Which Path Provides the Fastest Way To Circle the Central Black Hole? (Astronomy)

Black-hole spacetimes are known to possess closed light rings. Shahar Hod and colleagues using compact theorem in their present work revealed that these unique null circular geodesics provide the fastest way, as measured by asymptotic observers, to circle around spinning Kerr black holes.

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Fermat’s principle, also known as the principle of least time, asserts that among all possible null trajectories, the path taken by a ray of light between two given points A and B in a flat spacetime geometry is the path that minimizes the traveling time TA→B. This remarkably elegant principle implies, in particular, that the unique null trajectory taken by a ray of light between two given points is generally distinct from the straight line trajectory which minimizes the spatial distance dAB between these points.

In the present work, Shahar Hod highlighted an intriguing and closely related physical phenomenon which characterizes curved spacetime geometries. In particular, they raise the physically interesting question: Among all possible closed paths that circle around a black hole in a curved spacetime, which path provides the fastest way, as measured by asymptotic observers, to circle the central black hole?

Their compact theorem has revealed the physically intriguing fact that the equatorial null circular geodesics (closed light rings), which characterize the curved black-hole spacetimes, provide the fastest way to circle around spinning Kerr black holes. In particular, they have explicitly proved that, in analogy with the Fermat principle in flat spacetime geometries, the unique curved trajectories r = rfast(M, a) [refer Eq. 1 given below]

Equation 1

which minimize the traveling times T of test particles around central black holes are distinct from the tangential trajectories r = r+(M, a) [refer Eq. (2) given below] which could minimize the traveling distances around the black holes.

Equation 2

“What we find most intriguing is the fact that the spin-dependent radii rfast(M, a) of the fastest circular trajectories, as given by the functional expression (1), exactly coincide with the corresponding radii rγ(M, a) of the null circular geodesics which characterize the spinning Kerr black-hole spacetimes. One therefore concludes that co-rotating null circular geodesics (closed light rings) provide the fastest way, as measured by asymptotic observers, to circle around generic Kerr black holes.”, told Shahar.

This research is supported by the Carmel Science Foundation.

Reference: Shahar Hod, Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea, “Fermat’s principle in black-hole spacetimes”, International Journal of Modern Physics D, Vol. 27, No. 14, pp. 1-5, 1847025 (2021). https://www.worldscientific.com/doi/abs/10.1142/S0218271818470259?journalCode=ijmpd https://doi.org/10.1142/S0218271818470259

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