Questions: Frame Dragging (Lense-Thirring Effect)

Question 1

Frame dragging is the gravitational analog of which electromagnetic phenomenon?

• A. The electric field of a point charge
• B. The magnetic field generated by a current loop
• C. Electromagnetic induction (Faraday's law)
• D. The Coulomb force between static charges

Answer: (B)

In the gravitoelectromagnetic analogy of linearized GR, mass-energy density is analogous to charge density (producing a gravitoelectric field — ordinary Newtonian gravity), and mass-energy current (momentum) is analogous to electric current (producing a gravitomagnetic field — frame dragging). A rotating mass is a mass-energy current loop, and the gravitomagnetic field it produces causes frame dragging — just as a current loop produces a magnetic field. The precession of a gyroscope in the gravitomagnetic field is analogous to the precession of a magnetic dipole in a magnetic field.

Question 2

Gravity Probe B measured two distinct precession effects on its gyroscopes: geodetic precession and Lense-Thirring (frame-dragging) precession. The geodetic effect is about 170 times larger.

• T. True
• F. False

Answer: True

In the GP-B orbit (642 km altitude, polar orbit), the geodetic precession (due to the curvature of spacetime around Earth, even for a non-rotating Earth) is about 6606 milliarcseconds/year, while the Lense-Thirring precession (due to Earth's rotation dragging spacetime) is about 39 milliarcseconds/year — roughly 170 times smaller. Both are tiny effects, requiring gyroscopes with drift rates below 0.5 milliarcseconds/year, which GP-B achieved using superconducting niobium-coated quartz spheres. The geodetic effect was confirmed to 0.28% precision and the frame-dragging effect to about 19% precision.

Question 3

Explain why frame dragging makes it impossible for an observer to remain stationary (relative to distant stars) inside the ergosphere of a Kerr black hole.

Model answer: Frame dragging tilts the light cones in the direction of the black hole's rotation. Inside the ergosphere, the dragging is so strong that all future-directed timelike worldlines must have a positive angular velocity component in the direction of rotation. A 'stationary' observer (fixed angular coordinate φ) would need a four-velocity with dφ/dt = 0, but this requires the worldline to be outside the future light cone — it becomes spacelike, which is physically impossible. The static limit (outer boundary of the ergosphere) is defined as the surface where the Killing vector ∂/∂t becomes null; inside it, ∂/∂t is spacelike, so 'static' (constant φ) worldlines are spacelike and unphysical.

The ergosphere demonstrates frame dragging at its most extreme. Unlike the event horizon (from which nothing escapes), the ergosphere permits escape — an observer can enter and leave, as long as they co-rotate with the black hole. This is what makes the Penrose process possible: the negative-energy orbits within the ergosphere exist precisely because the timelike Killing vector is spacelike there.

Question 4

Calculate the order of magnitude of the Lense-Thirring precession rate for a gyroscope in low Earth orbit, given Earth's angular momentum J_⊕ ≈ 7 × 10³³ kg⋅m²/s and orbital radius r ≈ 7 × 10⁶ m.

Model answer: The Lense-Thirring precession rate is of order Ω_LT ~ GJ/(c²r³). Plugging in: Ω_LT ~ (6.67×10⁻¹¹)(7×10³³)/((3×10⁸)²(7×10⁶)³) = (4.7×10²³)/(3.1×10²⁵) ≈ 1.5 × 10⁻² arcsec/year, or about 15 milliarcseconds per year. This is consistent with the GP-B measurement of about 39 mas/yr (the exact value depends on orbital inclination and the precise formula). The extraordinary smallness of this effect — a precession rate of milliarcseconds per year — illustrates why frame-dragging required decades of technological development to measure directly.

The Lense-Thirring effect scales as GJ/(c²r³), falling off as r⁻³ — faster than the gravitoelectric (Newtonian) force which falls off as r⁻². Near rapidly rotating compact objects (neutron stars, Kerr black holes), the effect becomes dramatic: the ISCO (innermost stable circular orbit) shifts, and the ergosphere opens up.