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Mathematics for Physics

203 Spacetime

A four-dimensional set of events that is used to label occurrences by three dimensions of space and one of time.

The four-dimensional continuum. Every physical occurrence is a point of one four-dimensional continuum with coordinates \((x, y, z)\) and time \(t\). A four-dimensional continuum is the arena in which those occurrences take place. This principle is used to treat position and time as labels of one geometric structure.

The spacetime coordinates of an event are

\[ (x, y, z, t) \]

where

  • \((x, y, z)\) are the spatial coordinates of the event.
  • \(t\) is the time of the event.

Frame-dependent splitting of space and time. The split of spacetime into space and time depends on the observer. A Lorentz transformation is the linear map that converts \((x, y, z, t)\) in one inertial frame into the coordinates of another inertial frame in uniform relative motion. This principle is used to show that simultaneity, length, and elapsed time change from one inertial observer to another.

Invariance of the interval. Distinct observers disagree on \(\Delta x\) and \(\Delta t\) yet agree on the spacetime interval. A spacetime interval is the invariant four-dimensional squared separation of two events. This principle is used to give an observer-independent measure of the separation of two events.

The spacetime interval is

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

where

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

Causal structure. Relative to one event, spacetime splits into an absolute past, an absolute future, and an elsewhere, bounded by a light cone. A light cone is the surface swept by all light rays through that event. This principle is used to decide which events can influence one another by signals no faster than light.

Note: Also called space-time. An individual point in spacetime is called an event.

203.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — spacetime coordinates of an event.
  2. Carroll, S. M. Spacetime and Geometry. Cambridge University Press, 2021. — spacetime as a four-dimensional set of events; light cones.
  3. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — unified spacetime; Lorentz mixing of coordinates; causal regions.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — invariant spacetime interval.