15: Special Relativity
- Page ID
- 7029
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The phrase “special” relativity deals with the transformations between reference frames that are moving with respect to each other at constant relative velocities. Reference frames that are accelerating or rotating or moving in any manner other than at constant speed in a straight line are included as part of general relativity and are not considered in this chapter.
- 15.1: Introduction to Special Relativity
- This page justifies incorporating special relativity into classical mechanics, defining classical mechanics as pre-quantum. It emphasizes mechanical, kinematic, and dynamical problems, providing a limited view of relativity without exploring its electromagnetic aspects. Special relativity involves transformations between constant velocity reference frames, while accelerating or rotating frames, relevant to general relativity, are not discussed.
- 15.2: Preparation
- Detailed discussion of the exact definitions of the units of time, distance and speed is part of the subject of metrology. That is an important and interesting subject, but it is only marginally relevant to the topic of relativity, and consequently, having quoted the exact value of the speed of light, we leave further discussion of metrology here.
- 15.3: Preparation
- This page introduces the speed ratio \( \beta \) in special relativity, explaining its limits and relation to the speed of light \( c\). It covers several functions associated with \( \beta\) such as \( \gamma\), \( k\), \( z\), \( \phi\), and \( \theta\) and illustrates their mathematical interconnections.
- 15.4: Speed is Relative - The Fundamental Postulate of Special Relativity
- This page explores the challenge of measuring the speed of a uniformly-moving reference frame through internal mechanical experiments, highlighting the limitations of methods like pendulum swings and ball throws. Observations from inside a moving train emphasize the ambiguity of relative motion. It introduces a theoretical speedometer based on electrostatic and magnetic concepts, inviting readers to consider its plausibility.
- 15.5: The Lorentz Transformations
- It is impossible to determine the speed of motion of a uniformly-moving reference frame by any means whatever, whether by a mechanical or electrical or indeed any experiment performed entirely or partially within that frame, or even by reference to another frame
- 15.6: But This Defies Common Sense
- This page explores the conflict between common sense and the principles of relativity, focusing on the invariance of light speed and Lorentz transformations. It critiques oversimplified comparisons of everyday experiences with scientific principles, highlighting the intricacies of distance and time.
- 15.7: The Lorentz Transformation as a Rotation
- This page covers the Lorentz transformation, presenting two formulations: one with imaginary components and the other with real components. It introduces 4-vectors and emphasizes the constant nature of the spacetime interval across reference frames. The relationship with hyperbolic functions and the determinant's unity is discussed, although it notes that the matrices are not orthogonal.
- 15.8: Timelike and Spacelike 4-Vectors
- A “light-year” is a unit of distance used when describing astronomical distances to the layperson, and it is also useful in describing some aspects of relativity theory. It is the distance travelled by light in a year.
- 15.9: The FitzGerald-Lorentz Contraction
- This page explains length contraction, or the FitzGerald-Lorentz contraction, within relativity. It details how a moving rod appears shorter to a stationary observer compared to its proper length, which is measured when the rod is at rest. The importance of simultaneous measurements across different reference frames is highlighted, emphasizing that the perceived length of an object depends on the observer's frame of reference rather than being a subjective interpretation.
- 15.10: Time Dilation
- The interval s between two events is clearly independent of the orientation any reference frames, and is the same when referred to two reference frames that may be inclined to each other. But the components of the vector joining two events, or their projections on to the time axis or a space axis are not at all expected to be equal.
- 15.11: The Twins Paradox
- During the late 1950s and early 1960s there was great controversy over a problem known as the “Twins Paradox”. The controversy was not confined to within scientific circles, but was argued, by scientists and others, in the newspapers, magazines and many serious journals. . The paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more.
- 15.12: A, B and C
- A , B and C were three characters in the Canadian humorist Stephen Leacock’s essay on The Human Element in Mathematics. “ A , B and C are employed to dig a ditch. A can dig as much in one hour as B can dig in two...”
- 15.13: Simultaneity
- If the time interval referred to one reference frame can be different when referred to another reference frame (and since time interval is merely one component of a four-vector, the magnitude of the component surely depends on the orientation in four space of the four axes) this raises the possibility that there might be a time interval of zero relative to one frame (i.e. two events are simultaneous) but are not simultaneous relative to another.
- 15.14: Order of Events, Causality and the Transmission of Information
- Maybe it is even possible that if one event precedes another in one reference frame, in another reference frame the other precedes the one. In other words, the order of occurrence of events may be different in two frames. This indeed can be the case, and Minkowski diagrams can help us to see why and in what circumstances.
- 15.15: Derivatives
- We’ll pause here and establish a few derivatives just for reference and in case we need them later.
- 15.16: Addition of Velocities
- This page covers relative velocity in special relativity, emphasizing that speeds are not simply additive due to Lorentz transformations. It details formulas for transforming velocities in various frames, illustrated through examples involving an ocean liner and a moving dachshund. The page also discusses length contraction and provides exercises for combining gamma factors and calculating apparent lengths and angles between velocities under relativistic conditions.
- 15.17: Aberration of Light
- This page explains the Apex of the Earth’s Way, detailing how Earth's movement affects the light's velocity components from a star observed at a specific angle. It covers the transformation equations ensuring the speed of light remains constant in different frames. Additionally, it presents the relationship between the apparent angle of the star and its original angle, highlighting how this relationship is influenced by the Earth's velocity direction on a given date.
- 15.18: Doppler Effect
- It is well known that the formula for the Doppler effect in sound is different according to whether it is the source or the observer that is in motion. So I shall try to explain why, physically, there is a difference. Then, when you have thoroughly understood that observer in motion is an entirely different situation from source in motion, and the formulas must be different, we shall look at the Doppler effect in light, and we’ll return to square one when we find that the formulas for source in
- 15.19: The Transverse and Oblique Doppler Effects
- This page explores the Doppler effect at relativistic speeds, focusing on two main phenomena: the transverse Doppler effect, where perpendicular motion causes redshift due to time dilation, and the oblique Doppler effect, which results in redshift from a source moving at an angle. It includes equations that demonstrate how frequency, speed, and motion angle interrelate for these effects.
- 15.20: Acceleration
- This page explains how a particle's acceleration changes between two reference frames, \( \Sigma\) and \( \Sigma'\), where \( \Sigma'\) moves at speed \( v\) relative to \( \Sigma\). It derives the \( x\)- and \( y\)-components of acceleration in \( \Sigma\) from those in \( \Sigma'\) using Lorentz transformations, highlighting the impact of relative motion and time dilation on these components. This illustrates the transformation of acceleration across different inertial frames.
- 15.21: Mass
- This page explores mass in relativity, distinguishing between rest mass and relativistic mass, and explaining how both are invariant while measurements change with relative motion. It discusses experimental methods for determining mass through inertia and gravitational force, emphasizing that relativistic mass increases with speed.
- 15.22: Momentum
- This page explains linear momentum as the product of mass and velocity, highlighting that the mass may not be the rest mass. It includes an example that calculates the momentum of a proton traveling at 99% of the speed of light, demonstrating the application of the concept.
- 15.23: Some Mathematical Results
- This page introduces essential mathematical results that simplify future physics discussions, focusing on the Lorentz factor \(\gamma\) and its differentiation related to velocity and the speed of light. It also briefly explores scalar products and vector magnitudes. These foundational concepts are necessary for easing the understanding and calculations of later physics topics.
- 15.24: Kinetic Energy
- This page covers the interplay between force, velocity, and kinetic energy within special relativity, detailing how work is done on a particle through the equation \(\dot{T}=\mathbf{F}\cdot\mathbf{u}\). It clarifies the notation for velocity and kinetic energy, particularly distinguishing between \(\mathbf{u}\) and \(\mathbf{v}\), and introduces the Lorentz factor \(\gamma\).
- 15.25: Addition of Kinetic Energies
- This page examines the kinetic energy of two particles at both nonrelativistic and relativistic speeds, noting that kinetic energy depends on the reference frame, being minimal in the center of mass frame for nonrelativistic speeds. It highlights the significant changes in kinetic energy calculations at relativistic velocities, especially near light speed, and presents formulas for combining various speed measures, stressing the complexities involved in relativistic scenarios.
- 15.26: Energy and Mass
- This page explores the relationship between kinetic energy and mass in classical and relativistic physics, detailing the nonrelativistic and relativistic formulas. It emphasizes rest-mass energy (\( m_{0}c^{2} \)) and presents total energy as a sum of kinetic and rest-mass energy. Additionally, it discusses the implications of \( E=mc^{2} \) in nuclear reactions, clarifying misconceptions about atomic bombs and explaining how energy possesses mass due to binding energies in nuclei.
- 15.27: Energy and Momentum
- This page explores the interconnection between energy, rest mass, and momentum in moving particles, introducing the equation \( E^{2}=(m_{0}c^{2})^{2}+(pc)^{2} \). It illustrates how energy relates to momentum and mass, noting that massless particles such as photons move at light speed.
- 15.28: Units
- This page explains the electron volt (eV) as a key energy unit in particle physics for calculating energy changes in charged particles. It details the relationship among energy, momentum, and rest mass, and discusses relevant equations and conventions. The text warns about ambiguities regarding energy references (kinetic vs. total) and highlights the necessity for clear communication in the discipline.
- 15.29: Force
- Force is defined as rate of change of momentum, and we wish to find the transformation between forces referred to frames in uniform relative motion such that this relation holds on all such frames.
- 15.30: The Speed of Light
- This page discusses the speed of light, defined as 2.997 924 58 x 10^8 m/s, which is constant for all observers. It distinguishes between speed (scalar) and velocity (vector), and explains that the exact value of light speed is linked to the historical definition of the meter, now tied to the SI second from caesium frequency. The constancy of the speed of light is fundamental to special relativity, a critical subject in this chapter, leading to further examination in later sections.
- 15.31: Electromagnetism
- This page highlights the limitations of concentrating only on mechanics without considering special relativity and electromagnetism, which are interlinked. It underscores the significance of accurately defining physical quantities and their transformations to maintain the consistency of physical laws.
Thumbnail: Light cone. (Public Domain; Sakurambo).