Black Hole Flyby

Sebastian J. Szybka

Obserwatorium Astronomiczne UJ I a disclaimer: this is not a research talk

I 1 of 5 problems — the general relativity exam 2020 (third year astronomy undergraduates)

I a simple 20-minutes problem −→ the article Black Hole Flyby, SJS, 2021, (accepted in American Journal of Physics) The mission of the American Journal of Physics (AJP) is to publish articles on the educational and cultural aspects of physics that are useful, interesting, and accessible to a diverse audience of physics students, educators, and researchers. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity Michael S. Morris and Kip S. Thorne, AJP 56, 395 (1988) Other physicists who published in the American Journal of Physics

I John Archibald Wheeler

I Roger Pernose

I Subrahmanyan Chandrasekhar

I Hermann Bondi

I Wolfgang Rindler

I Gary William Gibbons

I John Stachel

I Jürgen Ehlers

I Richard H. Price (editor)

I David J. Griﬃths (consulting editor)

I ...

I The Event Horizon Telescope the ﬁrst image of the black hole 2019

I the unstable circular photon orbit r = 3m (the Schwarzschild solution)

I the black hole shadow J. L. Synge, The escape of photons from gravitationally intense stars, Mon. Not. R Astr. Soc. 131, 463–466, 1966 the black hole shadow is often omitted in general relativity textbooks with notable exceptions

I P. T. Chruściel, Elements of General Relativity, (Birkhäuser Basel, 2019)

I V. P. Frolov and A. Zelnikov, Introduction to Black Hole Physics, (Oxford University Press, 2011)

I C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, (W. H. Freeman, 1973) the problem for an exam — ‘the black hole shadow’ for massive particles that are stationary at inﬁnity

I the Schwarzschild solution

2m 2m −1 ds2 = − 1 − dt2 + 1 − dr 2 + r 2dΩ2 r r

I Killing ﬁelds and conserved quantities

r uα = (−e, grr u , 0, l)

α I the normalization uαu = −1 I the problem with the pseudopotential

e2 − 1 1 E = = (ur )2 + V (r) 2 2 l where m l 2 ml 2 V (r) = − + − l r 2r 2 r 3 E

0.1

3 4 5 6 0. r/m E - 1 18 -0.1

-0.2

maxima inflection point -0.3 some exact formulas

I the minimal distance

m q r = (1 − 6E)2 + 32E − (1 − 6E) min 4E

I the critical angular momentum per mass

2 2m 2 lcrit = 2(3E + )rmin rmin the critical orbit with e = 1 — the exact solution √ √ √ r + 2 m ϕ(r) = 2 ln √ √ + ϕ r − 2 m 0 1r2r τ(r) = − (r + 12m) + 4mϕ(r) + τ 3 m 0 1r2r t(r) = − (r + 18m) + 8mϕ(r) + t 3 m 0 √ √ ! r + 2m − 2m ln √ √ r − 2m

I the minimal distance for a marginally over-critical orbit

√ √ 4m + δl √ p √ δr = δl δl + δl + 8m + δl ≈ 8mδl 4m M87*, Sagittarius A*, Cygnus X-1 the parameters of the ‘eye-shaped’ ﬂyby which starts at rO = 50m with vO = 0.2c summary

I the event horizon r = 2m

I the unstable circular photon orbit r = 3m (and the black hole shadow)

I the minimal distance in a free non-relativistic ﬂyby r = 4m

I the innermost stable circular orbit r = 6m