Kerr Metric - Frame Dragging

Frame Dragging

We may rewrite the Kerr metric in the following form:

$c^{2} d\tau^{2} = \left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2} + g_{rr} dr^{2} + g_{\theta\theta} d\theta^{2} + g_{\phi\phi} \left( d\phi + \frac{g_{t\phi}}{g_{\phi\phi}} dt \right)^{2}.$

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ, where Ω is called the Killing horizon.

$\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} r \alpha c}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} r \alpha^{2} \sin^{2}\theta}.$

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging, and has been tested experimentally. Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. An "ice skater", in orbit over the equator and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be torqued spinward. The arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. If she is already rotating at a certain speed when she extends her arms, inertial effects and frame-dragging effects will balance and her spin will not change. Due to the Principle of Equivalence gravitational effects are locally indistinguishable from inertial effects, so this rotation rate, at which when she extends her arms nothing happens, is her local reference for non-rotation. This frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. A useful metaphor is a planetary gear system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. Think Mach's principle.

Famous quotes containing the words frame and/or dragging:

“The warped, distorted frame we have put around every Negro child from birth is around every white child also. Each is on a different side of the frame but each is pinioned there. And ... what cruelly shapes and cripples the personality of one is as cruelly shaping and crippling the personality of the other.”
—Lillian Smith (1897–1966)

“There I was dragging the ocean, that knock-out,
in and out by its bottle-green neck, letting it chew
the rocks, letting it haul beach glass and furniture sticks
in and out.”
—Anne Sexton (1928–1974)