1 Uniform Acceleration

Home , Equivalence principle, Invariant (physics), Lorentz factor, Proper acceleration, Rest frame, Shapiro time delay

Physics 139 Relativity

Relativity Notes 2003

G. F. SMOOT

Oce 398 Le Conte

DepartmentofPhysics,

University of California, Berkeley, USA 94720

Notes to b e found at:

http://aether.lbl.gov/www/classes/p139/homework/homework.html

1 Uniform Acceleration

This material is to prepare a transition towards General Relativity via the Equivalence

Principle by rst understanding uniform acceleration.

The Equivalence Principle stated in a simple form: Equivalence Principle: A

uniform gravitational eld is equivalent to a uniform acceleration.

This is not very precise statement and one lesson wehave learned in Sp ecial

Relativity is the need to b e precise in our statements, de nitions, and use of

co ordinates. We will come to a more precise statement of the Equivalence Principle

in terms like at a space-time p oint with gravitational acceleration ~g there is a tangent

reference frame undergoing uniform acceleration that is equivalent. This is similar to

the instantaneous rest frame of Sp ecial Relativity in the case of an ob ject undergoing

acceleration.

We rst need to understand carefully what is a uniform acceleration reference

frame, whichwe will do in steps.

First imagine a reference frame { a rigid framework of rulers and clo cks,

our standard reference frame { undergoing uniform acceleration. In classical

nonrelativistic physics we can imagine a rigid framework to whichwe can apply a

force which will cause it to move with constant acceleration.

However, in Sp ecial Relativity no causal impulse can travel faster than the

sp eed of light, thus the frame work cannot b e in nitely rigid. When the force causing

the acceleration is rst applied, the p oint where the force is rst applied b egins to

accelerate rst and as the casual impulse moves out, the other p ortions join in the

acceleration.

Consider a simple long ro d as an example: If one pulls on a long ro d, it will

lengthen at rst as the end b eing pulled starts moving b efore the other end even

knows it is. Then as it gains sp eed, Lorentz-FitzGerald contraction will cause it to

shorten. If one pushes on the long ro d from b ehind, it will rst shorten as the end with

the force moves toward the other end which sits there unaware of the some to arrive

acceleration. All ob jects, however rigid, evidently display some degree of elasticity

during acceleration. It is clear that in Sp ecial Relativity no ro d can b e in nitely rigid

but must b e elastic at some level. Home work problem: prove that since the sp eed 1

of sound is less than or equal to the sp eed of light, that the rigidityofany material

is less than xxx?

As a b o dy accelerates, it moves in a continuous fashion from one inertial system

to another. If it is to retain its same rest length in its instantaneous rest system, then

it length relative to its original inertial system will have to decrease continuously

b ecause of Lorentz-FitzGerald length contraction. If, on the other hand, it retained

the same length relative to the original inertial system, then the Lorentz-FitzGerald

contraction would require its rest length to increase as its gains sp eed. This is not

very satisfactory.

Either way, the metric will dep end up on time. If wewant a direct comparison

to gravity,we need to require an accelerated co ordinate system to have a time

indep endent form.

1.1 Accelerating a Point Mass

A uniformly accelerating p oint mass is one that is sub ject to the same force in each

and every one of its instantaneous rest systems. I.e. a uniformly accelerating p oint

mass is sub ject to a constant force F = m ~g along the +x-axis in a co ordinate system

which is the inertial frame where its velo city is zero instantaneous rest frame.

Instantaneous

Lab oratory Frame S

Rest Frame S

ct = ct

x = x

Light Cone

Acceleration transforms as

2 0

d x a

a = = 1

3 x

3 0

so that in the instantaneous rest frame a = a . In the instantaneous rest frame

F = F .Nowwe can solve the equation of motion in either of twoways: from the

acceleration or from the force. In Problem Set 2 we solved the problem for a uniformly

accelerating ro cket using the acceleration transformation.

1 0 0

In ro cket frame the acceleration was a = g Note reversal of S and S compared to discussion

x 2

Here we use force transformation.

F = = m g = F

o x

d m c

= m g

dct

d = dct

g gt

= ct = 2

c c

where the constantofintegration is set equal to zero b ecause we de ne the time zero

to b e when = 0. This can b e turned into an equation for alone:

g gt

= ct = =

c c

2 0

1 c

gt =c

= 3

1+gt =c

So that, from the lab oratory, observing the test particle start from rest we rst see

its velo city increasing linearly with time as we classically exp ect for a particle under

uniform acceleration. Then as the velo city b egins to b e a signi cant fraction of

the sp eed of light, the term in the denominator b ecomes increasing imp ortant and

the velo city increases ever more slowly in time and only approaches the sp eed of

light asymptotically. The shap e of the tra jectory of a particle undergoing uniform

acceleration is a hyp erb ola and not the classical parab ola, but for lowvelo cities they

are indistinguishable conics.

Note also

gt 1

= 1+ 4 =

If we lo ok at the Lorentz factor ,we see that it is rst very nearly unity and then

as the velo city b egins to saturate, increases linearly with time. This is simply

conservation of energy, as the constant acceleration force in the instantaneous rest

frame is constantly doing work W = cF .

Now take a side step and evaluate in terms of proper time.

0 0

dt dt

d = =

3=2

2 2

in this section. Thus the acceleration in the Earth frame was a = 1 v =c g = dv =dt.

x x

3=2

2 2

Regrouping we had gdt = du = 1 v =c and integrating gives gt = v= 1 v =c or

1+g t=c . v=c = g t=c= 3

0 0

dt c gt

= = sinh

g c

or inverting the equation to nd t in terms of prop er time

g g

sinh 5 t =

c c

end aside

0 0 0

Nowwe can solve for x using the de nition of = dx =dct .

0 0

dx = dct

Z Z

0 0 0 0

ct =ct ct =ct

gt =c

0 0 0

x = x + dct =x + dct

o o

0 0

ct =0 ct =0

1+gt =c

c g

0 0

ct =ct 0

= 1+ ct j + x

t =0 o

g c

2 2

c c

+ x 6 = 1+gt =c

g g

De ne

0 0

x x 7

P o

Then our equation b ecomes

c g

0 0

ct x x = 1+

g c

2 2

g g

0 0 0

x x = 1+ ct

2 2

c c

2 2

g g

0 0 0

= 1 8 x x ct

2 2

c c

This last equation describ es a hyp erb ola. Note that we can nd a formula for x in

0 0

terms of given the last equation for x in terms of t .

2 0 2 2 2 0 0

gt =c =1 g=c x x

2 0 2 2 2 2 2 0 0

=1+gt =c =1+sinh g =c=cosh g =c g=c x x

2 2 2

c c g c g

0 0 0

x = x + = x cosh + cosh

P 0

g c g g c

Note that x is directly related to the x in the accelerating frame.

Because the world line is a hyp erb ola in Minkowski space, the world line of

the p oint mass approaches the light line asymptotically. This means all events on the 4

world line will have a space like relationship to all events to the left of the fo cal p oint

P 0;x .

0 0

x = x 9

P o

So that the distance b etween the rest p oint and fo cal p oint is prop ortional to the

inverse of the acceleration.

insert gure here showing frames with small acceleration and with

large accelerations.

gt =c g gt

0 0

= x x = 1+ 10

c c

Therefore

= = tan 11

x x

where is the horizontal angle.

insert gure here showing etc. The line from p ointP,0;x to

0 0

p ointct ;x isthexaxis in the instantaneous rest frame. De nes simultaneity

in instantaneous rest frame is changing constantly since the instantaneous rest frame

is continuously changing.

insert gure here showing world lines etc. and that P is a pivot

p oint.

The observer A no matter where along his world lines never knows the future

of the observer passing through the pivot p oint and ob jects to the left are never in

casual contact but if they did would app ear to move backward through time. ....

0 0 0

Now calculate the distance from eventP = 0;x toeventct ;x

0 0 2 02

=x x c t 12

combining that with the equation for yields

x x

2 0 0 2 02

x x = c t 13

Evaluate this for t = 0 to get the distance, x ,between event P and where A crosses

the x axis.

2 2 2

2 0 0 2 0 0 2 0 0

x = =x x x x = 1 x x 14

P P P

0 0 0

x x = x 15

P PA 5

Lorentz contraction x = x = .

It is easy to show that the distance from the pivot p ointtoany p oint on the

hyp erb olic tra jectory is the same. The accelerating system moves in suchaway

that the distance to the pivot p oint is increasing in inertial space by precisely its

instanteous gamma so that the Lorentz length contraction makes the distance to the

pivot p oint in its rest frame constant. I.e. if the line of simultaneityintersects A's

tra jectory at p oint B then from the hyp erb ola formula ab ove for all B wehave

0 0 0

x x = x 16

B P PA

Thus x x = x = x The distance from the pivot p ointevent0;x to the

B P PB P

mass p oint at B as measured in the accelerated co ordinate system is the same as the

distance from the pivot p ointevent0;x to the mass p oint when it was at rest or

any other p oint on its tra jectory. Therefore to an observer in the accelerated system

the p oint mass maintains a xed distance to the pivot p ointevent0;x throughout

its motion. Thus despite accelerating away continously the eternal moment remains

a xed distance away.

1.2 Uniformly Accelerated Reference Frame

We are now in a p osition to discuss a uniformly accelerated reference frame.

insert gure of two uniformly accelerating masses with same fo cal

p oint.

Consider two observers 1 and 2 b oth with the same fo cus p oint x and b oth

0 0 0

cross the x -axis at the same t = 0. Then there is always the same distance from x

and thus each other. As a result they will havetohave di erent accelerations b ecause

they have the same fo cus

2 0 2 0

a = g = c =x a = g = c =x 17

1 1 2 2

1 2

This is what one sees in the gure with the curves further away from the fo cal p oint

b eing atter. A straight line is a the world line for a non-accelerating particle.

One can make a uniformly accelerated frame, if the acceleration of each

p ointisinversely prop ortional to its distance from the fo cus p oint x . Actually

0 0 0

ct ;x =0;x .

An observer riding with a meter stick in this accelerated frame would sayit

maintained a constant length. An observer in an inertial frame e.g. our Lab frame

claims the ro d is shrinking in time as it accelerates away.However, as it approaches

the origin, it lengthens and slows down.

A ro d on the other side of the origin accelerates to the left rather than the

right.

This situation is called a Rindler Space.

Note that the co ordinate choices are di erent from our usual every day

conventions. Usually wechose the vertical axis to b e the z -axis and have the e ective 6

acceleration \downward" toward negative z . What wewould observe conventionally

from our inertial frame would b e an elevator rushing doward towards us at high sp eed

and decellerating at a rate g coming to a stop at a distance and then accelerating

upwards away retracing its path.

Note that the acceleration dep ends on distance away from pivot p oint. That

is the only system in which the clo cks in the instanteous rest frame do not haveto

continuously b e reset relative to each other.

1.2.1 Co ordinate System for a Uniformly Accelerating System

There are two natural frames in which to describ e a uniformly accelerating system.

They are the accelerating co ordinates the prop er rest frame time and related

co ordinates and the co ordinates of one inertial frame that matches the accelerating

frame at one instant.

The transformation b etween the co ordinate systems is given by

2 2

c c g

x = cosh + x +

g g c

c g

ct = x + sinh 18

g c

where g is standing in for the uniform 3-acceleration magnitude a. Note that this is

what one must exp ect for the form of the time transformation. At the pivot p oint

x = x = c =g time must b e frozen thus the name pivot p oint. If we put that

value in, then we nd ct = 0 indep endentof As one moves away from the pivot

p oint, the conversion to t must increase linearly with the distance from the pivot

p oint. Thus it must b e prop ortional to x + c =g .

1.2.2 The Metric for a Uniformly Accelerating System

Wewant to nd g

2 2

cd = ds = g dx dx

For an inertial frame wehave the Minkowski metric which in Cartesian co ordinates

is diagonal and constant

=[1;1;1;1]

How ab out expressing this in terms of the prop er time and the instanteous rest

frame co ordinates x, y , and z ?

0 1 2

dx = cdt = sinhg =cdx +1+g x=c coshg =cd

1 0 1 1 2

dx = dx = coshg dx dx +1+g x=c sinhg =cd

2 2 2 2 2 2

cd = ds =1+g x=c cd d~x

2 2 2 2

g = [1 + g x=c ; 1; 1; 1] g = [1 + g x=c ; 1; 1; 1]

Note the di erence b etween covariant and contravariant. 7

1.3 Alternate Discussion to b e integrated

We revisited the Lorentz transformation in the case of circular motion, that is, motion

with a uniform sp eed but continously changing direction, in the case that results in

Thomas precession. Nowwe consider the velo city transformation.

1.3.1 Instanteous Velo cityTransformation

The Lorentz transformations of space-time co ordinates

t = t x=c

x = x ct

y = y

z = z 19

and their converse primes exchanged with unprimes and = v=c with are

di erentiated with resp ect to t and used to nd the velo city

dx dy dz

~u = u ;u ;u = ; ;

1 2 3

dt dt dt

0 0 0

dy dz dx

0 0 0 0

; ; 20 ~u = u ;u ;u =

1 2 3

0 0 0

dt dt dt

u v u u

1 2 3

0 0 0

u = ; u = ; u = 21

1 2 3

2 2 2

1 u v=c 1 u v=c 1 u v=c

1 1 1

0 0 0

u v u u

1 2 3

u = ; u = ; u = 22

1 2 3

0 0

2 2 2

1 u v=c 1 u 1 u v=c v=c

1 1

No assumption as the uniformityof ~u or ~u has b een made. These equations apply

equally to the instantaneous velo city in non-uniform or circular motion.

Now consider the magnitudes u and u de ned as

2 2 2 2 02 02 02 02

u = u + u + u ; u = u + u + u 23

1 2 3 1 2 3

Nowwe can readily calculate the u transformation laws by factoring out the dt

0 2 2 2 2 0 2 0 2 0

and dt from cd =cdt d~r =cdt d~r and substituting in for u

2 2 2 0 2 2 02 2 2 2 2 2 02

dt c u =dt c u =dt v 1 u v=c c u : 24

2 2 2 2 2

c c u c v

2 02

25 c u =

2 2

c u v

0 0

u u u v u v

= v 1 = v 1+ 26

2 0 2

u c u c 8

Now note how simple this is in the instanteous rest frame:

u = v u

This should remind you of the rapidity formulation given in the homework. where

the rapidity is de ned as the rotation angle

u u

u=tanh ; tanhu= 27

c c

u= u+v 28

Di erentiating this with resp ect to time gives us a simple waytowork out the

acceleration transformation.

1.3.2 Acceleration Transformation

d d dt

u= u 29

dt dt dt

Since the derivative of the hyp erb olic tangent is the hyp erb olic secant

d 1 du

u= u 30

dt c dt

Since

0 0

u dt

= 31

dt u

Substituting we obtain the acceleration transformation formula

du du

3 0 3

u = u 32

dt dt

Under the Galilean transformation, the acceleration is invariant; but, acceleration is

not in Sp ecial Relativity.

We need to de ne the prop er acceleration

d du

= [ uu] 33 ja~ j = u

dt dt

where is measured in the instantaneous rest frame.

Now constant instanteous acceleration constant prop er acceleration is a

particularly simple case. Integrating and chosing u =0att= 0 or vice versa

one nds

t = uu 34

Thus at lowvelo city u increases linearly with t and as u ! c u grows linearly with

time. Squaring, solving for u, and integrating again, chosing zero as the constantof

integration yields

2 2 4 2

x ct = c = X 35

Thus, for obvious reasons, rectilinear motion with constant prop er acceleration is

called hyp erb olic motion. 9

1.4 Rindler Space, Symmetry and GR

The equivlance principle implies a new symmetry and thus asso ciated invariance.

With a realization and the uniqueness of solutions give a formulation to the theory

of gravity.

The strong and weak Equivalence Principle: The weak equivalence principle

is that gravitational and inertial masses are precisely equal also includes Lorentz

invariance. The strong equivalence principle applies to all laws of nature that no

exp eriment can distinguish b etween an accelerating frame of reference and a uniform

gravitational eld.

We can also use this symmetry approach to nd the Rindler space. Consider

an \generalized elevator" as a kind of ro cket ship in outer space far from the strong

in uence of Earth or any other b o dy.Now give the \elevator" a constant acceleration

g upwards. All inhabitants of the \elevator" will feel the pressure from the o or,

just as if they were living in the gravitational eld at the surface of the Earth or

equivalent. This is a metho d of constructing \arti cial" gravitational eld. Wenow

consider this arti cial gravitational eld more carefully.

Supp ose wewant this arti cial gravitational eld to b e constant in space and

time. We will nd that we can make the arti cial gravitational eld uniform in time

and two spatial directions but it must decrease in the direction of the eld itself. The

inhabitants will feel a constant acceleration.

Consider a co ordinate grid for an elevator free to accelerate uniformly or b e

in a uniform gravitational eld, whichwe taketobe inside the elevator, such that

p oints on the elevator wall and o or are given by and are constant. The zeroth

comp onent = c , where is the prop er time elapsed instanteous rest time in the

elevator. An observer in outer space uses a standard Cartesian grid x in an inertial

frame there. The motion of the elevator is describ ed by the function x .

That is the elevator is free to move only along one axis the \vertical" axis.

We designate the \vertical" direction to b e the z -axis. The origin of the co ordinates

is a p oint in the middle of the o or of the elevator, which for convenience coincides

with the origin of thex ~ co ordinates at t = = =0.Thus the co ordinates of

the origin center p oint of elevator o or will b e

=c ; 0; 0; 0 x~ =ct ; 0; 0;z 36

Time run at a constant rate for the observer inside the elevator.

! ! !

2 2 2

@x @ct @z

= = c : 37

@ @ @

The acceleration is set to b e ~g , which is the spatial p ortion of the four-acceleration:

@ x

a~ = = g : 38

@ 10

At =0 we can sp ecify that the velo city of the elevator is zero:

=c; 0 at =0: 39

We can make use of the di erential prop er time along anyworld line d = dt= .

Using the relation

1 gt

= = 1+ 40

we nd

dt g gt

= = sinh 41

c c

Inverting this equation we nd a relationship for t in terms of

gt g

= sinh 42

c c

This equation works for the origin. The acceleration dep ends up on lo cation so that

the more general formula b ecomes

c g

ct = + sinh 43

g c

2 2

c c g

3 3

z = x = + cosh 44

g c g

At that moment t and coincide, and if the acceleration ~g is to b e b e constant,

then at =0, @~g=@ = 0, so that

@ F @

g =F; 0 = x =0; 45

@ c @

where F is an unknown constant.

Now this equation is Lorentz covariant. So not only at = 0, but also at all

times we should have

@ F @

g = x 46

@ c @

Combining equations x and y gives

2 2 2

c g g F

x + A = x +A = x + ; g =

2 2

c c c g

x = B coshg =c+C sinhg =c A ; 47 11

F , A , B , and C are constants. F = g =c can b e found from the derivativeof

four acceleration evaluated at = 0. Then from equations 16, 17, and the b oundary

conditions:

1 1 0 0

0 1

2 2

C C B B

c c

0 0

C C B B

2 2

g = cF = g ; B = C ; C = C ; A = B ; 48 B B

A A @ @

0 0

g g

1 0

and since at = 0, the acceleration is purely spacelike. We nd that the parameter

g is the absolute value of the acceleration.

We notice that the p osition of the elevator o or at \inhabitant time" is

obtained from the p osition at =0 by a Lorentz b o ost around the p oint x = A .

This must imply that the entire elevator is Lorentz-b o osted. The b o ost is given by

the rotation matrix with angle = g =c. This observation immediately gives the

co ordinates of all other p oints in the elevator. Supp ose at =0,

~ ~

x 0; =0; 49

Then at other values

0 1

sinhg =c +

B C

~ B C

x c ; = 50

B C

@ A

2 2

c c

coshg =c +

g g

The 0 and 3 height comp onents of the co ordinates, imb edded in the x

co ordinates, are pictured in the next gure. The light cone de nes the b oundary

of the space at = 0 the co ordinates lie on the p ositive x axis in a very ordinary

3 3

way. Each x co ordinate follows a hyp erb ola in x and c that keep it in the right

quadrant in x - c plane. The description of the quadrant of space time in terms of

the co ordiates is called \Rindler space".

It should b e clear that an observer inside the elevator feels no e ects that

dep end explicity on his time co ordinate , since a transition for to t is nothing but

a Lorentz transformation.

We also notice some imp ortant e ects:

i Equal lines lines of simultaneity converge at the left at the x c origin.

@x =@ c .varies with It follows that the lo cal clo ck sp eed, whichisgiven by =

height x :

3 2

=1+g =c ; 51

ii The acceleration or gravitational eld strength felt lo cally is ~g , whichis

prop ortional to the distance to the p oint x = A . So even though the eld is

. constant in the transverse direction and with time, it decreases with heightx 12

iii The region of space-time describ ed by the observer in the elevator is only part

3 2 0

of all of space-time, where x + c =g > jx j. The b oundary lines are called past and

future horizons.

All of these are trypically relativistic e ects. In the non-relativistic limit

g ! 0 the co ordinates simplfy to

3 3 2 0

x = + g ; x = c : 52

According to the equivalence principle the relativistic e ects discovered here should

also b e features of gravitational elds generated by matter or energy. Let us insp ect

them individually.

Observation i suggest that clo cks will run slower, if they are deep down in a

gravitational eld. Indeed as one susp ects equation x will generalize to

= 1+x=c 53

where x is the gravitational p otential. This will b e true, provided that the

gravitational eld is stationary not time varying. This e ect is called the

gravitational redshift.

Relativistic e ect ii could have b een predicted by the following argument.

The energy density of a gravitational p otential is negative. Since the energy of two

masses M and M at a distance r apart is E = G M M =r ,we can calculate the

1 2 n 1 2

energy density of a eld ~g as T = 1=8G j~gj . If wehave normalized c =1,

00 n

this is also its mass density. But then this mass density in turn should generate a

gravitational eld! This would imply

@ ~g =4G T = j~g j

n 00

so that the eld strength should decrease with height. However, this reasoning is to o

simplistic, since the eld ob eys a di erential equation but without the co ecient 1/2.

The p ossible emergence of horizons iii turns out to b e a new feature of

relativistic gravitational elds. Under normal circumstances the elds are so weak

that no horizon will b e seen, but gravitational collapse may pro duce horizons. If this

happ ens, there will b e regions of space-time from which no signals can b e observed.

The most imp ortant conclusion to b e drawn is that in order to describ e a

gravitational eld, one mayhave to p erform a transformation from the co ordinates

that were used inside the elevator where one feels the gravitational eld, toward

co ordinates x that describ e empty space-time, in which freely falling ob jects move

along straight lines. Nowwe know that in an empty space without gravitational elds

the clo ck sp eeds and the lengths of the rulers are describ ed by a distance fuction c

or ` as

2 2

cd = d` = g dx dx ; where g = diag 1; 1; 1; 1 54

In tems of the co ordinates appropriate for the elevator, wehave for in nitesimal

displacement d ,

0 3 3 2

dx = sinhg =cd +1+g =c coshg =cd ;

3 3 3 2

dx = coshg =cd +1+g =c sinhg =cd : 55

This implies

2 2 3 2 2 2 2

cd = d` =1+g =c dc d : 56

If we write this in the form

2 2 3 2 2 2 2

cd = d` = g d d =1+g =c dc d : 57

then we see that all e ects that the gravitational eld have on rulers and clo cks can b e

describ ed in terms of space and time dep endent eld g . Only in the gravitational

eld of a Rindler space can one nd co ordinates x inerms of these the function g

takes the simple form shown. We will see that g is all that is ned to describ e the

gravitational eld completely.

Spaces in which the in nitesimal distance cd or d` is describ ed by a space

time dep endent fuction g are called curved or Riemann spaces. Space-time is

apparently a Riemann space.

We can write the metric more explicitly as

0 1 0 1

2 2

0 0 0 1= 0 0 0

B C B C

0 1 0 0 0 1 0 0

B C B C

g = B C g = B C 58

@ A @ A

0 0 1 0 0 0 1 0

0 0 0 1 0 0 0 1

3 2

where =1+g =c .

dx =cd ; d~r sx = cd ; d~r 59

Note since the metric has the form

2 2 3 2 2 2 2

cd = d` =1+g =c dct d~r : 60

then an ob ject stationary at xed ~r has prop er time d = dt. A particle moving

with velo city ~v = d~r =dt will have prop er time

2 2 2 2

d = dt d~r=c = dt = dt= 61

2 2

where =1= . 14

1.4.1 Uniformly Accelerating Clo cks - Gravitational Freqency Shift

We could have derived these results by simply considering two clo cks in a spaceship

\ro cket elevator" with constant acceleration or the Doppler e ect on a photon

emitted at one end of the spaceship and received at the other and comparing this

to our thought exp eriment ab out a photon in a gravitational eld. This approachis

much more physical but do es not show all the features of the Rindler space.

Consider that the inertial and by inference gravitational \mass" of a particle

is given by the sum of its rest energy plus all other energies divided by c .Thus the

2 2

inertial and gravitational mass of photon is E=c = h =c . If a photon changes its

gravitational p otential through simple propagation then it must change its energy by

and amountE =E g h=c where g is the acceleration of gravity and h is the height

change. Thus

E gh

= = 62

E c

The fractional frequency change is the change in gravitational p otential divided by

c .

For comparison consider a lab accelerating at rate g with the two clo cks

separated by a instanteous distance h along the acceleration direction. If the rst

clo ck sends a photon of frequency , then the second clo ck recieves a photon

source

observed at frequency we know that they are related by the Doppler formula

observed

= 1 + cos 63

observed source

Since the angle is either 0 or 180 ,

observed

= 1 64

source

The time it takes for the photon to get from the rst to the second clo ckis

approximately t = h=c and the velo citychange is v = g t = g h=c or = g h=c

Di erentially one has = =1+g x=c

2 1

The gravitational redshift was st measured directly in the lab oratory in 1960

byPound and Rebka where they let a 14.4 keV -ray, emitted in the radioactive decay

57 2 15

of Fe, to fall 22.6 meters down an evacuated shaft where g h=c =2:47 10 , and

they measured a fractional change in frequency of 2:57 0:26 10 ,thus verifying

to that level the equivalence principle.

One could anticipate that for a spherical mass M in an otherwise at space-

time that the rate of clo cks would vary as

GM dtr

=1 65

dt1 c r

1.5 Lo cal Co ordinates

It is sometimes b etter to use lo cal standard clo cks for the determination of velo city

and acceleration at each p oint, rather than referring to a single co ordinate clo ck 15

lo cated at the origin. The former run faster than the latter by the factor , so that

the lo cal velo city of an ob ject moving over co ordinate intervals dx and dt is given

d 1

i i

= x or = 66

L L

A second application of this time derivative op erator to gives the connection

between co ordinate and lo cal acceleration:

" !

d @ d dx 1

i i

_ _

= 67 = =

L L x

d d d

since @ =@ ; hence,

x x

1 g

i i

a = a 68

L x

Thus the lo cal acceleration of a free-falling b o dy is

g g g

x y x z

a = 1 ; a = ; a = 69

L L L

x y z

2 2 2

Thus the acceleration dep ends up on the lo cal velo city and the lo cal value of g at any

i i

p oint is found to b e g = g= with the lo cal velo city = = so that one can

L L

write

h i

a = g 1 ; a = g ; a = g 70

L L L L L L L L L L L

x x y x y z x z

Freee-falling lo cal acceleration app ear here exclusively in terms of lo cal

velo cities and the lo cal acceleration constant g .

When an ob ject falls vertically, its acceleration a ranges b etween g and

L L

0 dep ending n , rather than b etween g and +g as it do es at the origin.

1.6 Dynamics

The 4-D momentum is de ned in an accelerated system just as it is de ned in an

inertial frame.

p~ = m u~ p = m u =p ;~p=m 1; ; ; 71

0 0 0 x y z

2 0 2 2

where ~p = m c and p = m c where E = m c is the prop er energy. For

0 0 0

m = 0 in these equations one replaces m c by E . Because this is a 4-D vector

0 0 0

multipliedbyaninvariant, it can b e found either by using he known values of in the

accelerated system or by transforming the inertial 4-D momentum to the accelerated

system. 16

In its covariant form, it is

2 0

p = g p = p ;~p= p ;~p 72

so that

2 2

p = m c

0 0

There are evidently twoentirely di erent energies of an ob ject in the accelerated

system; the covariant and he contravariant energies.

If momentum and energy are conserved in a lo cal interaction in an inertial

system, then the momentum and the covariant and contravariant energies are

conserved in the accelerated system. p = 0 is an invariant.

However, if there a quantity is conserved over the path of a freely-falling

particle? The answer as we will show later is that the covariant energy is.

More generally, the covariant energy of an ob ject is constant for any time-

indep endent metric.

1.7 Gravitational Redshift

The Equivalence principle leads directly to twointeresting predictions ab out the

b ehavior of light in the presence of gravity. The rst e ect is that as light climbs up

a gravitational gradient, its frequency decreases. The second is that light is de ected

by a gravitational eld.

These e ects are obvious, if one knows that light consists of photons where

E = h is the relation b etween the photon's kinetic energy E and the photon's

frequency . Einstein's formula relating inertial mass m to energy E = m c . The

I I

weak Equivalence Principle states m = m .For the work done by a gravitational

I G

eld with p otential on a particle of gravitational mass m as it traverses a p otential

di erence dism d. This must equal DE , the gain in the particle's kinetic energy.

For a photon, dE = hd , and so

E h

hd = m d=m d= d= d; 73

G I

2 2

c c

and thus

d d

74 =

Integrating this equation over a nite path from A to B, one nes

B A

75 = e =

As for light b ending in a gravitational eld, imagine a ray of light as a stream

of photons; since these photons have inertial and gravitational mass, we exp ect them

to ob ey Galileo's principle and follow a curved path just like a Newtonian bullet 17

traveling at velo city c. That wold make, for example, the downward curvature of

a horizontal b eam in the earth's gravitational eld with x horizontal and z vertical

equal to

2 2

d z d z g

= = : 76

2 2 2 2

dx c dt c

In units of years and light-yeas c = 1, and it so happ ens that g 1.

1.8 Static and Stationary SpaceTimes

A stationary eld is one that do es not change in time and a static one is one where

the sources do not move. The most imp ortant prop erty of stationary spacetimes is

that they admit a preferred time. The metric of every static eld can b e brougt to

the canonical form

2 2 2=c 2 2 2 2 2 2

cd = ds = e c dt d~r = cdt d~r 77

where the last part uses our previous notation. We can calculate the elapsed prop er

time

2 2 2 2=c 2 2 2

78 = dt e d = dt

2 2=c 2 2

d = dt = dt e = dt= 79

2 2=c 2

In the weak eld limit=c << 1, e ' 1+2=c .

2 2 2 2 2

ds ' 1 + 2=c c dt dl

For a particle-worldline b etween twoevents P and P ,wehave

1 2

1=2

Z Z Z

P t t

2 2 2

2 ds v

1+ ds = dt = c dt; 80

2 2

dt c c

P t t

1 1 1

where v = dl =dt is the co ordinate velo city of the particle. The binomial approximation

gives

! !

1=2

Z Z Z Z

2 2

t P t t

2 2 2 2

2 v 1 1 v 1

1+ ds = c 1+ dt = c dt = C T T v dt:

1 2

2 2 2 2

c c c 2 c c 2

t P t t

1 1 1 1

The condition that ds b e maximal is therefore equivalent to the last integral b eing

minmal. That is exacatly Hamilton's Principle.

One consequence whichwe can read o immediately is what is called the

Shapiro time delay. A light-ray satis es ds = 0 and thus e cdt = dl , the two signs

corresp onding to the two p ossible directions of travel. Consequently a radar, or other

light signal, re ected from a distant ob ject will return to its emission p oint after a

co ordinate time

t =2 e dl 82

has elapsed there, where the integration is p erformed over the path of the signal. 18