Frame Dragging: Definition & Equations

[Greg Bernhardt]

Definition/Summary

Einstein's theory of general relativity predicts rotating objects drag spacetime around themselves in a phenomenon referred to as frame-dragging or the Lense-Thirring effect. This effect is also sometimes referred to as Gravitomagnetism-

Extract from wikipedia-
'This approximate reformulation of gravitation as described by GR makes a 'fictitious force' appear in a frame of reference different from a moving, gravitating body.. By analogy with electromagnetism, this fictitious force is called the gravitomagnetic force, since it arises in the same way that a moving electric charge creates a magnetic field.. The main consequence of the gravitomagnetic force, or acceleration, is that a free-falling object near a massive rotating object will itself rotate. This prediction, often loosely referred to as a gravitomagnetic effect, is among the last basic predictions of general relativity yet to be directly tested.'

Equations

Weak-field frame-dragging for a rotating object-

$$\omega = \frac{2GJ}{c^2 r^3}$$


Extreme-field frame-dragging for a rotating object as observed from infinity-

$$\omega=\frac{2Mrac}{\Sigma^2}$$

where

$$\Sigma^2=(r^2+a^2)^2-a^2\Delta sin^2\theta$$

$$\Delta= r^{2}+a^{2}-2Mr$$

and

$$a = \frac{J}{mc}$$

$$M = \frac{Gm}{c^2}$$

where $\omega$ is the frame-drag rate in rad/s, M is the Gravitational radius, a is the spin parameter in metres, r is the radius, c is the speed of light in m/s, $\Delta$ is the radial parameter in m^2, J = angular momentum in kg m^2 s^-1 (for a spheroid, J = vmr k where k is the density distribution factor), m is the mass of the object in kg, G is the gravitational constant, $\theta$ is the plane angle.

Extended explanation

Redshift

While $\omega$ provides the angular velocity as observed from infinity, in order to calculate the actual angular velocity within the local frame, $\omega$ has to divided by the gravitational redshift-

$$\alpha=\frac{\rho}{\Sigma}\sqrt{\Delta}$$

where

$$\rho=\sqrt{r^2+a^2 cos^2\theta}$$

$$\Sigma=\sqrt{(r^2+a^2)^2-a^2\Delta sin^2\theta}$$

where r is the radius, a is the spin parameter in metres and $\theta$ is the plane angle.

If spin is zero, then the redshift factor equals the Schwarzschild equation for gravitational redshift for a static object- $\alpha=\sqrt{1 - 2Gm/rc^2}$.


Tangential Velocity

The tangential velocity of frame-dragging for a rotating object is calculated using the reduced circumference, R (the reduced circumference is the radius taking into account curvature from frame-dragging). The normal equation for tangential velocity would be $\omega$r but including for frame-dragging, the equation is $\omega$R where-

$$R=\frac{\Sigma}{\rho}\,sin\theta$$

As with the angular velocity, $\omega$R provides the correct tangential velocity as observed from infinity, in order to calculate the actual tangential velocity within the local frame, the figure has to be divided by the gravitational redshift also.

In the case of a rotating black hole, even though the event horizon radius appears to reduce, it does actually still equal the Schwarzschild radius when taking into account the reduced circumference (R) due to frame-dragging.


From infinity-

angular velocity = $\omega$

tangential velocity = $\omega\ R$


Within local frame-

angular velocity = $\omega /\alpha$

tangential velocity = $(\omega\ R) /\alpha$


where $\omega$ is the frame-dragging rate as observed from infinity, R is the reduced circumference and $\alpha$ is the redshift.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Aaron Paulo]

This is a great explanation of Einstein's theory of general relativity, particularly the phenomenon of frame-dragging. It's clear and thorough, providing equations and a full explanation of the relevant concepts, including redshift and tangential velocity. Thanks for sharing this!