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

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

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. 31 235008 (2014)

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.

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).

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.

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.

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.

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 < 2 75 3.

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).

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.

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.

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.

Class 1: M = 1, r = 7M, H(t) = 23 t 1, L = 7/2 rhτl 8-10 20 6 15 4 10 1000 HHtHΤLL yhτl 0 10 5 5 2 0-10 0-5 500 1000 Τ 2000 1500-5 0 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.

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.

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.