Einstein’s theory of relativity turns 100 years old this week. Due to time dilation, how much time has passed elsewhere in the solar system?
Short answer: Almost exactly 100 years, give or take a few seconds.
Long answer: On November 25, 1915 Einstein presented the final form his field equations to Royal Prussian Academy of Sciences, introducing general relativity to the world. 
Among other weird consequences, Einstein’s theory of relativity says that time does not pass at the same rate everywhere. Put simply, people standing in different places or people moving around will measure different amounts of time passing. For example, experiments have been done with incredibly accurate atomic clocks to test this – two clocks initially synchronized can be brought out of sync by moving one relative to the other, such as by putting it on a plane. This isn’t some trick of broken clocks, or some kind of optical illusion. In a very real sense, this means less time passed for the moving clock. 
This kinematic time dilation isn’t the full story though. That was all predicted in Einstein’s special theory of relativity, which describes motion without gravity and was published 10 years earlier. Einstein’s general relativity extends the theory to motion in gravitational fields as well. While special relativistic time dilation is due to relative motion, general relativistic time dilation is gravitational. Proximity to a massive compact body (like the earth) will further slow time compared to more distant observers, so gravity slows time just like relative motion.
In the ‘clocks on a plane’ experiment the effect of gravity on the clocks is comparable to the effects of the plane’s motion and they mostly counteract each other, and while these effects are only a matter of nanoseconds per day (weird units, right?), they’re still very real.
With the centennial of general relativity upon us it might be fun to calculate how long these hundred years on earth have been elsewhere in the solar system using the combined effects of kinematic time dilation and gravitational time dilation.
How much time has passed in deep space, far from the effects of the earth gravity? Unlike the ‘clocks on a plane’ experiment the effects of relative motion and gravitational time dilation are going to work together in this thought experiment. Let’s just consider the effects of the following on time dilation between the earth and deep space:
- Gravitational time dilation due to the earth’s mass
- Gravitational time dilation due to the sun’s mass
- Kinematic time dilation due to orbital motion
General relativity is seldom easy, so the equation we find will be approximate – in reality the annual variation of the earth’s motion around the sun has an effect, as well as the rotation of the planet, as well as the gravity of the other planets… but these effects are hard to calculate directly. Fortunately, they’re all tiny compared to the three above.
Using the Schwarzchild metric, a specific solution to Einstein’s equations for a spherical gravitating body, we can get a simple equation for time dilation between an observer on the earth and a distant observer who is ‘looking down’ on the solar system and is at rest with respect to the sun:
Where dtE is the 100 years on earth, dtc is the deep space time, ME and RE are the mass and radius of the earth, M☉ and rE are the mass of the sun and the earth’s orbital radius, v is the earth’s orbital velocity, and c and G are constants. By remembering that orbital velocities are determined by the mass of the central body, we can simplify this expression:
Using this relation, we find that for exactly 100 years of earth time, there will be 100 years and 25.5 seconds of time passed in the distant frame . Were you expecting more? The things in the solar system are just too small and slow to get more than a tiny effect from general relativity.
Now that we know how much time has passed in a frame far from the solar system, we can use the equation above going the other direction to find how much time has passed on any given body in the solar system!
Where i is the body of interest. Let’s start with the Sun; that’s an easy example because we can drop that second term (that’s an orbital term, and the sun doesn’t orbit itself!), and due to the sun’s large mass it will experience the most gravitational time dilation of any body in the solar system. For 100 years on the earth, the sun will experience 100 years – 111 minutes. This means the sun loses an extra 66 seconds for every year on earth.
As another point of comparison, Mars has a slightly smaller mass than earth and has a slightly more distant orbit, so for every 100 hundred years on earth, Mars surface gets 100 years + 9.7 seconds. For every year Mark Watney spent on Mars, he only aged 97 milliseconds seconds more than someone on earth.
image credit: Wikimedia Commons
special thanks to /u/rantonels
asked by Lukas Karner
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