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Page 136

7.3 Maximum Clock Rates in Gravity Fields 95

in the limit of small velocities. The quantity ¢ is simply the potential
difference between the location of the events, and the point of reference.
A more careful computation gives us an expression good for all velocities:

ds = dt \oil + 2¢/c2. (7.2.3)

Again, we should recall that we cannot simply use the Newtonian po-
tentials in this expression; our definition of ¢ must be relativistically


Now that we have concluded that gravitational effects make clocks run
faster in regions of higher potential, we may pose an amusing puzzle. We
know that a clock should run faster as we move it up, away from the
surface of the earth. On the other hand, as we move it, it should lose
time because of the time dilation of special relativity. The question is,
how should we move it up or down near the surface of the earth so as
to make it gain as much time as possible? For simplicity, we consider the
earth to have a uniform field, and consider motion in only one dimension.
The problem clearly has a solution. If we move very fast, at the speed of
light, the clock does not advance at all, and we get behind earth time. If
we move the clock up a very small distance, and hold it there, it will gain
over earth time. It is clear that there is some optimum way of moving
the clock, so that it will gain the most in a given interval of earth time.
The rules are that we are to bring the clock back to compare it with the
stationary clock.

We shall give the answer immediately, although it would be a good
exercise to work it out in detail. To make the moving clock advance the
most, in a given interval of ground time, say 1 hour, we must shoot it up
at such a speed that it is freely falling all the time, and arrives back just
in time, one hour later. See Figure 7.4. The problem is more difficult if we
try to do it in more dimensions, but the same answer is obtained; if we
want to make the clock return one hour later, but to a different spot on
earth, we must shoot the clock up into the ballistic trajectory. The same
answer is obtained in.a nonuniform gravitational field. If we are to shoot
a clock from one earth satellite to another, the true orbit is precisely that
which gives the maximum proper time.

In working out these problems, some troubles arise because we have
not made accurate definitions. For example, the free-fall solutions are not
necessarily unique; the ballistic "cannonball" problem in general has two
solutions, that is, two angles and initial velocities will give maxima (the
satellite orbit may go the long way about the earth). Nevertheless, any of
these solutions correspond to maxima of the elapsed time for the moving

Page 137

96 Lecture 7

1- D 2 - D 3 - D


Figure 7.4

clock. Whether they are relative or absolute maxima is not so important
for our purposes-what is important is that these solutions suggest how
we may obtain mechanics from a variational principle.

To understand the significance of the maximal property of elapsed
times, we may consider what happens in the limit of small velocities. The
elapsed time is an integral of d§, which represents the ticking rate of the
clock. In the nonrelativistic limit, the integral to be maximized is



J1- v2/e2 �~� 1 - 2e2 '

The first term integrates to the time difference in the reference system,
(tl - to). The other two terms may be rearranged to look like something
which should be very familiar, by multiplying by the mass of the particle
and changing sign,

J ds = (h - to) - �~� it} dt (! mv2 - me/» . me to 2 (7.3.2)
To maximize this expression, for a fixed time interval (tl - to), we take
the minimum of the integral on the right. But this integral is nothing but
the classical action for a particle of mass m in a gravitational potential e/>.
We see that the requirement that the proper time should be a maximum
is equivalent to the principle of least action in the cIassicallimit.

Page 272


probability of graviton emission per
disintegration, 214
propagator, 38

should it be quantized?, 11
weakness, 6
without any sources (stress-energy)?

Hubble time, 165, 178
Hubble's constant, 7
Hubble's law, 172
hydrostatic equilibrium (Newtonian),


invariant space-time interval,
with and without gravity, 98

invariant volume element, 137

kaon gravitational experiment, 5
Klein-Gordon equation in a gravita-

tional field, 146
Kronecker 6, 83

large number hypothesis, 7

Mach's Principle, 70-71
in quantum mechanics, 72-73
validity and boundary conditions,

mass of the photon, 23
maximal clock rates and the principle of

least action, 95-97
metric tensor, 98

derivative of determinant, 139
determinant and invariant volume
element, 137
geometrical interpretation, 99
invariants = physics, 111
inverse, 108
is covariantly constant, 124
number of independent invariant
quantities of second derivatives, 105
relation to linearized gravitational
field, 57
relation to proper time interval, 98

Mobius strip, 130-131

neutrino exchange example, 24-28

Olbers paradox, 186
orbital motion of a particle,

equation of motion, 63, 80


in the Schwarzschild geometry, 201

parallel transport of a vector, 130
along a closed path, 129
and topological properties, 130-131
relation to curvature, 129

and spin, 34, 39
graviton, 37-38, 52-53
normalization, 47
photons, 34

poly trope, 192
precession of the perihelion of Mercury,

principle of equivalence, 90, 93

limitation to infinitesimal regions,

proper time interval,
relation to metric tensor, 98

quadrupole character of spin two, 38-39
quantum field theory,

static force from spinless meson, 24
spinless meson exchange energy, 24
2 neutrino exchange energy, 25-27

quantum gravity,
is one-loop finite, xxxvii-xxxviii
loop representation, xl
one-loop, xxviii-xxix
renormalizable?, xxxvi-xxxviii, 211-
should gravity be quantized?, 11
two-slit diffraction experiment, 12

quantum mechanics,
amps become probs for complex
processes, 14
and Mach's principle, 72
external vs. internal observer, 13-14
fail at large distances?, 12

radio galaxies, 186
Ricci tensor, 88, 121, 132
Riemann normal coordinates, 104, 115
Riemann tensor, 87-88, 119

Bianchi identity, 131
in Riemann normal coordinates, 117
number of independent components,
relation to noncommuting covariant
second derivatives, 127
relation to parallel transport about
a closed path, 129
symmetries, 120

Page 273


Robertson-Walker metric, 168
light path and frequency shift, 171

scalar curvature, 121
Schr6dinger's cat paradox, 13, 22
Schwarzschild metric, 156, 199

and the Bianchi identity, 159
curvatures at r = 2m, 157
"singularity" result of choice of
coordinates, 157, 200
with charge, 202

shmootrino, 183
stress in a moving field, 217
stress-energy tensor,

change under a coordinate transfor-
mation, 108
difficulties in computing in a gravi-
tational field, 141
energy conservation, 36, 132
for a particle, 57, 78, 142
for a particle in a gravitational
field, 142
for a perfect fluid, 162
for a perfect fluid applied to cosmo-
logical models, 173
for a static fluid applied to super-
stars, 190
for scalar matter (field), 50, 67, 143
for scalar matter (field) in a grav
field, 143, 146
is covariantly divergenceless, 79-80,
88, 140
non uniqueness for scalar fields in a
grav field, 144-145

stress-energy tensor density, 140
for a particle, 142


for scalar matter (field) in a grav
field, 146
is not covariantly divergenceless,

supergravity, xxxviii
superstars, xvi-xxii, 189

equation of state used, xix, 191
explosions?, 195
numerical algorithm, 192
numerical results, 194
stability, xx-xxii

superstrings, xxxviii-xxxix
supersymmetry, xxxviii

tensor densities, 138, 140
when automatically divergenceless,

3-sphere metric and spherical coordi-
nates, 168-169

time dilation in a gravitational field, 68
tree diagrams, 211
two-slit diffraction experiment, 12

weakness of gravity, 6
Wheeler's conjecture on electrons and

positrons, 204-205
wormholes, xxiv, 159-161, 202

lack of time-like geodesics traversing
the wormhole, 200

Yang-Mills, xxxii, 114, 211

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