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The spacetime interval

The one combination of space and time separations that every inertial observer agrees on — and how it sorts pairs of events into spacelike, timelike and lightlike.

Follows MIT 8.286 lecture 12MIT 8.286 Lecture 12: Non-Euclidean Spaces: Open Universes and the Spacetime MetricOn MIT OpenCourseWare: the lecture and its readings (opens in a new tab)

You need this first

Full reading path (20 pages)
  1. Vectors (8.01)
  2. Position and velocity in one dimension (8.01)
  3. The Doppler effect and special relativity (8.286)
  4. Acceleration in one dimension (8.01)
  5. Motion in two dimensions and projectiles (8.01)
  6. Newton's laws of motion (8.01)
  7. The universal law of gravitation (8.01)
  8. The homogeneous expanding universe (8.286)
  9. Why a static universe cannot work (8.286)
  10. The expanding sphere and the equation for a(t) (8.286)
  11. The Friedmann equation (8.286)
  12. Open, closed and flat universes (8.286)
  13. The closed universe as a cycloid (8.286)
  14. The age of the universe and the Big Bang singularity (8.286)
  15. The age of a closed universe from H and Ω (8.286)
  16. The open universe and the age of any matter universe (8.286)
  17. Non-Euclidean geometry (8.286)
  18. The metric of a sphere and of a closed universe (8.286)
  19. Curvature and expansion — the Robertson–Walker metric (8.286)
  20. The geometry of open universes (8.286)
On this page · 4 sections
  1. Events and the interval
  2. Three kinds of separation
  3. The infinitesimal interval
  4. Summary

General relativity is a theory of the geometry of spacetime, not just of space. So after the geometry of open, flat and closed universes, the end of Lecture 12 adds time. The starting point is a fact from special relativity, which Guth takes as given (an appendix to 8.286's Lecture Notes 5 derives it).

Events and the interval#

An event is a point in spacetime: a place and a time, like the snap of your fingers (ideally shrunk to a single point). For two events aa and bb define the interval:

The spacetime interval
sab2≡(xa−xb)2+(ya−yb)2+(za−zb)2−c2(ta−tb)2s_{ab}^2 \equiv (x_a - x_b)^2 + (y_a - y_b)^2 + (z_a - z_b)^2 - c^2(t_a - t_b)^2
In words

s sub ab squared is defined as (x sub a minus x sub b ) squared plus (y sub a minus y sub b ) squared plus (z sub a minus z sub b ) squared minus c squared (t sub a minus t sub b ) squared

Source: 8.286 Lecture 12 transcript, p. 23

Different inertial observers, related by Lorentz transformations, in general disagree about the four differences (a boost along xx changes Δx\Delta x and Δt\Delta t, for example). Yet they all compute the same sab2s_{ab}^2: the interval is Lorentz invariant. It is to spacetime what distance is to ordinary space.

Three kinds of separation#

The sign of sab2s_{ab}^2 is therefore something every observer agrees on, and it sorts pairs of events into three kinds.

Spacelike, s2>0s^2 > 0. There is a frame in which the two events happen at the same time. In that frame ss is just the ordinary distance between them, called the proper distance.

Timelike, s2<0s^2 < 0. There is a frame in which the two events happen at the same place — think of a strobe light moving at constant velocity, flashing twice. In that frame only time separates them:

Proper time
sab2=−c2 τab2s_{ab}^2 = -c^2\,\tau_{ab}^2
In words

s sub ab squared equals minus c squared tau sub ab squared

Source: 8.286 Lecture 12 transcript, p. 25

τab\tau_{ab} is the proper time: what a wristwatch carried from one event to the other at constant velocity would read.

Lightlike, s2=0s^2 = 0. A light pulse could travel from one event to the other (in one direction only). For such pairs there is no invariant notion of how far apart they are: observers moving along the light ray can make the separation look arbitrarily small or large.

Event a is at the origin, event b at x = 5, ct = 3 light-seconds: s² = 16, spacelike. Move the slider: an observer moving at β regards the events on the line ct = βx as simultaneous with a. At β = 0.6 the line passes through b — in that frame the two events happen at the same time, 4 light-seconds apart.
Worked example

Finding the special frame

Events aa at the origin and bb at x=5x = 5, ct=3ct = 3 (light-seconds). Show that they are spacelike, find the proper distance, and find the speed of the frame in which they are simultaneous.

Interval

s2=52−32=16>0s^2 = 5^2 - 3^2 = 16 > 0: spacelike, with proper distance s=4s = 4 light-seconds.

The frame

In a frame moving at β\beta along xx, c t′=γ(ct−βx)c\,t' = \gamma(ct - \beta x). Setting t′=0t' = 0 for event bb: 3−5β=03 - 5\beta = 0, so β=0.6\beta = 0.6.

Check

In that frame γ=1.25\gamma = 1.25 and x′=γ(x−βct)=1.25(5−1.8)=4x' = \gamma(x - \beta ct) = 1.25(5 - 1.8) = 4: exactly the proper distance, as promised.

The timelike twin

Swap the numbers, x=3x = 3, ct=5ct = 5: s2=9−25=−16s^2 = 9 - 25 = -16, timelike, with proper time τ=4\tau = 4 s, measured in a frame moving at β=3/5\beta = 3/5 in which both events happen at the same place.

The infinitesimal interval#

For two events infinitesimally close together, the interval becomes

The interval between nearby events
ds2=dx2+dy2+dz2−c2 dt2ds^2 = dx^2 + dy^2 + dz^2 - c^2\,dt^2
In words

ds squared equals dx squared plus dy squared plus dz squared minus c squared dt squared

Source: 8.286 Lecture 12 transcript, p. 27

This is the spacetime version of Gauss's metric: three space directions add, time subtracts. Mathematicians say it has three positive and one negative eigenvalue — it is not positive definite, unlike the metric of a space. (Some books use the opposite overall sign.) Combining this with the Robertson–Walker metric gives the spacetime metric of the expanding universe, which is where the next lecture begins.

Summary#

  • An event is a point in spacetime; the interval s2=Δx2+Δy2+Δz2−c2Δt2s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2\Delta t^2 is the same for every inertial observer.
  • s2>0s^2 > 0: spacelike (simultaneous in some frame; ss = proper distance). s2<0s^2 < 0: timelike (same place in some frame; s2=−c2τ2s^2 = -c^2\tau^2, τ\tau = proper time). s2=0s^2 = 0: lightlike.
  • The infinitesimal form ds2=dx2+dy2+dz2−c2dt2ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2 has signature (+, +, +, −) and is the starting point for spacetime metrics in general relativity.

Symbols on this page

Show the 8 symbols used here
β\beta beta
Speed as a fraction of the speed of light, β = v/c. Say it: “beta” (sounds like “bay-ta or bee-ta”).
γ\gamma gamma
The Lorentz factor, γ = 1/√(1 − β²). Say it: “gamma”.
Δ\Delta capital delta
A change or difference in a quantity, e.g. Δx = x₂ − x₁. Say it: “capital delta”.
τ\tau tau
Proper time: the time measured by a clock moving with the object. Say it: “tau” (sounds like “rhymes with cow”).
≡\equiv identical to
Is defined as (or is identically equal to). Say it: “is defined as”.
aa a
A label for an event (events a and b). Say it: “a”.
zz z
A Cartesian coordinate, with x, y (and w on the 3-sphere). Say it: “z”.
cc c
The speed of light, c ≈ 2.998 × 10⁸ m/s. Say it: “c”.

Tap or click a symbol in a formula to see the same explanation. All symbols: symbols and notation.

Where this leads

Pages that build on this

Check your understanding

  1. Two events have Δx=5\Delta x = 5 light-seconds and cΔt=3c\Delta t = 3 light-seconds. What kind of separation is this?
  2. For a timelike pair of events, s2=−c2τ2s^2 = -c^2\tau^2. What is τ\tau?
  3. Why is there no invariant 'distance' between two lightlike-separated events?
  4. What is the signature of the spacetime interval ds2=dx2+dy2+dz2−c2dt2ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2?
0 of 4 answered

Flashcards

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