proper time

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Proper time is an invariant quantity that does not depend on the reference frame. It equals the time measured by a clock moving with the object. For a stationary observer, the time of such an object runs slower, and the effect is stronger the higher the speed or the closer a massive body. In everyday life, the slowing is imperceptible, but it is detected by precise atomic clocks.

History

The idea of time dilation was proposed by Hendrik Lorentz, and Albert Einstein created the special theory of relativity in 1905, which explained this effect.

How it works

According to Einstein's theory, the speed of light is an absolute limit, and time is forced to slow down for fast objects so they do not overtake light. The closer the speed is to light speed, the more noticeable the difference.

💡 An astronaut who spends six months on the International Space Station ages 0.005 seconds less than his peers on Earth.

In special relativity, proper time τ is the length of a world line in Minkowski space, invariant under Lorentz transformations. In general relativity, the definition is generalized through the metric tensor: proper time is measured along a timelike geodesic and depends on the gravitational potential. For a Schwarzschild black hole, the rate of proper time slows down when approaching the event horizon. Physically, proper time is realized in any local periodic processes.

History

The concept of local time appeared in the works of Hendrik Lorentz (1895) and Henri Poincaré (1900) as an auxiliary quantity for electrodynamics. In 1905, Albert Einstein gave it fundamental status in special relativity, and Hermann Minkowski geometrized it as world line length in 1908. In 1915, in general relativity, proper time became the basis for describing gravity. Experimental confirmations were obtained in the Hafele–Keating experiments with atomic clocks (1971).

How it works

In special relativity, proper time τ is related to coordinate time t by dτ = dt √(1 – v²/c²). As v → c, proper time slows sharply. In general relativity, the metric tensor is used: dτ² = –g₀₀ dt², for example, in the exterior Schwarzschild solution g₀₀ = –(1 – rₛ/r), and as r → rₛ (event horizon), proper time tends to zero. Time dilation for moving particles is confirmed by the lifetime of muons in the atmosphere.

\tau = t \sqrt{1 - \frac{v^2}{c^2}}

τ — proper time, t — coordinate time, v — object speed, c — speed of light (c ≈ 3×10⁸ m/s)

d\tau = \sqrt{-g_{00}} \, dt

g₀₀ — temporal component of the metric tensor (for the Schwarzschild metric g₀₀ = –(1 – 2GM/(rc²)), G — gravitational constant, M — mass, r — distance to the center, c — speed of light)

Open problems & fun facts

Compatibility of proper time with quantum mechanics: should time be quantized at the Planck scale?; Are closed timelike curves possible without violating causality in quantum gravity?

💡 An experiment with cosmic-ray muons shows that due to relativistic time dilation, the particles manage to fly through the atmosphere, although by laboratory standards they should have decayed earlier.

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proper time

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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