Kerr black holes are among the most intriguing predictions of Einstein’s general relativity theory1, 2. These rotating massive astrophysical objects drag and intermix their surrounding space and time, deflecting and phase-modifying light emitted near them. We have found that this leads to a new relativistic effect that imprints orbital angular momentum on such light. Numerical experiments, based on the integration of the null geodesic equations of light from orbiting point-like sources in the Kerr black hole equatorial plane to an asymptotic observer3, indeed identify the phase change and wavefront warping and predict the associated light-beam orbital angular momentum spectra4. Setting up the best existing telescopes properly, it should be possible to detect and measure this twisted light, thus allowing a direct observational demonstration of the existence of rotating black holes. As non-rotating objects are more an exception than a rule in the Universe, our findings are of fundamental importance.
At a glance
In curved spacetime geometries, the direction of a vector is generally not preserved when parallel-transported from one event to another, and light beams are deflected because of gravitational lensing. If the source of the gravitational field also rotates, it drags spacetime with it, causing linearly polarized electromagnetic radiation to undergo polarization rotation similar to the Faraday rotation of light in a magnetized medium. This is known as the gravitational Faraday effect5. As a result of the rotation of the central mass, each photon of a light beam propagating along a null geodesic will experience a well-defined phase variation. As a result, the beam will be transformed into a superposition of photon eigenstates, each with a well-defined value s of spin angular momentum and ℓ of orbital angular momentum6. Patterns drawn by such beams experience an anamorphosis7 with polarization rotation due to the gravitational Faraday effect, accompanied by image deformation and rotation due to the gravitational Berry phase effect8, 9. Hence, light propagating near rotating black holes experiences behaviour analogous to light propagating in an inhomogeneous, anisotropic medium in which spin-to-orbital angular momentum conversion occurs10.
Whereas the linear momentum of light is connected with radiation pressure and force action, the total angular momentum J=S+L is connected with torque action. The spin-like form S, also known as spin angular momentum (SAM), is associated with photon helicity and hence with the polarization of light. The second form, L, is associated with the orbital phase profile of the beam, measured in the direction orthogonal to the propagation axis, and is also known as orbital angular momentum (OAM). This physical observable, which is present in natural light, finds practical applications in nanotechnology11, communication technology12 and many other fields. In observational astronomy, OAM of light13, 14, 15 can improve the resolving power of diffraction-limited optical instruments by up to one order of magnitude for non-coherent light16 and facilitate the detection of extrasolar planets17, 18.
That a photon carries an amount of SAM, quantized as S=σ,σ=±1,and can also carry an amount of OAM, quantized as L=ℓ,ℓ=0,±1,±2,…,±N, is well known from quantum electrodynamics19. The OAM of photons has been confirmed experimentally20, 21 and discussed theoretically22. Generally, it is not always possible to split the total angular momentum J of a photon into two distinct gauge-invariant observables S and L. However, when a paraxial beam of light propagates in vacuum along the z axis, one can project S and L onto this axis and obtain two distinct and commuting operators
where and are the creation/annihilation operators of the electromagnetic field, expressed in spin (σ), orbital (ℓ) and radial (p) angular momentum states of a helical beam. Each photon propagation state can be approximated by a Laguerre–Gaussian mode with indices ℓ and p, that is an electromagnetic field with amplitude
where the azimuthal index ℓ is the number of twists of the helical wavefront, p is the number of non-coaxial modes, w is a scale parameter, Lpℓ(x) is the associated Laguerre polynomial and ϑ is the azimuthal angle around the z axis.
In geometric units (G=c=1), Kerr spacetime, expressed in the Boyer–Lindquist coordinates (t,r,θ, ), is described by the line element1
where the quantities ρ2=r2+a2cos2θ and Δ=r2−2 Mr+a2 depend on the mass M and on the angular momentum per unit mass a (≤1), that is the rotation parameter of the Kerr black hole (KBH). In Kerr metric, orbits are not planar and the only way to calculate the null geodesic equations is to use either the Walker–Penrose conserved quantities or two constants of motion, λ and Q, that are related to the z component of the KBH angular momentum and to the square of the total angular momentum, respectively. Each null geodesic is identified by its impact parameters α and β that describe the direction with respect of the image plane of an asymptotic observer located at latitude θobs with respect to the KBH,
Accretion is thought to occur mainly in the equatorial plane of the KBH (ref. 3). To image, at infinity, the shape of equatorial orbits around a KBH, assumed to model a thin accretion disk, and to calculate the phase acquired by light emitted from that accreting matter, we solve numerically the null geodesic equations in strong gravity conditions by using the software described in ref. 3 and in the Supplementary Information. The phase variation map of photons emitted by source elements in a region of size 100RS×100RS, where RS is the Schwarzschild radius, is calculated by using standard projection techniques. From this map, it is straightforward to estimate the OAM spectrum emitted by the radiating matter in that region of the sky. The phase and OAM acquired are independent of both frequency and intensity and the OAM spectrum will be given by the convolution of the acquired OAM and the emission law of the accretion disk (see Supplementary Information). As a result of spacetime dragging (Lense–Thirring effect), all of the sources around the KBH are forced to rotate and the phase of light will change8, 9. In geometric optics, the analogy between the propagation of light in inhomogeneous media and in curved spacetimes is well established23. In this picture, the polarization and image rotation are attributed to SAM and OAM, respectively. The present Letter extends this to Kerr spacetimes.
Specifically, we have investigated the supermassive Galactic Centre KBH Sgr A*, whose rotation is still a matter of debate: 0.5≤a<0.9939−0.0074+0.0026 (see refs 24 and 25). Figure 1 shows the phase map of light emitted in the equatorial plane around the Sgr A* KBH, projected onto the observer’s sky plane of view, for the quasi-extremal case a=0.99 and an inclination of i=45° with respect to the observer. Owing to the asymmetric gravitational lensing distortion and the black-hole rotation, this light has quite a wide and structured OAM spectrum (Fig. 1, inset).
Figure 2 shows, for the representative cases a=0.99 (Fig. 2a,b) and a=0.5 (Fig. 2c,d), the effect of the Sgr A* rotation on the photon phase (Fig. 2a,c), normalized to a quasi-static (a=0.01) KBH, inclined by the same angle, and the corresponding OAM spectra (Fig. 2b,d). In both cases, the morphing effects due to a particular inclination of the Sgr A* equatorial plane have been removed to exhibit the pure Kerr metric effects. Our simulations show that the main contribution to the phase difference comes from the inner stable orbits that approach the Sgr A* event horizon, and that slowly rotating black holes give rise to uniformly decaying power distributions of OAM modes around a zero OAM value.
To detect rotating black holes with the technique described here, it is sufficient to use the best available telescopes, provided that they are equipped with proper OAM diagnostic instrumentation (for example, holographic detectors). As OAM of light is merely related to its spatial structure, only spatial coherence of the source is required18, 26. The long distances from the KBH to the observer ensure this spatial coherence, even if the electromagnetic fields generated at different points in the accretion disk are mutually incoherent (see Supplementary Information). Further information about the KBH rotation can be obtained by analysing the OAM spectra in different frequency bands of the electromagnetic radiation as the new gravitational effect described in this Letter is wavelength independent. As we are interested only in the gravitational effect, we have not taken into account the possibility that interactions with plasma turbulence may induce variations in the OAM spectrum of the light at certain frequencies27. However, these variations are likely to be much smaller than the gravitational effect. We also anticipate that the analysis of several different frequency bands will provide useful information about the turbulence of the disk corona and the traversed interstellar medium, as well as about the gravitational effect itself.
The observables associated with a Kerr metric considered here are invariant with respect to the mass of the KBH. Consequently, the photon phase ℓϑ and the concomitant OAM (spiral) spectrum4 will depend only on the rotation parameter a and on the inclination i of the equatorial plane relative to the observer.
Contributions from the secondary ghost images3, 8, 9, 28 generated by those light beams that wind their paths n times around a KBH are expected to widen the OAM spectra. However, as n increases, the secondary ghost images become fainter and more difficult to observe7. Moreover, higher-order correction terms would lead to photon spin precession, a negligible second-order effect in the Lagrangian8. An investigation into these effects is beyond the scope of this Letter. High OAM values may also occur in particular situations of radial polar accretion29 or from the gravitational lensing of coherent astrophysical sources. The OAM mechanism described here should be valid also for neutrino fields30. Our results can be extended to more general situations such as in the latest stages of black-hole/black-hole collisions or in generalizations of Kerr metrics.
- 1992). The Mathematical Theory of Black Holes (Oxford Univ. Press,
- Gravitational lensing by black holes. Gen. Rel. Grav. 42, 2269–2300 (2010).
- Relativistic emission lines from accretion disks around black holes. Mon. Not. R. Astron. Soc. 363, 177–182 (2005). &
- Digital spiral imaging. Opt. Express 13, 873–881 (2005). , &
- Gravitational Faraday-effect. Int. J. Theor. Phys. 7, 467–474 (1973).
- Twisted photons. Nature Phys. 3, 305–310 (2007). , &
- Extreme gravitational lensing near rotating black holes. Mon. Not. R. Astron. Soc. 359, 1217–1228 (2005). &
- Phase evolution of the photon in Kerr spacetime. Phys. Rev. D 46, 5407–5413 (1992). , , &
- Gravitomagnetism and the Berry phase of photon in a rotating gravitational field. Int. J. Mod. Phys. D 10, 961–969 (2001). &
- Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 96, 163905 (2006). , &
- A revolution in optical manipulation. Nature 424, 810–816 (2003).
- Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12, 5448–5456 (2004). et al.
- Utilization of photon orbital angular momentum in the low-frequency radio domain. Phys. Rev. Lett. 99, 087701 (2007). et al.
- Photon orbital angular momentum in astrophysics. Astrophys. J. 597, 1266–1270 (2003).
- Photon orbital angular momentum in astronomy. Astron. Astrophys. 492, 883–922 (2008).
- Overcoming the Rayleigh criterion limit with optical vortices. Phys. Rev. Lett. 97, 163903 (2006). , , &
- An image of an exoplanet separated by two diffraction beamwidths from a star. Nature 464, 1018–1020 (2010). , &
- Optical vortices with star light. Astron. Astrophys. 488, 1159–1165 (2008). , , , &
- 2nd edn,Vol. 4 (Butterworth-Heinemann, 1982). , & Quantum Electrodynamics
- Entanglement of the orbital angular momentum states of photons. Nature 412, 313–316 (2001). , , &
- Measuring the orbital angular momentum of a single photon. Phys. Rev. Lett. 88, 257901 (2002). , , , &
- Photon wave function: A covariant formulation and equivalence with QED. Phys. Rev. A 78, 052116 (2008). &
- Viewing the shadow of the black hole at the Galactic Center. Astrophys. J. 528, L13–L16 (2000). , &
- Near-infrared flares from accreting gas around the supermassive black hole at the Galactic Centre. Nature 425, 934–937 (2003). et al.
- X-ray flares reveal mass and angular momentum of the Galactic Center black hole. Astron. Astrophys. 417, 71–78 (2004). , , &
- Estimating the parameters of Sagittarius A*’s accretion flow via millimeter VLBI. Astrophys. J. 697, 45–54 (2009). , , &
- Photon orbital angular momentum and mass in a plasma vortex. Europhys. Lett. 90, 45001 (2010). , , &
- The gravity field of a particle. Proc. R. Soc. Lond. 249, 180–194 (1959).
- The effect of the Kerr metric on the plane of polarization of an electromagnetic wave. Astrophys. J. 238, 1111–1125 (1980). &
- Neutrino orbital angular momentum in a plasma vortex. Europhys. Lett. 84, 41001 (2008). &
The authors thank J. P. Torres, M. Calvani, A. Čadež and M. Berry for helpful comments and suggestions. F.T. gratefully acknowledges the financial support from the CARIPARO Foundation within the 2006 Program of Excellence and the kind hospitality of Uppsala University/Swedish Institute of Space Physics and ICFO during the writing of the manuscript. B.T. gratefully acknowledges financial support from the Swedish Research Council (VR) and the hospitality of the Nordic Institute for Theoretical Physics (NORDITA), the University of Padova, and the Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Vienna, where parts of this work were carried out.
- Supplementary Information (800k)