By kamran

2016-05-17 03:35:23 8 Comments

Today the NASA International Space Station started the 100000 orbit after 17 years in the space.
I just wonder if there were a team of astronauts which were in the Lab for all the duration of last 17 years, how much older they would be due to the combination of Einstein's especial theory of relativity and general theory of relativity?
Here is the link to NASA's youtube video.


@Mr Bubble Hubble 2016-05-17 15:54:18

The first answer has all the results, but I will try to show some calculations, cause I have been writing them since there was no answer.

It is known from General Theory of Relativity (GTR) that the closer you are to a massive object - the slower the time goes. On the other hand Special Theory of Relativity (STR) gives us the next statement: the faster you move - the slower the time goes.

I can give here only a half of an answer due to my low knowledge of GTR.

Let's compare relative speeds of the ISS and a person on the surface of the Earth (on the equator):

We know that the orbit of ISS is approximatele 400 km. Knowing this we can calculate its speed: $$\frac{mV_1^2}{r+400000}=G\frac{mM}{(r+400000)^2}$$ where $r$ is a radius of the Earth, $M$-its Mass, $m$-mass of the ISS, $G$-Gravitational constant. From this we get $V_1=\sqrt{\frac{GM}{r+400000}}$

Also we can approximate the speed of a person on the surface of the Earth:$$V_2=r\frac{2\pi}{T}$$

So from theese two equations we can calculate the time dilation per second relatively to the center of the Earth for each case: $$\tau_1=\frac{1}{\sqrt{1-\frac{V_1^2}{c^2}}}, \tau_2=\frac{1}{\sqrt{1-\frac{V_2^2}{c^2}}}$$

Then we can calculate the difference and multuply it on 17 years:$$|\Delta \tau|*17*365*24*60*60=0.174634$$

For better understanding we can look at the problem from different angle i.e. Lorentz contraction (effect of STR) or contraction of the distances in the direction of speed:

Let the orbit length be $l=2 \pi (r+400000)$ With every turn the ISS will cover the distance equal to $l_1=l \sqrt{1-\frac{V_1^2}{c^2}}$. Then we just calculate the differenc between 2 times $$T_1=\frac{100000l}{V_1}, T_2=\frac{100000l_1}{V_1}$$ We get that $\Delta T \approx 0.17$

Note:we have only paid respect to time dilation due to STR. The GTR requires more complex calculations but it will shift our answer a bit to the direction of zero, because it will slow the time more for those who are closer to the center of the Earth.

@kamran 2016-05-18 08:16:58

As @stephane said we have to take the effect of acceleration due to both gravity and orbital circulation! my guess is the time space of astronauts has 2 curves, possibly working at opposing directions. in an exaggerated way one can imagine when you floor the pedal in a car you are buying time! and if you go around circles you lose time?

@Mr Bubble Hubble 2016-05-18 08:35:49

@kamran I understood what you are talking about. As I can suppose that it is not true, cause time dialation depends only on the speed: a clocks with a high speed have slower timeflow than the clocks in the rest. And it doesnt depend on the moving in circles or in a straight line. You wanted to say not orbital circulation but orbital speed.

@kamran 2016-05-18 15:08:26

There is centripetal acceleration: A=r. omega^2. it has to be accounted for. According to GR space-time is shaped and modified by mass, momentum, acceleration, pressure...

@Mr Bubble Hubble 2016-05-18 16:34:56

@kamran ok. I am telling that I am not so deep in GTR. But I can say that the effect of it according to known figures and calculations I made is just approximately 0.02 sec in 17 years, whereas STR gives 0.15 sec.

@Faisal Iqbal Baba 2016-05-18 05:21:17

our aging is directly proportional to metabolism of body and division,growth and death of cells of body and if gravity has some effect on the rate of above aspects astronauts will be definitely younger

@Stéphane Rollandin 2016-05-17 10:02:26

This is anwered in Gravity on the International Space Station - General Relativity perspective, where we learn that time dilation in the ISS with respect to Earth equator is 1.00000000028655.

So after 17 years for us, the astronauts would come back younger by about 0.15 seconds than if they have stayed on the ground.

Note that a full GR treatment is required. Now it looks like NASA people (well, at least one of them) break the calculation into a SR part concerned with velocity and a GR part accounting for the difference in gravity. This approximation gives a result of about 0.16 seconds after 17 years, close but different.

@kamran 2016-05-17 19:24:30

Thank you very much. My only question is in a case like this if the astronauts use the Greenwich meridian as a marker for orbital coordinate angle when they just cross it for hundred k times, are they going to be 0.17 seconds too early?

@Stéphane Rollandin 2016-05-17 19:32:07

Since they are in an accelerating frame relatively to us, what they see as simultaneous spacelike separated events is not what we see as such, so I am not too sure about the proper way to formulate that question, and even less about how to answer it. General relativity is tricky.

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