Will my wrist watch slow down on planet Jupiter?

[Benjamin_harsh]

Homework Statement

Does my wrist watch slows down on planet Jupiter?

Relevant Equations

Does my wrist watch slows down on planet Jupiter?

Does my wrist watch slows down on planet Jupiter? I know Jupiter's gravity is greater than Earth's gravity.

 

[phinds]

Problem Statement: Does my wrist watch slows down on planet Jupiter?
Relevant Equations: Does my wrist watch slows down on planet Jupiter?

Does my wrist watch slows down on planet Jupiter? I know Jupiter's gravity is greater than Earth's gravity.

No. That is a common misconception. To a person on Earth it will look like your watch has slowed down but to you it will not because it HAS not. That misconception confuses gravitational time dilation (appearance of rate change) with differential aging. If you go to Jupiter, stay there a while, and then return to Earth you will be a bit younger than someone who just stays on Earth but that's differential aging, not an actual slowing of your clock or your biological processes. It will happen because you've taken a different path through space-time.

 

[jbriggs444]

I know Jupiter's gravity is greater than Earth's gravity.

It would appear that you are concerned with gravitational time dilation. You should know that gravitational time dilation does not scale with the local acceleration of gravity (e.g. Jupiter being approximately 3 g's). Instead, it scales with gravitational potential.

Gravitational potential at a planet's surface is given (in the Newtonian approximation) by $-\frac{g_s}{r}$ where $g_s$ is the surface gravity and $r$ is the planet's radius. The gravitational potential at Jupiter's surface is about 29 times Earth's potential at its surface.

This does not mean that time passes 29 times as slowly on Jupiter. It does mean that if you look at the time dilation rates relative to hypothetical clocks "at infinity" using a reasonable coordinate system you might find something like 1.0000001 at Earth and 1.0000029 at Jupiter. (Numbers made up).

 

[Filip Larsen]

Allow me to add some engineer hand-waving.

If the gravitational dilation factor $f = t_0/t_\infty$ (i.e. local clock rate relative to clock rate at infinite) at distance $r$ from the center of a spherical gravitational field of mass $m$ can be found as $$f = \sqrt{1-\frac{2 G m}{r c^2}}$$ then for $f$ near 1 we can, to ease the numerical work, use the approximation $$f \approx 1 - \frac{Gm}{r c^2} = 1 - k, $$ where we now can work with $k$ values instead.

Further, we can get the dilation factor for the surface of a planet around a star relative to infinity, for instance for Earth and Jupiter in orbit around the Sun, by multiplying the factor for position in the planets field with the factor for the position in the stars field.

If I punch those $k$-numbers into a spreadsheet then for Earth I get $k_E \approx 1.1\cdot10^{-8}$ (combined from a surface and orbit $k$ around $6.9\cdot10^{-10}$ and $9.8\cdot 10^{-9}$) and for Jupiter I get $k_J \approx 2.3\cdot 10^{-8}$ (combined from a surface and orbit $k$ around $2.1\cdot 10^{-8}$ and $1.9\cdot 10^{-9}$).

Dividing the dilation factors for Earth and Jupiter (both relative to infinity) means that the relative clock rate at Jupiter surface relative to Earth surface should be $t_J/t_E \approx 1-1.2\cdot 10^{-8}.$ That means a clock on Jupiter surface would loose around 1 sec every 930 days relative to a clock on Earth due to gravitational dilation alone.

 

[DaveC426913]

Allow me to add some engineer hand-waving.

Amusing turn of a phrase.
An oxymoron, I'd say.

 

[jedishrfu]

Amusing turn of a phrase.
An oxymoron, I'd say.

Except if the engineer is driving a train then hand waving may mean bye bye or get off the track depending on where you are standing.

 

[Filip Larsen]

An oxymoron, I'd say.

Indeed, but my intent was to qualify my hand-waving so it wasn't confused with the sometimes more rigorously hand-waving by physicists and especially mathematicians.

Except if the engineer is driving a train then hand waving may mean bye bye or get off the track depending on where you are standing.

Right, I could see that particular oppertunity for confusion arise if thread had employed a relativistic steam train, as such threads on relativity often tend to do. But, luckily, for once, there was no train involved, only clocks and I am not sure clocksmiths have any tradition of waving their hand in the course of their work, quite the opposite I suspect.