advanced mechanics and general relativity j franklin

I again chose general relativity as a target – ifgeodesics and geometry can provide an introduction to the motion side of GR inthe context of advanced mechanics, why not use the techniqu

Aimed at advanced undergraduates with background knowledge of classicalmechanics and electricity and magnetism, this textbook presents both the parti-cle dynamics relevant to general relativity, and the field dynamics necessary tounderstand the theory.

Focusing on action extremization, the book develops the structure and tions of general relativity by analogy with familiar physical systems Topics rangingfrom classical field theory to minimal surfaces and relativistic strings are covered in

predic-a consistent mpredic-anner Nepredic-arly 150 exercises predic-and numerous expredic-amples throughout thetextbook enable students to test their understanding of the material covered A ten-sor manipulation package to help students overcome the computational challengeassociated with general relativity is available on a site hosted by the author A link tothis and to a solutions manual can be found at www.cambridge.org/9780521762458

j o e l f r a n k l i n is an Assistant Professor in the physics department of ReedCollege His work spans a variety of fields, including stochastic Hamiltoniansystems (both numerical and mathematical), modifications of general relativity,and their observational implications

ADVANCED MECHANICS AND GENERAL RELATIVITY

JOEL FRANKLIN

Reed College

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,

São Paulo, Delhi, Dubai, Tokyo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-76245-8

ISBN-13 978-0-511-77654-0

© J Franklin 2010

2010

Information on this title: www.cambridge.org/9780521762458

This publication is in copyright Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

Cambridge University Press has no responsibility for the persistence or accuracy

of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York www.cambridge.org

eBook (NetLibrary) Hardback

Preface pagexiii

vii

2.2 Lagrangian 61

4.4 Curves and surfaces 137

6.2 Energy–momentum tensor for E&M 211

8 Gravitational radiation 296

Classical mechanics, as a subject, is broadly defined The ultimate goal of ics is a complete description of the motion of particles and rigid bodies To find

mechan-x(t) (the position of a particle, say, as a function of time), we use Newton’s laws,

or an updated (special) relativistic form that relates changes in momenta to forces

Of course, for most interesting problems, it is not possible to solve the resulting

second-order differential equations for x(t) So the content of classical mechanics

is a variety of techniques for describing the motion of particles and systems ofparticles in the absence of an explicit solution We encounter, in a course on classi-cal mechanics, whatever set of tools an author or teacher has determined are mostuseful for a partial description of motion Because of the wide variety of such tools,and the constraints of time and space, the particular set that is presented dependshighly on the type of research, and even personality of the presenter

This book, then, represents a point of view just as much as it contains tion and techniques appropriate to further study in classical mechanics It is theculmination of a set of courses I taught at Reed College, starting in 2005, thatwere all meant to provide a second semester of classical mechanics, generally tophysics seniors One version of the course has the catalog title “Classical Mechan-ics II”, the other “Classical Field Theory” I decided, in both instantiations of thecourse, to focus on general relativity as a target The classical mechanical tools,when turned to focus on problems like geodesic motion, can take a student prettyfar down the road toward motion in arbitrary space-times There, the Lagrangianand Hamiltonian are used to expose various constants of the motion, and applyingthese to more general space-times can be done easily In addition, most studentsare familiar with the ideas of coordinate transformation and (Cartesian) tensors,

informa-so much of the discussion found in a first semester of classical mechanics can bemodified to introduce the geometric notions of metric and connection, even in flatspace and space-time

xiii

So my first goal was to exploit students’ familiarity with classical mechanics

to provide an introduction to the geometric properties of motion that we find ingeneral relativity We begin, in the first chapter, by reviewing Newtonian gravity,and simultaneously, the role of the Lagrangian and Hamiltonian points of view,and the variational principles that connect the two Any topic that benefits fromboth approaches would be a fine vehicle for this first chapter, but given the ultimategoal, Newtonian gravity serves as a nice starting point Because students have seenNewtonian gravity many times, this is a comfortable place to begin the shift from

L= 1

2m v2− U to an understanding of the Lagrangian as a geometric object The

metric and its derivatives are introduced in order to make the “length-minimizing”role of the free Lagrangian clear, and to see how the metric dependence on coordi-nates can show up in the equations of motion (also a familiar idea)

Once we have the classical, classical mechanics reworked in a geometric fashion,

we are in position to study the simplest modification to the underlying geometry –moving the study of dynamics from Euclidean flat space (in curvilinear coordinates)

to Minkowski space-time In the second chapter, we review relativistic dynamics,and its Lagrange and Hamiltonian formulation, including issues of parametrizationand interpretation that will show up later on Because of the focus on the role offorces in determining the dynamical properties of relativistic particles, an adver-tisement of the “problem” with the Newtonian gravitational force is included in thischapter That problem can be seen by analogy with electrodynamics – Newtoniangravity is not in accord with special relativity, with deficiency similar in character

to Maxwell’s equations with no magnetic field component So we learn that ativistic dynamics requires relativistic forces, and note that Newtonian gravity isnot an example of such a force

rel-Going from Euclidean space in curvilinear coordinates to Minkowski time (in curvilinear coordinates, generally) represents a shift in geometry In thethird chapter, we return to tensors in the context of these flat spaces, introducingdefinitions and examples meant to motivate the covariant derivative and associatedChristoffel connection These exist in flat space(-time), so there is an opportunity

space-to form a connection between tensor ideas and more familiar versions found invector calculus To understand general relativity, we need to be able to characterizespace-times that are not flat So, finally, in the fourth chapter, we leave the physicalarena of most of introductory physics and discuss the idea of curvature, and themanner in which we will quantify it This gives us our first introduction to theRiemann tensor and a bit of Riemannian geometry, just enough, I hope, to keepyou interested, and provide a framework for understanding Einstein’s equation

At the end of the chapter, we see the usual motivation of Einstein’s equation,

as an attempt to modify Newton’s second law, together with Newtonian gravity,under the influence of the weak equivalence principle – we are asking: “under

what conditions can the motion of classical bodies that interact gravitationally, beviewed as length-minimizing paths in a curved space-time?” This is Einstein’s idea,

if everything undergoes the same motion (meaning acceleration, classically), thenperhaps that motion is a feature of space-time, rather than forces

At this point in the book, an abrupt shift is made What happened is that Iwas asked to teach “Classical Field Theory”, a different type of second semester

of classical mechanics geared toward senior physics majors In the back of mostclassical mechanics texts, there is a section on field theory, generally focused onfluid dynamics as its end goal I again chose general relativity as a target – ifgeodesics and geometry can provide an introduction to the motion side of GR inthe context of advanced mechanics, why not use the techniques of classical fieldtheory to present the field-theoretic (meaning Einstein’s equation again) end ofthe same subject? This is done by many authors, notably Thirring and Landauand Lifschitz I decided to focus on the idea that, as a point of physical model-building, if you start off with a second-rank, symmetric tensor field on a Minkowskibackground, and require that the resulting theory be self-consistent, you end up,almost uniquely, with general relativity I learned this wonderful idea (along withmost of the rest of GR) directly from Stanley Deser, one of its originators and earlyproponents My attempt was to build up enough field theory to make sense of thestatement for upper-level undergraduates with a strong background in E&M andquantum mechanics

So there is an interlude, from one point of view, amplification, from another,that covers an alternate development of Einstein’s equation The next two chap-ters detail the logic of constructing relativistic field theories for scalars (massiveKlein–Gordon), vectors (Maxwell and Proca), and second-rank symmetric tensors(Einstein’s equation) I pay particular attention to the vector case – there, if welook for a relativistic, linear, vector field equation, we get E&M almost uniquely(modulo mass term) The coupling of E&M to other field theories also shares sim-ilarities with the coupling of field theories to GR, and we review that aspect ofmodel-building as well As we move, in Chapter 6, to general relativity, I makeheavy use of E&M as a theory with much in common with GR, another favoritetechnique of Professor Deser At the end of the chapter, we recover Einstein’sequation, and indeed, the geometric interpretation of our second-rank, symmetric,relativistic field as a metric field The digression, focused on fields, allows us toview general relativity, and its interpretation, in another light

Once we have seen these two developments of the same theory, it is time (late

in the game, from a book point of view) to look at the physical implications ofsolutions In Chapter 7, we use the Weyl method to develop the Schwarzschildsolution, appropriate to the exterior of spherically symmetric static sources, toEinstein’s equation This is the GR analogue of the Coulomb field from E&M,

and shares some structural similarity with that solution (as it must, in the end,since far away from sources, we have to recover Newtonian gravity), and welook at the motion of test particles moving along geodesics in this space-time Inthat setting, we recover perihelion precession (massive test particles), the bending

of light (massless test particles), and gravitational redshift This first solution alsoprovides a venue for discussing the role of coordinates in a theory that is coordinate-invariant, so we look at the various coordinate systems in which the Schwarzschildspace-time can be written and its physical implications uncovered

Given the role of gravitational waves in current experiments (like LIGO), Ichoose radiation as a way of looking at additional solutions to Einstein’s equation

in vacuum Here, the linearized form of the equations is used, and contact is againmade with radiation in E&M There are any number of possible topics that couldhave gone here – cosmology would be an obvious one, as it allows us to explorenon-vacuum solutions But, given the field theory section of the book, and theview that Maxwell’s equations can be used to inform our understanding of GR,gravitational waves are a natural choice

I have taken two routes through the material found in this book, and it is thecombination of these two that informs its structure For students who are interested

in classical mechanical techniques and ideas, I cover the first four chapters, andthen move to the last three – so we see the development of Einstein’s equation,its role in determining the physical space-time outside a spherically symmetricmassive body, and the implications for particles and light If the class is focused

on field theory, I take the final six chapters to develop content Of course, strictadherence to the chapters will not allow full coverage – for a field theory class,one must discuss geodesic and geometric notions for the punchline of Chapter 7 tomake sense Similarly, if one is thinking primarily about classical mechanics, somework on the Einstein–Hilbert action must be introduced so that the Weyl method

in Chapter 8 can be exploited

Finally, the controversial ninth chapter – here I take some relevant ideas fromthe program of “advanced mechanics” and present them quickly, just enough towhet the appetite The Kerr solution for the space-time outside a spinning mas-sive sphere can be understood, qualitatively and only up to a point, by analogywith a spinning charged sphere from E&M The motion of test bodies can bequalitatively understood from this analogy In order to think about more exoticmotion, we spend some time discussing numerical solution to ODEs, with an eyetoward the geodesic equation of motion in Kerr space-time Then, from our workunderstanding metrics, and relativistic dynamics, combined with the heavy use ofvariational ideas throughout the book, a brief description of the physics of relativis-tic strings is a natural topic We work from area-minimization in Euclidean spaces to

area-minimization in Minkowski space-times, and end up with the standard tions of motion for strings.

equa-I have made available, and refer to, a minimal Mathematica package that

is meant to ease some of the computational issues associated with forming thefundamental tensors of Riemannian geometry While I do believe students shouldcompute, by hand, on a large piece of paper, the components of a nontrivialRiemann tensor, I do not want to let such computations obscure the utility of theRiemann tensor in geometry or its role for physics So, when teaching this material,

I typically introduce the package (with supporting examples, many drawn from thelonger homework calculations) midway through the course Nevertheless, I hope

it proves useful for students learning geometry, and that they do not hesitate to usethe package whenever appropriate

A note on the problems in this book There are the usual set of practice problems,exercises to help learn and work with definitions But, in addition, I have leftsome relatively large areas of study in the problems themselves For example,students develop the Weyl metric, appropriate to axially symmetric space-times,

in a problem The rationale is that the Weyl metric is an interesting solution toEinstein’s equation in vacuum, and yet, few astrophysical sources exhibit this axialsymmetry It is an important solution, but exploring the detailed physics of thesolution is, to a certain extent, an aside In the end, I feel that students learn bestwhen they develop interesting (if known) ideas on their own That is certainly thecase for research, and I think problems can provide an introduction to that process

In addition to practicing the techniques discussed in the text, working out long,involved, and physically interesting problems gives students a sense of ownership,and aids retention Another example is the verification that the Kerr solution toEinstein’s equation is in fact a vacuum solution Here, too, a full derivation ofKerr is beyond the techniques introduced within the book, so I do not considerthe derivation to be a primary goal – verification, however, is a must, and can bedone relatively quickly with the tools provided I have marked these more involvedproblems with a∗ to indicate that they are important, but may require additionaltools or time

As appears to be current practice, I am proud to say that there are no new ideas

in this book General relativity is, by now, almost a century old, and the classicalmechanical techniques brought to its study, much older I make a blanket citation toall of the components of the Bibliography (found at the end), and will point readers

to specific works as relevant within the text

I would like to thank my teachers, from undergraduate to postdoctoral: NicholasWheeler, Stanley Deser, Sebastian Doniach, and Scott Hughes, for their thought-ful advice, gentle criticism, not-so-gentle criticism, and general effectiveness inteaching me something (not always what they intended).

I have benefitted greatly from student input,1 and have relied almost entirely

on students to read and comment on the text as it was written In this context, Iwould like to thank Tom Chartrand, Zach Schultz, and Andrew Rhines Specialthanks goes to Michael Flashman who worked on the solution manual with me, andprovided a careful, critical reading of the text as it was prepared for publication.The Reed College physics department has been a wonderful place to carry outthis work – my colleagues have been helpful and enthusiastic as I attempted tofirst teach, and then write about, general relativity I would like to thank JohnnyPowell and John Essick for their support and advice Also within the department,Professor David Griffiths read an early draft of this book, and his comments andscathing criticism have been addressed in part – his help along the way has beenindispensable

Finally, Professor Deser introduced me to general relativity, and I thank him forsharing his ideas, and commentary on the subject in general, and for this book inparticular Much of the presentation has been informed by my contact with him –

he has been a wonderful mentor and teacher, and working with him is always alearning experience, that is to say, a great pleasure

1 The Oxford English Dictionary defines a student to be “A person who is engaged in or addicted to study” – from

that point of view, we are all students, so here I am referring to “younger” students, and specifically, younger students at Reed College.

xviii

The program of classical mechanics is to determine the trajectory of a particle

or system of particles moving under the influence of some force The connectionbetween force and motion is provided by Newton’s second law:

supplemented by appropriate initial or boundary conditions The forces are

pro-vided (F might be made up of a number of different forces) and we solve the above

for x(t), from which any measurable quantity can be predicted.

As an approach to solving problems, Newton’s second law can be difficult towork with Given a generic force and multiple particles, a direct and completesolution proceeding from (1.1) is often unattainable So we content ourselves with

supplying less information than the full x(t) (sometimes, for example, we can easily find ˙x(t), but cannot proceed to x(t)), or, we work on special classes of

forces for which alternate formulations of the second law are tractable It is withthe latter that we will begin our work on Newtonian gravity – following a shortreview of the Lagrangian formulation of the equations of motion given a forcederivable from a potential, we will see how the Lagrange approach can be used

to simplify and solve for the trajectories associated with the Newtonian central

1

potential From there, we will move on to the Hamiltonian formulation of the sameproblem.

1.2 The classical Lagrangian

Here we will define the Lagrangian formulation of the fundamental problem ofclassical mechanics: “Given a potential, how do particles move?” This sectionserves as a short review of Lagrangians in general, and the next section will spe-cialize to focus on Keplerian orbits in the classical setting – if we are to understandthe changes to the motion of particles in general relativity (GR), it behooves us torecall the motion in the “normal” case Our ultimate goal is to shift from the specificsorts of notations used in introductory cases (for example, spherical coordinates),

to a more abstract notation appropriate to the study of particle motion in generalrelativity

As we go, we will introduce some basic tensor operations, but there will be more

of this to come in Chapter3 We just need to become comfortable with summationnotation for now

1.2.1 Lagrangian and equations of motion

A Lagrangian is the integrand of an action – while this is not the usual definition,

it is, upon definition of action, more broadly applicable than the usual “kineticminus potential” form In classical mechanics, the Lagrangian leading to Newton’ssecond law reads, in Cartesian coordinates:1

point a to point b, etc.) Mathematically, we use the equations of motion derived

from the Lagrangian, together with the boundary conditions, to determine the curve

x(t) = x(t) ˆx + y(t) ˆy + z(t) ˆz through three-dimensional space.

1 I will refer to the “vector” (more appropriately, the coordinate differential is the vector) of coordinates as

x= x ˆx + y ˆy + z ˆz, and its time-derivative (velocity) as v = dx.

Extremization of an action

The Euler–Lagrange equations come from the extremization, in the variational

calculus sense, of the action:

S [x(t)]=



We imagine a path connecting two points x(0) and x(T ), say Then we define the

dynamical trajectory to be the unique path that extremizes S Suppose we have an

arbitrary x(t) with the correct endpoints, and we perturb it slightly via

x(t) −→ x(t) + η(t) In order to leave the physical observation of the endpoints

unchanged, we requireη(0) = η(T ) = 0 The action responds to this change:

The small changeη(t) is arbitrary, but once chosen, its time-derivative is fixed We

would like to write the integrand of ( 1.5 ) entirely in terms of the arbitrary trajectory perturbation,η(t), rather than quantities derived from this We can use integration by

parts on the second term to “flip” the t-derivative onto the L-derivative:

The first term, as a total time-derivative, gets evaluated at t = 0 and t = T where

η vanishes We can use (1.6 ) to make the replacement under the integral in ( 1.5 ):

Now extremization means S = 0, and the arbitrary value of η(t) allows us to set the

term in parentheses equal to zero by itself (that’s the only way to get S= 0 for arbitraryη(t)).

As a point of notation, we use the variational derivative symbol δ to indicate that

we have performed all appropriate integration by parts, so you will typically see ( 1.8 ) written as:

where δx replaces η – this tells us that it is the variation with respect to x that is

inducing the change in S For actions that depend on more than one variable that can

be varied, the notation makes it clear which one is being varied In addition to this δS,

δxshift, we will also refer to the Euler–Lagrange equations from variation with

the “variational derivative” of S with respect to x Extremization is expressed by

δS= 0, or equivalently in this case,δS

δx = 0.

Variation provides the ordinary differential equation (ODE) structure of interest,

a set of three second-order differential equations, the Euler–Lagrange equations ofmotion:

d dt

∂L

∂v −∂L

In Cartesian coordinates, with the Lagrangian from (1.2), the Euler–Lagrange

equations reproduce Newton’s second law given a potential U :

The advantage of the action approach, and the Lagrangian in particular, is thatthe equations of motion can be obtained for any coordinate representation of thekinetic energy and potential Although it is easy to define and verify the correctness

of the Euler–Lagrange equations in Cartesian coordinates, they are not necessary

to the formulation of valid equations of motion for systems in which Cartesiancoordinates are less physically and mathematically useful

The Euler–Lagrange equations, in the form (1.11), hold regardless of our

associ-ation of x with Cartesian coordinates Suppose we move to cylindrical coordinates

{s, φ, z}, defined by

then the Lagrangian in Cartesian coordinates can be transformed to cylindricalcoordinates by making the replacement for {x, y, z} in terms of {s, φ, z} (and

associated substitutions for the Cartesian velocities):

L (s, φ, z) = L(x(s, φ, z)) = 1

2m (˙s

2+ s2φ˙2+ ˙z2)− U(s, φ, z). (1.14)But, the Euler–Lagrange equations require no modification, the variational proce-dure that gave us (1.11) can be applied in the cylindrical coordinates, giving threeequations of motion:

1.2.2 Examples

In one dimension, we can consider the Lagrangian L= 1

2m x˙2−1

2k (x − a)2,

appropriate to a spring potential with spring constant k and equilibrium spacing a.

Then the Euler–Lagrange equations give:

d dt

choice in terms of boundaries in time at t = t0 and t = t f (particle starts at 1 m

from the origin at t = 0 and ends at 2 m from the origin at t = 10 s), or as an

initial position and velocity (particle starts at equilibrium position with speed

5 m/s) – there are other choices as well, depending on our particular experimentalsetup

In two dimensions, we can express a radial spring potential as:

giving us two equations of motion:

Suppose we want to transform to two-dimensional polar coordinates via x=

s cos φ and y = s sin φ – we can write (1.18) in terms of the derivatives of s(t) and φ(t) and solve for ¨s and ¨ φ to get:

We can take advantage of the coordinate-independence of the Lagrangian to

rewrite L directly in terms of s(t) and φ(t), where it becomes:

∂L

∂ ˙s −∂L

∂s = m ¨s − m s ˙φ2+ k (s − a) = 0

d dt

case, the φ coordinate does not appear in the Lagrangian at all, only ˙ φ shows up.Then we know from the Euler–Lagrange equations of motion that:

d dt

∂L

∂ ˙ φ = 0 −→ ∂L

When possible, this type of observation can be useful in actually solving theequations of motion Finding (or constructing) a coordinate system in which one

or more of the coordinates do not appear is one of the goals of Hamilton–Jacobitheory

From the equations of motion (and the implicit definition of the potential), provide a

physical interpretation for the constants A and B.

Problem 1.2

The Euler–Lagrange equations come from extremization of the action So we expect

the “true”, dynamical trajectory to minimize (in this case) the value of S= L dt.

For free particle motion, the Lagrangian is L= 1

2m v2 (in one dimension, for a

particle of mass m) Suppose we start at x(0) = 0 and at time T , we end up at

x (T ) = x f The solution to the equation of motion is:

x (t)=x f t

(a) Compute S= T

0 L dtfor this trajectory.

(b) Any function that goes to zero at the endpoints can be represented in terms of the

sine series, vanishing at t = 0 and t = T : α j sin

j π t T



Find the value of the action S = T

0 L dtfor this arbitrary trajectory, show

(assuming α j ∈ IR) that the value of the action for this arbitrary trajectory is greater than the value you get for the dynamical trajectory.

Problem 1.3

Take a potential in cylindrical coordinates U (s) and write out the Euler–Lagrange

equations ( 1.15 ) for the Lagrangian written in cylindrical coordinates ( 1.14 ) Verify that you get the same equations starting from Newton’s second law in Cartesian

coordinates and transforming to cylindrical coordinates, then isolating ¨s, ¨ φ and ¨z.

1.3 Lagrangian for U (r)

We want to find the parametrization of a curve x(t) corresponding to motion under

the influence of a central potential Central potentials depend only on a particle’s

distance from some origin, so they take the specific form: U (x, y, z) = U(r) with

r2≡ x2+ y2+ z2 We know, then, that the associated force will be directed, from

some origin, either toward or away from the particle (the force is−∇U ∼ ˆr, the

usual result familiar from electrostatics, for example) Refer to Figure 1.1

The Lagrangian for the problem is:

The equations of motion for this Lagrangian are the usual ones:

is interesting Notice that there is only one term that we would associate with thephysical environment (only one term involves the potential), the rest are somehowresiduals of the coordinate system we are using

More interesting than that is the structure of the equations of motion – everythingthat isn’t ¨X(or∂U ∂r ) looks like f (r, θ ) ˙ X ˙ Y (here, X, Y ∈ {r, θ, φ}) That is somewhat

telling, and says more about the structure of the Lagrangian and its quadraticdependence on velocities than anything else Setting aside the details of sphericalcoordinates and central potentials, we can gain insight into the classical Lagrangian

by looking at it from a slightly different point of view – one that will allow us togeneralize it appropriately to both special relativity and general relativity We willreturn to the central potential after a short notational aside

1.3.1 The metric

And so, innocuously, begins our journey Let’s rewrite (1.26) in matrix–vectornotation (we’ll take potentials that are arbitrary functions of all three coordinates),the kinetic term is the beneficiary here:

coordinate systems, the infinitesimal distance (squared) between the two points can

be written in Cartesian or spherical coordinates:

ds2= dx2+ dy2+ dz2

These are just statements of the Pythagorean theorem in two different coordinate

systems The distance between the two points is the same in both, that can’t change,

but the representation is different

These distances can also be expressed in matrix–vector form:

by t in an infinitesimal interval dt, the answer is provided by:



dy

dt dt

2+

the new coordinates and use that to find g µν (actually, we rarely bother with theformal name or matrix, just transform kinetic energies and evaluate the equations

of motion)

Label the vectors appearing in (1.34) and (1.35) dx µ, so that the three

com-ponents associated with µ= 1, 2, 3 correspond to the three components of thevector:

for example Then we can define the “Einstein summation notation” to expresslengths Referring to (1.34) for the actual matrix–vector form, we can write:

The idea behind the notation is that when you have an index appearing twice, as

in the top line, the explicit is redundant The prescription is: take each repeated

index and sum it over the dimension of the space Rename x = x1, y = x2, z = x3,then:

to be the matrix defined in (1.35) and x1= r, x2= θ, x3= φ.

In Einstein summation notation, we sum over repeated indices where one is up,

one is down (objects like g µν dx µare nonsense and will never appear) The repeatedindex, because it takes on all values 1−→ D (in this case, D = 3 dimensions) has

no role in labeling a component, and so can be renamed as we wish, leading

to statements like2 (we reintroduce the summation symbols to make the pointclear):

Finally, the explicit form of the metric can be recovered from the “line element”

(just ds2written out) If we are given the line element:

2 There are a few general properties of the metric that we can assume for all metrics considered here 1.

The metric is symmetric, this is a convenient notational device, we have no reason to expect dxdy

in a line element 2 It does not have to be diagonal 3 It can depend (as with the spherical metric) on position.

then we know that the metric, in matrix form, is:

That’s sensible, and I don’t even have to tell you which coordinates I mean We can

vary the action associated with this Lagrangian as before to find the equations ofmotion (alternatively, we can appeal directly to the Euler–Lagrange equations) We

have to be a little careful, it is possible (as in the spherical case) that g µν (x µ), i.e.the metric depends on the coordinates The equations of motion, written in tensorform, are:

where α goes from 1 to 3, covering all three coordinates.

Our goal is to obtain a general explicit form for these, written in terms of x α,

its first and second derivatives, the metric, and U (x, y, z) For starters, we need to

take a ˙x αderivative of the kinetic term:

Turning to the derivative ∂x ∂L α:

Now for a little sleight-of-hand which you will prove in Problem 1.7 – notice in thislast line that the second term has the factor ˙x ν x˙γ , which is symmetric in ν ↔ γ

Using the result from Problem 1.7, we have:

and we can write the equations of motion as:

m g αν x¨ν + m ˙x ν

˙

x γ

12

The term in parentheses appears a lot, and is given a special name – it is called

giving a name and symbol to a particular combination of derivatives of the metric:

αγ ν = 12

and we’ll see the significance of this object in Chapter3

We have, in their final form, the equations of motion for any coordinate choice:

m g αν x¨ν ανγ x˙ν x˙γ = −∂U

where the index α appearing only once in each term on the left and right is an

“open” index – there are three equations here, one for each value α= 1, 2, and 3.The terms in (1.59) that are not explicitly second derivatives of the coordinates,

or derivatives of U , are quadratic in the first derivatives ˙ x µ, just as we saw explicitlyfor spherical coordinates in Section1.3(in particular (1.30))

We went slow, but we have made some progress, especially in terms of our

later work Setting U = 0 in (1.59), we have the so-called “geodesic” equation for

a generic metric g µν The geodesic equation has solutions which are interpreted

as “straight lines” in general Certainly in the current setting, if we take U = 0

ανγ = 0 (since the metric does not, inthis case, depend on position), the solutions are manifestly straight lines Later on,

in special and general relativity, we will lose the familiar notion of length, but if

we accept a generalized length interpretation, we can still understand solutions tothe geodesic equation as length-extremizing curves (the natural generalization ofstraight lines)

In developing (1.59), we made no assumptions about dimension or form for themetric (beyond symmetric, invertible, and differentiable), and these equations hold

for arbitrary dimension, coordinates, and potential The geodesic form (U = 0) isprecisely the starting point for studying particle motion in general relativity – there,the moral is that we have no forces (no potential), and the curved trajectories (orbits,

for example) we see are manifestations of the curvature of the space-time (expressed

ανγ term that approximates theforcing we would normally associate with a Newtonian gravitational source

Lagrangian associated with a central potential in cylindrical coordinates, with x1= s,

x2 = φ, x3= z From the Lagrangian itself (most notably, its kinetic term), write the

metric associated with cylindrical coordinates.

Problem 1.5

(a) For the metric g µν in spherical coordinates, with x1= r, x2= θ, x3= φ, find the

r component of the equation of motion (i.e α= 1):

(b) Starting from the Lagrangian in spherical coordinates, calculate the r equation of

motion directly from:

d dt

into a symmetric (S µν = S νµ ) and antisymmetric part (A µν = −A νµ) via

T µν = S µν + A µν The symmetric portion is often denoted T (µν), and the

(c) Using the decomposition above, show that, as in (1.56 ):

that metrics are symmetric, g µν = g νµ

1.4 Classical orbital motion

After that indexing tour-de-force, we are still stuck solving the problem of orbitalmotion The point of that introduction to the “covariant formulation” (meaningcoordinate-independent) of the equations of motion will become clear as we pro-ceed Now we take a step back, dangling indices do not help to solve the problem

in this case It is worth noting that we have taken coordinate-independence to anew high (low?) with our metric notation – in (1.59), you really don’t know what

coordinate system you are in

So we will solve the equations of motion for the gravitational central potential in achosen set of coordinates – the standard spherical ones There is a point to the wholeprocedure – GR is a coordinate-independent theory, we will write statements thatlook a lot like (1.59), but, in order to “solve” a problem, we will always have to intro-

duce coordinates That is the current plan After we have dispensed with Keplerianorbits, we will move on and solve the exact same problem using the Hamiltonianformulation, and for that we will need to discuss vectors and tensors again

We have the following basic mechanics problem shown in Figure 1.2: given

a body of mass M generating a Newtonian gravitational potential φ(r)= −M G

r ,

how does a particle of mass m move under the influence of φ(r)? In particular, we

are interested in the orbital motion, and we’ll tailor our discussions to this form,namely, the target elliptical orbits

Ellipse

Going back to the Lagrangian for a generic central potential (we will have U (r)=

m φ (r) eventually), in abstract language, we have:

φ(r) = − M G

r

Figure 1.2 A central body of mass M generates a gravitational field, given by the potential φ(r) A “test” mass m responds to the force of gravity.

We will transform the radial coordinate: let ρ = r−1, then the metric (specified

equivalently by the associated line element) becomes:

with the new coordinate differential dx α = (dρ, dθ, dφ)˙ T

The potential is spherically symmetric, meaning that there are no preferred

directions, or functionally, that it depends only on r (or, equivalently, ρ) We can

set θ = π

2 and ˙θ = 0 to put the motion in a specific plane (the horizontal plane – for

Cartesian coordinates in their standard configuration, this is the x − y plane) Our

choice here is motivated by the symmetry of the potential – since the potential

depends only on r, the particular plane in which the motion occurs cannot matter.

In fact, if we went through the equations of motion, we would find that θ = π

2

and ˙θ = 0 do not limit our solution That’s all well and good, but we have to becareful – when we use information about a solution prior to variation, we can lose

the full dynamics of the system As an example, consider a free particle classical

Lagrangian – just L f = 1

2m x˙2– we know that the solutions to this are vectors of

the form x(t)= x0+ v t If we put this into the Lagrangian, we get L f = 1

2m v2,just a number We cannot vary a number and recover the equations of motion, so

we have lost all dynamical information by introducing, in this case, the solutionfrom the equations of motion themselves That may seem obvious, but we havedone precisely this in our proposed specialization to planar motion In this case, itworks out okay, but you might ask yourself why you can’t equally well take the

motion to lie in the θ = 0 plane? We will address this question later on when wediscuss the Hamiltonian

Putting θ = π

2 reduces the dimensionality of the problem We may now consider

a two-dimensional metric with dx α = (dρ, dφ)˙ T, and:

At this point, we could find and solve the equations of motion, but it is easiest to

note that there is no φ dependence in the above, i.e φ is an ignorable coordinate.

From the equation of motion, then:

d dt

We can reparametrize – rather than finding the time development of the ρ(t) and

φ (t) coordinates, the geometry of the solution can be uncovered by expressing ρ,

and later r, in terms of φ That is, we want to replace the functional dependence

of ρ on t with ρ(φ) To that end, define ρ ≡ dρ

dφ and use change of variables torewrite ˙ρand ¨ρ:

So far, the analysis applies to any spherically symmetric potential We now

specialize to the potential for Newtonian gravity, U = −G m M

What type of solution is this? Keep in mind that A ≡ G M m2

Let’s agree to start with r (φ= 0) = 0, this amounts to starting with no radial

velocity, and tells us that β = 0:

and this familiar solution is shown in Figure1.3

That’s the story with elliptical orbits We used the Lagrange approach to find a

first integral of the motion (J ), then we solved the problem using φ as the parameter

Figure 1.3 Ellipse in r(φ) parametrization.

for the curve (r(φ), φ) There are a couple of things we will be dropping from our analysis Set G= 1, which just changes how we measure masses We can also set

the test mass m = 1, it cannot be involved in the motion – this choice rescales J z

We make these simplifying omissions to ease the transition to the usual choice of

units G = c = 1.

Problem 1.8

We have been focused on bound, elliptical, orbits, but one can also approach a

massive object along a straight line, this is called radial infall.

(a) From our radial equation for the φ-parametrized ρ(φ) = 1/r(φ) curve, we had, for arbitrary U (ρ):

J z2m



ρ (φ) + ρ(φ)= −dU (ρ)

Can this equation be used to develop the ODE appropriate for radial infall with

the Newtonian point potential? (i.e A particle falls inward from r(t = 0) = R with ˙r(t = −∞) = 0 toward a spherically symmetric central body with mass M sitting at r = 0.) If not, explain why If so, prepare to solve the relevant ODE for

r (t) in the next part.

(b) Solve the radial infall problem with initial conditions from part a i.e Find r(t)

appropriate for a particle of mass m falling toward r= 0 along a straight line –

assume a spherically symmetric massive body is located at r = 0 with mass M.

dφ and use change of variables torewrite ˙ρand ¨ρ:

So far, the analysis applies to any spherically symmetric... r(φ= 0) = 0, this amounts to starting with no radial

velocity, and tells us that β = 0:

and this familiar solution is shown in Figure1.3

That’s the story with... analysis Set G= 1, which just changes how we measure masses We can also set

the test mass m = 1, it cannot be involved in the motion – this choice rescales J z

We