Particle and photon orbits in McVittie spacetimes. Brien Nolan Dublin City University Britgrav 2015, Birmingham

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1 Particle and photon orbits in McVittie spacetimes. Brien Nolan Dublin City University Britgrav 2015, Birmingham

2 Outline Basic properties of McVittie spacetimes: embedding of the Schwarzschild field in a Robertson-Walker (RW) universe. Global structure - Class I and II Circular photon (and particle) orbits Bound particle and photon orbits Summary: apply some results from dynamical systems to the interpretation of an interesting (?) and important (??) solution of the Einstein equations. Class. Quantum Grav (2014)

3 Black holes in an expanding universe How can we model massive bodies embedded in expanding, isotropic universes? Test particle approximation; matching - cf. extensive literature on Einstein-Straus; exact or approximate solutions of the Einstein equations with appropriate boundary/asymptotic conditions. McVittie s metric (1933) takes this last approach and provides an intriguing solution of the Einstein equations.

4 The metric The following metric arises uniquely under certain conditions (Einstein equations for spherically symmetric perfect fluid, conditions on Misner-Sharp mass, asymptotic conditions) - see McVittie (1933), Nolan (1999), Nandra et al (2012): ds 2 = (f r 2 H 2 )dt 2 2rHf 1/2 dtdr + f 1 dr 2 + r 2 dω 2, f = 1 2m r, H(t) = S (t) S(t).

5 Basic properties The energy density and pressure are given by 8πρ = 3H 2 Λ, 8πp = 2H (1 2m r ) 1/2 3H 2 + Λ. Setting m = 0 yields the k = 0 RW universe with Hubble function H(t). Setting H(t) = 0 yields Schwarzschild spacetime. Setting H(t) = H 0 yields Schwarzschild-de Sitter spacetime with cosmological constant Λ = 3H 2 0. Except in these trivial cases, there is a curvature singularity at r = 2m.

6 Class 2: lim t 0 + S(t) = 0, lim t + H(t) = 0 r = 2m, t + r +, t + r +, t + r = 2m, t (0, + ) Penrose-Carter diagram for Class 2 McVittie. The singular boundary at r = 2m forms (a) the past boundary, cutting of the big bang at t = 0 and (b) an inner boundary at finite affine distance but at t +. The boundaries at infinity are at infinite affine distance. There are slight variations depending on the value of κ (a parameter related to sound speed at zero density) (cf. Nolan (1999, 2014)). No extendibility results across r = 2m; question is open. The singularity is gravitationally weak.

7 Class 1: lim t 0 + S(t) = 0, lim t + H(t) = H 0 r = r, t + r +, t + r = 2m, t (0, + ) Penrose-Carter diagram for Class 1 McVittie. The singular boundary at r = 2m forms the past boundary. There is an inner horizon at r = r, the inner horizon of the corresponding Schwarzschild-de Sitter spacetime. The boundaries at infinity are at infinite affine distance. Structure is universal for all Class 1 (cf. Kaloper et al (2010); Lake & Abdelqader (2011); Nolan (2014)). Kaloper et al conjectured, and Lake & Abdelqader demonstrated extendibility to Schwarzschild-de Sitter across the inner horizon.

8 Orbits - non-radial causal geodesics The effects of cosmological expansion and the central mass/black hole compete with one another. Is it possible for these to strike a balance and allow (i) circular orbits? or (ii) bound orbits? Black holes generically admit (a) a unique circular photon orbit (CPO), at r = 3m, which is unstable and (b) families of circular particle orbits, with the characteristic existence of innermost stable circular orbits (ISCO). Schwarzschild: r ISCO = 6m. Schwarzschild-de Sitter: r ISCO = f (m, H 0 ), and we must have mh 0 <

9 Circular photon orbits Proposition A McVittie metric with mass parameter m admits a CPO at radius r = a if and only if the Hubble function and background scale factor have the form H(t) = H 0 tanh(ah 0 t), S(t) = (cosh(ah 0 t)) 1/A where A = (1 3m a ), H 0 = a 1 1 2m 1 2m a. a McVittie metric has one parameter, m and a function H(t). There is only a 2-parameter family of McVittie metrics with a CPO. No big bang. Similar results hold for particle orbits (timelike geodesics).

10 Bound orbits Proposition All McVittie metrics of Class 1 and Class 2 admit bound photon and particle orbits i.e. future complete null and timelike geodesics along which r remains finite. Along the bound photon orbits, we have lim r(s) = 3m, s and along the bound particle orbits, we have lim r(s) = r, s where r is the radius of a circular particle orbit in the corresponding Schwarzschild-de Sitter (Class 1) or Schwarzschild (Class 2) spacetime.

11 Sample sketch proof: particle orbits in Class 2 Lemma There exists a C 2 function w : R R, with w(0) = 0, such that the geodesic equations read H = w(h)u(x), (1) ṙ = p, (2) ṗ = V (r) + rh 2 + rf 1/2 w(h)u(x) 2. (3) The C 2 function u is defined in a neighbourhood of x 0 = (0, r, 0) and ṫ = u(x). Key point: x 0 is a non-hyperbolic equilibrium point of the flow.

12 Schwarzschild dynamics McVittie dynamics Recall that H(t) 0 as t. {x : H = 0} is an invariant manifold of the flow, corresponding to particle motion in Schwarzschild spacetime: r + V m,l (r) = 0. Then W = 1 2ṙ2 + V m,l (r) V m,l (r ) is a Lyapunov function for the flow. McVittie dynamics: W is no longer conserved, but its evolution can be controlled by a Gronwall-type argument. Along the geodesic, H(t(s)) undergoes power-law decay, with overall power-law decay to a stable circular orbit of Schwarzschild spacetime.

13 Class 1: M = 1, r = 7M, H(t) = 23 t 1, L = 7/2 rhτl HHtHΤLL yhτl Τ xhτl 5 10 Plot of the area radius along the time-like geodesic. Note that r 2m+ at finite proper time in the past. The geodesic is future complete. Spacetime representation of the geodesic; t and τ increase downwards. H 0 as τ, and the trajectory oscillates around r = r = 7M.

14 Conclusions The singular surface {r = 2m, t > 0} forms a past boundary of the spacetime: all causal geodesics meet this surface at finite affine/proper time in the past. Global structure results extend to all Class 1 and Class 2 spacetimes. The central region can capture photons and particles, maintaining them in future complete, bound orbits. Class 1 and 2 McVittie spacetimes admit ISCO s, indicating possibility of formation of thin accretion disks. This is a black hole-like quality of these spacetimes.

15 Cosmological models? The k = 0 RW class arises as a limit. A potential application: Determine H(t) in the usual way, with k = 0 as a prior. Redo CMB calculation, allow for corrections due to m in the angular diameter distance D A. Corrections to H(t), and future evolution... Existence of k = 1 solution has been proven: metric in terms of elliptic integrals in comoving coordinates. Area radial might be more tractable.