# Can you solve a time dilation math problem?

March 10, 2012 1:32 PM Subscribe

I'm writing a story and it has to do with time dilation and I don't want it to be one of those stories where people roll their eyes and say, "Well... It's just a story." I want accurate math. I suck at math.

Here are the parameters. The story I want to tell is about a man who travels away from the Earth at near-light speed and returns almost exactly 50 years later. His total journey time from his perspective should be roughly 18 months there and back. It isn't really important how long he stays at his destination, so for the sake of argument, he can spend as little as 30 minutes before he starts his journey home.

Can he make it to ANY body/object outside the solar system?

Ideally the answer I'm looking for is this:

1 Earth to destination at .999 the speed of light is X days.

2 Hang out at destination is Y days.

3 Destination to Earth at .999 the speed of light is Z days. (I suspect X=Z but you tell me)

So X+Y+Z= 18 months(roughly) and Earth is now 50 years older(roughly).

Here are the parameters. The story I want to tell is about a man who travels away from the Earth at near-light speed and returns almost exactly 50 years later. His total journey time from his perspective should be roughly 18 months there and back. It isn't really important how long he stays at his destination, so for the sake of argument, he can spend as little as 30 minutes before he starts his journey home.

Can he make it to ANY body/object outside the solar system?

Ideally the answer I'm looking for is this:

1 Earth to destination at .999 the speed of light is X days.

2 Hang out at destination is Y days.

3 Destination to Earth at .999 the speed of light is Z days. (I suspect X=Z but you tell me)

So X+Y+Z= 18 months(roughly) and Earth is now 50 years older(roughly).

You're not going fast enough. According to this site the Relativistic Change Factor for .999 of the speed of light is only 22.36627204212937.

Crunch the numbers and you realize that 18 months of .999 speed of light travel is equal to 33 years 6 months and 18 days.

You'd need to travel faster than 0.99955 times the speed of light to get the Relativistic Change Factor up to 33 1/3 so that 18 months = 50 years.

Someone else can answer this.

posted by zinon at 1:52 PM on March 10, 2012 [1 favorite]

Crunch the numbers and you realize that 18 months of .999 speed of light travel is equal to 33 years 6 months and 18 days.

You'd need to travel faster than 0.99955 times the speed of light to get the Relativistic Change Factor up to 33 1/3 so that 18 months = 50 years.

*Can he make it to ANY body/object outside the solar system?*Someone else can answer this.

posted by zinon at 1:52 PM on March 10, 2012 [1 favorite]

Looks like you've got a good handfull of stars he can easily make it to but still travel far enough. I don't know how interesting any of them are.

posted by cmoj at 2:18 PM on March 10, 2012

posted by cmoj at 2:18 PM on March 10, 2012

You may also want to consider whether the mechanism your character has for light speed travel also requires acceleration and deceleration at either end, as this will lower the average speed and affect the time dilation to a potentially significant extent.

posted by fearnothing at 4:52 PM on March 10, 2012

posted by fearnothing at 4:52 PM on March 10, 2012

All excellent and helpful.

"Magic handwave" is an apt term for this story. The character does not understand the science involved and uses the ship much in the way an average person uses their car. For this reason, I'll also avoid the issues of propulsion and mass gain.

You'll also need to excuse me, I used .999 as shorthand to express that I know he will not be and can't break the speed of light. I didn't mean to imply that I had already chosen the speed he will be traveling.

posted by CarlRossi at 5:45 PM on March 10, 2012

"Magic handwave" is an apt term for this story. The character does not understand the science involved and uses the ship much in the way an average person uses their car. For this reason, I'll also avoid the issues of propulsion and mass gain.

You'll also need to excuse me, I used .999 as shorthand to express that I know he will not be and can't break the speed of light. I didn't mean to imply that I had already chosen the speed he will be traveling.

posted by CarlRossi at 5:45 PM on March 10, 2012

The best resource for hard sci-fi writers is Project Rho. The section on Slower than Light Travel: Relativity has plenty of the math worked out for different mass ratios, thrusts and time dilation.

Do be wary of adding too much Handwavium to make the numbers work. The Universe is a harsh mistress when it comes to writing science fiction that respects science.

posted by autopilot at 6:06 PM on March 10, 2012 [2 favorites]

Do be wary of adding too much Handwavium to make the numbers work. The Universe is a harsh mistress when it comes to writing science fiction that respects science.

posted by autopilot at 6:06 PM on March 10, 2012 [2 favorites]

*You may also want to consider whether the mechanism your character has for light speed travel also requires acceleration and deceleration at either end, as this will lower the average speed and affect the time dilation to a potentially significant extent.*

Near instantaneous acceleration to light speed seems to be a common popular SF trope... I don't think anybody would fault you for following in the tradition.

posted by eddydamascene at 6:56 PM on March 10, 2012

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Assuming he has magic handwave rockets that don't spend much time accelerating or decelerating, the ratio of the traveler's proper time (the time he experiences) to Earth's is γ=1/sqrt(1 - v^2/c^2). You need a travel speed of at least .9996 the speed of light to get enough time dilation for your scenario.

There is no plausible propulsion scheme for the spaceship, so don't even try to describe it unless some aspect of it is really relevant to the story.

This category of travel is sometimes called NAFAL (nearly as fast as light) among SF worldbuilding geeks, so you might search for that term.

posted by hattifattener at 1:48 PM on March 10, 2012 [1 favorite]