An Introduction to Black Holes, Gravitational Waves, and ... · General Relativity: An Introduction to Black Holes, Gravitational Waves, and Cosmology Michael J W Hall Chapter 1 Concepts

General Relativity An Introduction to Black Holes, Gravitational Waves, and Cosmology

Michael J.W. HallThis book is based on a set of 18 class-tested lectures delivered to fourth-year physics undergraduates at Griffith University in Brisbane, and the book presents new discoveries by the Nobel-prize winning LIGO collaboration. The author begins with a review of special relativity and tensors and then develops the basic elements of general relativity – a beautiful theory that unifies special relativity and gravitation via geometry — with applications to the gravitational deflection of light, global positioning systems, black holes, gravitational waves, and cosmology.

The book provides readers with a solid understanding of the underlying physical concepts; an ability to appreciate and in many cases derive important applications of the theory; and a solid grounding for those wishing to pursue their studies further. General Relativity: An Introduction to Black Holes, Gravitational Waves, and Cosmology also connects general relativity with broader topics. There is no doubt that general relativity is an active and exciting field of physics, and this book successfully transmits that excitement to readers.

About Concise Physics Concise Physics™ publishes short texts on rapidly advancing areas or topics, providing readers with a snapshot of current research or an introduction to the key principles. These books are aimed at researchers and students of all levels with an interest in physics and related subject areas.

General Relativity: An Introduction to Black Holes, Gravitational Waves, and Cosm

ology - Hall

General Relativity: An Introductionto Black Holes, GravitationalWaves, and Cosmology

General Relativity: An Introductionto Black Holes, GravitationalWaves, and Cosmology

Michael J W HallGriffith University, Queensland, Australia

andAustralian National University, Canberra, Australia

Morgan & Claypool Publishers

Copyright ª 2018 Morgan & Claypool Publishers

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ISBN 978-1-6817-4885-6 (ebook)ISBN 978-1-6817-4882-5 (print)ISBN 978-1-6817-4883-2 (mobi)

DOI 10.1088/978-1-6817-4885-6

Version: 20180301

IOP Concise PhysicsISSN 2053-2571 (online)ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise PhysicsPublished by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA,94901, USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Contents

Preface viii

About the author ix

List of symbols x

1 Concepts in special relativity 1-1

1.1 Galilean relativity 1-1

1.2 Inertial frames 1-3

1.3 Special relativity 1-4

1.4 Velocity addition, length contraction, and time dilation 1-8

1.5 Questions 1-10

References 1-11

2 Tensors in relativity 2-1

2.1 Motivation 2-1

2.2 General tensors and their basic properties 2-2

2.3 Lorentz tensors 2-4

2.4 Example: 4-momentum and force 2-7

2.5 Example: Doppler effect 2-9

2.6 Questions 2-11

Reference 2-14

3 The equivalence principle and local inertial frames 3-1

3.1 Inertial versus gravitational mass 3-1

3.2 Einstein’s equivalence principle 3-2

3.3 Local inertial frames 3-6

3.4 Questions 3-9

References 3-9

4 The motion of freely falling particles in general relativity 4-1

4.1 Local inertial frames and the geodesic equation 4-1

4.2 The metric tensor 4-3

4.3 Gravity as geometry 4-5

4.4 The Newtonian limit 4-8

4.5 Questions 4-10

v

5 The Schwarzschild metric and black holes 5-1

5.1 Spherical symmetry and the Schwarzschild metric 5-1

5.2 Geodesics in spherically symmetric spacetimes 5-3

5.3 Particle geodesics in a Schwarzschild spacetime 5-5

5.4 Deflection of light by the Sun 5-6

5.5 Falling into a black hole 5-8

5.6 Questions 5-12

References 5-14

6 Tensors and geometry 6-1

6.1 Covariant derivatives 6-1

6.2 Basic properties of covariant derivatives 6-4

6.3 Riemann and Ricci tensors 6-5

6.4 Symmetries and Bianchi identities 6-7

6.5 Questions 6-9

Reference 6-11

7 Einstein’s field equations 7-1

7.1 Overview 7-1

7.2 Energy–momentum tensor and conservation laws 7-2

7.2.1 Conservation of electric charge 7-2

7.2.2 Conservation of energy–momentum 7-4

7.2.3 The energy–momentum tensor 7-5

7.3 The field equations for general relativity 7-8

7.4 The cosmological constant 7-9

7.5 Questions 7-13

References 7-15

8 Solving the field equations: vacuum solutions 8-1

8.1 The vacuum field equations 8-1

8.2 The Schwarzschild–de Sitter solution 8-2

8.2.1 Vacuum field equations for static spherically symmetricmetrics

8-2

8.2.2 Deriving the Schwarzschild–de Sitter metric 8-4

General Relativity: An Introduction to Black Holes, Gravitational Waves, and Cosmology

vi

8.3 Gravitational waves 8-5

8.3.1 Weak-field approximation 8-6

8.3.2 Harmonic gauge 8-7

8.3.3 Plane waves and polarisation 8-9

8.3.4 Detection of gravitational waves 8-11

8.4 Questions 8-15

References 8-16

9 Solving the field equations: cosmological solutions 9-1

9.1 The cosmological principle 9-1

9.2 The Friedmann-Robertson-Walker metric 9-1

9.2.1 Checking homogeneity and isotropy 9-2

9.2.2 Galaxies, distances, and the cosmological redshift 9-3

9.3 Friedmann-Robertson-Walker universes 9-4

9.3.1 A perfect (fluid) world 9-5

9.3.2 Local conservation of energy and momentum 9-6

9.3.3 Cosmic microwave background 9-7

9.3.4 Our accelerating Universe 9-8

9.4 Questions 9-10

References 9-11

Appendices

A Derivation of Lorentz transformations A-1

B Derivation of Einstein’s field equations B-1

C Remarks on selected questions C-1


vii

Preface

This book begins with a review of special relativity and tensors. It then develops thebasic elements of general relativity—a beautiful theory that unifies special relativityand gravitation via geometry—with applications to the gravitational deflection oflight, global positioning systems, black holes, gravitational waves, and cosmology.

The aims of the book are to give the reader a good understanding of theunderlying physical concepts, an ability to appreciate and in many cases deriveimportant applications of the theory, and a solid grounding for those wishing topursue their studies further. I have also included occasional ‘asides’ in somechapters, that make tangential connections between general relativity and othertopics.

I have based the book on a set of 18 lectures delivered for five consecutive years(2013–2017), to fourth-year physics undergraduates at Griffith University inBrisbane. A gravitational wave section was added in 2015 (in anticipation of theirimminent detection), which I have had the pleasure of updating each year thereafterin the light of discoveries by the Nobel-prize winning LIGO collaboration. The finalchapter of the book closes with another recent Nobel-prize winning discovery—ouraccelerating Universe. There is no doubt that general relativity is an active andexciting field of physics, and I hope these pages will transmit some of that excitementto the reader.

Each chapter ends with several questions, many of which were originally assign-ment and exam questions. Those marked with an asterisk are particularly recom-mended for spending time on, both to develop understanding and because they aretypically relevant to later chapters. The others are optional, and aimed at thosewishing to delve a little deeper. Further discussion of some questions is given at theend of the book.

Be aware that different sources use different sign conventions for the metric andRiemann tensors—for example, this book defines the metric tensor with signature(+,−,−,−), so that time-like intervals, corresponding to physical motion, arepositive.

I wish to thank Malcolm Anderson for sharing his deep knowledge of this subject,Howard Wiseman for encouraging me to give lectures on it, and the many studentswho asked excellent questions along the way.

Michael HallBrisbane, Australia

viii

About the author

Michael J W Hall

Michael Hall gained his PhD from the Australian NationalUniversity in 1989. Since then he has worked in various areas of thefoundations of physics, from both inside and outside academia(including a year as a Humboldt Fellow in Ulm, and sixteen as apatent examiner). For the last six years he has been a researchfellow in the Centre for Quantum Dynamics at Griffith Universityin Brisbane, and has recently taken up an honorary appointment at

the Australian National University in Canberra.His research interests include quantum optics and information, time observables,

quantum locality, relativistic Hamiltonians, the consistent coupling of classicalspacetime to quantum matter, and the many-interacting-worlds approach toquantum mechanics. He is a member of the editorial boards of the Journal of PhysicsA and Physical Review A.

ix

List of symbols

α,β,γ,… Greek indices run from 0 to 3

αβg , αβg metric tensor, with signature (+,−,−,−), and its matrix inverse

ηαβ Minkowski metric tensorαx spacetime coordinates

ξα spacetime coordinates in a local inertial frame

≡α AA A( , )0 4-vector αA with temporal component A0 and spatial component A

∂ ≡μα

μαA A , partial derivative of αA with respect to coordinate μx

μαA ; covariant derivative of αA with respect to coordinate μx

c speed of light in vacuum in local inertial frames (2.99 792 458 × 108 m s−1)

G Newton’s gravitational constant (6.674 × 10−11 m3 kg−1 s−2)

H0 Hubble parameter in present epoch (70 km s−1 Mpc−1 ≈ 3 × 10−18 s−1)

Λ cosmological constant (2 × 10−52 m−2)

⊕M mass of Earth (5.97 × 1024 kg)

⊙M mass of Sun (1.99 × 1030 kg)

⊕R mean radius of Earth (6.731 × 106 m)

⊙R mean radius of Sun (6.597 × 108 m)

x

IOP Concise Physics

General Relativity: An Introduction to Black Holes, Gravitational

Waves, and Cosmology

Michael J W Hall

Chapter 1

Concepts in special relativity

Special relativity is a fascinating and important subject. Its main concepts arerecalled in this book only to the extent needed to formulate and understand the basicelements of general relativity. However, there are many textbooks which explorefurther details of special relativity, of which one by Rindler is particularlyrecommended for further reading [1].

1.1 Galilean relativityIn 1632 a brilliant Italian physicist, GalileoGalilei, published a book that was to end upgetting him into a lot of trouble. It was calledDialogueConcerning the TwoChiefWorldSystems [2], and gave strong arguments for the consistency of the Copernican (Sun-centred)model of the solar system. The arguments also supported the superiority of thismodel over thePtolemaic (Earth-centred)model, which iswhat caused the later trouble.

At that time, one of the arguments against the Copernican model was that if theEarth is rotating to the east, then objects that are thrown or dropped will tend to fallbehind, to the west. For example, a stone dropped from the mast of a stationary shipshould land westward of the mast. Galileo had already contented himself byexperiment that this was not so—but also that, contrary to the very basis of theargument, that even if the ship was sailing, no matter in which direction, the stonealways landed at the foot of the mast. The motion of a (smoothly) moving ship couldnot be detected in such a way! In his book he further developed this idea [2]:

Shut yourself up with some friend in the main cabin below decks on some largeship and have with you there some flies, butterflies, and other small flyinganimals. Have a large bowl of water with some fish in it; hang up a bottle thatempties drop by drop into a narrow vessel beneath it. With the ship standingstill, observe carefully how the little animals fly with equal speed to all sides of

doi:10.1088/978-1-6817-4885-6ch1 1-1 ª Morgan & Claypool Publishers 2018

https://doi.org/10.1088/978-1-6817-4885-6ch1

the cabin. The fish swim indifferently in all directions; the drops fall into thevessel beneath; and, in throwing something to your friend, you need throw itno more strongly in one direction than another, the distances being equal;jumping with your feet together, you pass equal spaces in every direction.

When you have observed all these things carefully (though there is no doubtthat when the ship is standing still everything must happen in this way), havethe ship proceed with any speed you like, so long as the motion is uniform andnot fluctuating this way and that. You will discover not the least change in allthe effects named, nor could you tell from any of them whether the ship wasmoving or standing still [emphasis added].

In jumping, you will pass on the floor the same spaces as before, nor willyou make larger jumps toward the stern than toward the prow even though theship is moving quite rapidly, despite the fact that during the time that you arein the air the floor under you will be going in a direction opposite to yourjump. In throwing something to your companion, you will need no more forceto get it to him whether he is in the direction of the bow or the stern, withyourself situated opposite. The droplets will fall as before into the vesselbeneath without dropping toward the stern, although while the drops are in theair the ship runs many spans. The fish in their water will swim toward the frontof their bowl with no more effort than toward the back, and will go with equalease to bait placed anywhere around the edges of the bowl. Finally thebutterflies and flies will continue their flights indifferently toward every side,nor will it ever happen that they are concentrated toward the stern, as if tiredout from keeping up with the course of the ship, from which they will havebeen separated during long intervals by keeping themselves in the air.

The idea in the middle paragraph above (in italics), illustrated in figure 1.1, is thatthe laws of physics are the same in frames of reference moving at constant velocities

Figure 1.1. Galileo’s thought experiment: an observer below deck cannot determine, by local experiments,whether the ship is motionless on the water or sailing smoothly in some direction.


1-2

with respect to one another. This is now called Galilean relativity. It may besharpened in Newtonian mechanics to invariance of the laws of motion under anycoordinate transformation of the form

′ = − +′ = +

x x v aR tt t b

,(1.1)

for any rotation R, velocity boost v, and translations a and b. For example,Newton’s universal law of gravitation,

¨ = − −−

xx X

x Xm

GmM( ), (1.2)3

takes precisely the same form in the transformed coordinates:

¨′ = − ′ − ′′ − ′

xx X

x Xm

GmM( ), (1.3)3

as may easily be checked.The above ‘Galilean’ transformations form a ten-parameter group. Note that

they separately preserve spatial and temporal distances, i.e., writing ≡x x y z( , , )and ≡X X Y Z( , , ),

′ − ′ + ′ − ′ + ′ − ′ = − + − + −′ − ′ = −

x X y Y z Z x X y Y z Zt T t T

( ) ( ) ( ) ( ) ( ) ( ) ,.

(1.4)2 2 2 2 2 2

1.2 Inertial framesFor Galileo’s principle of relativity to apply to Newtonian mechanics, one has tobegin with a coordinate system in which Newtonian mechanics is itself valid. Inparticular, Newton’s first law of motion, that free particles remain at rest or move ina straight line, must hold. That is, a free particle trajectory must satisfy

=xddt

0.2

2

This would not be the case, for example, if the coordinate system was fixed relative toa spinning roundabout (try rolling a ball to a friend while on one of these).

Newton’s first law of motion is often called the law of inertia, and hencecoordinate systems which respect it are called ‘inertial frames of reference’, or,more simply, inertial frames. Thus, in any inertial frame free particles move instraight lines. Galilean relativity may be broadly stated in the form that the laws ofphysics have the same form in all inertial frames.

Note that Newton did not regard inertial frames as fundamental. He believed thatspace and time are ‘absolute’, with each point in space and time having meaningfulfixed coordinate values. He considered the use of relative coordinates, defined withrespect to some arbitrary spatial and temporal origin, to be a mere convenience.


1-3

Galilean relativity is not incompatible with this belief, but it does imply that noexperiment can distinguish such an ‘absolute’ frame of reference from any otherinertial frame.

For this reason, Galileo’s principle of relativity was put in doubt by earlyinterpretations of James Clerk Maxwell’s equations for the electromagnetic field,which postulated an ‘ether’ through which electromagnetic waves propagated,analogously to the propagation of sound waves through air. The ether was thoughtto be at rest with respect to absolute space. Surprisingly, however, all experimentsmade to try and detect the motion of the Earth through this ether—most notably theMichelson–Morley experiment—failed, consistent with Galilean relativity.

Nevertheless, Maxwell’s equations still raised a difficulty for Galilean relativity,at least when embedded into Newtonian mechanics. For example, the equationspredicted (consistently with the Michelson–Morley experiment) that the speed oflight in free space is a fixed constant in all inertial frames, despite the light source(e.g., a lamp) having different speeds with respect to different frames. This cannot bereconciled with the form of the Galilean transformations in equation (1.1), underwhich Newtonian mechanics is invariant. It was this incompatibility that led AlbertEinstein to his ‘Special Theory of Relativity’, published in 1905 [3, 4].

1.3 Special relativityPostulates

The basic elements of Einstein’s special relativity can be formulated as follows.First, inertial frames are defined via the motion of free particles:

Postulate 1: Free particles move in straight lines in inertial frames.

Thus, an inertial frame is a spacetime coordinate system in which Newton’s first lawof motion is valid (the other laws need to be slightly modified).

Second, motivated by the prediction of Maxwell’s equations in the previoussection, it is further assumed that:

Postulate 2: The speed of light in a vacuum is a constant, c, in all inertial frames,when measured using a standard set of clocks and rulers at rest in each frame.

Standard time and length standards are required because otherwise the speed of lightcould be numerically different in different directions, even in the same inertial frame—e.g., if coordinates were measured in the x-direction and the y-direction using rulershaving different units. Postulate 2 implies that only agreement on a standard clock isactually needed, as the spatial distance between two points at rest in a given inertialframe can then be defined via the time for light to propagate between them.

The first and second postulates still allow the possibility that standard length andtime scales can only be defined up to a common scale factor, since straight-linemotion and the speed of light are invariant under such a rescaling. If this were so,then it would not be physically meaningful, for example, to compare the rates of


1-4

clocks at rest in different inertial frames, as there would be no ‘absolute’ rate even forclocks at rest. There are various choices for removing this possibility, a suitable onebeing:

Postulate 3: If a standard clock at rest in a first inertial frame moves at velocity vwith respect to a second inertial frame, then a standard clock at rest in the secondframe moves at velocity −v with respect to the first frame, and the moving clockin each frame ticks at the same rate.

It could alternatively be required that the volume of a given spacetime region is thesame for all observers, or that the laws of physics are not scale-invariant (implyingthe existence of a standard time scale for each inertial frame), as discussed inappendix A.

Finally, we cannot in fact fully adopt the broad notion of Galilean relativity insection 1.2, that the laws of physics have the same form in all inertial frames. Apostulate is needed to rule out invariance of physical laws between inertial framesrelated by transformations such as spatial reflections and reversing the direction oftime (under which the laws of physics are known not to be invariant, from particledecay experiments), or related by swapping the time coordinate with a spatialcoordinate (which converts subluminal motion to superluminal motion and viceversa):

Postulate 4: The laws of physics are invariant in inertial frames connected by acontinuous sequence of intermediate coordinate transformations.

In the language of group theory, this implies that the physically relevant group oftransformations between inertial frames is continuously connected to the identity. Itmakes plausible the idea that systems and observers can, by continuous operations,move from being at rest in a first inertial frame to being at rest in a second inertialframe, with no change in the laws of physics.

The invariant spacetime interval

The above four postulates are sufficient to derive the group of spacetime coordinatetransformations that link inertial frames (see appendix A). These are called (properorthochronous) Lorentz transformations, after their first discoverer, HendrikLorentz [4], and have some significant differences from the Galilean transformationsin (1.1).

The derivation in appendix A shows, in particular, that transformations betweeninertial frames preserve the quantity

τΔ ≔ Δ − Δ − Δ − Δc c t x y z( ) ( ) ( ) ( ) ( ) (1.5)2 2 2 2 2 2 2

linking any two points t x y z( , , , ) and + Δ + Δ + Δ + Δt t x x y y z z( , , , ) in space-time, where c is the invariant speed of light in vacuum. This may be directlycompared with (1.4) for Galilean transformations, under which the spatial and


1-5

temporal distances are separately preserved. In special relativity only the abovecombination of the spatial and temporal distances is invariant, and is called theinvariant spacetime interval. The two points are said to be time-like separated, light-like separated, and space-like separated for values of τΔ( )2 that are, respectively,positive, zero, and negative (see figure 1.2). Note that τΔ has units of time.

Any particle travelling at the speed of light in some inertial frame has τΔ = 0from (1.5), and hence must travel at the speed of light in all inertial frames. Thus thelightcones in figure 1.2 are invariant.

Further, if t x y z( , , , ) and + Δ + Δ + Δ + Δt t x x y y z z( , , , ) refer to successivespacetime locations of a free particle, then in the rest frame of the particle one hasΔ = Δ = Δ =x y z 0, and so

τΔ = Δ >t( ) ( ) 0.2 2

Thus, a free particle follows a time-like trajectory, and the duration τΔ is equal to theelapsed time experienced by the particle in moving between the two spacetimelocations, relative to its own rest frame. Since τΔ has the same value in all inertialframes, it has the same physical meaning in all inertial frames, and is called the

light signal (lightlike) particle trajectory (timelike)

spacelike curve

Figure 1.2. Spacetime diagrams and causal structure. Each point in spacetime has an invariant past and futurelightcone, defined by the regions from which light signals can be directly received and transmitted. Theinteriors of the lightcones define the absolute past and absolute future of each point. Particle trajectories aretime-like, and hence move from the absolute past to the absolute future of any given point. All events outsidethe lightcones of a point are space-like separated from the point, and so no (subluminal) signal or particle canbe sent or received from them. In the Galilean limit c → ∞ the past and future lightcones of a given point (t ,x,y, z) flatten and merge into a single hyperplane, corresponding to the set of spatial positions at a fixed time t.


1-6

invariant or proper time. Note, it also follows that any particle that has a rest framecannot move faster than the speed of light, as this would require τΔ( )2 to change,from a positive value to a negative value.

Finally, if t x y z( , , , ) and + Δ + Δ + Δ + Δt t x x y y z z( , , , ) refer to the twoendpoints of a body such as a ruler, then in the rest frame of the body one hasΔ =t 0at any given time. Hence τΔ <( ) 02 from (1.5) (implying τΔ is imaginary). Hence theendpoints of a body at rest are space-like separated, and the invariant interval

τΔ = − Δ − Δ − Δ ≕ −c x y z D( ) ( ) ( ) ( )2 2 2 2 2 2

defines the distance D between these endpoints in its rest frame, called the properdistance or rest length.

Lorentz transformations

General Lorentz transformations are those coordinate transformations betweeninertial frames that preserve the invariant interval τΔc ( )2 2 in (1.5). They have thelinear form (see appendix A)

′′′′

= + =

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ctxyz

L

ctxyz

a L GL G, , (1.6)T

for some 4 × 4 matrix L and spacetime translation vector a, where G denotes the 4 ×4 matrix ≔ − − −G diag[1, 1, 1, 1]. The second equation ensures that τΔc ( )2 2 has thesame value in all inertial frames. Postulate 4 above further requires that =Ldet 1and ⩾L 100 , corresponding to the subgroup of proper orthochronous Lorentztransformations (see appendix A).

The proper orthochronous Lorentz transformations form a ten-parameter group,similarly to the Galilean transformations in equation (1.1). Six of these parameterscorrespond to spatial rotations and translations, and one to time translations, just asfor the Galilean group. However, the transformations between two frames having anonzero relative velocity are different in the two cases (as they must be, for the speedof light to be invariant). In particular, a velocity boost v in the x-directioncorresponds to the Lorentz transformation

γ γγ γ=

−− =

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟L

v cv c a

0 00 0

0 0 1 00 0 0 1

, 0, (1.7)

where

γ ≔−

⩾v c

1

11. (1.8)

2 2


1-7

Thus, the x and t coordinates transform as

γ γ′ = − ′ = −x x vt t t vx c( ), ( ), (1.9)2

which reduces to the corresponding Galilean boost ′ = −x x vt, ′ =t t in the limit→v c/ 0. Hence special relativity is well approximated by Galilean relativity for

frames having small relative speeds. It is easy to check that =L GL GT , as requiredby (1.6). The generalisation to an arbitrary velocity boost is considered in question1.3 at the end of this chapter.

1.4 Velocity addition, length contraction, and time dilationVelocity addition

To check that the boost in (1.9) does indeed correspond to two frames with relativevelocity v in the x-direction, consider an infinitesimal portion of a spacetime trajectorybetween ξ and ξ ξ+ d , where ξ denotes the spacetime vector ct x y z( , , , ). Now, takingthe inverse of the transformation in (1.9) (equivalent to replacing v by−v), one has

γ γ= ′ + ′ = ′ + ′dx dx vdt dt dt vdx c( ), ( ).2

Hence, if the trajectory velocity in the two frames is denoted by =u dx dt/ and′ = ′ ′u dx dt/ , respectively, it follows that

γγ

= = ′ + ′′ + ′

= ′ ′ ++ ′ ′

= ′ ++ ′

udxdt

dx vdtdt vdx c

dx dt vv dx dt c

u v

( )

( ) 1 ( )

1.

(1.10)

u v

c

2 2

2

Hence, a particle at rest in the primed coordinates, i.e., with ′ =u 0, has velocity u =v in the unprimed coordinates. That is, the primed frame moves at velocity v withrespect to the unprimed frame. Note also that a particle at rest in the unprimedcoordinates, i.e., with u = 0, has velocity ′ = −u v in the primed coordinates, asrequired by postulate 3 in section 1.3.

The velocity addition formula in (1.10) may further be used to check theinvariance of the speed of light, as required by postulate 2. In particular, taking′ =u c immediately gives u = c also, independently of the relative velocity v.

Length contraction

If ′x1 and ′x2 denote the endpoints of a ruler in its rest frame, then the distancebetween them, is given by = ∣ − ∣′ ′D x x0 1 2 . As noted previously, this is called the‘proper distance’ or ‘rest length’. Further, in a frame where the ruler is moving atspeed v in the x-direction, let x1 and x2 be the positions of the endpoints at equaltimes = =t t t1 2 . Equation (1.9) then yields

γ γ= − = − − − = −′ ′D x x x vt x vt x x( ) ( ) .0 1 2 1 1 2 2 1 2

Hence, defining the ‘length’ D of the ruler in a given frame to be the distancebetween its endpoints at equal times, one has


1-8

γ= = −−D D D v c1 . (1.11)10 0

2 2

Since <D D0 for ≠v 0, this is referred to as length contraction.Note that length contraction should not be confused with the apparent length of a

moving rod. This is because one does not actually ‘see’ the endpoints simultaneouslyin general, as light will take different times to reach the eye from each endpoint ifthey are located at different distances. It may be shown, for example, that the outlineof a moving sphere is circular for any inertial observer, rather than contracted in thedirection of relative motion [1]. In relativity, what you get is not necessarily whatyou see! Relativistic optics will not be reviewed further here.

Time dilation

If ′t1 and ′t2 denote the times of successive ticks of a clock in its rest frame, then its restperiod is given by = −′ ′T t t0 2 1. In a frame where the clock is moving at speed v in thex-direction, the ticks will be separated by = −T t t2 1, and by a spatial distance

− = −x x v t t( )2 1 2 1 . Hence, using (1.9), one has

γ γ γ= − = − − − = − − =′ ′ −T t t t t v x x c t t v c T[( ) ( ) ] ( ) (1 ) .0 2 1 2 1 2 12

2 12 2 1

The period relative to the moving frame is therefore given by

γ= = −T T T v c1 . (1.12)0 02 2

Since >T T0 for ≠v 0, this is referred to as time dilation. That is, moving clocks tickslowly.

Like length contraction, time dilation should not be confused with the apparentperiod of a moving clock. In particular, since the clock will be at different distancesbetween successive ticks, these ticks will take different lengths of time to arrive to anobserver at rest. This leads to the relativistic Doppler effect, discussed in chapter 2.


1-9

1.5 QuestionsReaders are encouraged to attempt questions marked with an asterisk (*), both todevelop understanding and because they are typically relevant to later chapters.Other questions are aimed at those wishing to explore more deeply.

Question 1.1* Bert and Ernie started out at the spacetime point (0, 0, 0, 0) in someinertial frame. Bert moved in a straight line to (5, 3, 0, 0), while Ernie first moved to(3, 2, 0, 0) and then on to (5, 3, 0, 0). Who had aged the most when they met upagain? (units are chosen such that c = 1).Hint: calculate the elapsed proper times via (1.5).

Question 1.2* Show that the defining property of Lorentz transformations in (1.6),i.e., =L GL GT for = − − −G diag[1, 1, 1, 1], ensures that the interval τΔ( )2 in (1.5)has the same value in all inertial frames.

Question 1.3 (General form of Lorentz boost)Check that the Lorentz transformation in (1.7), corresponding to a velocity boost

=v v( , 0, 0) in the x-direction, can be generalised to the block-matrix form

γ γ

γ γ=

−

− + −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟v

v

v vvL

c

cI

v

( )1

, (1.13)

T

T2

where I is the 3 × 3 identity matrix. Show for a general 3-vector v, with ∣ ∣ = <v v c,that a particle at rest in the unprimed frame (i.e., with Δ =x 0) has velocity−v in theprimed frame. Check that = −− v vL L( ) ( )1 . How are these properties related topostulate 3 in section 1.3?

Question 1.4 (Topological twin paradox)Note that there is no paradox arising from question 1.1 above: Bert behaves as a freeparticle, moving in a single straight line (relative to any inertial frame), whereasErnie changes direction and speed (relative to any inertial frame). Thus there is afundamental physical asymmetry in their motions, which underlies the differentamounts by which they age.

Consider, however, the case of a cylindrical spacetime, with one spatial dimensionthat is periodic in some inertial frame, so that x is identified with x + L. Suppose Bertand Ernie again start from spacetime point (0, 0, 0, 0), but with Bert remaining atrest in this frame and Ernie moving in the x-direction at speed v. Thus both move ina straight line, and the physical asymmetry of question 1.1 above is no longerpresent. Indeed, from Ernie’s point of view, he is at rest and Bert is moving in the


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x-direction with speed −v. Check that, nevertheless, Bert will have aged the mostwhen they meet again at time =t L v/ . Is this a true twin paradox?

[The resolution within special relativity is discussed by Dray [5]].

Question 1.5 An arrow having rest length A0 is shot horizontally, and flies atconstant speed into the open end of a cylinder having rest length <C A0 0.

(i) When the tip of the arrow reaches the other end of the cylinder, is the arrowwholly within the cylinder relative to:

(a) the rest frame of the arrow?(b) the rest frame of the cylinder?

Is this paradoxical?(ii) If the far end of the cylinder is closed by an immovable solid wall, that

prevents the arrow from travelling further, at what minimum speed must thearrow travel at to guarantee that it is enclosed by the cylinder relative toboth rest frames?Hint: the feathered end of the arrow cannot be affected by the presence ofthe wall prior to some physical signal or shockwave, generated by the tiphitting the wall, being transmitted at light speed or less along the length ofthe arrow.

References[1] Rindler W 1991 Introduction to Special Relativity (Oxford: Clarendon Press)[2] Galilei G 1967 Dialogue Concerning the Two Chief World Systems, Ptolemaic And Copernican

(Berkeley, CA: University of California Press)[3] Einstein A 1905 Zur Elektrodynamik bewegter Körper Ann. Phys. 322 891–921[4] Lorentz H A, Einstein A, Minkowski H, Weyl H and Sommerfeld A 1952 The Principle of

Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity(New York: Dover) This book contains translations of a number of important papers inrelativity theory originally published in German

[5] Dray T 1990 The twin paradox revisited Am. J. Phys. 58 822–5


1-11

https://doi.org/10.1002/andp.19053221004

https://doi.org/10.1119/1.16373

IOP Concise Physics

General Relativity: An Introduction to Black Holes, Gravitational

Waves, and Cosmology

Michael J W Hall

Chapter 2

Tensors in relativity

2.1 MotivationIt is convenient, in both mathematics and physics, to be able to write down relationsthat have the same form in a wide variety of contexts. For example, in Euclideangeometry, vector relations such as = +p q r, ∣ ∣ = ∣ ∣p q , and = ×p q r are invariantunder the group of rotations. Hence, all geometrical relations between Euclideanvectors can be expressed without having to specify a preferred rotational orientation(see also the ‘aside’ box on the next page).

A simple physical example is the invariance of Newton’s laws of motion underGalilean transformations. For example, for any Galilean transformation one has¨′ = ¨x xR from (1.1). Hence, noting force is a vector, Newton’s second law = ¨F xm istransformed into

′ = = ¨ = ¨′F F x xR R m m( ) ,

and so has the same form in the transformed coordinates. Newton’s second law thusrespects Galilean relativity (as do the others).

In special relativity, the relevant group of coordinate transformations is theproper orthochronous Lorentz group, corresponding to those linear transformationsthat preserve the invariant interval in (1.5) and are continuously connected to theidentity. The requirement that the laws of physics have the same form in all inertialframes connected by such transformations (postulate 4 in section 1.3), is guaranteedwhenever they can be expressed in terms of Lorentz tensors, as will be seen below.

More generally, tensors are very useful for formulating physical laws that areautomatically invariant under a given group of coordinate transformations. Inparticular, if the components of two tensors are equal in one coordinate system, thenthey are equal in all transformed coordinate systems. Thus, the invariance of a givenlaw is assured if it can be written in tensor form. This is particularly useful in generalrelativity, where the corresponding group encompasses all possible (differentiable)

doi:10.1088/978-1-6817-4885-6ch2 2-1 ª Morgan & Claypool Publishers 2018

https://doi.org/10.1088/978-1-6817-4885-6ch2

transformations. It is worth noting that there are other similarly useful objects inaddition to tensors, called spinors, but they are not needed in this book.

2.2 General tensors and their basic propertiesNotation Spacetime coordinates with respect to a general coordinate system will bedenoted by μx , where Greek indices run from 0 to 3. The time coordinatecorresponds to x0, and the spatial coordinates to ≡x x x x( , , )1 2 3 . The coordinateswith respect to a second coordinate system will typically be denoted by either ′μx or

μ′x , where associating the prime with the index will be seen to be a useful practice.The transformation from the first coordinate system to the second will be denoted by

→ ′μ μx x or →μ μ′x x . For example, a linear transformation has the form

′ = Λ +μνμ ν μx x a , (2.1)

where Λ νμ and μa are constants. Here we use the standard Einstein summation

convention, of summing over a repeated upper and lower index (thusΛ ≡ ∑ Λν

μ νν ν

μ ν=x x0

3 ). Note that a Lorentz transformation as per (1.6) is of thistype, with =x ct0 and Λ =ν

μμνL .

An aside: geometry and invariants

In 1872 the mathematician Felix Klein suggested a new way of thinking aboutgeometry, as the set of properties of a given space that is invariant under the actionof a given group of transformations. For example, Euclidean plane geometrycorresponds to those properties that remain invariant under the group of rotations,reflections and translations on the plane. A circle remains a circle under suchtransformations, triangles remain triangles, and distances and areas are preserved,and hence such properties form part of Euclidean geometry.Generalisations of Euclidean geometry can be found by extending the group. If scalingtransformations λ λ→x y x y( , ) ( , ) are added, for example, then circles, triangles, andsimilar triangles remain invariant concepts but distances do not (although ratios suchas π remain invariant). More generally, extending the group to all invertible lineartransformations on the plane leads to affine geometry, in which, for example, straightlines remain straight lines and convex figures remain convex, but circles are no longerdistinguishable from ellipses and the concept of similar triangles is lost.Extending further to all transformations on the plane that are continuously connectedto the identity leads to ‘rubber-sheet’ geometry, or topology. Now even the concept ofconvexity is lost, and there is no invariant distinction between straight and curved lines.Nevertheless, geometric properties such as whether a point lies inside or outside aclosed loop, or whether two lines intersect, remain invariant under such transforma-tions, and hence form part of two-dimensional topology. General relativity isanalogous to topology, in that its physical concepts are invariant with respect toarbitrary coordinate systems. Many of these concepts, corresponding to invariantphysical laws, can be expressed using tensors, and may be thought of as geometric incharacter (see also chapter 6).


2-2

Definition of tensorsA tensor is an object with a set of components, labelled by upperand lower indices, that obeys a simple transformation law with respect to a specifiedgroup of coordinate transformations. The choice of group is important. Generalrelativity, for example, requires invariance under the full group of all possible(differentiable) coordinate transformations, whereas special relativity only requiresinvariance under the group of (proper orthochronous) Lorentz transformations.

In particular, a spacetime-dependent object T, with components ν ν νμ μ μ

⋯⋯T

nm

1 21 2 , is a

tensor with respect to a given group of coordinate transformations if and only if ittransforms as

= ∂∂

⋯ ∂∂

∂∂

⋯ ∂∂ν ν ν

μ μ μμ

α

μ

α

β

ν

β

ν β β βα α α

′ ′⋯ ′′ ′⋯ ′

′ ′

′ ′ ⋯⋯

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥T

xx

xx

x

x

x

xT , (2.2)

n

mm

m

n

n n

m

1 2

1 21

1

1

1 1 2

1 2

under any coordinate transformation →μ μ′x x belonging to the group. Repeatedindices are summed over, in accordance with the summation convention. Thus, it isseen each partial derivative term acts like a matrix, which multiplies the correspond-ing index of T. One can alternatively denote the components of the transformedtensor ′T by ′ν ν ν

μ μ μ⋯⋯T

nm

1 21 2 . However, putting the primes on the indices helps to remember

the transformation law correctly.The upper indices are called contravariant, and the lower indices are called

covariant (note ‘co’ rhymes with ‘below’). A tensor with m contravariant and ncovariant indices is called an m n( , )-tensor, and its rank is defined to be m + n.

Some basic properties There are various ways to form new tensors from existingtensors. The sum of two tensors of the same type is defined by the addition of theircorresponding components, where this sum will clearly also obey (2.2). The productof two arbitrary tensors, not necessarily of the same type, is defined by simplymultiplying their components together—e.g., the product of a (0, 2)-tensor A and a(1, 0)-tensor B is given by ≔μν

λμν

λT A B , and is easily checked to be a (1, 2)-tensor.New tensors can also be formed by contraction, i.e., by setting an upper and lowerindex equal to each other and summing over it. For example, if μν

λT is a (1, 2)-tensor,then μν

νT is a (0, 1)-tensor, as shown further below.The tensor transformation rule (2.2) respects the group properties of the

associated group of coordinate transformations, essentially because the 4 × 4

Jacobian matrices with coefficients ∂∂

μα′x

xform a representation of the group. For

example, for a second coordinate transformation →μ μ′ ″x x one has, using the chain

rule1 ≡∂∂

∂∂

∂∂α

βα βx

yx y

, the group closure property

∂∂

= ∂

∂

∂∂

μ

α

μ

β

β

α

″ ″

′

′xx

x

x

xx

. (2.3)

1 The partial derivative of a function αf u( ) is defined via δ δ δ≔ + − ≕α α α αα

∂∂

f f u u f u u( ) ( ) fu

for arbitrary

infinitesimal variations δ αu . Repeated application gives δ δ δ δ= = =αα

ββ

α βα β

∂∂

∂∂

∂∂

∂∂

x f y xfx

f

y

yx

f

y, and the chain rule

follows by equating the first and last expressions for arbitrary δ αx and f.


2-3

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