Geometric creation of quantized vorticity by frame dragging Michael R.R. Good∗ and Chi Xiong† Institute of Advanced Studies, Nanyang Technological University, Singapore 639673
Alvin J.K. Chua‡ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
Kerson Huang§ Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA 02139 and Institute of Advanced Studies, Nanyang Technological University, Singapore 639673
arXiv:1407.5760v1 [gr-qc] 22 Jul 2014
We consider a complex scalar field in background metrics that exhibit frame-dragging, in particular the BTZ metric in 2+1 dimensional spacetime, and the Kerr metric in 3+1 dimensional spacetime. In the equation of motion for the scalar field, which is a nonlinear Klein-Gordon equation, we identify and isolate the terms arising from frame-dragging that correspond to the Coriolis force and the centrifugal force, which produce local rotation in the field. The field physically describes a superfluid, which can rotate only through the creation of quantized vortices. We demonstrate such vortex creation through numerical simulation of a simplified metric that exhibits frame-dragging.
In general relativity, the Lense-Thirring effect [1], or “frame-dragging”, refers to a special distortion of spacetime geometry, caused by a rotating mass, that is equivalent to a local rotating frame. A test particle in the neighborhood of the rotating mass will undergo precession as it moves along a geodesic. This is a very small effect when caused by ordinary heavenly objects, but it has been detected [2]. We know, on the other hand, that the vacuum is not empty but filled with at least one complex scalar field, namely the Higgs field postulated to generate mas in the standard model of particle physics. Its existence has found experimental support in the discovery of the associated field quantum, the Higgs boson [3,4]. The phase dynamics of such a field, on a macroscopic scale, leads to superfluidity, and in this sense the universe is a superfluid. There are proposals that associate many cosmological phenomena with this superfluidity, chief among which are dark energy, dark matter, and inflation [5-8]. Here, we explore the possibility that the precessional effect of frame dragging can create quantized vortices in this cosmic superfluid. Quantized vorticity is the only means through which a superfluid can rotate, and as such stands as a signature of superfluidity. We consider a generic complex scalar field Φ, not particularly tied to any vacuum field in particle theory, described phenomenologically by a nonlinear Klein-Gordon equation (NLKG)
and λ, F0 are parameters. The nonlinear self-interaction spontaneously breaks the global gauge invariance, and maintains a nonzero vacuum field F0 . As shown in [8], this vacuum field is stable against collapse due to selfgravitation, i.e., the equivalent Jeans length is greater than the radius of the universe. In a phase representation Φ = F eiσ , the superfluid velocity is given by vs = ξs ∇σ, where ξs = c2 (−∂σ/∂t)−1 is a spacetime dependent factor that makes |vs | < c. In the non-relativistic limit ξs → ~/m, where m is a mass scale [8,9]. In practice, it is simpler to work with ∇σ. A quantized vortex is a solution to the NLKG with I ∇σ · ds =2πn (n = 0, ±1, ±2, . . .) , (2)
Φ + λ(|Φ|2 − F02 )Φ = 0, (1) h i 1/2 −1/2 µ where Φ ≡ (−g) ∂ (−g) gµν ∂ ν Φ , gµν is the metric through which frame-dragging effects would enter,
ds2B = gtt dt2 + grr dr2 + gφφ dφ2 + 2gtφ dtdφ,
∗ Electronic
address: address: ‡ Electronic address: § Electronic address: † Electronic
[email protected] [email protected] [email protected] [email protected]
C
over some spatial closed loop C. Such quantized vortices have been extensively studied in liquid helium and BoseEinstein condensates of cold trapped atoms, both experimentally and theoretically [8-11]. We want to extract specific terms in the NLKG (1) arising from the metric that would induce vortex creation. For orientation, we consider first the BTZ metric [12] in 2+1 dimensional spacetime, which is a vacuum solution to Einstein’s equation in the region outside of a rotating “star” (a central distribution that has both mass and angular momentum). The line element in spatial polar coordinates is given by (3)
with −1 2 16G2 J 2 8GM r c2 r 2 + − , gtt = 8GM − 2 , grr = a a2 c6 r 2 c2 4GJ (4) gφφ = r2 , gtφ = − 2 , c where G is the gravitational constant, and M and J are the mass and angular momentum of the central star, respectively. The metric does not approach that of flat
2 spacetime at infinity, but blows up like r2. For this reason, it is not quite physical, and we use it only for illustrative purposes. The frame-dragging component gtφ implies a local rotational angular velocity ΩB = −
4GJ gtφ = 2 2, gφφ c r
(5)
which is independent of the mass M. In order to recognize frame rotation when we see it, we consider a comparison metric that describes rotation in flat 2+1 dimensional spacetime, with angular velocity Ω0 : ds2comparison = −c2 dt2 + dr2 + r2 (dφ − Ω0 dt)2 .
(6)
The effect of rotation enters the NLKG only through Φ, which separates into a non-rotational term plus terms that can be identified with the Coriolis force and centrifugal force: Φ = (0) Φ + RCoriolis + Rcentrifugal,
(7)
where (0) Φ is that for the Minkowski metric, and RCoriolis = −
2Ω0 ∂ 2 Φ , c2 ∂t∂φ
Rcentrifugal = −
Ω20 ∂ 2 Φ . c2 ∂φ2
(8)
The BTZ metric formally reduces to the comparison metric for the special values M = −c2 /8G, a2 = −c2 /Ω20 . In general, we can identify the effective Coriolis force and centrifugal force arising from the BTZ metric by expanding the latter in powers of J, or equivalently ΩB , to second order: B RCoriolis =
2ΩB ∂ 2 Φ , gtt ∂t∂φ
B Rcentrifugal =
Ω2B ∂ 2 Φ . gtt ∂φ2
(9)
There are of course higher order terms in Φ; but the presence of these terms indicates that the field Φ does feel the rotation of the star through spacetime geometry. These terms depend on r, because ΩB and gtt depend on r. This is to be expected, since frame-dragging is spacetime dependent. The terms in (9) approach the forms (8) when gtt → −c2 , in the limit M → 0 and r → a, or M → −c2 /(8G) and a → ∞. To explicitly demonstrate vortex creation, we solve the NLKG numerically in the comparison metric, which retains frame-dragging and is simpler to handle numerically. For various choices of model parameters, a vortex lattice emerges in the superfluid. Although the rotational effects are contained in Φ, the nonlinear self-interaction is necessary to produce vortices. The situation here is very similar to that for the NLKG with a rotating star as source, as discussed in [8,9], however the numerical computation here is more challenging, owing to the cross derivative in the Coriolis force. Fig.1 shows an example of a vortex lattice, with a contour plot of the phase σ, showing phase discontinuities due to the vortices.
FIG. 1: Left panel: Lattice of 100 quantized vortices. This is a contour plot of the modulus of a complex scalar field that satisfies a nonlinear Klein-Gordon equation in 2+1 dimensional curved space-time, with frame-dragging. Right panel: Contour plot of the phase of the complex scalar field, showing the “strings” across which the phase jumps by 2π.
For a more realistic example, we turn to the Kerr metric [13] in 3+1 dimensional space-time, which describes the region outside of a rotating star of mass M and angular momentum J. It reduces to the Schwarzschild metric when J = 0. The line element is, in spatial spherical coordinates, ds2K = gtt dt2 +grr dr2 +gθθ dθ2 +gφφ dφ2 +2gtφ dtdφ, (10) with gtt = −c2 ∆ − α2 sin2 θ Σ−1 , grr = Σ∆−1 , h 2 i −1 Σ , gθθ = Σ, gφφ = − sin2 θ α2 ∆ sin2 θ − r2 + α2 −1 2 2 2 gtφ = −cα sin θ r + α − ∆ Σ , (11)
where
∆ ≡ α2 + r2 − rrs , Σ ≡ α2 cos2 θ + r2 , rs ≡ 2GM c−2 , α ≡ J/M c.
(12)
The local angular velocity is given by [14] ΩK = −
gtφ α(r2 + α2 − ∆)c = 2 . gφφ (r + α2 )2 − ∆α2 sin2 θ
(13)
For a comparison metric in 3D, the Coriolis and centrifugal terms have the same form as in the 2D case given in (8), except that φ is to be identified with the azimuthal angle in 3D. We can calculate Φ using the Kerr metric, and extract the Coriolis and centrifugal terms by expanding in powers of J, or equivalently in ΩK , to second order: K RCoriolis =
2ΩK ∂ 2 Φ Ω2 ∂ 2 Φ K . , Rcentrifugal = K gtt ∂t∂φ gtt ∂φ2
(14)
These are formally the same as in the the BTZ metric. We recover (8) when gtt → −c2 , in the limit r ≫ rs and c → ∞. These quantities vanish for the
3 Schwarzschild metric, for which ΩK = 0. In our calculations, we use the potential approximation to the unperturbed Schwarzschild metric [15]. To actually create vortices, frame-dragging might have to be very strong, and this may be the case in the neighborhood of a black hole with ultra-high angular momentum. In fact, the so-called “non-thermal filaments” observed near the center of the Milky Way (where there are black holes) may be due to quantized vorticity, as has been pointed out in [6]. As the angular momentum increases, one expects that vortex filaments would stick closer to the black-hole surface, forming a sort of hydrodynamic boundary layer, and this may be relevant to the problem of black-hole collapse. It is well-known in hydrodynamics that such a boundary layer can accommodate any necessary boundary conditions for the surrounding fluid. One of the outstanding theoretical problems in general relativity is to extend the Oppenheimer-Snyder solution [16] of blackhole collapse in the Schwarzschild metric, which carries no angular momentum, to the Kerr metric. A main obstacle appears to be the lack of an appropriate generalization of the interior metric, which in the OppenheimerSnyder case is just the non-rotating Robertson-Walker metric. When a hydrodynamic boundary layer is present, perhaps the outside Kerr metric could be joined onto any interior metric. We thank Prof. Weizhu Bao and Dr. Yong Zhang of the National University of Singapore for helpful discussions and suggestions on numerical computations, and Mr. Yulong Guo for assistance in computer programming.
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