Violation of the weak equivalence principle due to gravity-matter - - PowerPoint PPT Presentation

Violation of the weak equivalence principle due to gravity-matter entanglement

Nikola Paunkovi´ c1,2, Francisco Pipa3 and Marko Vojinovi´ c4

1 Department of Mathematics, IST, University of Lisbon 2 Security and Quantum Information Group (SQIG), Institute of Telecommunications, Lisbon 3 Department of Physics, IST, University of Lisbon 4 Group for Gravitation, Particles and Fields (GPF), Institute of Physics, University of Belgrade

Weak Equivalence Principle

The local effects of particle motion in a gravitational field are indis- tinguishable from those of an accelerated observer in flat spacetime.

Consequence:

A particle in a gravitational field should follow the geodesic, since this is how the straight line in flat space looks like from the accelerated frame.

Deriving WEP (geodesic motion) in GR

Single-pole approximation: T µν(x) =

• C

dτ Bµν(τ)δ(4)(x − z(τ)) √−g . (1) Conservation of stress-energy tensor (assuming the local Poincar´ e in- variance for both SG[g] and SM[g, φ]): ∇νT µν = 0 . (2) Replacing (2) into (1), we obtain the geodesic equation, with uµ ≡ dzµ(τ)

dτ

and uµuµ ≡ −1 (Mathisson and Papapetrou [2, 3]; see also [4]): uλ∇λuµ = 0 . Using Cristoffel symbols, d2zλ(τ) dτ 2 + Γ λ

µν

dzµ(τ) dτ dzν(τ) dτ = 0 .

Quantising gravity

Fundamental gravitational degrees of freedom ˆ g and ˆ πg: ∆ˆ g∆ˆ πg ≥ 2 , ∆ˆ φ∆ˆ πφ ≥ 2 . Separable state (|g and |φ – coherent states of gravity and matter): |Ψ = |g ⊗ |φ . Effective classical metric and stress-energy tensors: gµν ≡ Ψ|ˆ gµν|Ψ , Tµν ≡ Ψ| ˆ Tµν|Ψ .

Violation of WEP due to entanglement

Entangled state (perturbation |˜ Ψ = |˜ g ⊗ |˜ φ, with coherent classical states |˜ g and |˜ φ): |Ψ Ψ Ψ = α|Ψ + β|˜ Ψ . “Entangled” metric: g g gµν = Ψ Ψ Ψ|ˆ gµν|Ψ Ψ Ψ = gµν + β hµν + O(β2) . The perturbation is evaluated to be: hµν = 2 Re

• Ψ|ˆ

gµν|˜ Ψ − Ψ|˜ Ψgµν

• .

“Entangled” geodesic equation with the manifestly covariant correc- tion: d2zµ(τ) dτ 2 + Γ Γ Γ µ

ρν

dzρ(τ) dτ dzν(τ) dτ = 0 , uλ∇λuµ + β

• ∇ρhµ

ν − 1

2∇µhνρ

• uρuν + O(β2) = 0 .

Bibliography

[1] N. Paunkovi´ c and M. Vojinovi´ c, (2017), arXiv:1702.07744 [2] M. Mathisson, Acta Phys. Polon. 6, 163 (1937) [3] A. Papapetrou, Proc. R. Soc. A 209, 248 (1951) [4] M. Vasili´ c and M. Vojinovi´ c, JHEP , 0707, 028 (2007), arXiv:0707.3395

5

THANK YOU!