aa r X i v : . [ g r- q c ] J a n This essay received an Honorable Mention in the 2018 Essay Competition of the Gravity Research Foundation
Fermat’s principle in black-hole spacetimes
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: January 19, 2021)
Abstract
Black-hole spacetimes are known to possess closed light rings. We here present a remarkably compacttheorem which reveals the physically intriguing fact that these unique null circular geodesics provide the fastest way, as measured by asymptotic observers, to circle around spinning Kerr black holes.Email: [email protected] minimizes the traveling time T A → B . This remarkablyelegant principle implies, in particular, that the unique null trajectory taken by a ray of lightbetween two given points is generally distinct from the straight line trajectory which minimizesthe spatial distance d AB between these points.In the present Essay we would like to highlight an intriguing and closely related physical phe-nomenon which characterizes curved spacetime geometries. In particular, we here raise the phys-ically interesting question: Among all possible closed paths that circle around a black hole in acurved spacetime, which path provides the fastest way, as measured by asymptotic observers, tocircle the central black hole?We first note that, in flat spacetimes, the characteristic orbital period T ⊙ of a test particlethat moves around a spatially compact object of radius R is trivially bounded from below by thecompact relation [3] T ⊙ ≥ T flat ⊙ min = 2 πR , (1)where the equality sign in (1) is attained by massless particles that circle the compact object onthe shortest possible (tangential) trajectory with r fast = R .It should be emphasized, however, that the simple lower bound (1) is not valid in realisticcurved spacetimes. In particular, it does not take into account the important time dilation (red-shift) effect which is caused by the gravitational field of the central compact object [4]. In addition,the flat-space relation (1) does not take into account the well-known phenomenon of dragging ofinertial frames by spinning compact objects in curved spacetimes [4].As we shall explicitly show below, due to the influences of these two interesting physical effects,the shortest possible orbital period T ⊙ of a test particle around a central compact object, asmeasured by asymptotic observers, is larger than the naive flat-space estimate (1). In particular,we shall prove that, in generic curved spacetimes, the unique circular trajectory r = r fast thatminimizes the traveling time T ⊙ around a central Kerr black hole is distinct from the tangentialtrajectory with r = r short which could minimize the traveling distance around the spinning blackhole. The fastest circular orbit around a spinning Kerr black hole.—
We shall analyze the physicaland mathematical properties of equatorial circular trajectories around spinning Kerr black holes.In Boyer-Lindquist coordinates ( t, r, θ, φ ), the asymptotically flat black-hole spacetime can be de-2cribed by the curved line element [4] ds = − ∆ ρ ( dt − a sin θdφ ) + ρ ∆ dr + ρ dθ + sin θρ [ adt − ( r + a ) dφ ] , (2)where M is the black-hole mass, J ≡ M a is its angular momentum, ∆ ≡ r − M r + a , and ρ ≡ r + a cos θ . The black-hole (event and inner) horizons are determined by the spatial zerosof the metric function ∆: r ± = M ± ( M − a ) / . (3)We would like to identify the unique circular trajectory which minimizes the orbital period T ⊙ ,as measured by asymptotic observers, around the central black hole. We shall therefore assumethe relation v/c → − for the velocity of the orbiting test particle [5]. The corresponding radius-dependent orbital periods T ⊙ ( r ) of the test particles can easily be obtained from the characteristicblack-hole curved line element (2) with ds = dr = dθ = 0 and ∆ φ = ± π [6–8]. This yields thecompact functional relation T ⊙ ( r ) = 2 π · √ r − M r + a − Mar − Mr (4)for the orbital periods of co-rotating test particles around the central spinning black hole.The physically interesting co-rotating circular orbit with r = r fast , which is characterized by theshortest possible orbital period T ⊙ min = min r { T ⊙ ( r ) } around the central spinning black hole, isdetermined by the functional relation dT ( r = r fast ) /dr = 0. This yields the characteristic algebraicequation r − M r + 2 a + 2 a p r − M r + a = 0 for r = r fast . (5)Remarkably, this equation can be solved analytically to yield the simple functional relation r fast = 2 M · { − ( − a/M )] } . (6)for the unique orbital radius r fast ( M, a ) which characterizes the fastest co-rotating circular trajec-tory (the closed circular path with the shortest possible orbital period) around the central spinningKerr black hole.What we find most intriguing is the fact that the spin-dependent radii r fast ( M, a ) of the fastestcircular trajectories, as given by the functional expression (6), exactly coincide with the correspond-ing radii r γ ( M, a ) of the null circular geodesics [9] which characterize the spinning Kerr black-holespacetimes. One therefore concludes that co-rotating null circular geodesics (closed light rings)3rovide the fastest way, as measured by asymptotic observers, to circle around generic Kerr blackholes.It is physically interesting to define the dimensionless ratio [see Eqs. (1) and (3)]Θ(¯ a ) ≡ T ⊙ min πr + ; ¯ a ≡ a/M , (7)which characterizes the unique closed circular trajectories [with r = r fast (¯ a )] that minimize theorbital periods around the central spinning black holes. As emphasized above, a naive flat-spacecalculation predicts the relation Θ flatmin ≡ T flat ⊙ min / πR = 1 [see Eq. (1)]. However, substituting Eqs.(4) and (6) into (7), one finds the characteristic inequality [10]Θ Kerr (¯ a ) > a ∈ [0 , Kerr (¯ a ) exhibits a non-trivial (non-monotonic) functional dependence onthe dimensionless black-hole rotation parameter ¯ a with the property [11]min ¯ a { Θ Kerr (¯ a ) } ≃ − − √ a Kerrmin ≃ − − √ . (9) Summary.—
Fermat’s principle asserts that, in a flat spacetime geometry, the path taken by aray of light is unique in the sense that it represents the spatial trajectory with the shortest possibletraveling time between two given points [1, 2]. This intriguing principle implies, in particular, thatthe paths taken by light rays are generally distinct from the straight line trajectories which couldminimize the traveling distances between two given points.In the present short Essay we have highlighted an intriguing and closely related phenomenonin curved black-hole spacetimes. In particular, we have raised the physically interesting question:Among all possible trajectories that circle around a spinning Kerr black hole, which closed tra-jectory provides the fastest way, as measured by asymptotic observers, to circle the central blackhole?Our compact theorem has revealed the physically intriguing fact that the equatorial null circulargeodesics (closed light rings), which characterize the curved black-hole spacetimes, provide thefastest way to circle around spinning Kerr black holes. In particular, we have explicitly proved that,in analogy with the Fermat principle in flat spacetime geometries, the unique curved trajectories r = r fast ( M, a ) [see Eq. (6)] which minimize the traveling times T ⊙ of test particles around centralblack holes are distinct from the tangential trajectories r = r + ( M, a ) [see Eq. (3)] which couldminimize the traveling distances around the black holes.4
CKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren,Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for helpful discussions. [1] A. Lipson, S. G. Lipson, H. Lipson,
Optical Physics G = c = 1.[4] S. Chandrasekhar, The Mathematical Theory of Black Holes , (Oxford University Press, New York,1983).[5] In the present essay we consider geodesic as well as non-geodesic trajectories of test particles aroundthe central Kerr black holes. It is well known that the null circular geodesic of the black-hole spacetime,on which massless particles that move exactly at the speed of light can circle the central black hole, ischaracterized by a well defined radius r γ = r γ ( M, a ). It is important to stress the fact that, by usingman-made rockets which are based on non-gravitational forces, massive particles can also circle thecentral black hole on non-geodesic trajectories with orbital velocities that, in principle, may approacharbitrarily close to the speed of light and with orbital radii that may differ from the unique radius r γ ( M, a ) of the black-hole null circular geodesic.[6] Here the upper/lower signs correspond respectively to co-rotating/counter-rotating trajectories of thetest particles around the central spinning black holes.[7] In the present essay we are interested in circular trajectories that minimize the orbital periods T ⊙ around the central spinning black holes as measured by asymptotic observers. We shall therefore focuson co-rotating circular orbits.[8] S. Hod, Phys. Rev. D , 104024 (2011).[9] See equation (2.18) of J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astrophys. Jour. , 347(1972).[10] In particular, one finds Θ Kerr (¯ a → → √ / Kerr (¯ a → → Kerr (¯ a ) = 2 + √ √ − · √ ǫ + (14 / − √ · ǫ + O ( ǫ / ) inthe near-extremal 0 ≤ ǫ ≡ − ¯ a ≪1 regime.