##### Document Text Contents

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

accurate.

7.3 MAXIMUM CLOCK RATES IN GRAVITY FIELDS

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

Earth

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

(7.3.1)

as

v2

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

Index

probability of graviton emission per

disintegration, 214

propagator, 38

gravity,

should it be quantized?, 11

weakness, 6

without any sources (stress-energy)?

133

Hubble time, 165, 178

Hubble's constant, 7

Hubble's law, 172

hydrostatic equilibrium (Newtonian),

193

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,

184-185

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

231

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

polarization,

and spin, 34, 39

graviton, 37-38, 52-53

normalization, 47

photons, 34

poly trope, 192

precession of the perihelion of Mercury,

65,80

principle of equivalence, 90, 93

limitation to infinitesimal regions,

92

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-

212

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,

120

relation to noncommuting covariant

second derivatives, 127

relation to parallel transport about

a closed path, 129

symmetries, 120

Page 273

232

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

Index

for scalar matter (field) in a grav

field, 146

is not covariantly divergenceless,

140

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,

139

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

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

accurate.

7.3 MAXIMUM CLOCK RATES IN GRAVITY FIELDS

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

Earth

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

(7.3.1)

as

v2

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

Index

probability of graviton emission per

disintegration, 214

propagator, 38

gravity,

should it be quantized?, 11

weakness, 6

without any sources (stress-energy)?

133

Hubble time, 165, 178

Hubble's constant, 7

Hubble's law, 172

hydrostatic equilibrium (Newtonian),

193

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,

184-185

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

231

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

polarization,

and spin, 34, 39

graviton, 37-38, 52-53

normalization, 47

photons, 34

poly trope, 192

precession of the perihelion of Mercury,

65,80

principle of equivalence, 90, 93

limitation to infinitesimal regions,

92

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-

212

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,

120

relation to noncommuting covariant

second derivatives, 127

relation to parallel transport about

a closed path, 129

symmetries, 120

Page 273

232

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

Index

for scalar matter (field) in a grav

field, 146

is not covariantly divergenceless,

140

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,

139

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