At first glance this may seem like a dumb question (well pretty much every question in this paper is, but that is besides the point), how far away you are from your place of birth depends on who you are and where you are. But it actually a bit more complicated than that (actually a lot more complicated). That’s because the earth orbits around the sun. so if you were born on June 25th, and its currently December 25th, then you are quiet far from where you were born.
But it gets even worse, because the sun itself “orbits” the milky way galaxy at a whopping speed of 230 km/s. AND the universe is expanding, “faster” than the speed of light which makes you even MORE further away from your birthplace. Now of course to get to any specific number, we will need to know your age, and we are going to suppose you are a 13 year old teen with nothing better to do than to know how far away your birthplace is. In our calculation, we will ignore the aforementioned extra distance caused by the orbit of earth around the sun, as the values we are going to be calculating are so darn big, those distances are negligible. So if you are 13 year old, that means you are going:
230 km in 1 second
= 828000 km in 1 hour
= 19872000 km in 1 day
= 7253280000 km in 1 year
= 94292640000 km in 13 years
km far from your birthplace in 13 years, or ~630.3 au. For context, 1 au is the average distance between the sun and the earth, and our solar system is 180 au wide. But we are not done yet as the milky way is moving 600 km/s in reference to other galaxies. So after 13 years, using the same method as we did before, we would be ~1644.28 au from the place of birth. Now we can add the previous values to get ~2274.58 au, since the sun is orbiting in the same direction the milky way is moving. But we are still not done as we need to include the expansion of the universe, as it pulls stuff away from each other. Now the universe is expanding at a rate of 67.5 km/s/mpc (note: this is pronounced as kilometer per second per megaparsec). A parsec is a unit of distance roughly equivalent to ~3.27 light years. A megaparsec is 1 million parsecs, or 3,262,000 light years. The rate, 67.5 km/s/mpc means that every second, things that are 1 megaparsec away from each other is going to be 67.5 kilometer further away from each other. This value is called the Hubble’s constant. Now we can find the new distance between those two points by using the formula:
new distance= d × eH×t
Where:
d is the distance, without accounting for the expansion of the universe
e is Euler’s number approximately equal to 2.71828…
H is the aforementioned Hubble’s constant which is equal to 67.5 km/s/mpc
t is time
Now it is incredibly important to keep the units consistent, and since we are working with au, km and seconds we have to convert the Hubble’s constant to km/s/au. Converting we get, 3.27249 x 10-10 km/s/au. So plugging the values into the equation we get:
new distance=2274.58 × e(3.2749 × 10^(-11)×4.1 × 10^8 )
= 2601.36 au
So in total if you are a 13 year old, you are 2601.36 au from your place of birth or 389,157,917,000 (389 billion, 157 million) km away from your place of birth. Needless to say, you’re not getting back there. One thing is that these values are approximate and could off by a few couple au’s, but this calculation should give you a good idea for how large and weird the universe is.
But at the start, and in pretty much any science textbook, it is said that the universe is expanding faster than the speed of light, but 67.5 km/s/mpc is definitely not faster than the speed of light which is 299792.458 km/s, so did they lie to us? The answer is technically both yes and no. the reason is, things that are 1 mpc away from each other will move apart at a relative speed of 67.5 km every second, but this separation will increase as the distance increases. So things that are 2 mpc away will move 135 km away from each other every second and so on. So if you do some calculations, you will find that things that are 46.5 billion light-years apart will move apart at the speed of light, and anything above that distance will move "faster" than the speed of light. Now the distance 46.5 billion light-years just so happens to be the radius of the observable universe, and this is no coincidence. This is because, light will travel at the speed of light, at all times, and nothing can go faster than that. So light emitted by some star further away than the radius of the universe, can’t ever reach us, since its moving apart, faster than it is travelling. This essentially gives the universe a border, beyond which we don’t know what exist, and the best part is, we cannot know what exists. There could be even more universe, aliens, raining toilets, heaven/hell or just simply nothing, we don’t know. Now one last thing is that nothing can travel faster than the speed of light, so what is happening here? The reason this happens is because the objects themselves aren’t moving, but the space in between them is getting bigger, and so to us it seems like it’s moving. As space can grow at whatever speed it likes, and since nothing is truly moving, only getting further apart, which isn’t exactly intuitive, but it this is getting long enough as is, so Einstein can rest easy knowing his laws are still true (for now).