Here’s a clean way to connect the 1997 measurement to an energy scale, then compare it to what a traversable wormhole would demand.
1) “Express the Casimir force as an energy value” (Lamoreaux, 1997)
Lamoreaux’s landmark 1997 experiment measured the Casimir attraction using a sphere–plane setup (gold-coated sphere of radius R=11.3 cm facing a flat plate) at separations
a ≈ 0.6–6 μm. In the proximity-force limit, the sphere–plane force is related to the
parallel-plate Casimir
energy per unit area |E|/A by F(a) ≈ 2πR (E/A).
For ideal metals at zero temperature:
|E|/A = − π²ℏc/720a³ and P(a) = − π²ℏc/240a⁴,
where P is the pressure. Lamoreaux explicitly uses this relation in deriving the sphere–plane expression and reports agreement with theory at the 5% level over a = 0.6 – 6 μm.
(
Physical Review Journals)
Plugging in the closest reported separation a = 0.6 – 6 μm:
- Energy density (per area): ∣E∣/A ≈ 2.0×10⁻⁹ J m⁻².
- Pressure magnitude: ∣P∣ ≈ 1.0×10⁻² Pa.
So even a
full square metre of perfectly parallel plate at 0.6 μm gap would store only ∼2.0×10⁻⁹ joules of (negative) Casimir energy.
(Using the same formula at 1 μm gives ∣E∣/A ≈ 4.3×10⁻¹⁰J m⁻² and ∣P∣≈1.3×10⁻³ Pa.) (
Physical Review Journals)
2) Why this is nowhere near enough to hold a wormhole open
Traversable Morris–Thorne wormholes require
exotic matter that violates the null energy condition. Quantum-field-theory “quantum inequalities” derived by Ford & Roman show that negative energy can only occur in tiny amounts, for very short times, and must be compensated by larger positive energy—severely limiting any macroscopic engineering. (
arXiv,
Physical Review Links)
A widely cited estimate (Visser, summarized in
Scientific American) puts the requirement into perspective: a wormhole with a
1-metre throat would need negative energy of order the
total energy output of ~10 billion stars for one year—i.e., ∼10⁴⁴ joules. The same article notes the negative energy would need to be confined to an
extremely thin band around the throat (down to fantastically small thicknesses in some models), which Casimir setups do not provide at macroscopic scales. (
JSTOR,
webhome.phy.duke.edu)
Compare scales:
- Casimir (lab, a = 0.6 μm): ∼ 2×10⁻⁹ J per square metre.
- Wormhole (~1 m throat, Area throat = 4π(0.5)²≈ 3.14 m²): 10⁴⁴/3.14 m² ∼ 3.2×10⁴³ J m⁻² plus stringent localization/time-duration limits from quantum inequalities.
Even if you tiled many square metres of plates, you’d be short by a factor of about
3.2×10⁴³ J m⁻² / 2×10⁻⁹ J m⁻² ∼ 1.6×10⁵²,
and you would still fail the quantum-inequality constraints on how negative energy can be distributed in space and time. In short: the
measured Casimir energy densities are real but
astronomically too small—and too delocalized under QI limits—to keep a traversable wormhole open. (
Physical Review Journals,
arXiv,
JSTOR)
Sources
• S.K. Lamoreaux, “Demonstration of the Casimir Force in the 0.6 to 6 µm Range,”
Phys. Rev. Lett. 78, 5–8 (1997). Includes the sphere–plane formula and experimental details. (
Massachusetts Institute of Technology)
• A. Lambrecht, S. Reynaud, review context for E/A and P(a) relation F(a) ≈ 2πR (E/A). (
ScienceDirect)
• G.L. Klimchitskaya et al.,
Rev. Mod. Phys. 81, 1827 (2009): standard formulas |E|/A = − π²ℏc/720a³ and P(a) = − π²ℏc/240a⁴.(
Physical Review Journals)
• L.H. Ford & T.A. Roman, “Quantum inequalities” (constraints on negative energy). (
arXiv,
Physical Review Links)
• M. Visser summary in
Scientific American (quantitative wormhole energy estimate, 1-m throat ≈ 10 billion stars/year). (
webhome.phy.duke.edu,
JSTOR)
