Gravitational Casimir Effect...B. S. DeWitt, “Superconductors and Gravitational Drag”, Phys....

Gravitational Casimir EffectJAMES Q. QUACH

Quach, PRL 114, 081104 (2015)

Hendrik Casimir (1948)

𝑃𝐼 𝑎 = −𝜋2

240ℏ𝑐𝑎4

𝐸𝑛 = ℏ𝜔(𝑛 +12)

𝐸0 =ℏ𝜔2

Casimir Effect

�Motivation�Gravitational Casimir Effect

�Ordinary Materials�Superconductors

Outline

QM & GR Interface

Hawking Radiation

Detectors

BA

DC

T

g

COW Experiment

QM & GR InterfaceDetectors

BA

DC

T

g

COW Experiment

A. Overhauser, R. Colella, S. Werner


BA

DC

T

g

Colella, Overhauser, Werner, PRL 1975

COW Experiment


BA

DC

T

g

Colella, Overhauser, Werner, PRL 1975

𝐻 =𝒑2

2𝑚𝑖+ 𝑚𝑔𝒈𝒓 − 𝝎𝑳

Δ𝛽𝑔 ≈𝑚𝑔𝑚𝑖𝒈𝑲𝑇𝑇′

COW Experiment

Lammerzahl, Gen. Rel. Grav.1996

WEP violation in QMExample: Dirac Equation

𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)

𝑖ℏ𝛾𝛼𝐷𝛼 −𝑚𝑐 𝜓 = 0

𝐷𝛼 = ℎ𝛼𝑖 𝐷𝑖, 𝐷𝑖 = 𝜕𝑖 +𝑖4𝜎𝛼𝛽𝜔𝑖

𝛼𝛽

Obukhov, PRL, 2001

There is no experiment that can be done, in a smallconfined space, which can detect the differencebetween a uniform gravitational field and anequivalent uniform acceleration.


𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)



𝛼𝛽

𝑉 = 1 +𝒂 ∙ 𝒙𝑐2, 𝑊 = 1

𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑝2

2𝑚 + 𝛽𝑚𝒂 ∙ 𝒙 +ℏ2𝑐 𝚺 ∙ 𝒂 +

ℏ2𝑚𝑐2 𝛽𝚺 ∙ (𝒂 × 𝒑)

(ii) Accelerated frame

Obukhov, PRL, 2001

ℏ𝑔𝑐= 2 × 10−23eV

𝑉 = 1 −𝐺𝑀2𝑐2𝑟

1 +𝐺𝑀2𝑐2𝑟

−1

, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟

2


2𝑚− 𝛽𝑚𝒈 ∙ 𝒙 −

ℏ2𝑐𝚺 ∙ 𝒈 −

ℏ𝑚𝑐2𝛽𝚺 ∙ (𝒈 × 𝒑)

(i) Gravitational field 𝒈 = −𝐺𝑀𝒓𝑟3

(Foldy–Wouthuysen transformation)


𝑑𝑠2 = 𝑉2 𝑑𝑥0 2 −𝑊2(𝑑𝒙 ∙ 𝑑𝒙)



𝛼𝛽

𝑉 = 1 +𝒂 ∙ 𝒙𝑐2, 𝑊 = 1


2𝑚 + 𝛽𝑚𝒂 ∙ 𝒙 +ℏ2𝑐 𝚺 ∙ 𝒂 +

ℏ2𝑚𝑐2 𝛽𝚺 ∙ (𝒂 × 𝒑)

(ii) Accelerated frame

Obukhov, PRL, 2001

ℏ𝑔𝑐= 2 × 10−23eV

𝑉 = 1 −𝐺𝑀2𝑐2𝑟


−1

, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟

2


2𝑚− 𝛽𝑚𝒈 ∙ 𝒙 −

ℏ2𝑐𝚺 ∙ 𝒈 −


(i) Gravitational field 𝒈 = −𝐺𝑀𝒓𝑟3

(inertial spin-orbit coupling)

(gravitational spin-orbit coupling)

𝑉 = 1 +𝒂 ∙ 𝒙𝑐2 , 𝑊 = 1

𝐻𝑎 = 𝛽𝑚𝑐2 + 𝛽𝑚𝒂 ∙ 𝒙 + 𝛽𝑝2

2𝑚+ℏ2𝑐𝚺 ∙ 𝒂 +

ℏ2𝑚𝑐2𝛽𝚺 ∙ (𝒂 × 𝒑)

(i) Accelerated frame

𝑉 = 1 −𝐺𝑀2𝑐2𝑟


−1

, 𝑊 = 1 +𝐺𝑀2𝑐2𝑟

2

𝐻𝑎 = 𝛽𝑚𝑐2 − 𝛽𝑚𝒈 ∙ 𝒙 + 𝛽𝑝2

2𝑚−ℏ2𝑐𝚺 ∙ 𝒈 −


(ii) Gravitational field

ℏ𝑔𝑐= 2 × 10−23eV

Altschul, et al., Advances in Space Research (2015) 𝜂 = 2𝑎𝐴 − 𝑎𝐵𝑎𝐴 + 𝑎𝐵

Eötvös ratio:

𝒈 = −𝐺𝑀𝒓𝑟3

WEP violation in QM

B. S. DeWitt, “Superconductors and Gravitational Drag”, Phys. Rev. Lett. 16 , 1092 (1966).

G. Papini, “London moment of rotating superconductors and lense-thirring fields of general relativity”, Nuovo Cimento B 45 , 66 (1966).

G. Papini, “Particle Wave Functions in Weak Gravitational Fields”, Nuovo Cimento B, 52, 136–41 (1967).

G. Papini, “Detection of inertial effects with superconducting interferometers”, Phys. Lett. A 24, 32–3 (1967).

G. Papini, “Superconducting and Normal Metals as Detectors of Gravitational Waves”, Lettere Al Nuovo Cimento 4, 22 1027 (1970).

J. Anandan, “Gravitational and Inertial Effects in Quantum Fluids”, Phys. Rev. Lett. 47 , 463 (1981).

R. Y. Chiao, “Interference and inertia: A superfluid-helium interferometer using an

Enhanced gravitational interaction with quantum systems

Gravitational Casimir Effect

𝛻 ∙ 𝑬 = 𝜅𝝆(𝐸)𝛻 ∙ 𝑩 = 𝜅𝝆 𝑀

𝛻 × 𝑬 = −𝜕𝑩𝜕𝑡− 𝜅𝑱 𝑀

𝛻 × 𝑩 =𝜕𝑬𝜕𝑡+ 𝜅𝑱 𝐸

𝜅 =8𝜋𝐺𝑐4

𝐸𝑖𝑗 = 𝐶0𝑖0𝑗𝐵𝑖𝑗 = ⋆ 𝐶0𝑖0𝑗

𝐽𝜇𝜈𝜌 = −𝑇𝜌 𝜇,𝜈 +13𝜂𝜌[𝜇𝑇,𝜈]

𝜌𝑖𝐸 = −𝐽𝑖00

𝜌𝑖𝑀 = − ⋆ 𝐽𝑖00𝐽𝑖𝑗𝐸 = 𝐽𝑖0𝑗𝐽𝑖𝑗𝑀 = ⋆ 𝐽𝑖0𝑗

Gravitoelectromagnetism (GEM)

𝐸0 =ℏ4𝜋 0

∞𝑘∥𝑑𝑘∥

𝑛

𝜔𝑛+ + 𝜔𝑛× 𝜎

Δ 1 +𝜅𝜒2𝑬𝑇𝑇 = 0

Δ𝑩𝑇𝑇 = 0

𝑬𝑇𝑇 =𝛼(1 −

sin2𝛾2) 𝛽cos𝛾

𝛽cos𝛾 −𝛼(1 −sin2𝛾2)𝑒𝑖(𝒌∙𝒓−𝜔𝑡)

𝑩𝑇𝑇 =−𝛽(1 −

sin2𝛾2) 𝛼cos𝛾

𝛼cos𝛾 𝛽(1 −sin2𝛾2)𝑒𝑖(𝒌∙𝒓−𝜔𝑡)


(smoothness principle)


𝛼′ 1 +𝜅𝜒2 1 −

𝑆′

2 𝑒−𝑞′𝑎/2 = 𝛼 1 −

𝑆2

2 𝑒−𝑞𝑎/2 + 𝛼′′ 1 −

𝑆2

2 𝑒𝑞𝑎/2

𝑆 = sin𝛾𝐶 = cos𝛾

𝛼′′′ 1 +𝜅𝜒21 −𝑆′

2𝑒−𝑞′𝑎/2 = 𝛼 1 −

𝑆2

2𝑒𝑞𝑎/2 + 𝛼′′ 1 −

𝑆2

2𝑒−𝑞𝑎/2

𝛼′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−

𝑞𝑎2 − 𝛼′′𝐶𝑒𝑞𝑎/2

-𝛼′′′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−

𝑞𝑎2 − 𝛼′′𝐶𝑒−𝑞𝑎/2

𝑞2 = −𝑘𝑧2 = 𝑘∥2 −𝜔2

𝑐2

𝑞′2 = −𝑘𝑧′2 = 𝑘∥2 − (1 + 𝜅𝜒)

𝜔2

𝑐2

𝛼′ 1 +𝜅𝜒2 1 −

𝑆′

2 𝑒−𝑞′𝑎/2 = 𝛼 1 −

𝑆2

2 𝑒−𝑞𝑎/2 + 𝛼′′ 1 −

𝑆2

2 𝑒𝑞𝑎/2

𝑆 = sin𝛾𝐶 = cos𝛾

𝛼′′′ 1 +𝜅𝜒21 −𝑆′

2𝑒−𝑞′𝑎/2 = 𝛼 1 −

𝑆2

2𝑒𝑞𝑎/2 + 𝛼′′ 1 −

𝑆2

2𝑒−𝑞𝑎/2



-𝛼′′′𝐶′𝑒−𝑞′𝑎2 = 𝛼𝐶𝑒−

𝑞𝑎2 − 𝛼′′𝐶𝑒−𝑞𝑎/2

Δ+ = 𝑒−𝑎𝑞′{ 𝐶′ 𝑆2 − 2 − 1 +𝜅𝜒2𝐶 𝑆′2 − 2

2𝑒−𝑎𝑞 − 𝐶′ 𝑆2 − 2 + 1 +

𝜅𝜒2𝐶 𝑆′2 − 2

2𝑒𝑎𝑞 = 0


𝑛

𝜔𝑛(𝑞) =12𝜋𝑖 𝑖∞

−𝑖∞𝜔(𝑞) 𝑑lnΔ(q) +

𝐶+𝜔(𝑞)𝑑lnΔ(q)

𝐸0 =ℏ4𝜋 0


𝑛

𝜔𝑛+ + 𝜔𝑛× 𝜎

𝐸 𝑎 =𝐸0𝜎− lim𝑎→∞

𝐸0𝑎

𝐸 𝑎 =ℏ4𝜋2 0


0

∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉

𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆

′2 − 2)

𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)

, 𝑟× =1 + 𝜅𝜒2 𝐶

′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)

1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)


𝐸 𝑎 =ℏ4𝜋2 0


0

∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉

𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆

′2 − 2)

𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)

, 𝑟× =1 + 𝜅𝜒2 𝐶

′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)

1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)

𝛼′ 1 +𝜅𝜒21 −𝑆′

2𝑒−𝑞′𝑎/2 = 𝛼 1 −

𝑆2

2𝑒−𝑞𝑎/2 + 𝛼′′ 1 −

𝑆2

2𝑒𝑞𝑎/2



𝛼′′

𝛼=−𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆

′2 − 2)

𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)

𝑒𝑞𝑎


𝐸 𝑎 =ℏ4𝜋2 0


0

∞ln 1 − 𝑟+2𝑒−2𝑞𝑎 + ln 1 − 𝑟×2𝑒−2𝑞𝑎 𝑑𝜉

𝑟+ =𝐶′ 𝑆2 − 2 − 1 + 𝜅𝜒2 𝐶(𝑆

′2 − 2)

𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶(𝑆′2 − 2)

, 𝑟× =1 + 𝜅𝜒2 𝐶

′ 𝑆2 − 2 − 𝐶(𝑆′2 − 2)

1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶(𝑆′2 − 2)

𝛼′′

𝛼=−𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶 𝑆

′2 − 2

𝐶′ 𝑆2 − 2 + 1 + 𝜅𝜒2 𝐶 𝑆′2 − 2

𝑒𝑞𝑎 = |𝑟+|

𝛽′′

𝛽=− 1 + 𝜅𝜒2 𝐶

′ 𝑆2 − 2 + 𝐶 𝑆′2 − 2

1 + 𝜅𝜒2 𝐶′ 𝑆2 − 2 + 𝐶 𝑆′2 − 2

𝑒𝑞𝑎 = r×


Ordinary material

𝑛 = 1 +2𝜋𝐺𝜌𝜔2

𝜒 =1 − 𝑛 𝜔 2

𝜅𝑐2

𝐺 = 6.67 × 10−11m3kg−1s−2

Peters, PRD, 1974

𝑂 𝜌 = 𝑂[𝑐2

𝐺] ≈ 1027kg/m3

𝑂 𝜌 = 104kg m−3

𝑃 𝑎 = −𝜕𝐸 𝑎𝜕𝑎

𝑃(10−6) ≈ 10−21nPa

Superconductor𝐻 =𝒑 − 𝑞𝑨 −𝑚𝒉 2

2𝑚+ 𝑉

𝒋𝐺 =1𝑚Re 𝜓∗

ℏ𝑖𝛻 − 𝑞𝑨 −𝑚𝒉 𝜓

=1𝑚−𝑞𝑨 −𝑚𝒉 𝜓∗𝜓

𝒋𝐺 = 𝜎𝐺𝑬𝐺

𝑟𝐺 = 1 +2𝛿2

𝑐𝑑𝜉−1

𝑟𝐸 = 1 +2𝜆𝛿2

𝑐𝑑2𝜉−1

Minter, Wegter-McNelly, Chiao, Physica E, 2010

Heisenberg-Coulomb effect

𝐻 =𝒑 − 𝑞𝑨 −𝑚𝒉 2

2𝑚+ 𝑉

𝒋𝐺 =1𝑚Re 𝜓∗

ℏ𝑖𝛻 − 𝑞𝑨 −𝑚𝒉 𝜓

=1𝑚−𝑞𝑨 −𝑚𝒉 𝜓∗𝜓

𝒋𝐺 = 𝜎𝐺𝑬𝐺

𝑟𝐺 = 1 +2𝛿2

𝑐𝑑𝜉−1

𝑟𝐸 = 1 +2𝜆𝛿2

𝑐𝑑2𝜉−1𝑑 = 2 nm 𝛿 = 37 nm 𝜆 = 83 nm

Superconductor

Gravitational Casimir Effect...B. S. DeWitt, “Superconductors and Gravitational Drag”, Phys....

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