Robert McNees
The acceleration due to gravity near the surface of the Earth is about 9.8 m/s^2.
Our bodies are basically bags of fluid adapted to doing stuff under conditions that want us to experience this acceleration.
(Happy Valentine's day, bag of fluid! Love, gravity)
A delightful numerical coincidence (and it's nothing more than that) is that this acceleration, when expressed in light-years / year^{2}, happens to be very, very close to 1.
g = 1.03 light-years / year^2
Robert McNees
Why is this delightful?
This came up just now with some students I'm working with.
Imagine engineering a spacecraft that could take you to a very distant destination. It accelerates at g for the first half of the trip, then flips over and decelerates at g for the second half. It does the same thing on the way home.
(This is completely impractical, don't take it seriously.)
That would be ideal. You – the fluid bag who likes Earth gravity – would have natural conditions the whole trip.
Robert McNees
Because of the rules of special relativity, you'd spend much of the trip traveling at a speed very close to the speed of light. (You'd never reach it, though.)
As a result, very little time would pass in your frame of reference, as opposed to my frame of reference back here on Earth.
So I stay here on Earth. In my frame of reference, the time it takes for you to complete a round trip to a destination D light-years away, in years, is
t = 2 D \sqrt{ 1 + 4 /(1.03 D) }
Robert McNees
If D is 4 light years, I say it takes you about 11.2 years to get there and make it back.
For a destination 10 light years away, I say the round trip takes you about 23.6 years of my life.
Above that, it's basically just 2 D, in years.
But for *you*, the amount of proper time that passes on your journey is much less than what I experience. The amount of time that you experience on the round trip, in years, is
\tau = (4/1.03) ArcCosh( 1 + 1.03 D / 2 )
Robert McNees
For short distances, since g is so close to 1, this is roughly 4 \sqrt{D} years. But for large distances the ArcCosh behaves like a log, and we get
\tau ~ 3.88 log (D)
So if we could engineer a trip like this (we can't), you would make a trip of 10^5 light-years (the diameter of the Milky Way) and back in about 45 years.
Robert McNees
Because log(D) grows so slowly with distance, it means you could travel so far in a human lifetime that special relativity would no longer be a good approximation. We'd have to pay attention to the large scale dynamics of the Universe.
The Geoff
@mcnees I played with this from a science fiction angle and worked out that much of the observable universe is within a human lifetime from the naive version of the maths. I also like to fall asleep imagining the pressure from the mattress isn't gravity, but a spaceship's drive. It's surprising how far you can travel on 8 hours sleep...
Robert McNees
@_thegeoff Yeah, 100 years is about a 1/H_0 round trip if you ignore other stuff.
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