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- Herausgeber: John Wiley & Sons
- Kategorie: Lebensstil
- Sprache: Englisch
Introducing General Relativity An accessible and engaging introduction to general relativity for undergraduates In Introducing General Relativity, the authors deliver a structured introduction to the core concepts and applications of General Relativity. The book leads readers from the basic ideas of relativity--including the Equivalence Principle and curved space-time--to more advanced topics, like Solar System tests and gravitational wave detection. Each chapter contains practice problems designed to engage undergraduate students of mechanics, electrodynamics, and special relativity. A wide range of classical and modern topics are covered in detail, from exploring observational successes and astrophysical implications to explaining many popular principles, like space-time, redshift, black holes, gravitational waves and cosmology. Advanced topic sections introduce the reader to more detailed mathematical approaches and complex ideas, and prepare them for the exploration of more specialized and sophisticated texts. Introducing General Relativity also offers: * Structured outlines to the concepts of General Relativity and a wide variety of its applications * Comprehensive explorations of foundational ideas in General Relativity, including space-time curvature and tensor calculus * Practical discussions of classical and modern topics in relativity, from space-time to redshift, gravity, black holes, and gravitational waves * Optional, in-depth sections covering the mathematical approaches to more advanced ideas Perfect for undergraduate physics students who have studied mechanics, dynamics, and Special Relativity, Introducing General Relativity is an essential resource for those seeking an intermediate level discussion of General Relativity placed between the more qualitative books and graduate-level textbooks.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Preface
Constants and Symbols
About the Companion Website
Chapter 1: Introducing General Relativity
Chapter 2: A Special Relativity Reminder
2.1 The need for Special Relativity
2.2 The Lorentz transformation
2.3 Time dilation
2.4 Lorentz–Fitzgerald contraction
2.5 Addition of velocities
2.6 Simultaneity, colocality, and causality
2.7 Space–time diagrams
Problems
Note
Chapter 3: Tensors in Special Relativity
3.1 Coordinates
3.2 4‐vectors
3.3 4‐velocity, 4‐momentum, and 4‐acceleration
3.4 4‐divergence and the wave operator
3.5 Tensors
3.6 Tensors in action: the Lorentz force
Problems
Notes
Chapter 4: Towards General Relativity
4.1 Newtonian gravity
4.2 Special Relativity and gravity
4.3 Motivations for a General Theory of Relativity
4.4 Implications of the Equivalence Principle
4.5 Principles of the General Theory of Relativity
4.6 Towards curved space–time
4.7 Curved space in two dimensions
Problems
Notes
Chapter 5: Tensors and Curved Space–Time
5.1 General coordinate transformations
5.2 Tensor equations and the laws of physics
5.3 Partial differentiation of tensors
5.4 The covariant derivative and parallel transport
5.5 Christoffel symbols of a two‐sphere
5.6 Parallel transport on a two‐sphere
5.7 Curvature and the Riemann tensor
5.8 Riemann curvature of the two‐sphere
5.9 More tensors describing curvature
5.10 Local inertial frames and local flatness
Problems
Notes
Chapter 6: Describing Matter
6.1 The Correspondence Principle
6.2 The energy–momentum tensor
Problems
Chapter 7: The Einstein Equation
7.1 The form of the Einstein equation
7.2 Properties of the Einstein equation
7.3 The Newtonian limit
7.4 The cosmological constant
7.5 The vacuum Einstein equation
Problems
Note
Chapter 8: The Schwarzschild Space–time
8.1 Christoffel symbols
8.2 Riemann tensor
8.3 Ricci tensor
8.4 The Schwarzschild solution
8.5 The Jebsen–Birkhoff theorem
Problems
Notes
Chapter 9: Geodesics and Orbits
9.1 Geodesics
9.2 Non‐relativistic limit of geodesic motion
9.3 Geodesic deviation
9.4 Newtonian theory of orbits
9.5 Orbits in the Schwarzschild space–time
Problems
Notes
Chapter 10: Tests of General Relativity
10.1 Precession of Mercury's perihelion
10.2 Gravitational light bending
10.3 Radar echo delays
10.4 Gravitational redshift
10.5 Binary pulsar PSR 1913+16
10.6 Direct detection of gravitational waves
Problems
Note
Chapter 11: Black Holes
11.1 The Schwarzschild radius
11.2 Singularities
11.3 Radial rays in the Schwarzschild space–time
11.4 Schwarzschild coordinate systems
11.5 The black hole space–time
11.6 Special orbits around black holes
11.7 Black holes in physics and in astrophysics
Problems
Notes
Chapter 12: Cosmology
12.1 Constant‐curvature spaces
12.2 The metric of the Universe
12.3 The matter content of the Universe
12.4 The Einstein equations
Problems
Note
Chapter 13: Cosmological Models
13.1 Simple solutions: matter and radiation
13.2 Light travel, distances, and horizons
13.3 Ingredients for a realistic cosmological model
13.4 Accelerating cosmologies
Problems
Notes
Chapter 14: General Relativity: The Next 100 Years
14.1 Developing General Relativity
14.2 Beyond General Relativity
14.3 Into the future
Advanced Topic A1: Geodesics in the Schwarzschild Space–Time
A 1.1 Geodesics and conservation laws
A 1.2 Schwarzschild geodesics for massive particles
A 1.3 Schwarzschild geodesics for massless particles
Problems
Advanced Topic A2: The Solar System Tests in Detail
A 2.1 Newtonian orbits in detail
A 2.2 Perihelion shift in General Relativity
A 2.3 Light deflection
A 2.4 Time delay
Problems
Note
Advanced Topic A3: Weak Gravitational Fields and Gravitational Waves
A 3.1 Nearly‐flat space–times
A 3.2 Gravitational waves
A 3.3 Sources of gravitational waves
Problems
Notes
Advanced Topic A4: Gravitational Wave Sources and Detection
A 4.1 Gravitational waves from compact binaries
A 4.2 The energy in gravitational waves
A 4.3 Binary inspiral
A 4.4 Detecting gravitational waves
Problems
Notes
Bibliography
Background
Undergraduate
Postgraduate/researcher
Answers to Selected Problems
Index
Wiley End User License Agreement
List of Tables
Chapter 10
Table 10.1 The orbital parameters for Mercury relevant for the evaluation o...
Table 10.2 Non‐relativistic contributions to the perihelion precession of M...
Table 10.3 Orbital parameters of PSR 1913+16, as determined from pulse arri...
Chapter 11
Table 11.1 Approximate masses, radii, and Schwarzschild radii
for selected...
Chapter 13
Table 13.1 The cosmological parameter values of our Universe, as according ...
List of Illustrations
Chapter 2
Figure 2.1 A frame
moving relative to another frame
, with velocity
alo...
Figure 2.2 Light pulses are sent out from the origin in the
direction in f...
Figure 2.3 A rod of length
is stationary in a frame with coordinates
, wi...
Figure 2.4 Diagram of Minkowski space–time, showing how the light cone divid...
Chapter 4
Figure 4.1 The Einstein Equivalence Principle implies that experiments such ...
Figure 4.2 A spaceship of rest length
is accelerating upwards, while a lig...
Figure 4.3 A laboratory in free‐fall towards the Earth, containing two parti...
Figure 4.4 The circumference
of a circle of radius
on a sphere of radius...
Chapter 5
Figure 5.1 In order to define covariant differentiation of a vector field
...
Figure 5.2 Parallel transport of a vector on a two‐sphere. A vector
pointi...
Figure 5.3 When a vector is parallel transported from point
to point
in ...
Chapter 6
Figure 6.1 Left:
stationary particles in a volume
, with average density
Figure 6.2 Forces exerted by adjacent volume elements on each other, when no...
Chapter 9
Figure 9.1 In order to define the derivative of the tangent vector at point
Figure 9.2 Geodesic deviation: in a curved space–time, nearby geodesics may ...
Figure 9.3 The orbit of a particle in a central potential, according to Newt...
Figure 9.4 Orbit of a particle in a central potential with
, according to G...
Figure 9.5 Orbit of a photon in a central potential, according to General Re...
Chapter 10
Figure 10.1 Einstein's photon has further to go. Radar signals between Earth...
Figure 10.2 Sketch of the orbit of the pulsar, showing the angles
(orbital...
Chapter 11
Figure 11.1 A schematic illustration of the behaviour of the four solutions ...
Figure 11.2 Particle motion in the Schwarzschild space–time, displayed in Kr...
Chapter 13
Figure 13.1 An all‐sky map of the CMB temperature from the final 2018 data r...
Chapter 14
Figure 14.1 Albert Einstein photographed in 1921 by Ferdinand Schmutzer. (© ...
Advanced Topic A2
Figure A 2.1 The Keplerian elliptical orbit predicted by Newtonian gravity i...
Advanced Topic A3
Figure A 3.1 The effect of a gravitational wave passing through a ring of fr...
Advanced Topic A4
Figure A 4.1 The geometry of a binary system, with observer located at
at ...
Figure A 4.2 LIGO waveforms of the first direct gravitational wave detection...
Figure A 4.3 Estimate of the strain amplitude at the Earth of the gravitatio...
Figure A 4.4 Diagram of a gravitational wave detector based on a Michelson i...
Figure A 4.5 Pulsar timing as a detection method for gravitational waves. A ...
Figure A 4.6 The LISA orbit. LISA will consist of three spacecraft in an equ...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Constants and Symbols
About the Companion Website
Begin Reading
Advanced Topic A1: Geodesics in the Schwarzschild Space–Time
Advanced Topic A2: The Solar System Tests in Detail
Advanced Topic A3: Weak Gravitational Fields and Gravitational Waves
Advanced Topic A4: Gravitational Wave Sources and Detection
Bibliography
Answers to Selected Problems
Index
Wiley End User License Agreement
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Introducing General Relativity
Mark Hindmarsh
University of SussexBrighton, UK andUniversity of HelsinkiHelsinki, Finland
Andrew Liddle
University of EdinburghEdinburgh, UK andPerimeter Institute for Theoretical PhysicsWaterloo, Canada andUniversidade de LisboaLisboa, Portugal
This edition first published 2022© 2022 John Wiley and Sons Ltd
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The right of Mark Hindmarsh and Andrew Liddle to be identified as the authors of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data applied for
ISBN: 9781118600719
Cover Design: WileyCover Images: © ESA/Hubble & NASA, S. Jha;Acknowledgment: L. Shatz
Preface
Find a physicist. Sit them down, treat them to a coffee, and ask for their opinion as to the most beautiful theory that the subject has to offer. The chances are that their answer will be General Relativity.
Einstein's General Theory of Relativity, to give it its full and proper name, is over a hundred years old. Yet it offers a defiantly modern viewpoint, defining principles of how the Universe ought to work and establishing a mathematical framework upon them. It gives a radical, even shocking, reconception of a fundamental force, gravity. And it has maintained an exquisite agreement with observational data, making predictions of such subtlety that one of its key implications, the existence of gravitational waves, took over a hundred years to be directly verified.
This book is based on a lecture course at the University of Sussex, given by each of us at various times. The course is taken by final‐year physics undergraduates, and has no special prerequisites, so we have attempted to limit the coverage of topics and to be as explicit as possible. For many undergraduates, a lecture course on General Relativity is the pinnacle of their theoretical education, and their main exposure to the modern methodology of physics as based on principles and symmetries.
Our aim is to make that experience as enjoyable as possible, while accepting that the pleasure comes not just from the astonishing physical implications, such as black holes, singularities, and gravitational waves, but from the elegance of the underlying mathematical structure. By the standards of textbooks on the topic, we have sought to create something that is genuinely introductory, yet which provides the mathematical tools to see the theory work in a quantitative way. We hope that something of the elegance of the theory emerges from the technical difficulty, along with an understanding of the physical predictions that have been so beautifully confirmed by decades of experiment and observation. Enjoy the challenge!
Mark Hindmarsh and Andrew LiddleHelsinki and Lisbon, October 2021
Constants and Symbols
Some Fundamental Constants and Astronomical Values
Newton's gravitational constant
Speed of light
Reduced Planck constant
Boltzmann constant
Solar mass
Solar radius
Parsec
pc
Electron volt
eV
Commonly‐Used Symbols
coordinates
defined on page
4
speed of light
5
velocity (usually the relative velocity between two frames)
6
Lorentz boost factor,
7
or
space–time distance (or interval) between two events
7, 19
velocity (of an object measured in a frame)
11
coordinates (as a 4‐vector)
18
Lorentz transformation
18
Minkowski metric (for Special Relativity)
19
metric of space–time
20, 50
Kronecker delta
21
proper time
24
4‐velocity
24
4‐momentum
25, 83
4‐acceleration (this notation is also used for a generic 4‐vector)
26
4‐derivative
27
D'Alembertian
27
Newton's gravitational constant
37
mass (usually of a gravitating body)
37
mass (usually of a test body near a larger mass
)
37
gravitational potential
39
infinitesimal interval in space–time
50
covariant differential
60
connection/Christoffel symbols
60
partial derivative
63
covariant derivative
63
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/hindmarsh/introducingGR
The website includes: Instructor’s Manual
Chapter 1Introducing General Relativity
It is now more than a hundred years since Albert Einsteinpresented the final form of the General Theory of Relativity to the Prussian Academy of Sciences, in November 1915. Since then, it has migrated from an extraordinary achievement at the frontiers of physics, reputedly understood by only a very few, to a standard advanced undergraduate course. General Relativity (as it is usually called, commonly shortened to simply GR) is essential for the understanding of the Universe as a whole, wherever gravity is strong, and also whenever precise time measurements are made. The Global Positioning System (GPS), now built in to billions of devices around the world, would not work without the General Relativistic prediction that clocks run more slowly on Earth than in the satellites defining the GPS reference frame.
Part of the fascination of General Relativity lies in the personality of Einstein, and the way he is often presented as a lone genius working for years in isolation, finally to reemerge with the fully formed and beautiful theory we know today. In reality, he was in constant communication with other scientists, and others were working towards a relativistic theory of gravitation. The first such theory was actually written down by Gunnar Nordström in 1913, who attempted a direct relativistic generalisation of the Newtonian gravitational potential. Einstein was the first to understand that the appropriate dynamical quantity is the space–time metric itself, but the geometric aspect of General Relativity was probably first appreciated by the mathematicians Marcel Grossmann and David Hilbert. Einstein worked with his friend Grossmann, and had crucial correspondence with Hilbert before coming up with the final and correct formulation. Einstein himself took several wrong turnings doing the years between 1907 and 1915 when he was working most intensively on the theory. The lone genius is a myth, but it is fair to consider that General Relativity is Einstein's own, and crowning, achievement.
The technical complexity of General Relativity comes from several sources. The fundamental objects of relativity are tensors, because the relativity principles (both special and general) are statements about the properties of physical laws under transformations between coordinates. Thus any General Relativity course must start with tensor calculus. General Relativity is a geometrical theory, treating space–time as a manifold, describing its dynamics in terms of geometrical quantities. Thus in approaching General Relativity the basic geometrical concepts developed by Bernhard Riemann and others from the mid 1800s — of connection, geodesic, parallel transport, and curvature — must be introduced. Tensor calculus and Riemannian geometry are not part of the standard mathematical equipment of a physics undergraduate. This was also true in Einstein's undergraduate career, although there were courses on offer. It has been speculated that had he gone to an advanced geometry course, he would have later saved himself several years’ work.
A final difficulty is the one of translating the mathematical concepts into physical observables. General Relativity rethinks the fundamentals of space and time, which take part in physical processes rather than being a framework on which things happen. So deciding what is observable, rather than simply an artefact of a particular choice of coordinates, is difficult. Indeed, Einstein changed his mind a couple of times as to whether gravitational waves were real or not, and it took about fifty years for a unanimous view to emerge.
Gravitational waves, an early prediction of General Relativity, are amongst the hottest topics in physics following their direct detection by the LIGO/Virgo collaboration, announced in 2016. Their real significance is not so much as a triumphant vindication of Einstein's theory; there was no serious doubt that gravitational waves existed following the careful measurements of the orbital decay of a binary pulsar system discovered by Russell Hulse and Joseph Taylor in the 1970s. Rather, the detection signals the beginning of a new branch of astronomy, which has the prospect of detecting violent astronomical events right back to the very earliest stages of the Big Bang. New detectors, similar to LIGO and Virgo, are being built in Japan and India, and a space‐based gravitational wave detector called LISA is planned for the early 2030s. General Relativity will continue to be at the forefront of scientific research in the 21st century, as it was throughout the 20th.
Chapter 2A Special Relativity Reminder
The Special Theory unites space and timeshorter lengths and longer timesseeing it with diagrams
Before launching into our account of General Relativity, we give a brief reminder of the main characteristics of its predecessor theory, the Special Theory of Relativity. This was introduced by Einstein in 1905, and is usually referred to by the shorthand Special Relativity. These theories have a rather different status to traditional physics topics, such as electromagnetism or atomic physics, which seek to understand phenomena of a particular type or within a certain domain. Instead, the relativity theories set down principles which apply to all physical laws and restrict the ways in which they can be put together. Whether those principles are actually true is something that needs to be tested against experiment and observation, but the assumption that they do hold has far‐reaching implications for how physical laws can be constructed. In particular, the role of symmetries of Nature is highlighted, which is a defining feature of how modern physics is constructed; as such the relativity theories often give students the first glimpse of how contemporary theoretical physics is done.
Both the theories focus on how physical phenomena are viewed in different coordinate systems, with the underlying principle that the outcome of physical processes should not depend on the choice of coordinates that we use to describe them. Special Relativity restricts us to so‐called inertial frames, where the term frame means a set of coordinates to be used for describing physical laws. As we will see, this restricts us to coordinate transformations which are linear in the coordinates, corresponding to coordinate systems moving relative to one another with constant velocity, and/or rotated with respect to one another. This turns out to be a suitable framework for considering all known physical laws except for those corresponding to gravity.
Einstein's remarkable insight, leading to the General Theory of Relativity, was that allowing arbitrary non‐linear coordinate transformations would allow gravity to be incorporated. Indeed, if we want to allow non‐linear transformations, we have to include gravity. Understanding the motivations for, and implications of, this extraordinary statement is the purpose of this book. But for now, we place the focus on Special Relativity, emphasising those features that will later generalise.
2.1 The need for Special Relativity
In Newtonian dynamics, the equations are invariant under the Galilean transformation which takes us from one set of coordinates to another according to the rule
where is the relative speed between the two coordinate systems, which have been aligned so that the velocity is entirely along the direction. [NB primes are not derivatives!] Each coordinate frame is idealised as extending throughout space and time, providing the scaffolding that lets us locate physical processes in space and time. We introduce an event as something which happens at a specific location in space and at a specific time, such as the collision of two particles.
Typically any observer will want to choose a coordinate system to describe events, and will be located somewhere within the coordinate system. Commonly, though not always, observers will decide to choose coordinate frames that move along with them as a natural way to describe the phenomena as they see them, and so it can be useful to sometimes think of a coordinate system as being associated to a particular observer who carries the coordinate system along with them. For instance, we might consider two different observers moving at a constant velocity with respect to one another, and ask how they would describe the same physical process from their differing points of view.
When we refer to invariance of a physical quantity, we mean that a physical quantity expressed in the new coordinates is identical to the same quantity expressed in the old ones. That means that observers in relative motion agree on its value.
In particular, acceleration is invariant in Newtonian dynamics; it depends on second time derivatives of the coordinates of, for example, a moving particle, and the second time derivatives of and of are equal. An everyday example is that an object dropped in a train moving at constant velocity appears, to an observer in the carriage, to follow exactly the same trajectory as it would were the train stationary.
The Galilean transformation is characterised by a single universal time coordinate that all observers agree upon. Combining relative velocities in each of the coordinate directions means that generally , , and , but always remains equal to . The idea of a universal time sits in good agreement with our everyday experience. However, our own direct perceptions of physical laws probe only a very restrictive set of circumstances. For example, we are unaware of quantum mechanics in our day‐to‐day life, because quantum laws such as Heisenberg's Uncertainty Principle are significant only on scales far smaller than we can personally witness. Hence, we cannot immediately conclude that invariance under the Galilean transformation should apply to all physical laws.
Indeed, it was already known in Einstein's time that Maxwell's equations, describing electromagnetic phenomena including the propagation of light waves, are not consistent with Galilean invariance. For example, they state that the speed of light is independent of the motion of a source, whereas the Galilean transformation would predict that light would emerge more rapidly from a torch if its holder were running towards you. In a famous thought experiment (i.e. an experiment carried out only in the mind, not in the laboratory), Einstein tried to envisage what would happen if one tried to catch up with a light wave by matching its velocity, knowing that Maxwell's equations would not permit a stationary wave.
One possible resolution of this would be if there were special frame of reference in which Maxwell's equations were valid, a frame that came to be known as the aether. However, since the Earth revolves around the Sun, it cannot always be stationary with respect to this aether. In the late 1880s, Albert Michelson and Edward Morley sought to detect the motion of the Earth relative to this aether, using an interferometer experiment. It should have had the sensitivity to easily see the effect, given the known properties of the Earth's orbit, yet no signal was found, putting the existence of the aether in doubt.
From the viewpoint of wanting a unified view of physical laws, it makes little sense that different types of physical laws should respect different invariance properties. After all, electromagnetic phenomena lead to dynamical motions. This incompatibility posed a stark problem for physics.
Einstein's 1905 paper resolved this seeming paradox decisively in favour of electromagnetism. Based on his thought experiments, he demanded that physical laws satisfied two postulates:
1. The laws of physicsare the same in all inertial frames.
2. The speed of light, denoted
,is the same in all inertial frames and independent of the motion of the source.
As remarked above, inertial frames are those which move with a constant velocity with respect to one another. The requirement that the laws of physics be the same in each is inherited from the Galilean transformation, which also requires it. Another way of expressing this first postulate is to say that there is no possible experiment an observer can carry out to measure their absolute velocity.
But the second postulate then requires that the coordinate transformation between frames must mix space and time, as we are about to see. It is inconsistent with the notion of a universal time coordinate, and requires that invariance under the Galilean transformation be abandoned. If Nature's laws are to be invariant under coordinate transformations, the invariance must be of another type.
2.2 The Lorentz transformation
Hendrik Lorentz, in 1904, had already discovered a transformation that left Maxwell's equations invariant, and it now bears his name. We will derive it under the assumption that the transformation is linear, like a Galilean transformation, and reduces to a Galilean transformation in the limit of relative velocities much less than that of light.
Consider a frame, which we call , moving relative to the original frame with velocity along the ‐axis, so that we can assume and .1 This is shown in Figure 2.1. Linearity lets us write
where , , , and are constants. Now, the origin of is moving relative to at velocity , so corresponds to , implying . So the second of the above equations becomes
By symmetry, the same equation must hold for transforming back from to , exchanging , so
Figure 2.1 A frame moving relative to another frame , with velocity along the ‐axis. Demanding that the transformation relating the coordinates to is linear and preserves the speed of light , uniquely fixes it to be the Lorentz boost, equation (2.7).
Now, we use the assumption of a constant speed of light. At the instant when the two coordinate systems agree, send out a pulse of light along the ‐axis. Then in the original frame, and in the new one. Substitute this in to equation (2.3) to get
and into equation (2.4) to get
Consistency of these two requires . This rules out the Galilean transformation, which would have required to be equal to one.
We write this as , where the famous ‐factor is . This measures the strength of relativistic effects. Again by symmetry, . Combining these gives which we can rearrange to get . We can finally write the Lorentz boost (in the ‐direction) as
Notice the strong symmetry if we think of as a coordinate. The Lorentz boost is the unique linear transformation relating time and space coordinates satisfying the postulates of Special Relativity. It also reduces to the Galilean transformation in the limit .
If we define the distance of a point in space and time from the origin as
this quantity, which is known as the space–time interval, is invariant under the Lorentz boost (see Problem 2.1). Hence, all inertial observers agree on the interval between two events, even though they will not generally agree on how much of that interval is in the space direction and how much in the time direction. Invariant quantities such as this will play an important role throughout our book. Indeed, a more modern and fundamental point of view would be to start with the principle that the interval is to be invariant, and to show that the most general linear transformation consistent with that property is the Lorentz transformation.
2.3 Time dilation
Figure 2.2 Light pulses are sent out from the origin in the direction in frame , bouncing off a mirror at a distance , and returning to the origin at time later. As soon as they are received another pulse is sent. In frame , the path is longer, and the arrival times back at the detector correspondingly later. If we think of the arrival times of the pulses as the ticking of a clock, the conclusion is that the moving clock runs slower, by a factor (see equation (2.9)).
A consequence of the Lorentz transformation is that moving clocks run more slowly. Consider a train of pulses of light sent out from the origin of a frame , reflected from a mirror a distance away, and returning a time later. As soon as a pulse returns another is sent.
Consider now how this looks in frame , moving parallel to the mirror with speed (see Figure 2.2). In the latter frame, the path taken by the light is longer, and due to the constancy of the speed of light in all frames, the time taken to return is longer.
In Frame , the pulse return time is . In Frame , it is , where . Combining these gives
Consider the train of light pulses as the ticking of a clock in the frame. As , the observer (who sees a moving clock) notices that its ticks are slower. The situation is symmetric, so whenever we look at a moving clock, we will see it running slower than our own.
A classic example of time dilation comes from muons created in the upper atmosphere through impact of highly energetic cosmic rays. Muons are unstable particles whose lifetime, as measured in the laboratory, is only seconds, which suggests that they could travel at most m before decaying. But in practice, muons created high in the atmosphere are detected at the Earth's surface. The reason is that they are moving close to the speed of light and hence, from our perspective, their evolution is heavily time‐dilated with the apparent lifetime much greater than the rest‐frame lifetime.
2.4 Lorentz–Fitzgerald contraction
A complementary phenomenon to time dilation is that moving objects appear shorter, known as the Lorentz–Fitzgerald contraction. To derive it, take a rod that is stationary in frame with ends at and , and choose a moving frame such that the spatial origins coincide at . In this frame, the rod appears to move with speed in the direction. What is the length of the rod in the moving frame?
To answer the question, we need to be a bit more precise about how we determine lengths. We can define the length of the object to be the difference in the coordinate between the ends, when measured at the same time. In the frame, where the problem was set up, this is trivially , at any time .
However, in the frame, the calculation is not so trivial, as we need to establish the positions of the ends of the rod at the same value of . It is convenient to take this time to be , as we set up the coordinate systems so that their origins coincide at this time. The Lorentz transformation (2.7) tells us that measurements taken at time must happen at times in the frame . Hence, the measurement the position of the end of the rod () at in the frame occurs at time in frame (see Figure 2.3).
Figure 2.3 A rod of length is stationary in a frame with coordinates , with its left‐hand end at the origin. A frame is moving at speed in the direction, with coordinates coinciding with the original frame at and . A measurement of the position of the end of the rod at time happens at time in the original frame. Knowing that the end of the rod is at , the position of the end of the rod in the moving frame at time , and hence its length , can be calculated from the Lorentz transform (2.7). The conclusion is that , i.e. a moving rod is shorter.
The Lorentz transformation (2.7) of the coordinates corresponding to the measurement tells us that it happened at coordinate
Hence, the length of the rod in the moving frame is
The rod is therefore shorter, when measured in the moving frame.
Returning to the example of the atmospheric muons of the previous section, in the muon rest‐frame the lifetime to decay is indeed seconds. Nevertheless, muons can still reach the ground from the upper atmosphere, which from their point of view is because the atmosphere is approaching them at relativistic speeds and is Lorentz–Fitzgerald contracted. Ultimately, the physical outcome, muons reaching the ground and being detected there, has to be the same from either our point of view or the muons. Hence, we see that time dilation and Lorentz–Fitzgerald contraction are really two different sides of the same coin; it couldn't make sense to have one without the other.
2.5 Addition of velocities
We now ask how velocities change between moving frames. Consider an object moving with velocity along the axis in frame . What is the velocity measured in frame , which is moving with velocity along the axis?
In Newtonian dynamics, where the Galilean transformation applies, velocities simply add. But we can already expect that this will not hold in Special Relativity, because of the postulate that the speed of light is constant in all frames regardless of their relative velocity.
In the frame, the component of velocity along the ‐axis is given by
where the distance is travelled in a time . Using the Lorentz boost given by equation (2.7), we can write (the factors immediately cancel out)
Since the component of velocity measured in frame is , this becomes
Going between frames in the opposite direction simply swaps for in the Lorentz transformation, so we can also write the equivalent equation
In the non‐relativistic limit and , we recover the Galilean law that applies in our everyday experience. Taking , we find that regardless of the value of , confirming successful implementation of the frame‐independence of the speed of light. With a little more work, we can show that as long as both and are less than , then so is . There is no way to generate a speed greater than that of light from speeds which are less, so the speed of light forms a barrier that cannot be crossed.
Some further useful properties of the Special Relativistic velocity composition law are explored in Problem 2.4.
2.6 Simultaneity, colocality, and causality
We begin with a few definitions. We recall that an event refers to something which takes place at a specific location in space at a particular time. The set of all possible events is the space–time. Events are said to be simultaneous if they happen at the same time, and to be colocal if they happen at the same location. Note that different observers do not agree on which events are simultaneous or colocal. Coincident events happen at the same time and place, and everyone agrees on that.
Consider two events. Unless they are simultaneous, there is always a Galilean transformation which makes them happen at the same location, while in contrast there is no such transformation that can make two events appear simultaneous unless they are so in all frames. In Special Relativity, the situation is more complicated.
Consider the following question. Suppose an observer
