How slow would you age on a double gravity planet?
Gravity of the planet A is 10 units. If you are taken from planet A to planet B where gravity is double that of A, i.e. 20 units. How slowly would you age as compared to being on planet A, and how?
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asked Aug 7, 2021 at 22:53
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$\begingroup$ BTW, gravitational time dilation on a planet is quite small. Eg, the difference in the time rate between the Earth's surface and its centre is $\approx 3.48×10^{-10}$, so over a century a clock at the centre of the Earth would be slow by ~1.1 seconds compared to a clock on the surface. $\endgroup$– PM 2RingCommented Aug 8, 2021 at 0:03
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2 Answers
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A simple way to calculate the gravitational time dilation on the surface of a planet comes by assuming that
- the planet is spherically symmetric
- other sources of gravitational potential (a star, a moon) are negligible on the surface of the planet
In that case, the time experienced by the person on planet 1 is related to the time experienced by the person on planet 2 by
$${\Delta \tau_1 \over \Delta \tau_2} = \frac{\sqrt{1-\frac{2GM_1}{c^2R_1}}}{\sqrt{1-\frac{2GM_2}{c^2R_2}}}$$
On a planet, the factor ${2GM \over c^2R}$ is usually very small, so the formula can be approximated to
$${\Delta \tau_1 \over \Delta \tau_2} = (1-\frac{GM_1}{c^2R_1})(1+\frac{GM_2}{c^2R_2})$$
This is valid if the two planets are still with respect to each other. If they are moving at a substantial speed also SR time dilation should be considered. In any case, the effect is really small.
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1$\begingroup$ Just to extend this a little, if the density of the planets is the same, then surface acceleration is proportional to R, (and mass is proportional to R^3). Thus if the second planet has double the gravity, it would have double the radius, and eight times the mass. Plugging in the values for Earth mass and radius gives a ratio of 1.00000000209 times slower on the planet with more gravity. $\endgroup$– James KCommented Aug 8, 2021 at 19:38
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Gravitational time dilation depends on the gravitational potential, not the gravitational acceleration. It can't depend on the gravitational acceleration, because by the equivalence principle that can have any value you like, including zero for a free-falling observer.
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1$\begingroup$ Would be helpful if you can mention how can we calculate. Any equation? A little bit detail. Thanks. $\endgroup$ Commented Aug 7, 2021 at 23:34
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2$\begingroup$ @SazzadHissainKhan this answer to Time dilation between Mars and Earth due to different mass may be helpful. $\endgroup$– uhohCommented Aug 7, 2021 at 23:44
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1$\begingroup$ @SazzadHissainKhan Also see physics.stackexchange.com/a/69048/123208 & the various questions linked there. $\endgroup$– PM 2RingCommented Aug 7, 2021 at 23:58