Chapter 7: Minkowski Spacetime
"Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." — Hermann Minkowski, 1908
Spacetime as Geometry
Minkowski realized that special relativity becomes beautifully simple when viewed geometrically. Instead of separate 3D space and 1D time, we have a unified 4D spacetime.
3D Space + Time (Old View)
- • Space and time are separate entities
- • Time is absolute (same for everyone)
- • Distances are preserved (Euclidean geometry)
- • Galilean transformations mix x and t
4D Spacetime (New View)
- • Space and time are unified
- • The spacetime interval is invariant
- • Minkowski geometry (pseudo-Euclidean)
- • Lorentz transformations = rotations in spacetime
In Minkowski's view, Lorentz transformations are simply rotations in spacetime—analogous to rotating axes in ordinary space. Just as spatial rotations preserve distances, Lorentz transformations preserve the spacetime interval.
The Spacetime Interval
In ordinary space, the distance between two points is invariant under rotations:
\( \Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 \)
(Euclidean distance—same in all rotated frames)
In spacetime, the invariant quantity is the spacetime interval:
\( \Delta s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \)
The minus signs make spacetime "pseudo-Euclidean"
Key property: All observers agree on Δs², even though they disagree on Δt and Δx separately. This is the fundamental invariant of special relativity—the "length" in spacetime that everyone measures the same.
Types of Spacetime Intervals
The sign of Δs² classifies the relationship between events:
Timelike (Δs² > 0)
\( c^2\Delta t^2 > \Delta x^2 \)
- • Events can be causally connected
- • One event can influence the other
- • All observers agree on time order
- • A clock can travel between them
Lightlike/Null (Δs² = 0)
\( c^2\Delta t^2 = \Delta x^2 \)
- • Connected by a light ray
- • On the boundary of causality
- • Proper time = 0 along path
- • Light cones in spacetime diagrams
Spacelike (Δs² < 0)
\( c^2\Delta t^2 < \Delta x^2 \)
- • Events cannot be causally connected
- • No signal can reach between them
- • Time order is frame-dependent
- • "Elsewhere" in spacetime
Light Cones
At any event in spacetime, we can draw a light cone—the surface formed by all light rays passing through that event.
The Structure of a Light Cone
Future Light Cone: All events that CAN be reached from the origin by signals traveling at or below light speed. These are in the "absolute future."
Past Light Cone: All events that COULD HAVE influenced the origin. These are in the "absolute past."
Elsewhere: Events outside both cones. Cannot be causally connected to the origin. Different observers may disagree on whether these are "past" or "future."
Light cone structure is absolute! While observers disagree on simultaneity within the "elsewhere" region, they all agree on what's inside the past and future light cones. Causality is preserved.
Worldlines
The trajectory of an object through spacetime is called its worldline.
At Rest
A vertical line (only moving through time, not space). This is the worldline of a stationary particle.
Constant Velocity
A straight line tilted from vertical. The slope equals v/c. Must be less than 45° (inside light cone).
Accelerating
A curved line. The slope (velocity) changes continuously. Must always remain inside the light cone.
Light Ray
A line at exactly 45° (slope = 1 when using ct for time axis). Always on the light cone surface.
Proper Time Along a Worldline
The proper time τ elapsed along a worldline is the time measured by a clock traveling along that path:
\( d\tau = \sqrt{1 - v^2/c^2} \, dt = dt/\gamma \)
The Minkowski Metric
The geometry of spacetime is encoded in the metric tensor:
\( \eta_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \)
(Using the +−−− "mostly minus" convention)
The interval can be written compactly as:
\( ds^2 = \eta_{\mu\nu} \, dx^\mu \, dx^\nu = (cdt)^2 - dx^2 - dy^2 - dz^2 \)
Preview of General Relativity: In GR, the metric tensor gμν becomes position-dependent, encoding the curvature of spacetime caused by mass and energy. The Minkowski metric ημν is the special case of flat spacetime.
Lorentz Boosts as Hyperbolic Rotations
In ordinary space, rotations mix x and y while preserving \( x^2 + y^2 \). In spacetime, boosts mix ct and x while preserving \( (ct)^2 - x^2 \).
Spatial Rotation
\( x' = x\cos\theta - y\sin\theta \)
\( y' = x\sin\theta + y\cos\theta \)
Uses circular functions (sin, cos)
Lorentz Boost
\( ct' = ct\cosh\phi - x\sinh\phi \)
\( x' = -ct\sinh\phi + x\cosh\phi \)
Uses hyperbolic functions (sinh, cosh)
The Rapidity φ
The parameter φ is called the rapidity: \( \tanh\phi = v/c = \beta \)
Unlike velocities, rapidities add linearly: \( \phi_{total} = \phi_1 + \phi_2 \)
This makes rapidity the natural "angle" for Lorentz transformations.
Summary: The Geometry of Special Relativity
• Space and time form a unified 4D continuum: spacetime
• The spacetime interval Δs² is invariant under Lorentz transformations
• Events are classified as timelike, lightlike, or spacelike
• Light cones separate causally connected events from "elsewhere"
• Worldlines trace paths through spacetime; their "length" is proper time
• The Minkowski metric ημν encodes spacetime geometry
• Lorentz boosts are hyperbolic rotations in spacetime
The Big Picture: Minkowski's geometric view transforms special relativity from a collection of strange effects (time dilation, length contraction) into a coherent geometric theory. This geometric perspective is essential for understanding general relativity, where spacetime becomes curved rather than flat.