Special Relativity Toolkit

Relativistic wave equations inherit every sign and factor from their spacetime conventions. This chapter fixes those choices before introducing scalar or spinor amplitudes. The toolkit connects the metric ledger to Lorentz-covariant objects and the free mass shell, then develops the measures, causal geometry, currents, and symmetry generators used throughout the volume.

Reading sequence

1. Metric and Units fixes $(+---)$, $x^0=ct$, derivatives, $\epsilon^{0123}$, the plane-wave phase, signed charge, four-potential, and SI/natural-unit translation.
2. Four-Vectors explains what a Lorentz transformation law means and applies it to coordinates, momentum, derivatives, currents, and potentials.
3. Energy–Momentum Relation derives $p^2=m^2c^2$, distinguishes its frequency branches, and computes its low- and high-momentum limits.

If index notation or boosts need reinforcement, insert Spacetime Notation and Lorentz Transformations between the metric ledger and Four-Vectors. The notation page is a worked companion to the ledger; the transformation page develops rapidity and group composition.

From the common kinematic base, select the next tool by its use:

Question	Page	Capability to take forward
How should momentum states be integrated?	Relativistic Phase Space	Derive the one-particle invariant mass-shell measure.
Which events can influence one another?	Causality and Light Cones	Distinguish causal response, wavefunction tails, and correlations.
What is conserved on a tilted time slice?	Relativistic Currents	Integrate a current over a spacelike surface and test positivity.
Why do mass and spin label particles?	Poincaré Group	Read the generator algebra and its Casimir invariants.

Checks to carry forward

Before combining a relativistic formula with later pages, verify:

• spatial covariant components have the sign implied by the mostly-minus metric;
• $p\cdot x=Et-\mathbf p\cdot\mathbf x$;
• $i\hbar\partial_\mu e^{-ip\cdot x/\hbar}=p_\mu e^{-ip\cdot x/\hbar}$;
• mass, rest energy, and natural-unit mass dimension have not been conflated;
• an active boost has not been interpreted with a passive sign;
• a four-component object has a declared transformation law.

Carry the toolkit into wave equations

The Klein–Gordon Equation applies the mass shell to a scalar amplitude. Gamma Matrices linearizes the same mass shell, and the Covariant Dirac Equation introduces a spinor transformation law. Compare their conserved currents using the geometric positivity test rather than inferring probability from conservation alone.

One useful readiness check is to explain all three statements at once: positive-energy momenta remain positive under proper orthochronous boosts; a conserved current need not have positive density; and a nonzero spacelike correlation need not transmit a signal. They constrain different objects and are mutually compatible.

References

• W. Rindler, Relativity: Special, General, and Cosmological, 2nd ed., Oxford University Press, 2006.
• E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed., W. H. Freeman, 1992.
• S. Weinberg, Gravitation and Cosmology, Wiley, 1972.