Special Relativity and Minkowski Spacetime

Special relativity replaces Newtonian absolute time with an invariant causal geometry. Its basic object is not a preferred coordinate grid but Minkowski spacetime: an affine space of events equipped with a Lorentzian metric and a time orientation. In inertial Cartesian coordinates,

$$\eta_{\mu\nu}=\operatorname{diag}(-1,+1,+1,+1), \qquad ds^2=\eta_{\mu\nu}dx^\mu dx^\nu.$$

Helpful background. Calculus and Linear Algebra for Relativity supplies the linear maps and contractions used throughout; Variational Mechanics and Symmetries supplies the particle-action and conservation-law language used near the end of the chapter.

Working conventions

We use $c=1$ throughout. Thus time and distance have the same units, future-directed timelike four-velocities satisfy $u^\mu u_\mu=-1$, and a massive particle obeys $p^\mu p_\mu=-m^2$. When dimensions clarify an experimental statement, restore $x^0=ct$ before setting $c=1$ again.

Coordinates label events; they are not intrinsic observables. Different inertial charts assign different components to the same geometric vector. Agreement comes from scalars, causal relations, and observer-defined projections. For example, an observer represented by a future-directed unit timelike vector $u^\mu$ assigns energy

$$E_{(u)}=-u_\mu p^\mu$$

to a four-momentum $p^\mu$. That contraction is a scalar once both the momentum and observer are specified. A bare component such as $p^0$ becomes an energy only in the inertial frame whose time axis is $u^\mu$.

What this chapter builds

The six pages form one chain from geometry to conservation:

• Events, Intervals, and Minkowski Geometry defines events, invariant separation, light cones, and causal classes.
• Lorentz Transformations, Rapidity, and Velocity Addition derives a signed boost, explains collinear rapidity addition, and marks the noncollinear boundary.
• Proper Time, Worldlines, and the Clock Effect compares clock readings by integrating along complete worldlines rather than invoking acceleration slogans.
• Four-Vectors and Lorentz-Invariant Decomposition separates tensors from their components and projects them relative to an observer.
• Relativistic Energy, Momentum, and Mass Shells distinguishes particle rest mass from the invariant mass of a system.
• Poincaré Symmetry and Relativistic Conservation Laws connects spacetime symmetries to particle and field charges.

Hyperbolic functions appear only as a natural parametrization of boosts; the relevant identities are derived where used.

Scope

The background in this chapter is flat, and inertial coordinates cover all of it. Worldlines may accelerate, but accelerated coordinate systems, horizons, and rotating frames are deferred. Gravity and curved Lorentzian geometry are also excluded: proper time will later retain its local definition while the constant matrix $\eta_{\mu\nu}$ is replaced by a position-dependent metric.

Continue to Relativistic Measurement, Particles, and Collisions to turn invariant geometry and four-momentum into quantities measured by specified observers.

Check your understanding

Two events have inertial-coordinate separation $\Delta x^\mu=(5,3,0,0)$. Classify their separation, find the proper time recorded by an inertial clock joining them, and determine that clock’s speed in the stated chart.

Solution

With the chapter’s signature,

$$\Delta s^2=-(\Delta t)^2+|\Delta\mathbf x|^2=-25+9=-16.$$

The separation is timelike, so one inertial clock can pass through both events. It records $\Delta\tau=\sqrt{-\Delta s^2}=4$, while its chart speed is $v=|\Delta\mathbf x|/\Delta t=3/5$. Indeed, $\gamma=(1-v^2)^{-1/2}=5/4$ gives $\Delta t/\gamma=4$. A Lorentz boost can change the time and spatial components, but it cannot change the timelike classification or the clock reading.

Tong 2013, Chapter 7, printed pp. 107–160 supplies a useful parallel development. It uses the opposite signature $(+---)$, so interval and mass-shell signs are reversed here. Hughes 2024, MIT 8.033 Lectures 4–9, printed pp. 27–78 (PDF) uses observer-centered four-vector methods that align closely with this chapter’s emphasis.

References

• Hughes, Scott A. Introduction to Relativity and Spacetime Physics. MIT 8.033 Lectures 4–9, printed pp. 27–78 (PDF), Fall 2024.
• Tong, David. Dynamics and Relativity. Chapter 7, printed pp. 107–160 (PDF), University of Cambridge, Lent Term 2013. Signature translated from $(+---)$ to $(-+++)$.