Math 2.0

Relativity and gravitational theory Definition

Proper time

Builds on Minkowski spaceWorld lineSpacetime interval

Definition

Let MM be Minkowski space with metric η\eta of signature (−,+,+,+)(-, +, +, +), and let c>0c > 0 be the speed of light. Let σ ⁣:[a,b]→M\sigma \colon [a, b] \to M be a smooth parametrisation of a timelike world line. The proper time along σ\sigma from σ(a)\sigma(a) to σ(b)\sigma(b) is Δτ=1c∫ab−η(σ′(s),σ′(s)) ds.\Delta\tau = \frac{1}{c} \int_a^b \sqrt{-\eta(\sigma'(s), \sigma'(s))} \, ds. In an inertial frame, with the world line written as t↦(ct,𝐱(t))t \mapsto (ct, \mathbf{x}(t)) for t∈[t1,t2]t \in [t_1, t_2] and velocity 𝐯(t)=d𝐱/dt\mathbf{v}(t) = d\mathbf{x}/dt, this is Δτ=∫t1t21−∣𝐯(t)∣2c2 dt.\Delta\tau = \int_{t_1}^{t_2} \sqrt{1 - \frac{\abs{\mathbf{v}(t)}^2}{c^2}} \, dt.

Remarks

Example: for the straight world line σ(t)=(ct,vt,0,0)\sigma(t) = (ct, vt, 0, 0) with v=0.6cv = 0.6 c and t∈[0,10]t \in [0, 10], Δτ=∫0101−0.36 dt=8\Delta\tau = \int_0^{10} \sqrt{1 - 0.36} \, dt = 8. For a straight world line from pp to qq, Δτ=−s2/c\Delta\tau = \sqrt{-s^2}/c, where s2s^2 is the spacetime interval between pp and qq. Non-example: for the spacelike curve σ(s)=(0,s,0,0)\sigma(s) = (0, s, 0, 0) the integrand −η(σ′,σ′)=−1\sqrt{-\eta(\sigma', \sigma')} = \sqrt{-1} is not real, so proper time is not defined; such curves have a proper length instead. Proper time does not depend on the parametrisation, by the change of variables formula, and it is invariant under Poincaré transformations. Along a null world line the same integral is 00.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Minkowski space, concept, no evidence levelMinkowski spaceconceptSpacetime interval, concept, no evidence levelSpacetime intervalconceptWorld line, concept, no evidence levelWorld lineconceptProper time, definition, no evidence levelProper timedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Minkowski space concept
  2. Spacetime interval concept
  3. World line concept
  4. Proper time definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Proper_time
About this page
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Kinddefinition
WritingW0 stub
MSC83A05, 53B30
DefinesProper time
UsesMinkowski space, World line, Spacetime interval
Changed3 November 2026

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