Lecture 35–38 — Special Relativity: Noether’s Theorem, Four-Vectors, and the Lorentz Group#
Source: NPTEL Classical Physics, Mod-01 Lec-35 (Noether’s theorem; Special relativity, Part 1), Lec-36 (Part 2), Lec-37 (Part 3), and Lec-38 (Part 4), Prof. V. Balakrishnan.
A handful of lectures (Lec-31–34, continuing critical phenomena and laying the group-theory groundwork for Noether’s theorem) sit between Lecture 28–30 and this page and aren’t transcribed here yet. This is, however, the last lecture of Balakrishnan’s course — it ends, fittingly, by handing off to quantum mechanics.
Prelude: symmetry implies conservation#
Lec-35 opens with Noether’s theorem, the precise statement of an idea used informally all through this course: every continuous symmetry of a Lagrangian hands you a conserved quantity for free. Make an infinitesimal change of coordinates \(q_i \to q_i + \epsilon f_i(q,t)\) (schematically \(\delta q_i\)) and compute the resulting change in the Lagrangian two independent ways. Using the Euler–Lagrange equations directly,
(the first term became \(\tfrac{d}{dt}p_i\,\delta q_i\) via the equations of motion, and \(\tfrac{d}{dt}\delta q_i = \delta\dot q_i\) since the variation and the time derivative commute). If, independently — without invoking the equations of motion — the transformation also happens to change \(L\) by nothing more than a total time derivative, \(\Delta L = \tfrac{d}{dt}\big(\epsilon f(q,t)\big)\), then the equations of motion themselves don’t change (the action is invariant), and equating the two expressions gives
Two familiar examples fall out immediately. Translate every particle by a constant vector \(\vec\epsilon\): a free particle’s Lagrangian doesn’t change at all (\(\Delta L\equiv 0\), so \(f=0\)), and the theorem hands back \(\vec p\cdot\vec\epsilon = \text{const}\) for arbitrary \(\vec\epsilon\) — linear momentum is conserved because space is homogeneous. Rotate the coordinate system instead, by an infinitesimal angle about axis \(\hat n\) (\(\delta\vec r = \epsilon\,\hat n\times\vec r\)): a central potential \(V(r)\) depends only on \(|\vec r|\), so again \(\Delta L\equiv 0\), and the theorem gives \(\vec p\cdot(\hat n\times\vec r) = \hat n\cdot(\vec r\times\vec p) = \text{const}\) for every \(\hat n\) — angular momentum is conserved because space is isotropic. (A third case, where \(f\ne 0\), appears when the value of the constant of motion depends explicitly on time — dropping an object under gravity, invariant under \(z\to z+\epsilon\) since adding a constant to a potential changes nothing physical, hands back \(p = -mgt + \text{const}\), i.e. exactly the equation for the initial velocity. The “trivial” translation symmetry, worked through honestly, re-derives free fall.)
Every symmetry considered so far has been a symmetry of space alone, with time sitting outside, flowing uniformly and absolutely in the background. What happens once the symmetry mixes space and time together — the way motion itself does?
The postulate that broke Newton’s clock#
Newtonian mechanics already assumed a great deal of invariance: translate the origin, rotate the axes, shift the zero of time, or view everything from a frame moving at constant velocity, and the laws of motion keep their form. Together these transformations — 3 translations, 3 rotations, 1 time shift, 3 boosts — form a ten-parameter group, the Galilean group. Newton’s one questionable extra ingredient was to single out a preferred set of these frames (“fixed relative to the distant stars”); the better, coordinate-free definition of an inertial frame is simply one in which Newton’s first law holds — a particle left alone moves in a straight line at constant speed.
Einstein’s insight was to add exactly one further postulate to this list: there exists a finite speed, the same in every inertial frame, independent of the motion of source or observer. Light in vacuum happens to travel at this fundamental speed \(c\) — a fortunate physical accident that made the postulate discoverable at all — but the statement itself is purely about spacetime, not about light specifically.
The consequence is derived exactly the way a rotation is derived: by demanding that some quadratic form stay invariant. Send out a spherical light pulse from a common origin at \(t=t'=0\); in frame \(S\) it satisfies \(c^2t^2 = r^2\), and by the postulate, in frame \(S'\) moving at velocity \(v\) along \(x\), the same pulse satisfies \(c^2t'^2=r'^2\) with the same \(c\). Newtonian mechanics could only preserve \(c^2t^2 - r^2 = c^2t'^2-r'^2\) by taking \(t=t'\) and \(c\to\infty\); relativity preserves it instead by letting \(t\) itself transform. Requiring this quadratic form to be invariant — exactly as requiring \(x^2+y^2+z^2\) to be invariant forces rotations to be orthogonal matrices — forces the transformation between frames to be the Lorentz transformation:
Four-vectors and the geometry of spacetime#
Bundle space and time into a single object, the four-vector \(x^\mu \equiv (ct,\,x,\,y,\,z)\), \(\mu=0,1,2,3\). Anything that transforms the same way under Lorentz transformations — rotations and boosts together — is a four-vector by definition. The invariant quadratic form is packaged into the metric tensor,
three minus signs for the three space directions, one plus sign for time — a pseudo-Euclidean signature, in contrast to an ordinary (positive-definite) Euclidean metric. (This is the flat-spacetime special case of a much more general idea: on a curved manifold like the surface of a sphere, \(ds^2 = g_{ij}\,dx^i dx^j\) with \(g_{ij}\) varying from point to point is exactly how general relativity encodes curvature. Special relativity is the case where the \(g_{\mu\nu}\) happen to be constants everywhere — genuinely flat.)
The minus signs are the whole reason four-vectors come in two flavors. Given \(a^\mu\) (upper index, “contravariant”), define the covariant partner \(a_\mu \equiv g_{\mu\nu}a^\nu\) — numerically \(a_0=a^0\), but \(a_i = -a^i\) for the three space components. Every rule of the notation exists to hide the minus signs automatically:
Contraction rule: a repeated index must appear once up, once down, and is then summed (Einstein convention) and gone; a free index must match on both sides of an equation; an index appearing three times means an error.
Dot product: \(a\cdot b \equiv a^\mu b_\mu = a^0b^0 - \vec a\cdot\vec b\) — the spatial minus sign is now built into the notation rather than written by hand every time.
Raising and lowering: \(g^{\mu\nu}a_\nu = a^\mu\) and \(g_{\mu\nu}g^{\nu\rho} = \delta_\mu^\rho\) (the metric is its own inverse here).
The gradient reverses the pattern: \(\partial_\mu \equiv \partial/\partial x^\mu = (\tfrac1c\partial_t,\,+\vec\nabla)\) while \(\partial^\mu = (\tfrac1c\partial_t,\,-\vec\nabla)\) — exactly backwards from \(x^\mu=(ct,\vec r)\) vs. \(x_\mu = (ct,-\vec r)\), so that \(\partial_\mu x^\mu = 4\) comes out consistent, as it must.
Spacetime geometry: light cones and the mass shell#
The invariant \(x\cdot x = c^2t^2 - r^2\) splits every point of spacetime, relative to the origin, into three causal classes: timelike (\(x\cdot x>0\), reachable at less than light speed — always in the definite past or future of the origin, in every frame), spacelike (\(x\cdot x<0\), unreachable without exceeding \(c\) — no consistent time-ordering exists, and different observers can disagree on which event happened first), and lightlike (\(x\cdot x=0\), on the light cone itself). Because \(x\cdot x\) is Lorentz-invariant, a Lorentz transformation can never move an event from one class to another — it can, however, freely slide a spacelike pair of events back and forth in time-order (this is the relativity of simultaneity), or a timelike pair back and forth in space (never reordering which came first).
The same invariance, applied to momentum instead of position, gives \(p\cdot p = E^2/c^2 - \vec p^{\,2} = m^2c^2\) — every free particle of mass \(m\) is confined to a fixed hyperbola (the mass shell) in energy-momentum space, however it’s boosted.
Drag the slider: on the left, the S′ axes tilt symmetrically toward the light cone but never cross it — a boost is a “rotation” that can approach, but never reach, \(45°\). On the right, the same rapidity slides the particle’s energy-momentum point along the mass-shell hyperbola. Both panels are driven by exactly the same number because both are exactly the same transformation, once written in the right variable.
Rapidity: the variable that actually adds up#
That “right variable” deserves its own name. Ordinary velocities famously don’t add: boost by \(v_1\) and then by \(v_2\) (same direction) and the result is \(v = (v_1+v_2)/(1 +v_1v_2/c^2)\), not \(v_1+v_2\). But \(v/c\) ranges only over \((-1,1)\), which is exactly the range of \(\tanh\) — suggesting rapidity \(u \equiv \tanh^{-1}(v/c)\) as the natural variable. In terms of \(u\), \(\gamma=\cosh u\) and \(\gamma v/c = \sinh u\), so the Lorentz boost matrix takes the manifestly rotation-like (but hyperbolic, matching the pseudo-orthogonal metric) form
and the composition law becomes trivial: \(u = u_1+u_2\). Rapidities add exactly like Newton would have expected velocities to — it’s rapidity, not velocity, that Galileo’s intuition was reaching for.
(One caution the composition law hides: this simple addition only works for boosts in the same direction. Two boosts in different directions compose into a boost plus a rotation — boosts alone don’t form a subgroup of the Lorentz group, only rotations do. This extra rotation, applied to a spinning object dragged through a sequence of non-collinear boosts, is the relativistic effect known as Thomas precession.)
Covariant electrodynamics in one line#
The four-vector machinery earns its keep immediately on Maxwell’s equations. Package charge density and current into a four-current \(j^\mu=(c\rho,\vec\jmath)\); the continuity equation collapses to \(\partial_\mu j^\mu = 0\) — manifestly the same statement, hence charge conservation, in every frame. Package the scalar and vector potentials into \(A^\mu = (\phi/c,\vec A)\); the Lorenz gauge condition becomes \(\partial_\mu A^\mu=0\) (as opposed to the Coulomb gauge \(\vec\nabla\cdot\vec A=0\) alone, which is not Lorentz-covariant — imposing it in one frame doesn’t preserve it in another). In this gauge the wave equations for \(\phi\) and \(\vec A\) merge into one four-dimensional statement, \(\Box A^\mu = \mu_0 j^\mu\), where \(\Box \equiv \partial_\mu\partial^\mu = \tfrac1{c^2}\partial_t^2-\nabla^2\) is the d’Alembertian — all four of Maxwell’s equations, in one line. (Setting the source to zero and translating \(E,\vec p \to i\hbar\partial_t, -i\hbar\vec\nabla\) turns this into the Klein–Gordon equation for a relativistic particle — a striking aside, though a detour from where this lecture is headed.)
The electric and magnetic fields themselves combine into the antisymmetric rank-2 field tensor \(F^{\mu\nu} \equiv \partial^\mu A^\nu - \partial^\nu A^\mu\): 16 components, but the diagonal vanishes and antisymmetry ties the lower triangle to the upper, leaving exactly 6 independent numbers — precisely \(E_x,E_y,E_z,B_x,B_y,B_z\).
Two, and only two, independent Lorentz scalars can be built from \(F^{\mu\nu}\): \(F_{\mu\nu}F^{\mu\nu} \propto B^2 - E^2/c^2\) (up to normalization, this is the electromagnetic Lagrangian density) and \(F_{\mu\nu}\tilde F^{\mu\nu} \propto \vec E\cdot\vec B\) (built with the dual tensor \(\tilde F^{\mu\nu}\), which swaps the roles of \(\vec E\) and \(\vec B\)). Both survive unchanged under any boost — which is exactly why “is this radiation transverse, with \(\vec E\perp\vec B\)” is a frame-independent question, even though \(\vec E\) and \(\vec B\) individually are not frame-independent at all.
Boosting a field: how E and B mix#
That individually-not-invariant behavior has an exact formula. Split a field into components parallel and perpendicular to the boost velocity \(\vec v\):
The components along the boost don’t change at all; the transverse components mix into each other with a factor of \(\gamma\). Watch what this means physically: stand next to a charge in its rest frame and you see a pure electric field, no magnetism at all. Walk past it (equivalently, boost the field) and a magnetic field appears out of nothing — magnetism is what electricity looks like from a moving frame.
Start with pure \(\vec E\) (\(B_0=0\)) and drag the slider: \(\vec E'\) grows by \(\gamma\) while an entirely new \(\vec B'\), perpendicular to both the boost and \(\vec E\), grows right alongside it — and the title tracks \(E'^2-c^2B'^2\) recomputed at every frame, landing on exactly \(E_0^2\) throughout. That’s the invariant from the previous section, verified rather than merely asserted: electricity and magnetism were never two different fields, only one field looked at from different states of motion.
Relativistic dynamics of a charged particle#
The free-particle Lagrangian that reproduces the correct (nonlinear) relativistic momentum \(\vec p = m\vec v/\sqrt{1-v^2/c^2}\) is \(L = -mc^2\sqrt{1-v^2/c^2}\) — check it directly: \(\partial L/\partial\vec v\) gives exactly that \(\vec p\), and \(\partial L/\partial\vec r=0\) (free particle) makes the Euler–Lagrange equation say \(\vec p\) is conserved, as it must be. Coupling to an electromagnetic field costs nothing beyond the same interaction term used non-relativistically, \(L_{\text{int}} = e\vec A\cdot\vec v - e\phi\) (covariantly, \(\propto j^\mu A_\mu\)) — the simplest Lorentz scalar linear in both the current and the potential, called minimal coupling. Every consequence of switching on the field turns out to be the single replacement
applied uniformly to the free-particle energy–momentum relation, the canonical momentum, and the Hamiltonian alike. Grinding through the Euler–Lagrange equation with this Lagrangian reproduces the exact relativistic equation of motion,
the same Lorentz force as always, now acting on the relativistic momentum rather than \(m\vec v\) — which, worked through to an explicit equation for \(\dot{\vec v}\), picks up an extra \(\vec v(\vec v\cdot\vec E)/c^2\) term that vanishes smoothly in the \(v\ll c\) limit, recovering the non-relativistic Lorentz force exactly.
Closing: the Poincaré group, and where mass and spin come from#
Pull every symmetry of flat spacetime together — 4 translations (space and time alike) plus the 6-parameter Lorentz group \(SO(3,1)\) (3 rotations, 3 boosts) — and you get the full Poincaré group, 10 parameters in all. It isn’t simply a direct product of the two pieces: translating and then Lorentz-transforming is not the same as Lorentz-transforming and then translating by the same amount, because the Lorentz transformation acts on the translation too. (Composing \((\Lambda_1,a_1)\) then \((\Lambda_2,a_2)\) gives \((\Lambda_2\Lambda_1,\ \Lambda_2 a_1+a_2)\) — the translations form an abelian normal subgroup that the Lorentz part acts on, making the whole thing a semidirect product \(SO(3,1)\ltimes T(4)\).)
This is where the story closes the loop back to Noether’s theorem at the very beginning. Every one of the Poincaré group’s 10 generators is a conserved quantity: the 4 translation generators are the four-momentum \(P^\mu\) (energy and momentum — literally the same conservation laws derived by hand in the prelude, now promoted to spacetime), and the 6 Lorentz generators \(J^{\mu\nu}\) package ordinary angular momentum together with the boost generators. A group like this has Casimir invariants — special combinations of the generators that commute with everything in the algebra, and therefore label its representations, the same way \(\vec J^2\) commuting with every \(J_i\) is why hydrogen-atom states carry a definite total angular momentum quantum number. The Poincaré group has exactly two: \(P^\mu P_\mu\), whose eigenvalue is (up to \(c^2\)) the square of the particle’s rest mass, and a more elaborate combination \(W^\mu W_\mu\) built from both \(J\)’s and \(P\)’s, whose eigenvalue fixes the particle’s spin.
Mass and spin, in other words, are not extra ingredients bolted onto quantum particles — they are literally the labels of an irreducible representation of the symmetry group of spacetime itself. Every other quantum number a particle carries (charge, baryon number, and the rest) has to come from somewhere else; mass and spin come from nothing more than requiring the laws of physics to look the same to every inertial observer. It is a fitting place to end a course that opened with orders of magnitude and the regimes of physics, and ends by discovering that geometry alone — the honest insistence that no frame is special — is enough to conjure up two of the most basic properties any particle can have. What comes next is to see how all of this reads once matter is described not by trajectories but by wavefunctions: quantum mechanics.