Lecture 9

We considered extending our program of making physical laws Lorentz-covariant to gravity. We noted the special connection of gravity to accelerated reference frames, and hence to the notion of inertial reference frames. We introduced both Mach's principle and the Equivalence Principle.

We first noted the basic Newtonian problem with inertial frames: that Newton's laws are valid only in inertial frames. In accelerated frames, requiring Newton's laws to be true makes us introduce ``fictitious'' forces (centrifugal, Coriolis, etc) which label the effects of the frame acceleration as constituent ``forces''. Yet other than consistency, we have no indication which frames should be inertial, nor why out of all possible sets of reference frames, Nature should have picked one particular set to be inertial.

Newton claimed space is absolute, unaffected by the matter it contains, with motion with respect to this absolute grid determining the inertial properties of matter. Our experience in applying Newtonian dynamics to increasing large astrophysical systems suggests the contrary. It seems that, as we wish to make predictions on larger and larger scales, we must take an inertial frame at rest with respect to larger and larger domains of matter in the universe. This ultimately leads to Mach's principle, that matter itself defines the inertial frame, which is at rest with respect to the average motion of matter in the universe. (This would be better called Mach's conjecture.)

We then introduced the principle of equivalence. Inertial and gravitational mass are empirically equivalent (tested to within a part in $10^{12}$ ). This means that the same parameter controls both the force a particle experiences in a gravitational field, and its inertial resistance to converting any applied force into acceleration. The result is that, unlike other fundamental forces, gravity causes all test particles to experience the same acceleration. This connection between gravity, and a single prescribed acceleration, leads to a stronger restatement of the Equivalence Principle. It means that an observer cannot distinguish between two cases: 1) he is in an inertial frame with constant gravitational field $\vec{g}$ downward; 2) he is in a frame with acceleration $\vec{a} = -\vec{g}$ upward, and no gravitational field.

This ability to disguise a constant gravitational field $\vec{g}$ as an accelerated reference frame suggests that we can make $\vec{g}$ unobservable, by choosing a reference frame with particular acceleration. Such a reference frame is freely falling, accelerating downward with $\vec{a} = \vec{g}$ , so that particles and the frame have no relative acceleration. We then showed the Equivalence Principle: that Newton's laws, excluding gravitational forces, are obeyed in all local freely falling frames (which are thus called locally inertial).

This resolves some of our initial confusion; gravity has singled out which frames are inertial (with respect to nongravitational forces). However we noted that this can only be done locally, in regions where the gravitational field is effectively constant. We showed how large regions with varying gravitational fields have tidal forces indicating the absolute presence of gravity, which can't be canceled by a choice of any single accelerated frame. Thus all the complication of gravity has been removed locally, but deferred to the problem of patching together local inertial frames to make predictions about widely separated events.

This patching problem is the classic problem of Differential Geometry, where curvature effects mean that as we stitch together flat patches sometimes we get total surfaces which can no longer be flattened. We will soon study this topic, after a closer look at consequences of the Equivalence Principle which are local (true even within one patch). We noted that we still have many choices of freely falling local inertial frames, all related by Lorentz transformations. Thus the Lorentz covariance of physical law is still something we demand of physical laws within our local patches.

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2004-02-16