Can Time Dilation Be Calculated for a Hypothetical Relativistic Trip?

[DaveC426913]

I'm a bit of a spazz when it comes to higher-order math, but I want to understand how to calc a realistic (yet still hypothetical) relativistic trip.

By my calcs, at 1g, it would take 312 days to accelerate to 90% of c. The final 9.999% gets trickier as the Lorentz transform becomes great enough to affect each additional 10m/s. With my math, I'd have to do it by brute force. Then they immediately flip over and decelerate.

Is there a better way to calc the time dilation for this trip?

To make a trip of much shorter length, I put my occupants in immersion tanks, and accelerate them at 10gs. Now it only takes 31 days.


I want to massage the numbers (such as how close they get to c, and how long they spend there) and create a scenario that demonstrates time dilation on a storyline scale (i.e. months). The distance would be interstellar (a couple of light years).

Help?

 

[pervect]

Are you aware of http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html?

This has the formulas for a constant-acceleration trip. If it's what you want, and you need some help with the hyperbolic trig functions, let us know. If it's not what you want, let us know. (For instance you might want to see more of the derivation).

 

[DaveC426913]

Yeah, I've got that page bookmarked. Sigh - I'm getting really frustrated with having only a high school level math.

 

[Wallace]

Which part of the maths are you having trouble with specifically?

 

[DaveC426913]

Which part of the maths are you having trouble with specifically?

Getting started. I guess once I'm determined to squeeze some hard numbers out of it, I'll be able to sit down and plough through it.

 

[George Jones]

Converting the hyperbolic trig to exponential and natural log gives something that is more easily punched into a calculator. In the stuff below, I use years ($y$) as the unit of time and lightyears ($ly$) as the unit of distance. In these units, $g = 1.032 ly/y^2$.

Let: $T$ be the time elapsed according to a clock on the ship; $d$ be distance traveled in the inertial frame of the starting and end points; $a$ be the accleration of the ship. I get (I could easily have messed up the calculation, though.)

$$T = \frac{2}{a} \mathrm{ln} \left[a \left(1 + \frac{d}{2} + \sqrt{\left( 1 + \frac{d}{2} \right)^2 - \frac{1}{a^2}} {} \right) \right]$$

for a ship that accelerates for half the distance and decelerates for half the distance.

Taking $a = g = 1.032$ and $d = 2$, I get that $T = 2.6 y$.

Try playing with other acclerations and distances.

If what I have written doesn't make sense, just fire back with more questions.

 

[JesseM]

Getting started. I guess once I'm determined to squeeze some hard numbers out of it, I'll be able to sit down and plough through it.

Is it just solving the equations, or are you having trouble setting up the right equation for the scenario you have in mind?

 

[Athiril]

Thats funny I swore I posted in this thread, with an offer of writing some simple code.

 

[DaveC426913]

OK, I'll plough my way through it.

I'm trying to work it into a story where the trip takes a short time but still demonstrates time dilation. I may have two trips of the same distance with differing times.

I figure 8gs and then 10gs is about the maximum I can push the human body (without the aid of artifiical gravity compensators) - I'll immerse them in fluid. (Then all I have to do is figure out how I can keep my passengers in glorified bathubs for a 2-month-long trip without having them go insane).

As you can see, the story is going to be highly contrived and somewhat fantastical to wrap its plotline around the real physics.