Gravitational Time Dilation Calculator

In the Schwarzschild solution to Einstein's field equations, the metric for a non-rotating, uncharged, spherically symmetric mass is:

$$ds^2 = \left(1 - \frac{r_s}{r}\right)c^2\,dt^2 - \left(1 - \frac{r_s}{r}\right)^{-1}dr^2 - r^2\,d\Omega^2$$

where $$r_s = \frac{2GM}{c^2}$$ is the Schwarzschild radius. For a stationary observer at radius $$r$$, the relationship between proper time $$t_0$$ (time measured locally) and coordinate time $$t$$ (time measured by a distant observer) is:

$$t_0 = t\sqrt{1 - \frac{r_s}{r}}$$

Rearranging gives the observed (coordinate) time:

$$t = \frac{t_0}{\sqrt{1 - \frac{2GM}{rc^2}}}$$

Key physical insights:

• Closer to mass → slower clocks: As $$r$$ decreases toward $$r_s$$, the square root factor approaches zero, and the dilation factor diverges — time effectively stops at the event horizon.
• Weak-field approximation: For $$r \gg r_s$$ (like Earth's surface), $$\Delta t \approx t_0 \cdot \frac{GM}{rc^2}$$, giving tiny but measurable differences.
• Schwarzschild radius: The critical radius $$r_s = 2GM/c^2$$ defines the event horizon of a black hole. For Earth, $$r_s \approx 8.87\,\text{mm}$$; for the Sun, $$r_s \approx 2.95\,\text{km}$$.
• GPS correction: Clocks on GPS satellites (altitude ~20,200 km) run faster by about 45 μs/day due to weaker gravity, partially offset by ~7 μs/day of special-relativistic time dilation from orbital velocity.

The Observed Time is the duration measured by an observer far from the gravitational source. It is always longer than the proper time $$t_0$$, meaning a clock deep in a gravitational well runs slower from an outside perspective. The Time Difference shows the absolute offset. The Dilation Factor gives the multiplicative ratio — values very close to 1 indicate weak-field conditions (planets), while values significantly greater than 1 indicate extreme gravity (neutron stars, black holes). If the radial distance approaches the Schwarzschild radius, the dilation factor grows without bound, signaling the event horizon.