REL-05

Gravitational time dilation

Δt = Δτ / √(1 − 2GM/(r c²)) for a static observer.

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General relativityEinstein 1915Pound–Rebka

Governing equation

\dfrac{\Delta t}{\Delta\tau}=\dfrac{1}{\sqrt{1-2GM/(rc^2)}}

$r \in \left[3, 1\times 10^{7}\right]$

where

M: Mass (suns) (M_\odot)
r: Radius (km)
\Delta\tau: Proper tick (s)
\Delta t: Coordinate time (s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\dfrac{\Delta t}{\Delta\tau}=\dfrac{1}{\sqrt{1-2GM/(rc^2)}}

$r \in \left[3, 1\times 10^{7}\right]$

$r = 3$ $\Delta t = 8.018$

Along the curve

②

Domain of the variable — blue

r_{\min} = 3,\quad r_{\max} = 1\times 10^{7}

$r \in \left[3, 1\times 10^{7}\right]$

$r = 3$ $r = 3$

Along the curve

③

Response curve — green

\dfrac{\Delta t}{\Delta\tau}=\dfrac{1}{\sqrt{1-2GM/(rc^2)}}

$r \in \left[3, 1\times 10^{7}\right]$

$r = 3$ $\Delta t = 8.018$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta t}{\mathrm{d}r}

$r \in \left[3, 1\times 10^{7}\right]$

$r = 62896.1$ $\Delta t^{\prime} = -5.58\times 10^{-5}$

Along the curve

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-05 — Gravitational time dilation) is the form associated with Einstein 1915 · Pound–Rebka. Working symbols: $M$ , $r$ , $\Delta\tau$ $\rightarrow$ $\Delta t$ . In Schwarzschild spacetime the metric coefficient gtt redshifts clocks deeper in a gravitational well, confirmed by Pound–Rebka and GPS.

Purpose

Purpose: compute

\Delta t

from

M

r

\Delta\tau

in Relativity via

\dfrac{\Delta t}{\Delta\tau}=\dfrac{1}{\sqrt{1-2GM/(rc^2)}}

Δt = Δτ / √(1 − 2GM/(r c²)) for a static observer. Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 1.000\,\mathrm{M_odot}

r = 7000.000\,\mathrm{km}

\Delta\tau = 1.000\,\mathrm{s}

, the governing relation

\dfrac{\Delta t}{\Delta\tau}=\dfrac{1}{\sqrt{1-2GM/(rc^2)}}

yields

\Delta t = 1.000211\,\mathrm{s}

. Static spherical mass, r > Rs. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass (suns) M1 M_\odotRadius r7000 kmProper tick \Delta\tau1 s

Outputs

Coordinate time \Delta t1.000211 s

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Narration of this film

Static spherical mass, r > Rs.

In Schwarzschild spacetime the metric coefficient gtt redshifts clocks deeper in a gravitational well, confirmed by Pound–Rebka and GPS.

Reading speed

Watch on YouTube