Whereas in the case of the static metric (9.5) the 3-metric of the lattice can be read off directly as (minus) its spatial part, the same is not true in the general case (9.2). Why? Because there the sections i = const cut obliquely across the worldlines i = var of the lattice points; and that means (switch to SR geometry locally!) that the lattice is measured from a moving frame, and thus with length contraction. The way to find the lattice-metric (and more) is via what we shall call the canonical form of the metric, which results when we 'complete the square' in (9.2) to absorb the time-space cross-terms:
ds2 = e2$/c2 (c di - -2wi dxi)2 - kij dX dxj, (9.13)
with w1 = -(c/2)Ae-2$/c2, etc. Of course, $ is the clock-rate function as it has been all along, and wi and kij are newly arising time-independent coefficients. We shall presently recognize in wi an analog of the Maxwellian vector potential w and in kij the metric of the lattice.
Our synchronization process of Section 9.2 leaves little leeway for the time coordinate i: we can change all clock rates by a constant factor k > 0, and we can change the zero points by a continuous function of position:
It is easy to see (cf. Exercise 9.7) that this is, in fact, the most general time transformation that leaves the metric (9.2) and hence also the metric (9.13) form-invariant; of course, the lattice coordinates can be transformed arbitrarily: xl ^ xl = xl (xi). Under each of these transformations the two summands of (9.13) remain separate, and, in particular, under (9.14) we find:
$' = $ - c2logk, w' = k(wi + cfi), k'j = kij, (9.15)
where once again we use the derivative notation introduced in (7.7). We shall refer to (9.14) and its concomitants (9.15) as gauge-transformations and note that physically meaningful quantities must be 'gauge-invariant'. Examples of gauge-invariant quantities are:
At any given lattice point we can always achieve $ = wi = 0 by a suitable gauge-transformation and thus reduce (9.13) to ds2 = c2di2 — kij dx' dxj, which establishes kij as the metric of the lattice [cf. after (9.5)].
A stationary metric (9.13) under certain circumstances (bad choice of clock settings) can be transformed into a static metric with the same lattice. This is the case whenever we can transform wi away; that is, whenever there exists an f such that (with c = 1) wi = —fi (w = —grad f), which happens (in simply connected spaces) whenever wj — wjj ('curl w') vanishes. In fact, we shall see presently that the second quantity in (9.16) describes the local rotation rate of the lattice: static lattices are characterized by non-rotation. (This is directly related to the 'hypersurface-orthogonality' of the lattice worldlines.)
To elucidate the physical significance of wi, let us repeat our earlier weak-field slow-orbit calculation that led from (9.6) to (9.7), but this time with w. We approximate the metric (9.13) by ds2 = (' + IrX1 — ^W?1) c2i'2 — d'2
writing w • v for w dx1 /dt and once again dl2 for the metric of the lattice. This is equivalent to replacing the $ of (9.6) by $ — (w • v)/c and thus leads to the following generalization of (9.7):
The geodesic requirement on the orbit, namely that / ds be maximal, is once again equivalent to the integral on the RHS being minimal. But an identical variational principle is known to hold in an electromagnetic field with scalar potential $ and vector potential w for the non-relativistic (that is, slow) motion of a particle whose mass and charge are numerically equal (m = q).1 In gravitation, the mass and 'charge' of a particle are always equal (mj = mG). Soin a stationary field a free particle moves like a particle of equal charge and mass subject to a Lorentz-type force law
1 See, for example, J. D. Jackson, Classical Electrodynamics, 2nd edn, John Wiley, New York, 1975, eqn (12.9).
per unit mass. For it is this law, in conjunction with Newton's f = ma, that is equivalent to the above variational principle when dw/dt = 0. [Cf. Jackson, loc. cit., eqn (6.31).] But now $ and w are the relativistic potentials of the gravitational field, defined by the canonical metric (9.13). (The term 'gravimagnetic potential' is also used for w.) The slow general-relativistic motion of particles in weak stationary fields is thus seen to have pronounced Maxwellian features, which will be seen to carry over even to the field equations. (One corollary: moving matter gravitates differently from static matter.)
Now we know from classical mechanics that if a point P of a rigid reference frame L travels at acceleration a through an inertial frame while L rotates about P at angular velocity Q, then a free particle of unit mass at P moving relative to L at velocity v experiences relative to L an 'inertial' force f = -a + 2v x Q, (9.19)
where the last term is the well-known Coriolis force. But according to Einstein, inertial and gravitational forces are indistinguishable. Comparison of (9.19) with (9.18) thus leads to the conclusion that the lattice at any of its points P moves relative to the local inertial frame (LIF) with acceleration a « grad $ (9.20)
and angular velocity
Conversely (according to the present approximative calculation) the acceleration of the LIF and its rotation rate relative to the lattice are -a and -Q respectively. The first, of course, is what would be recognized as the gravitational field relative to the lattice, and the second is the rotation rate of a 'gyrocompass' suspended at a lattice point. The exact calculation leads to the following (gauge-invariant!) expressions:
|Q|= (proper-time) rotation rate of gyrocompass
= —Ue$/c2[^^(w^j - wjj)(wkj - whk)]1/2, (9.23) 2 2c where the magnitude of the gravitational field is defined as that of the proper acceleration of a lattice point. [For the proofs, see eqn (10.50) and Exercise 10.11.] We have seen $ in its role as clock-rate function, and, almost equivalently, with that as frequency-shift function. And both earlier and now again we have recognized its role as scalar potential. This latter, however, is to be understood as determining the whole field relative to the lattice—or what in Newtonian language would be the sum of the gravitational and the inertial force on a unit mass. In a rotating frame, for example, that includes the centrifugal force and corresponds to what on earth is called the 'geopotential'. Einstein, in his famous 1905 paper, in which he already predicted time dilation by a y -factor, suggested that this might be tested by comparing clock rates at the equator with those at the pole. His equivalence principle and with it the recognition of gravitational time dilation was still two years away. As a result, we know today (also from measurements) that there is no rate-discrepancy at all between clocks at the pole and clocks at the equator: the surface of the oblate earth is practically an equipotential surface $ = const! This can be seen by imagining the earth as having cooled from a rigidly rotating liquid ball; if its surface were not equipotential, the total field would have a component in the surface causing the liquid to move sideways, thus violating the assumption of equilibrium (rigid motion).
Lastly, let us take one more look at eqn (9.17) for an arbitrary path. It shows that the proper-time increment c-1 f ds measured by a slowly transported standard clock (like one of those flown by Hafele and Keating around the world—cf. penultimate paragraph of Section 3.5) differs from the coordinate-time increment (t2 - t1) by: (i) the usual special-relativistic time dilation - J 2(v2/c2) dt [cf. eqn (3.3)]; (ii) the general-relativistic 'altitude' correction - f ($/c2) dt; and (iii) in non-static fields also a 'source-motion' correction - f (w ■ v/c3) dt.
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