits an oscillatory propagating behavior: where again f 1 t , f 2 t are arbitrary functions. Switching from our ansatz to this spatially flat gauge is rather simple, as it simply requires the definition of the tortoise coordinate Our purpose in this paper is to learn the guiding ideas that could lead us to a solution of self-confining gravitational solutions in 4D gravity that could eventually be detected in our real universe. We need to extend our approaches to deal with the possibilities in 4D Einstein gravity. In [5,9], they already suggested the possibility that the gravitational geon can be attained. We hope that from what we have learned here, we can find self-confining gravitational waves in our expanding universe, either in Friedman-Lemaitre-Robertson-Walker (FLRW) metric or in the particular case of de Sitter geometries. If such solutions are obtained, it would be important to ask whether we can observe the gravitational waves trapped in space by gravitational interactions or find traces of such trapping by measuring the diffused remnants. Our metric ansatz is for spaces with 'horizons,' which are found at the points p(r) = 0 in our representation. If these GW are as in FIG 4,FIG 5, and FIG 6 then the GW may be trapped behind a "black hole" horizon and may be unobservable. The most promising example is that of FIG 7 and FIG 8. We may interpret the background geometry as the Schwarzschild -de Sitter space, according to [14]. In this case, the gravitational waves are trapped between the two horizons, i.e., outside of the "black hole" and inside the dS horizon in the expanding dS-like space, and the solution is meta-stable. In such a case, there are good chances that such an object may be observable. Still, it seems extremely difficult to construct the gravitational waves fairly localized as a geon particle. We hope the analysis presented here will be useful to tackle the full 4D problem.