Spherically symmetric spacetimes in f(r) gravity Daniel Sunhede University of Jyväskylä K Kainulainen JYU, J Piilonen JYU, V Reijonen HIP
Introduction Solar System constraints / Post-Newtonian parameter ds 2 g µν x µ x ν = e A(r) dt 2 + e B(r) dr 2 + r 2 dω 2 ds 2 = (1 2GM r )dt 2 + (1 + 2γ PPNGM r )dr 2 + r 2 dω 2 γ PPN 1 B A 1 = (2.1 ± 2.3) 10 5 Cassini time delay Bertotti et al Nature 425 (2003) 374 Debate γ PPN =? 1/2 Chiba, Phys Lett B575 (2003) 1 1 Faraoni, Phys Rev D74 (2006) 023529 1/2 Erickcek et al, Phys Rev D74 (2006) 121501 1 Zhang, Phys. Rev D76 (2007) 024007 Goal: Compute the continuous metric by solving the full field equations
Metric vs Palatini Beyond Einstein-Hilbert action: Same action yields different EOM:s depending on choice of free variables S = 1 16πG g µν Metric formalism d 4 x gf(r) + S m (g, ψ), independent Γ ρ µν { ρ µν } g R g µν R µν (Γ) F R µν 1 2 fg µν µ ν F + g µν F = 8πGT µν F f R
Metric vs Palatini Beyond Einstein-Hilbert action: Same action yields different EOM:s depending on choice of free variables S = 1 16πG g µν Metric formalism d 4 x gf(r) + S m (g, ψ), independent g µν, Γ ρ Γ ρ µν { µν } ρ µν g Palatini formalism R g µν R µν (Γ) independent Larger space of allowed spacetime manifolds Min may be found outside Levi-Civita subspace F R µν 1 2 fg µν µ ν F + g µν F = 8πGT µν ρ ( gf (R)g µν ) = 0 Γ ρ µν { ρ µν } F g F f R
Trace equation Metric case F + 1 3 (F R 2f) = 8πG 3 T For the static and spherically symmetric case A = 1 1 + γ B = 1 eb r ( 1 e B r + reb F reb F 8πGp + reb 2 8πG reb (2ρ + 3p) + 3 6 ) ( f ) 4γ R + F r ( f ) R + γa F F + 2 r F + A B F + eb 2 3 (F R 2f) = eb 3 8πGT Kainulainen, Piilonen, Reijonen, and Sunhede, Phys Rev D76 (2007) 024020 Together with the conservation law and a given eqn of state, these form a complete generalization of the Tolman-Oppenheimer-Volkov eqns p = A 2 γ rf 2F (ρ + p), p = p(ρ)
Metric case Weak field limit A, B 1, Newtonian approx Example: f(r) = R µ 4 /R, µ 2 = 4Λ/ 3 d + 2 r d µ2 (1 3d) 3 d Neglect p ρ d F 1 = 8πG 3 ρ d = 2GM + d 0 Boundary condition
Metric case Weak field limit A, B 1, Newtonian approx Example: f(r) = R µ 4 /R, µ 2 = 4Λ/ 3 d + 2 r d µ2 (1 3d) 3 d Neglect p ρ d F 1 = 8πG 3 ρ d = 2GM + d 0 Boundary condition A = B r 2d + µ2 r d 1 + d (rb) = 16πG r 2 (µr) 2 ρ + 3 3 d(1 + d) A = 8GM B = 4GM (rb) 0 + A 0 + (rb) 0
Metric case Weak field limit A, B 1, Newtonian approx Example: f(r) = R µ 4 /R, µ 2 = 4Λ/ 3 d + 2 r d µ2 (1 3d) 3 d Neglect p ρ d F 1 = 8πG 3 ρ d = 2GM + d 0 Boundary condition A = B r 2d + µ2 r d 1 + d (rb) = 16πG r 2 (µr) 2 ρ + 3 3 d(1 + d) A = 8GM B = 4GM (rb) 0 + A 0 + (rb) 0 For any finite value of γ PPN = B A = 4GM + 3(rB) 0 = 1 8GM + 3(rB) 0 2 B 0
Metric case A, B, d dvac CASE I: d 0 > 10 6, non-linear term is neglible and expected results follow 5 0 5 Metric components and d d vac 10 6 d d vac B A γppn 1.5 1 0.5 PPN parameter GR f(r) 0 0 2 4 6 8 10 r/r Results independent of boundary conditions (set at the center of the star) Possible background such as DM and solar wind has no effect on the solution 0 2 4 6 8 10 r/r Kainulainen, Piilonen, Reijonen, and Sunhede, Phys Rev D76 (2007) 024020
CASE II: d 0 < 10 6, non-linear term dominant and solution gets trapped in Palatini track d + 2 r d µ2 (1 3d) 3 d Neglible If in this limit also = 8πG 3 ρ F R 2f = 8πGT F 1 R 8πGρ Metric case d, A A0 0.8 0.6 0.4 0.2 1 10 61 0 0 2 4 6 8 10 10 28 r/r GR-like limit exists! However f(r) = R µ 4 /R solution unstable in time Stabilized +αr 2 solution not compatible with obs d Dolgov & Kawasaki, Phys Lett B573 (2003) 1 Nojiri & Odintsov, Phys Rev D68 (2003) 123512 Kainulainen, Reijonen, and Sunhede Phys Rev D76 (2007) 043503
For constant density, Palatini case R is also constant F (R)R 2f(R) = 8πGT R = const. G µν + Λ ρ g µν = 8πG T µν, Λ ρ 1 ( R ρ f ) ρ F ρ 2 F ρ Schwarzschild-dS exterior, M = r 0 dr 4πr2 ρ + Λ ρ Λ 0 F ρ 6G r3 γ PPN = 1. However Kainulainen, Reijonen, and Sunhede Phys Rev D76 (2007) 043503 For f(r) = R µ 4 /R R ρ = 1 2 (8πGρ ± (8πGρ) 2 + 12µ 4 ) 8πGρ F = 1 + µ4 R 2 1 OK! Kainulainen, Piilonen, Reijonen, and Sunhede, Phys Rev D76 (2007) 024020
For constant density, Palatini case R is also constant F (R)R 2f(R) = 8πGT R = const. G µν + Λ ρ g µν = 8πG T µν, Λ ρ 1 ( R ρ f ) ρ F ρ 2 F ρ Schwarzschild-dS exterior, M = r 0 dr 4πr2 ρ + Λ ρ Λ 0 F ρ 6G r3 γ PPN = 1. However Kainulainen, Reijonen, and Sunhede Phys Rev D76 (2007) 043503 For f(r) = R µ 4 /R R ρ = 1 2 + αr 2 (8πGρ ± ) (8πGρ) 2 + 12µ 4 8πGρ F = 1 + µ4 + R 2 2αR 1 1 NOT OK! Kainulainen, Piilonen, Reijonen, and Sunhede, Phys Rev D76 (2007) 024020
Summary METRIC Non-linear terms negligible for major part of parameter space and γ PPN = follows Small region exists for which the solution follows a Palatini track. It appears that, as long as is not dominated by a cosmological constant, this solution is unstable or incompatible with obs. PALATINI Exterior solution always Schwarschild-dS, Relation between density and apparent mass non-standard, only functions δf(r) increasing slower than R will yield GR gravitational field 1 2 δf(r) f(r) R Kainulainen & Sunhede, in progress γ PPN = 1