For Undergraduates

Special Theory of Relativity

A complete, self-contained exposition covering kinematics, dynamics, four-vector formalism, relativistic optics and the electromagnetic field transformations.

Introduction

The Special Theory of Relativity (STR), published by Albert Einstein in 1905, revolutionised our understanding of space and time. It rests on two deceptively simple postulates and yields profound consequences: the relativity of simultaneity, length contraction, time dilation, the equivalence of mass and energy, and the unification of electric and magnetic fields.

This document presents a systematic, book-level treatment of the theory. Mathematical derivations are given in full, physical interpretations are emphasised, and interactive visualisations are provided where they aid intuition (Minkowski diagrams, velocity addition, Doppler shift, etc.).

Notation conventions
  • Units where $c=1$ are frequently used; $c$ is restored when needed for clarity.
  • Greek indices $\mu,\nu,\dots = 0,1,2,3$; Latin indices $i,j,\dots = 1,2,3$.
  • Metric signature $(+,-,-,-)$ (mostly-plus convention is also common; we adopt mostly-minus).
  • Four-vectors are written $A^\mu = (A^0,\mathbf{A})$.

Video Lectures — Continuum: The Special Theory of Relativity

Classic educational film (National Film Board of Canada, 1979) that visually develops the core ideas of special relativity: the constancy of the speed of light, Lorentz transformations, time dilation, length contraction, and the geometric structure of spacetime.

Full Film (both reels)

Continuum: The Special Theory of Relativity (Reel 1 + Reel 2 combined)

If the embed does not load, you can watch it directly on the Internet Archive:
https://archive.org/details/continuumthespecialtheoryofrelativity

1. Michelson–Morley Experiment and its Outcome

1.1 Historical context

In the late 19th century the luminiferous aether was postulated as the medium that carries light waves. The Earth was assumed to move through this aether, producing an “aether wind”. Albert A. Michelson (1881) and later Michelson & Edward Morley (1887) designed an interferometric experiment to detect this wind.

1.2 Principle of the experiment

A beam of light is split into two perpendicular paths of equal length $L$. After reflection the beams recombine and interfere. If the apparatus moves at velocity $v$ relative to the aether, the travel times along the parallel and perpendicular arms differ:

$$ t_\parallel = \frac{2L}{c}\frac{1}{1-\beta^2},\qquad t_\perp = \frac{2L}{c}\frac{1}{\sqrt{1-\beta^2}} $$
$\beta = v/c$

The expected time difference is

$$ \Delta t = t_\parallel - t_\perp \approx \frac{L}{c}\beta^2 $$

which produces a fringe shift of order $\delta = (L/\lambda)\beta^2$. For the 1887 apparatus $L\approx 11\,\mathrm{m}$ and the expected shift was about 0.4 fringe — easily detectable.

1.3 Outcome

The experiment yielded a null result: the fringe shift was consistent with zero to within experimental error (upper limit $\lesssim 0.01$ fringe). Subsequent repetitions with greater precision (Illingworth, Joos, Kennedy–Thorndike, modern laser interferometers) confirmed the null result to extraordinary accuracy.

Physical implication
The speed of light is independent of the motion of the Earth (and, by extension, of any inertial observer). There is no detectable aether wind. This experimental fact is one of the two pillars of special relativity.

2. Two Events in Two Different Inertial Frames

An event is a point in spacetime specified by four coordinates $(ct,x,y,z)$. Consider two inertial frames $S$ and $S'$ related by a boost of velocity $v$ along the common $x$-axis (standard configuration). The origins coincide at $t=t'=0$.

Let event $A$ have coordinates $(ct_A,x_A,y_A,z_A)$ in $S$ and $(ct'_A,x'_A,y'_A,z'_A)$ in $S'$. Likewise for event $B$. The coordinate differences

$$ \Delta t = t_B-t_A,\quad \Delta x = x_B-x_A,\quad\dots $$

transform according to the Lorentz transformation (derived in §5). The correspondence between the two descriptions is therefore completely determined once the relative velocity and the orientation of the axes are known.

Invariant interval
Although the individual $\Delta t$ and $\Delta x$ change from frame to frame, the combination $$ (\Delta s)^2 = c^2(\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2 $$ is the same in every inertial frame. This is the geometric foundation of special relativity.

3. Simultaneity and Order of Events

3.1 Relativity of simultaneity

Two events that occur at the same time ($\Delta t=0$) but at different locations ($\Delta x\neq 0$) in frame $S$ are not simultaneous in $S'$:

$$ \Delta t' = \gamma\Bigl(\Delta t - \frac{v}{c^2}\Delta x\Bigr) = -\gamma\frac{v}{c^2}\Delta x \neq 0 $$

Thus simultaneity is frame-dependent. This is the deepest conceptual novelty of special relativity.

3.2 Order of events and causality

The temporal order of two events can be reversed by a Lorentz transformation if and only if the interval between them is spacelike ($(\Delta s)^2<0$). For timelike or lightlike intervals the order is absolute and coincides with the causal order: a cause always precedes its effect in every inertial frame.

No causal paradoxes
Because information and physical influences cannot travel faster than light, the region of spacetime that can be influenced by a given event (its future light-cone) is the same for all observers. Hence the relativity of simultaneity never produces causal contradictions.

4. Postulates of the Special Theory of Relativity

Einstein’s 1905 paper rests on two postulates:

Postulate I — Principle of Relativity
The laws of physics take the same form in all inertial frames of reference. No experiment can detect absolute uniform motion.
Postulate II — Invariance of the speed of light
The speed of light in vacuum, $c$, is the same for all inertial observers, independent of the motion of the source or the observer.

From these two statements alone the entire kinematic structure of special relativity follows. The Lorentz transformations are the unique linear coordinate transformations that leave the speed of light invariant and reduce to the Galilean transformations when $v/c\to 0$.

5. Lorentz Transformations

5.1 Derivation (standard boost)

Consider a light signal emitted at the common origin at $t=t'=0$. Its world-line satisfies

$$ c^2t^2 - x^2 - y^2 - z^2 = 0 = c^2t'^2 - x'^2 - y'^2 - z'^2. $$

Assuming linearity and the standard configuration ($S'$ moves at velocity $v$ along the positive $x$-axis of $S$), the unique transformations that preserve the light-cone and reduce to the identity when $v=0$ are

$$ \begin{aligned} ct' &= \gamma\bigl(ct - \beta x\bigr),\\ x' &= \gamma\bigl(x - \beta ct\bigr),\\ y' &= y,\\ z' &= z, \end{aligned} \qquad \gamma = \frac{1}{\sqrt{1-\beta^2}},\quad \beta = \frac{v}{c}. $$

The inverse transformations are obtained by $v\to -v$ (or $\beta\to -\beta$).

5.2 Matrix form

In four-vector notation the boost is represented by the matrix

$$ \Lambda^\mu{}_\nu = \begin{pmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

5.3 Interactive Lorentz transformation visualiser

World-lines of the light-cone (yellow) and a few events (coloured dots) as seen in $S$ (solid) and $S'$ (dashed). Drag the slider to change the relative velocity.

6. Lorentz (Length) Contraction

Consider a rod at rest in $S'$ with proper length $L_0 = x'_2 - x'_1$ (measured at equal times $t'$). In frame $S$ the length is the difference of the simultaneous ($t=\mathrm{const}$) coordinates of the ends:

$$ L = x_2 - x_1 = \frac{L_0}{\gamma} = L_0\sqrt{1-\beta^2}. $$

The rod appears shorter when it is moving. Only the dimension parallel to the boost is contracted; transverse dimensions are unchanged.

Important caveat
Length contraction is not a dynamical compression of the rod; it is a consequence of the relativity of simultaneity. The two ends of the rod are located at different times in the rest frame when they are measured simultaneously in the laboratory.

7. Time Dilation

A clock at rest in $S'$ ticks with proper time interval $\Delta\tau = \Delta t'$. In the laboratory frame $S$ the same two ticks occur at the same location $x'=\mathrm{const}$, so

$$ \Delta t = \gamma\Delta\tau = \frac{\Delta\tau}{\sqrt{1-\beta^2}}. $$

Moving clocks run slow. This has been verified to high precision with atomic clocks on aircraft, muons in the atmosphere and in storage rings, and GPS satellite clocks.

8. The Twin Paradox

Twin A stays on Earth; twin B travels at high speed to a distant star and returns. Upon reunion B is younger than A. The apparent paradox (“each sees the other’s clock running slow”) is resolved by noting that the situations are not symmetric: B undergoes acceleration (or equivalently changes inertial frames) while A remains inertial throughout.

The proper time along a world-line is

$$ \tau = \int\sqrt{1-\beta(t)^2}\,dt. $$

The inertial path maximises the proper time between two events; any accelerated path yields a smaller $\tau$. Hence the travelling twin ages less.

Resolution
There is no paradox once the asymmetry of the world-lines is recognised. The result is a direct consequence of the geometry of Minkowski spacetime.

9. Proper Time and Proper Length

9.1 Proper time

The proper time $\tau$ between two events on a timelike world-line is the time measured by a clock that travels along that world-line:

$$ d\tau = \frac{ds}{c} = dt\sqrt{1-\beta^2}. $$

It is a Lorentz scalar and is therefore the same for all observers.

9.2 Proper length

The proper length of an object is its length measured in the inertial frame in which the object is at rest. It is the maximum length that any observer will attribute to the object.

10. Relativistic Transformation of Velocity and Frequency

10.1 Velocity transformation

If a particle has velocity $\mathbf{u}=(u_x,u_y,u_z)$ in $S$, its velocity $\mathbf{u}'$ in $S'$ is

$$ \begin{aligned} u'_x &= \frac{u_x-v}{1-u_xv/c^2},\\ u'_y &= \frac{u_y}{\gamma(1-u_xv/c^2)},\\ u'_z &= \frac{u_z}{\gamma(1-u_xv/c^2)}. \end{aligned} $$

10.2 Frequency transformation (preview of Doppler effect)

The four-wave-vector $k^\mu=(\omega/c,\mathbf{k})$ transforms as a four-vector. Consequently the observed frequency depends on both the relative velocity and the angle of observation (full treatment in §22–23).

11. Timelike, Spacelike and Lightlike Intervals; Causality

The invariant interval $(\Delta s)^2$ classifies the separation of any two events:

TypeConditionCausal relationFrame dependence of order
Timelike $(\Delta s)^2 > 0$ Can be connected by a subluminal signal Order absolute
Lightlike (null) $(\Delta s)^2 = 0$ Connected by a light signal Order absolute
Spacelike $(\Delta s)^2 < 0$ Cannot be causally connected Order can be reversed

The light-cone of an event divides spacetime into its absolute future, absolute past, and the elsewhere (spacelike region). Causality is preserved because physical influences propagate only inside or on the light-cone.

12. Addition of Velocities and Lorentz Transformations

The relativistic velocity addition law follows directly from the Lorentz transformation of differentials. For collinear velocities

$$ w = \frac{u+v}{1+uv/c^2}. $$

If $|u|

Interactive velocity addition

Comparison of classical ($u+v$) and relativistic velocity addition. The relativistic sum never exceeds $c$.

13. Variation of Mass with Velocity

In older literature one defines a velocity-dependent relativistic mass

$$ m(v) = \gamma m_0 = \frac{m_0}{\sqrt{1-v^2/c^2}}. $$

Modern practice prefers to keep mass as the Lorentz-invariant rest mass $m_0$ and to write the energy and momentum as

$$ E = \gamma m_0 c^2,\qquad \mathbf{p} = \gamma m_0\mathbf{v}. $$

Both descriptions are mathematically equivalent; the invariant-mass language is cleaner when four-vectors are used.

14. Massless Particles and Mass–Energy Equivalence

14.1 Rest energy

Even a particle at rest possesses energy

$$ E_0 = m_0 c^2. $$

This is the most famous relation of special relativity. Any change in the internal energy of a system (chemical, nuclear, thermal, \ldots) changes its rest mass by $\Delta m = \Delta E/c^2$.

14.2 Massless particles

For a particle with $m_0=0$ (photon, graviton, gluon in the perturbative regime) one has

$$ E = |\mathbf{p}|c,\qquad v = c. $$

The energy–momentum relation for a general particle reads

$$ E^2 = \mathbf{p}^2 c^2 + m_0^2 c^4. $$

15. Four-Vector Formalism

A four-vector $A^\mu$ transforms under a Lorentz transformation exactly as the coordinate differentials $dx^\mu$:

$$ A'^\mu = \Lambda^\mu{}_\nu A^\nu. $$

The Minkowski inner product

$$ A\cdot B = A^\mu B_\mu = A^0 B^0 - \mathbf{A}\cdot\mathbf{B} $$

is a Lorentz scalar. Important four-vectors include:

16. Minkowski Diagram

A Minkowski diagram is a spacetime diagram in which the vertical axis is $ct$ and the horizontal axis is $x$. Light-rays travel at $45^\circ$. A Lorentz boost tilts the $ct'$ and $x'$ axes toward the light-cone by the same angle $\theta$ with $\tanh\theta=\beta$.

Interactive Minkowski diagram. Yellow lines = light-cone; blue = $S$ axes; magenta = $S'$ axes; green dots = sample events. The unit hyperbolae (proper time / proper length) are also shown.

17. The Four-Velocity

The four-velocity of a particle is the derivative of its position four-vector with respect to proper time:

$$ u^\mu = \frac{dx^\mu}{d\tau} = \gamma(c,\mathbf{v}). $$

It is normalised:

$$ u^\mu u_\mu = c^2. $$

18. Four-Acceleration

The four-acceleration is

$$ a^\mu = \frac{du^\mu}{d\tau}. $$

Because $u\cdot u = c^2$ is constant, one has the orthogonality relation

$$ a^\mu u_\mu = 0. $$

In the instantaneous rest frame $a^\mu = (0,\mathbf{a}_{\mathrm{proper}})$.

19. Four-Force and Four-Momentum

The four-momentum is

$$ p^\mu = m_0 u^\mu = \bigl(E/c,\mathbf{p}\bigr). $$

The four-force is defined by

$$ f^\mu = \frac{dp^\mu}{d\tau} = m_0 a^\mu $$ (for constant rest mass). Its components are related to the ordinary three-force $\mathbf{F}$ by

$$ f^\mu = \gamma\bigl(\mathbf{F}\cdot\mathbf{v}/c,\mathbf{F}\bigr). $$

20. Conservation of Four-Momentum

In any collision or decay process that occurs in isolation, the total four-momentum is conserved:

$$ \sum_{\mathrm{in}} p^\mu_i = \sum_{\mathrm{out}} p^\mu_f. $$

This single four-vector equation encodes both energy conservation and three-momentum conservation, and is valid in every inertial frame.

21. Energy Conservation

The zeroth component of four-momentum conservation is the relativistic energy conservation law

$$ \sum_i \gamma_i m_i c^2 = \sum_f \gamma_f m_f c^2 $$ (including possible rest-mass changes). In nuclear and particle physics the conversion of rest energy into kinetic energy (or vice versa) is routinely observed.

22. Relativistic Doppler Effect

The relativistic Doppler formula for a source moving at velocity $\mathbf{v}$ relative to the observer, with angle $\theta$ between the velocity and the line of sight (in the observer’s frame), is

$$ \omega = \omega_0\frac{\sqrt{1-\beta^2}}{1-\beta\cos\theta}. $$

When the source approaches ($\theta=0$) one obtains a blueshift; when it recedes ($\theta=\pi$) a redshift. Even at $\theta=\pi/2$ a residual redshift remains (transverse Doppler effect).

23. Longitudinal & Transverse Doppler Effect and Aberration

23.1 Longitudinal Doppler effect

$$ \frac{\omega}{\omega_0} = \sqrt{\frac{1\pm\beta}{1\mp\beta}} \quad\text{(approach / recession)}. $$

23.2 Transverse Doppler effect

$$ \frac{\omega}{\omega_0} = \sqrt{1-\beta^2} = \frac{1}{\gamma} \quad(\theta=\pi/2). $$

This pure time-dilation redshift was first measured by Ives & Stilwell (1938).

23.3 Aberration of light

The angle of a light ray transforms as

$$ \cos\theta' = \frac{\cos\theta-\beta}{1-\beta\cos\theta}. $$

Consequently a source that emits isotropically in its rest frame appears strongly forward-beamed in the laboratory when $\gamma\gg 1$ (relativistic beaming / headlight effect).

Interactive Doppler calculator

Frequency ratio $\omega/\omega_0$ versus observation angle $\theta$ for the selected $\beta$. The transverse value ($\theta=90^\circ$) is marked.

24. Decay Processes

Consider the two-body decay of a particle of rest mass $M$ into two particles of rest masses $m_1$ and $m_2$. In the rest frame of $M$ energy-momentum conservation fixes the energies:

$$ E_1 = \frac{M^2 + m_1^2 - m_2^2}{2M}c^2,\qquad E_2 = \frac{M^2 + m_2^2 - m_1^2}{2M}c^2. $$

The decay products emerge with equal and opposite three-momenta. Boosting to the laboratory frame yields the characteristic anisotropic angular distribution and energy spectrum observed in particle physics.

25. Transformation of $\mathbf{E}$ and $\mathbf{B}$ Fields

The electromagnetic field-strength tensor $F^{\mu\nu}$ transforms as a rank-2 tensor. For a boost in the $x$-direction the field components mix as follows:

$$ \begin{aligned} E'_x &= E_x,\\ E'_y &= \gamma(E_y - v B_z),\\ E'_z &= \gamma(E_z + v B_y),\\[0.6em] B'_x &= B_x,\\ B'_y &= \gamma(B_y + \tfrac{v}{c^2} E_z),\\ B'_z &= \gamma(B_z - \tfrac{v}{c^2} E_y). \end{aligned} $$

A pure electric field in one frame acquires a magnetic component in another, and vice versa. This is the relativistic origin of magnetism.

26. Invariance of Maxwell’s Equations

Maxwell’s equations can be written in manifestly covariant form:

$$ \partial_\mu F^{\mu\nu} = \mu_0 j^\nu, \qquad \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0. $$

Because both $F^{\mu\nu}$ and $j^\nu$ transform as tensors, the equations retain exactly the same form in every inertial frame. This is the relativistic invariance of classical electrodynamics and was one of the principal motivations for Einstein’s 1905 paper.

Historical remark
Lorentz, Poincaré and others already knew that Maxwell’s equations are invariant under the transformations that now bear Lorentz’s name. Einstein’s decisive step was to elevate the invariance to a universal principle that applies to all of physics, not merely to electromagnetism.

Summary of Key Formulae

QuantityExpression
Lorentz factor$\gamma=(1-\beta^2)^{-1/2}$
Time dilation$\Delta t=\gamma\Delta\tau$
Length contraction$L=L_0/\gamma$
Velocity addition (collinear)$w=(u+v)/(1+uv/c^2)$
Energy–momentum$E^2=p^2c^2+m^2c^4$
Rest energy$E_0=mc^2$
Four-velocity$u^\mu=\gamma(c,\mathbf{v})$
Doppler shift$\omega/\omega_0=\sqrt{1-\beta^2}/(1-\beta\cos\theta)$
Invariant interval$ds^2=c^2dt^2-d\mathbf{x}^2$

References & Further Reading

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