Special relativity — postulates and Lorentz transformations

Intuition [Beginner]

Newtonian mechanics assumes that velocities add: if you walk at 5 km/h on a train moving at 100 km/h, you move at 105 km/h relative to the ground. This rule is called the Galilean transformation and it matches everyday experience. But it is wrong.

In 1905 Einstein proposed two postulates. First: the laws of physics are the same in every inertial frame (a frame that moves at constant velocity with no acceleration). Second: the speed of light in vacuum, $c$, is the same in every inertial frame, regardless of how the light source moves.

The first postulate extends the Galilean principle of relativity to all of physics. The second postulate is the radical one. It says that if you chase a light beam at $0.9c$, the beam still recedes from you at $c$ — not $0.1c$. Velocities do not add the way Newton assumed.

These two postulates force three consequences that rewrite your understanding of space and time.

Time dilation. A moving clock runs slow. If your friend flies past you at high speed, you see their watch ticking more slowly than yours. The factor by which it slows is the Lorentz gamma factor:

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

At everyday speeds ($v$ much less than $c$), $\gamma$ is almost exactly 1 and you recover Newtonian physics. As $v$ approaches $c$, $\gamma$ grows without bound.

Length contraction. A moving object is shortened along its direction of motion. A 10-metre spaceship flying past you at high speed measures less than 10 metres long in your frame.

Relativity of simultaneity. Two events that happen at the same time in one frame may happen at different times in another. There is no universal "now" — simultaneity is frame-dependent.

The Lorentz transformation replaces the Galilean transformation. It is the coordinate change between inertial frames that keeps the speed of light the same in every frame. All three consequences above follow from it.

Worked example [Beginner]

A spaceship travels at $v=0.8c$ relative to Earth. The trip, as measured by the ship's own clock, takes 10 years. How long does the trip take according to Earth clocks?

Step 1. Compute the gamma factor. With $v=0.8c$, the ratio $v^2/c^2=0.64$, so:

$$\gamma =\frac{1}{\sqrt{1-0.64}}=\frac{1}{\sqrt{0.36}}=\frac{1}{0.6}=\frac{5}{3}\approx \mathrm{1.667.}$$

Step 2. Apply time dilation. The ship clock measures the proper time — the time interval in the frame where the departure and arrival happen at the same place. Earth sees this interval dilated by $\gamma$:

$$\Delta t_{\text{Earth}}=\gamma \Delta t_{\text{ship}}=\frac{5}{3}\times 10=\frac{50}{3}\approx 16.7\text{ years}.$$

The Earth observer says the trip took about 16.7 years. The astronaut says 10 years. Both are correct — time intervals depend on the frame.

Step 3. Check length contraction (as a cross-check). In the Earth frame, the distance to the destination is $d=v\times \Delta t_{\text{Earth}}=0.8c\times 16.7\text{ yr}\approx 13.3$ light-years. In the ship frame, this distance is contracted by $\gamma$: $d_{\text{ship}}=d/\gamma =13.3/(5/3)=8.0$ light-years. At speed $0.8c$, the ship covers 8 light-years in $8/0.8=10$ years. Consistent.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $S$ be an inertial frame with coordinates $(t,x,y,z)$ and let $S^{\prime }$ be a second inertial frame moving with velocity $v$ along the $x$-axis of $S$, with coordinates $(t^{\prime },x^{\prime },y^{\prime },z^{\prime })$. Assume the origins coincide at $t=t^{\prime }=0$.

The Lorentz transformation between $S$ and $S^{\prime }$ is

$$\boxed{t^{\prime }=\gamma \left(t-\frac{vx}{c^2}\right),x^{\prime }=\gamma (x-vt),y^{\prime }=y,z^{\prime }=z,}$$

where $\gamma =(1-v^2/c^2{)}^{-1/2}$. The inverse transformation (solving for unprimed in terms of primed) is obtained by replacing $v$ with $-v$:

$$t=\gamma \left(t^{\prime }+\frac{v{x}^{\prime }}{c^2}\right),x=\gamma (x^{\prime }+v{t}^{\prime }),y=y^{\prime },z=z^{\prime }.$$

Define the four-vector $x^{\mu }=(ct,x,y,z)$ with Greek index $\mu =0,1,2,3$. The Lorentz transformation becomes a linear map $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$ where

$${\Lambda }^{\mu }{}_{\nu }=\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\beta :=\frac{v}{c}.$$

The Minkowski metric (with the $+---$ sign convention) is ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$. The invariant interval between two events is

$$(\Delta s{)}^2=c^2(\Delta t{)}^2-(\Delta x{)}^2-(\Delta y{)}^2-(\Delta z{)}^2={\eta }_{\mu \nu }\Delta x^{\mu }\Delta x^{\nu }.$$

This quantity has the same value in every inertial frame: ${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }={\eta }_{\alpha \beta }$, which is the defining property of a Lorentz transformation.

The interval classifies the causal relationship between events. If $(\Delta s{)}^2>0$ the separation is timelike — one event can influence the other. If $(\Delta s{)}^2=0$ it is lightlike (null) — the events are connected by a light ray. If $(\Delta s{)}^2<0$ it is spacelike — no signal at or below $c$ can connect them. The set of all null intervals from a given event forms the light cone, the boundary between causally accessible and causally forbidden regions.

The proper time along a worldline is the time measured by a clock that follows that worldline:

$$d{\tau }^2=\frac{d{s}^2}{c^2}=d{t}^2\left(1-\frac{v^2}{c^2}\right),\text{so}d\tau =\frac{dt}{\gamma }.$$

Proper time is maximised along inertial (straight) worldlines — a clock at rest between two events records the longest elapsed time.

Velocity addition

If an object has velocity $u$ in frame $S$, and $S^{\prime }$ moves at velocity $v$ relative to $S$ (both along the $x$-axis), then the velocity of the object in $S^{\prime }$ is

$$u^{\prime }=\frac{u-v}{1-uv/c^2}.$$

This reduces to the Galilean $u^{\prime }=u-v$ when $uv\ll c^2$. For $u=c$, the formula gives $u^{\prime }=c$ regardless of $v$ — the speed of light is the same in every frame, as the second postulate demands.

Relativistic Doppler effect

For a source moving directly toward (or away from) the observer at speed $v$, the observed frequency $f_{\mathrm{obs}}$ relates to the emitted frequency $f_0$ by

$$f_{\mathrm{obs}}=f_0\sqrt{\frac{1+\beta }{1-\beta }}(\text{approaching}),f_{\mathrm{obs}}=f_0\sqrt{\frac{1-\beta }{1+\beta }}(\text{receding}).$$

The transverse Doppler effect ($f_{\mathrm{obs}}=f_0/\gamma$ for motion perpendicular to the line of sight) is a purely relativistic effect — a direct consequence of time dilation.

Counterexamples to common slips

• The Lorentz transformation applies to inertial frames only. Accelerated frames require differential-geometric methods (Rindler coordinates for uniform acceleration; general relativity for gravitation).
• "Moving clocks run slow" is frame-dependent language. Each inertial observer finds the other's clock running slow. There is no contradiction because the two observers disagree about simultaneity.
• The Lorentz contraction does not mean an object is physically squeezed. In the object's own rest frame it has its rest length. The contraction is a property of the coordinate description in a different frame.
• The invariant interval $(\Delta s{)}^2$ is not a norm: it can be negative. The Minkowski metric is indefinite, which is what makes the causal classification work.

Key theorem with proof [Intermediate+]

Theorem (Lorentz transformations preserve the Minkowski interval). Let ${\Lambda }^{\mu }{}_{\nu }$ be a Lorentz boost along the $x$-axis with velocity $v$. Then for any two events with separation $\Delta x^{\mu }$, ${\eta }_{\mu \nu }\Delta x^{\prime \mu }\Delta x^{\prime \nu }={\eta }_{\alpha \beta }\Delta x^{\alpha }\Delta x^{\beta }$.

Proof. The transformed separation is $\Delta x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }\Delta x^{\nu }$. Compute the transformed interval:

$${\eta }_{\mu \nu }\Delta x^{\prime \mu }\Delta x^{\prime \nu }={\eta }_{\mu \nu }{\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }\Delta x^{\alpha }\Delta x^{\beta }.$$

It suffices to show ${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }={\eta }_{\alpha \beta }$ for each pair $(\alpha ,\beta )$. Write out the non-zero off-diagonal components. The $\Lambda$ matrix has ${\Lambda }^0{}_0=\gamma$, ${\Lambda }^0{}_1=-\gamma \beta$, ${\Lambda }^1{}_0=-\gamma \beta$, ${\Lambda }^1{}_1=\gamma$, with ${\Lambda }^2{}_2={\Lambda }^3{}_3=1$ and all other entries zero.

For $(\alpha ,\beta )=(0,0)$:

$${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_0{\Lambda }^{\nu }{}_0=({\Lambda }^0{}_0{)}^2-({\Lambda }^1{}_0{)}^2={\gamma }^2-{\gamma }^2{\beta }^2={\gamma }^2(1-{\beta }^2)=1={\eta }_{00}.$$

For $(\alpha ,\beta )=(0,1)$:

$${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_0{\Lambda }^{\nu }{}_1={\Lambda }^0{}_0{\Lambda }^0{}_1-{\Lambda }^1{}_0{\Lambda }^1{}_1=\gamma (-\gamma \beta )-(-\gamma \beta )\gamma =0={\eta }_{01}.$$

For $(\alpha ,\beta )=(1,1)$:

$${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_1{\Lambda }^{\nu }{}_1=({\Lambda }^0{}_1{)}^2-({\Lambda }^1{}_1{)}^2={\gamma }^2{\beta }^2-{\gamma }^2=-{\gamma }^2(1-{\beta }^2)=-1={\eta }_{11}.$$

The spatial components $(2,2)$ and $(3,3)$ give $-1$ by direct substitution. Mixed components with different indices vanish. Hence ${\eta }_{\mu \nu }{\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }={\eta }_{\alpha \beta }$ for all $\alpha ,\beta$. $\square$

Corollary. The Lorentz transformation preserves the speed of light. If $\Delta x^{\mu }$ is null (${\eta }_{\mu \nu }\Delta x^{\mu }\Delta x^{\nu }=0$), then ${\eta }_{\mu \nu }\Delta x^{\prime \mu }\Delta x^{\prime \nu }=0$ as well. A light ray in one frame is a light ray in every frame.

Worked example at intermediate level: relativistic velocity addition

A spaceship moves at $v=0.6c$ relative to Earth. It fires a probe forward at $u=0.8c$ relative to the ship. What is the probe's speed relative to Earth?

The naive (Galilean) answer is $0.6c+0.8c=1.4c$. The relativistic velocity-addition formula gives:

$$u^{\prime }=\frac{u+v}{1+uv/c^2}=\frac{0.8c+0.6c}{1+(0.8)(0.6)}=\frac{1.4c}{1.48}\approx 0.946c.$$

The result is less than $c$. No composition of subluminal velocities reaches or exceeds $c$.

Exercises [Intermediate+]

The Lorentz group and its Lie algebra [Master]

The full Lorentz group $O(1,3)$ consists of all $4\times 4$ matrices $\Lambda$ with ${\Lambda }^{\mathrm{T}}\eta \Lambda =\eta$. It has four connected components, distinguished by the sign of $\det \Lambda$ (orientation) and the sign of ${\Lambda }^0{}_0$ (time orientation). Physical transformations — those continuously connected to the identity — lie in the proper orthochronous Lorentz group ${\mathrm{SO}}^{+}(1,3)$, defined by $\det \Lambda =+1$ and ${\Lambda }^0{}_0\ge 1$.

Every element of ${\mathrm{SO}}^{+}(1,3)$ can be written as a product of a rotation and a boost. The six generators decompose into three rotations $J_i$ and three boosts $K_i$:

$$J_1=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right),K_1=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

with $J_2,J_3$ and $K_2,K_3$ obtained by cyclic permutation. The Lie algebra $\mathfrak{so}(1,3)$ is spanned by $\{J_i,K_i\}$ with commutation relations

$$[J_i,J_j]={ϵ}_{ijk}{J}_k,[J_i,K_j]={ϵ}_{ijk}{K}_k,[K_i,K_j]=-{ϵ}_{ijk}{J}_k.$$

The minus sign in $[K_i,K_j]$ is the signature of the indefinite metric: boosts do not form a compact subgroup. A finite boost is the exponential $B_i(\varphi )=\exp (-\varphi K_i)$, reproducing the rapidity parametrisation of the previous section.

A general boost with velocity vector $\mathbf{v}$ has rapidity vector $\mathbf{\varphi }=\varphi \hat{\mathbf{v}}$ where $\varphi ={\tanh }^{-1}(v/c)$, and the boost matrix is $\Lambda =\exp (-\mathbf{\varphi }\cdot \mathbf{K})$. The composition of two non-collinear boosts is not a pure boost but a boost followed by a rotation — this is Thomas precession, responsible for the spin-orbit coupling correction in atomic physics.

Lorentz invariance of Maxwell's equations

Maxwell's equations in vacuum are

$${\partial }_{\mu }{F}^{\mu \nu }=0,{\partial }_{[\alpha }{F}_{\beta \gamma ]}=0,$$

where $F^{\mu \nu }$ is the electromagnetic field tensor. Under a Lorentz transformation $\Lambda$, the field tensor transforms as $F^{\prime \mu \nu }={\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }{F}^{\alpha \beta }$, and the derivatives transform as ${\partial }_{\mu }^{\prime }={\Lambda }_{\mu }{}^{\nu }{\partial }_{\nu }$. The contraction ${\partial }_{\mu }{F}^{\mu \nu }$ is then a Lorentz scalar in the free index $\nu$: ${\partial }_{\mu }^{\prime }{F}^{\prime \mu \nu }={\Lambda }^{\nu }{}_{\beta }{\partial }_{\alpha }{F}^{\alpha \beta }$. If ${\partial }_{\alpha }{F}^{\alpha \beta }=0$, then ${\partial }_{\mu }^{\prime }{F}^{\prime \mu \nu }=0$.

This is not a coincidence: Maxwell's equations are Lorentz-covariant because the electromagnetic action

$$S=-\frac{1}{4{\mu }_0}\int F_{\mu \nu }{F}^{\mu \nu }{d}^{4}x$$

is a Lorentz scalar. The connection runs deeper than covariance of the equations of motion: the requirement that the action be invariant under the Poincare group (Lorentz transformations plus spacetime translations) constrains the possible interaction terms and determines the minimal-coupling structure of electrodynamics. This is the starting point of unit 10.06.01 pending.

Connections [Master]

To general relativity 13.01.01 pending. The Lorentz group is the local symmetry group of spacetime in general relativity. At each point in a curved spacetime, the tangent space is Minkowski space, and the local inertial frames are related by Lorentz transformations. The tetrad (vierbein) formalism of GR makes this explicit: the metric is written $g_{\mu \nu }=e_{\mu }{}^a{e}_{\nu }{}^b{\eta }_{ab}$, where $e_{\mu }{}^a$ is the tetrad and ${\eta }_{ab}$ is the Minkowski metric in the local Lorentz frame. The equivalence principle says that at each point you can choose a freely-falling frame in which the laws of physics reduce to those of special relativity — the Lorentz transformations derived here become the local symmetry.

To quantum field theory. The representation theory of the Poincare group (Lorentz group plus translations) classifies all relativistic quantum fields. Wigner's classification (1939) associates each irreducible unitary representation with a particle species, labelled by mass and spin. The Casimir operators of the Poincare algebra are $P^{\mu }{P}_{\mu }$ (mass squared) and the Pauli-Lubanski vector $W^{\mu }{W}_{\mu }$ (spin squared). The entire particle content of the Standard Model is an instance of this classification.

To the philosophy of space and time 20.04.01 pending. The relativity of simultaneity undermines the presentist view that "only the present exists." In its place, the block-universe interpretation holds that all events — past, present, and future — coexist in a four-dimensional spacetime manifold, and what we call "now" is a frame-dependent slicing of this manifold. Whether this is a metaphysical commitment or merely a coordinate artifact is debated in the philosophy of physics.

To covariant electromagnetism 10.06.01 pending. The next unit rewrites Maxwell's equations in manifestly Lorentz-covariant form using the field-strength tensor $F^{\mu \nu }$ and the four-potential $A^{\mu }$. The Lorentz force law becomes $d{p}^{\mu }/d\tau =q{F}^{\mu \nu }{u}_{\nu }$. The electric and magnetic fields are unified as components of $F^{\mu \nu }$, and the distinction between them is frame-dependent — a purely electric field in one frame has magnetic components in another.

Historical and philosophical context [Master]

The Lorentz transformation predates Einstein. Lorentz (1904) and FitzGerald (1889) introduced the transformation and the length-contraction hypothesis as an ad hoc explanation for the null result of the Michelson-Morley experiment (1887). They viewed the contraction as a dynamical effect of the ether on moving bodies. Poincare (1905) showed that the transformations form a group and noted the connection to the principle of relativity.

Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" derived the same transformations from the two postulates alone — no ether, no dynamical contraction hypothesis. The conceptual shift was decisive: Lorentz asked "what happens to objects moving through the ether?" while Einstein asked "what does the principle of relativity require of space and time?" The answer is that simultaneity is conventional, lengths and times are frame-dependent, and $c$ is a conversion factor between space and time.

Minkowski's 1908 lecture "Space and Time" recast Einstein's kinematics as the geometry of a four-dimensional manifold with metric ${\eta }_{\mu \nu }$. This geometrical reformulation — not merely a calculational trick but a change in the ontology of physics — is the foundation of both special and general relativity.

Bibliography [Master]

• Einstein, A. "Zur Elektrodynamik bewegter Korper." Annalen der Physik 17, 891–921 (1905). The founding paper.
• Minkowski, H. "Raum und Zeit." Physikalische Zeitschrift 10, 104–111 (1909). The geometrical reformulation.
• Michelson, A. A. & Morley, E. W. "On the Relative Motion of the Earth and the Luminiferous Ether." American Journal of Science 34, 333–345 (1887). The null result.
• Lorentz, H. A. "Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than That of Light." Proceedings of the Royal Netherlands Academy of Arts and Sciences 6, 809–831 (1904).
• Wigner, E. P. "On Unitary Representations of the Inhomogeneous Lorentz Group." Annals of Mathematics 40, 149–204 (1939). The classification of relativistic particle states.
• Jackson, J. D. Classical Electrodynamics, 3rd ed. Wiley (1999). Ch. 11.
• Landau, L. D. & Lifshitz, E. M. The Classical Theory of Fields, 4th ed. (Course of Theoretical Physics Vol. 2). Butterworth-Heinemann (1975). Ch. 1.
• Griffiths, D. J. Introduction to Electrodynamics, 4th ed. Cambridge University Press (2017). Ch. 12.
• Susskind, L. & Friedman, A. Special Relativity and Classical Field Theory. Basic Books (2017). Lectures 7–8.
• Tong, D. "Lectures on Electromagnetism." §5. University of Cambridge (2015).