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Mathematics for Physics

204 Spacetime Interval

A scalar formed from the time and space separations of two events that is used to compare separations that all inertial observers agree on.

Invariance of the interval. The spacetime interval between two events has the same value for every inertial observer. An inertial observer is an experimenter at rest in an inertial frame. This principle is used to give a frame-independent measure of the separation of two events.

The spacetime interval is

\[ s^{2} = c^{2}(\Delta t)^{2} - (\Delta x)^{2} \]

where

  • \(\Delta t\) is the time separation of the two events.
  • \(\Delta x\) is the spatial separation of the two events.
  • \(c\) is the speed of light.
  • \(s\) is the spacetime interval.

Classification of separations. The sign of \(s^{2}\) sorts pairs of events into timelike, spacelike, and lightlike separations. A timelike pair can be joined by a slower-than-light signal. A spacelike pair cannot be joined by any causal signal. A lightlike pair can be joined only by light. This principle is used to classify which events can influence one another.

Protection of causal order. If two events are joined by a signal no faster than light, every inertial frame agrees on which happened first. Causality is the requirement that a cause precede its effect. This principle is used to keep cause and effect in the same order for every inertial observer.

Proper time. Along a timelike path the interval measures the time on a clock that travels that path. Proper time is the time read by a clock present at each event of the path. This principle is used to compute the elapsed time of a traveler between two events.

Note: Also called the invariant interval. Also called the interval. Also called the line element when written as \(ds^{2}\).

204.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(s^{2}=c^{2}(\Delta t)^{2}-(\Delta x)^{2}\) as an invariant; proper time.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — causal order of timelike and lightlike pairs.
  3. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — frame independence of the interval.
  4. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — classification of separations; proper time along a timelike path.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — invariant interval and causal regions.