general relativity - Time Dilation Effects from simply being on a spinning planet orbiting a star in a rotating galaxy in an expanding universe.

I am a layman, so take this with a grain of salt.

I saw a TV show the other day which showed a Russian Cosmonaut who had spent more time in space than any other human. The relativistic effects of the low gravity and extreme speeds at which he had spent a decent part of his life had pushed him a small, but surprisingly non-trivial fraction of a second into the "future" as compared to the rest of us observers here on earth. I want to say it was something like a 50th of a second.

What, if any, are the relativistic effects all of us experience in an average lifespan simply by being on the earth as it travels through space in orbit and as the galaxy rotates and the universe expands.

By effects I mean as compared to a hypothetical observer who is able to remain completely motionless in space? Is it even measurable?

I know gravity is not being taken into account, so is this question even answerable?

Thanks for your patience with my question.

Answer

As the comments say, you have to be precise about your reference point when you talk about time dilation. Time dilation is always relative to something else.

But there is an obvious interpretation to your question. Suppose you have an observer well outside the Solar system and stationary with respect to the Sun. For that observer your clock on Earth is ticking slowly for two reasons:

1. 
   
   

   you're in a gravitional well so there is gravitational time dilation.

   
   


2. 
   

   you're on the Earth which is hurtling round the Sun at about (it varies with position in the orbit) 30 km/sec. The Earth's surface is also moving as the Earth rotates, but the maximum velocity (at the equator) is only 0.46 km/sec so it's small compared to the orbital velocity and we'll ignore it.

   
   

As it happens the problem of combined gravitational and Lotentz time dilation has been treated in the question How does time dilate in a gravitational field having a relative velocity of v with the field?, but this has some heavy maths so let's do a simplified calculation here.

The gravitational time dilation, i.e. the factor that time slows relative to the observer outside the Solar System is:

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

where $M$ is the mass of the object and $r$ is the distance from it.

For the Sun $M = 1.9891 \times 10^{30}$ kilograms and $r$ (the orbital radius of the Earth) $\approx 1.5 \times 10^{11}$ so the time dilation factor is $0.99999999017$.

For the Earth $M = 5.97219 \times 10^{24}$ kilograms and $r$ (the radius of the Earth) $\approx 6.4 \times 10^{6}$ so the time dilation factor is $0.999999999305$.

The Lorentz factor due to the Earth's orbital motion is:

$$\begin{align} \frac{1}{\gamma} &= \sqrt{1 - v^2/c^2} \\ &= 0.999999995 \end{align}$$

And to a first approximation we can simply multiply all these factors together to get the total time dilation factor:

$$ \frac{t}{t_0} = 0.999999984 $$

To put this into context, in a lifetime of three score and ten you on Earth would age about 34 seconds less than the observer watching from outside.