Tides in Higher-Dimensional Newtonian Gravity

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1 Tides in Higher-Dimensional Newtonian Gravity Philippe Landry Department of Physics University of Guelph 23 rd Midwest Relativity Meeting October 25, 2013

2 Tides: A Familiar Example Gravitational interactions with the Sun and Moon cause ocean tides More generally, tides occur whenever a body is subject to an external, time-dependent gravitational field (the tidal potential) Given a tidal potential, what is the body s response?

3 The Tidal Deformation Problem The tidal potential V produces a deformation of the body s gravitational potential U V = l,m 4π 2l + 1 d lmr l Y lm δu = G 4π 2l + 1 I lmr (l+1) Y lm l,m We work under the assumptions of Static tides Perfect fluid body Initially spherically-symmetric configuration Small perturbations

4 Radau s Equation The perturbed internal structure of the body is governed by Radau s equation, which comes from the fluid equations rη l + η l (η l 1) + 6D(η l + 1) (l 1)l = 0 η l contains information on the displacement of the fluid elements (related to the displacement vector) The body s equation of state enters into D = 4 3 πr 3 ρ/m = ρ/ ρ

5 Love Numbers Matching conditions for δu at the boundary of the body give GI lm = 2k l R 2l+1 d lm Gravitational Love numbers k l contain dependence on internal structure of body k l = (l + 1) η l(r) 2 (l + η l (R)) Calculating the Love numbers for a body solves the tidal deformation problem once and for all

6 Beyond 3 Dimensions Why calculate Love numbers for bodies in N dimensions? Tidal deformation problem in higher-dimensional gravity Black holes and other objects in higher-dimensional gravity and string theory [e.g. Kol & Smolkin 2011] We want to compare Love numbers of exotic objects to those of more familiar ones (e.g. polytropes)

7 Generalization to N Dimensions The concept of Love numbers can straightforwardly be extended to higher dimensions (r, θ, φ) (r, θ 1, θ 2,..., θ n ) with n = N 1 4π Ω n = 2π(n+1)/2 Γ( n+1 2 ) Y lm, d lm, I lm Y lj, d lj, I lj D = ρ/ ρ Ω nr n+1 ρ (n + 1)m Radau s equation becomes rη l + η l(η l n + 1) + 2(n + 1)D(η l + 1) (l + n 1)l = 0

8 Love Numbers in N Dimensions The equation which determines the gravitational Love numbers generalizes to GI lj = 2k l R 2l+n 1 d lj Gravitational Love numbers pick up dependence on n k l = (l + n 1) η l(r) 2 (l + η l (R))

9 Polytropes in N Dimensions Polytropic stellar models have the equation of state P = Kρ Γ with Γ = 1 + 1/ν and central density ρ 0 Equation of hydrostatic equilibrium dp dr Mass gradient dm dr = Ω n ρr n = ρ Gm r n Introduce scaled density θ ν, radial variable ξ and mass µ N-dimensional Lane-Emden equations dθ dξ = µ ξ n, dµ dξ = ξn θ ν Equations can be integrated to determine the mass and density functions, which enter into D in Radau s equation

10 Love Numbers across Dimensions Gravitational Love numbers k 2 for uniform-density spheres and polytropes compared to Schwarzschild black holes (from Kol & Smolkin 2011) in N dimensions

11 Polytrope Density Profiles in 3D vs 4D Density profiles as a function of polytropic index ν in 3D

12 Polytrope Density Profiles in 3D vs 4D Density profiles as a function of polytropic index ν in 4D

13 Summary The theory of Love numbers in Newtonian gravity was generalized straightforwardly to N dimensions Love numbers for polytropes were computed The k l depend strongly on the compactness They depend also on the number of angular dimensions n Polytrope Love numbers were compared to those for black holes in higher dimensions Why are the black hole k l negative? Why are they zero in 3D but non-zero in other dimensions?