Wednesday, June 10, 2009

Rotating Inertial Frames

Suppose you are standing at the North Pole on a cloudy night. You happen to have a Foucault Pendulum with you. Having nothing better to do, you set up the Pendulum vertically at midnight and you watch it for 24 hours.

What would you see? You would see the plane of oscillation of the pendulum rotating a full 360 degrees.The plane of the pendulum is precessing i.e., the plane in which the pendulum would appear to be inertially at rest i.e. to NOT precess would be one that is rotating w.r.t. the Earth at the same speed as the precessive velocity of the pendulum.

Now, after the 24 hours are up, lets say the clouds clear away, leaving you with a clear view of the stars in the sky. You would observe that the distant star also rotate, even as the pendulum precesses i.e., the plane of the pendulum is stationary wrt the distant stars.

That is, the 'inertial frame' of the pendulum is definable as the frame containing the distant stars. Thus for practical applications, inertial frames are those frames in mechanics which are at rest w.r.t. the distant stars.

Although Ernst Mach first proposed this principle and it is eminently usable in navigation and other applications, General Relativistically speaking this is not a necessary principle i.e., GR/SR would not be compromised if this principle is discarded in its realm.

In GR/SR, the distant stars are assumed to be a rotating frame of reference i.e., the Universe is assumed to be the rotating entity. The precessing Foucault Pendulum would then be doing the bidding of inertia in rotating with the distant stars as soon as it is positioned.  The Foucault Pendulum's plane of oscillation is then seen as being dragged by the slow rotating massive body (i.e., the Universe). This slow dragging is analyzed using Schwarzchild geometry.

Using this method, 'frame dragging' has been calculated for many rotating objects. or example, the rotating Earth also produces a 'frame drag' worth about 0.3 radians per year. This means that the Foucault Pendulum would show not only the planar precession due to the distant stars but also a 0.3 radians per year change in its plane also due to the Earth's rotation. The gyroscope is then a test body that can be used to observe the dragging of inertial frame.

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