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The official notes give only “Part 1: as in textbooks” (PDF p. 13, handwritten p. 8). The requested derivation is supplied here.

Put a light source at the rod midpoint in its rest frame $S$ and emit two pulses simultaneously toward its ends, each a distance $\ell$ away. In $S$, the reflection events are $(t,x)=(\ell /c,\pm \ell )$. Invariance of light speed, the Einstein's principle of relativity and linearity give a Lorentz transformation $x^{\prime }=A(x-vt)$, $t^{\prime }=A(t-vx/c^2)$. Applying the inverse transformation with speed $-v$ gives $A^2(1-v^2/c^2)=1$, hence $A=\gamma$.

The transformed reflection events haveThese events are not simultaneous in $S^{\prime }$. Both rod ends move at $-v$ there, so extrapolating each endpoint to $t^{\prime }=0$ gives $x_{\pm }^{\prime }(0)=x_{\pm }^{\prime }+v{t}_{\pm }^{\prime }=\pm \ell /\gamma$. Measuring simultaneous endpoints therefore yields length contraction:

The Einstein's principle of relativity says that the laws of physics take the same form in every inertial frame: no inertial frame is physically privileged, and uniform absolute motion cannot be detected by an internal experiment.

Assume $|u|<c$. Direct expansion of the transformed coordinates givesSince $y^{\prime }=y$ and $z^{\prime }=z$, the full Minkowski interval is invariant. Points of the outgoing light shell therefore satisfyAt each fixed $t^{\prime }>0$ this is a sphere of radius $c{t}^{\prime }$. The future branch remains future-directed: for a shell point with $t>0$, $|x|\le ct$ gives $t^{\prime }=\gamma (t-ux/c^2)>0$.

A Lorentz transformation is a linear coordinate transformation preserving the Minkowski metric, equivalently ${\Lambda }^T\eta \Lambda =\eta$. The calculation establishes this property for all events, not only points on the light cone. The displayed map has determinant one and preserves time orientation, so it is the proper orthochronous boost along the $x$ direction.

One can derive a family containing the Lorentz transformations without assuming anything about light, but the Einstein's principle of relativity alone does not select that family member over the Galilean transformation. Extra assumptions must be stated: spatial and temporal translational symmetry, isotropy of SPACE, continuous transformations between equivalent inertial frames, reciprocity, and closure under composition. These are substantive restrictions, rather than consequences of the bare assertion that physical laws agree between inertial frames.

Here is the algebraic content. Translational symmetry makes the transformation between coincident origins a linear transformation. In one spatial direction the moving origin $x=vt$ fixes $x^{\prime }=a(v)(x-vt)$. Isotropy and the inverse transformation at $-v$ allow the standard calibrated formTo see why $K$ is universal, temporarily write the upper-right entry as $-b(v)$, leaving the lower-left entry $-v$. Equality of the two diagonal entries in a composite boost gives $b(u)v=ub(v)$, so $b(v)/v=K$ is constant for nonzero boosts. Inversion fixes $a(v{)}^2(1-K{v}^2)=1$, with the positive branch continuous at zero. This is the universality of an invariant speed under boost composition.

The resulting velocity-addition formula law and invariant quadratic expression areFor $K>0$, $C=K^{-1/2}$ is an invariant limiting speed; setting the empirically determined $C$ equal to the vacuum speed of light gives the standard Lorentz transformation. For $K=0$, $t^{\prime }=t$ and $x^{\prime }=x-vt$: an entirely consistent Galilean transformation satisfying the stated Einstein's principle of relativity. Thus the general assumptions do not establish that an invariant finite speed exists, much less identify it with light.

There is also a formal negative-$K$ branch, which has a rotation-like geometry in the time–space plane. Standard derivations exclude it using additional global assumptions, such as a preserved time orientation and unrestricted composition of admissible boosts. Without identifying those assumptions it is misleading to claim a two-way classification from the Einstein's principle of relativity alone. In three spatial dimensions isotropy requires the same constant in every direction.

The light postulate is therefore dispensable as the starting point for a conditional kinematical derivation; empirical input remains indispensable in identifying our world's value and sign of $K$. Albert Einstein's original construction uses the light postulate and Einstein synchronisation, making that empirical connection explicit.^[ref] Replacing the postulate by an independently established finite invariant speed changes the route, rather than deriving all of special relativity from relativity alone.

Historically, electromagnetism was central to Albert Einstein's 1905 argument. His opening example compares a moving magnet and a moving conductor. The same relative motion produces the observed induction, but a description using electric fields in one frame and the magnetic part of the Lorentz force in another seemed to divide the explanation according to which object was called stationary. The unsuccessful search for an observable preferred state of rest also motivated treating the laws of mechanics, optics and electromagnetism on a common footing. That establishes a strong historical connection, not a deduction that relativity applies only to electromagnetic systems.^[ref]

The logical starting points of the kinematical argument are Einstein's principle of relativity and an invariant speed of light in vacuum, together with the assumptions needed to compare homogeneous inertial frames and physical measuring standards. Einstein synchronisation assigns distant clock readings by light exchange. In standard coordinates, the resulting Lorentz transformation for a boost along $x$ isNeither the derivation of this transformation nor the deductions of relativity of simultaneity and time dilation require solving Maxwell equations for every particular material system. They constrain the relation between measurements made in different inertial frames.^[ref]

There is nevertheless an empirical commitment in identifying $c$ with the speed of vacuum light. It is not a freely chosen numerical label: the theory says different inertial observers, using the prescribed standards, obtain the same measured value. More generally, $c$ specifies the causal structure of Minkowski spacetime. A theory could contain a massive scalar field with a Lorentz invariant equation and no electromagnetic field at all. Its particles, proper times and causal propagation would still be governed by relativistic geometry. This gives a direct sense in which the framework is logically more general than electromagnetism.

Conversely, the presence of Maxwell equations alone would not establish the full relativity principle for all matter. One could append a law involving a physically preferred timelike vector field, giving a nonelectromagnetic sector a preferred frame while retaining the usual electromagnetic equations. The covariant equation with a preferred timelike vector is an explicit model of this distinction. Requiring every relevant interaction to respect Lorentz invariance adds content; writing its components in four-dimensional notation does not supply that content automatically.

The magnet–conductor puzzle is resolved because electric fields and magnetic fields are related parts of one electromagnetic field, with the Lorentz force transforming consistently. The measured current must be compared for the same physical arrangement, including the conductor's state of motion, rather than by demanding the same electric and magnetic components separately. This is an application of the general frame transformations to a particular interaction, not the definition of every possible relativistic interaction.

Albert Einstein's later distinction between a principle theory and a constructive theory clarifies what the 1905 argument provides. It places universal restrictions on admissible laws without constructing every rod and clock from a particular microscopic model. A constructive explanation can additionally investigate why actual material standards approximate the required behavior. Harvey R. Brown and Oliver Pooley emphasize this dynamical question; it does not make the kinematical transformations meaningless or confine them to electromagnetic phenomena.^[ref][ref]

The strongest account therefore distinguishes historical motivation, the operational use of light, and the theory's universal scope. The 1905 paper addresses an electromagnetic problem through a framework whose restrictions extend to every physical process claimed to obey special relativity.

There are two distinct explanatory questions. One asks why a proposed system of equations has a particular dynamical symmetry. The other asks why the common spacetime geometry is the right geometry for the behavior of physical rods, clocks and fields. A geometric account and a dynamical account give different directions of explanation, and their merits cannot be decided by a change of notation alone.

For the geometric account, Minkowski spacetime has a fixed Lorentzian metric; its spacetime isometries form the Poincaré group. If the matter equations use that metric tensor and no additional symmetry-breaking background structure, with couplings and boundary assumptions appropriate to the transformation, these spacetime isometries take solutions into solutions. For example, the Klein-Gordon equationin metric signature $(-,+,+,+)$ is invariant under the Poincaré group. This provides one common reason for the same transformation rules appearing in many laws. An individual solution, such as a particular plane wave, need not possess every symmetry of the equation.

Geometry alone is insufficient unless the assumed coupling to geometry is included. In the covariant equation with a preferred timelike vector,the fixed timelike $n$ selects a preferred rest frame. All the quantities can be transformed consistently under a coordinate transformation, but boosts that change $n$ are not symmetries with the physical background fixed. Thus general covariance, tensor notation and the mere presence of the Minkowski metric do not by themselves establish Einstein's principle of relativity.

A dynamical account instead starts with the common invariance of the actual interactions and asks why their composite rods and clocks measure a common interval. If their equations and suitable equilibrium solutions transform consistently under Lorentz transformations, their moving configurations can exhibit length contraction and their periodic processes time dilation. This is the explanatory strategy defended by Harvey R. Brown and Oliver Pooley, who deny that merely redescribing these effects geometrically supplies their microscopic explanation.^[ref] Such an account must still explain why independent interactions have the same relevant symmetry and why a particular clock is a good clock; it cannot simply infer those facts from the word “dynamics”.

The positions therefore need not disagree about experimental predictions. The geometric presentation makes a shared constraint explicit and permits economical deductions. The constructive presentation investigates why material systems realize that constraint. Defining geometry from universal dynamical behavior is a substantive proposal; deriving behavior from geometry also requires substantive matter-coupling assumptions. Neither direction is made compulsory by the formal equivalence of the descriptions.

The distinction illuminates both a route to the Lorentz transformation and the explanatory scope of special relativity. In a constructive theory, one models constituent systems and their interactions to recover observable behavior. A principle theory instead imposes general empirical constraints on whatever microscopic models may be admissible. Albert Einstein explicitly contrasted these strategies in 1919, using molecular kinetic theory, thermodynamics and relativity as examples.^[ref]

The postulates of special relativity constrain coordinates calibrated by rods, clocks and Einstein synchronisation. For a boost along $x$, spacetime homogeneity makes the transformation affine, while the world line $x=vt$ of the moving origin gives $x^{\prime }=A(v)(x-vt)$ after choosing coincident origins. Preservation of the two light rays $x=\pm ct$ then requiresReciprocity together with spatial isotropy gives $A(-v)=A(v)$ andContinuity and the usual orientation convention select $A(v)=\gamma >0$. Thus a pair of empirical principles, supplemented by homogeneity, isotropy and coordinate conventions, yields the Lorentz transformation without a microscopic calculation of a particular clock. Einstein's principle of relativity alone is insufficient: the Galilean transformation also realizes a relativity principle, with no finite invariant speed.^[ref]

The distinction also tells us what has not been derived. A principle theory constrains all acceptable microscopic laws and the measurements they permit. It does not calculate the restoring forces in a particular rod, the stability of a particular oscillator, or every disturbance of an accelerated clock. Those are questions for detailed dynamics. Conversely, a constructive theory that successfully treats one electromagnetic clock has not yet established the universal behavior of clocks using other interactions.

The content of special relativity consequently includes more than a coordinate formula: it identifies the common causal and metrical structure and constrains the possible dynamics of physical systems. Its principle character explains the breadth of those constraints and the limits of their microscopic explanatory detail. A constructive realization can complement them. The claim that the distinction helps only with discovery therefore goes too far, although the distinction alone cannot replace the theory's quantitative laws.

The two postulates of special relativity are: all nongravitational laws have the same form in every inertial frame, and light in vacuum has the same finite speed of light $c$ in every inertial frame, independently of the motion of its source. The first is Einstein's principle of relativity; the second supplies an experimentally identified invariant speed.

For the stated standard configuration, the Lorentz transformation isTake differentials along the particle's worldline, rather than comparing coordinate displacements at equal times in the two frames. Writing $u_i=d{x}^i/dt$, one obtainsDividing by the transformed time interval proves the velocity-addition formula:For a massive particle, $|\mathbf{u}|<c$ and $|v|<c$, so the denominator $1-v{u}_x/c^2$ is positive.

The first postulate is Einstein's principle of relativity: nongravitational laws have the same form in every inertial frame. The second states that light in vacuum propagates at the same finite speed of light $c$ in every inertial frame, independently of the source's motion. Together with the usual spacetime symmetries and calibrated coordinates, these postulates yield the Lorentz transformation. The relativity principle alone also admits the Galilean transformation with its infinite invariant-speed limit.

Special relativity describes physics in Minkowski spacetime, where inertial frames assign different space and time coordinates to the same spacetime events. Its two postulates are Einstein's principle of relativity and the same vacuum speed of light $c$ in every inertial frame, independent of the source's motion. Together they replace the Galilean transformation with the Lorentz transformation, coupling space and time rather than providing a universal clock. Einstein's 1905 paper developed this account through measuring rods, clocks, and light signals. ^[ref].

A spacetime event is an occurrence at a definite place and time: a pulse reaching a detector, for example. An inertial frame supplies a Cartesian coordinate system and clocks at rest throughout it. Einstein synchronisation sets a distant clock's reflection reading to the midpoint between a light signal's departure and return readings at the sending clock. The resulting coordinate time records the local synchronized clock at each spacetime event; it differs from the time when a distant observer receives its image. This distinction makes time dilation, length contraction, and the relativity of simultaneity precise measurements. ^[ref].

The beauty of special relativity is that these different measurements fit one geometry: all inertial frames agree on the Minkowski interval. It lets us translate an experiment between a laboratory and a moving apparatus while preserving the light-signal relations between spacetime events.