General Relativity: Homework 4 Solutions - Department of Physics ...

) To find an upper bound on the vacuum energy density, we will find what value is necessary to keep Pluto
bound to the solar system. We can write this as the requirement that the negative gradient of the potential
(i.e. the acceleration) be negative (so that the acceleration is towards the center of the solar system).
Isolating the vacuum energy density we arrive at
−∇Φ(r P luto ) = − GM ⊙
rP 2 + 4π
luto
3 Gρ Λr P luto < 0
ρ Λ < 3
4π
M ⊙
r 3 P luto
= 3 1.989 × 10 30 kg ( 1GeV )(1.97 × 10 −16 m
4π (6 × 10 12 m) 3 1.78 × 10 −27 kg GeV −1 ) 3
ρ Λ < 9.44 × 10 −30 GeV 4 ≈ 10 −29 GeV 4
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