Calculating Time taken to reach speed with Time Dilation

[TheStick]

TL;DR Summary

Basically seeing if the code I wrote correctly calculates how long it would take for a constantly accelerating ship to reach certain speeds, focusing mostly on time dilation and how that would slow it down as opposed to the time we might otherwise think it would take.

So I've learned about time dilation and how getting close to the speed of light can have affects of time dilation. Looking into a bit more I see that when you are really close to the speed of light, say getting to the point of 99.9999999999% the speed of light, then it starts to become that accelerating a single meter would increase time dilation by quite a bit.

using Lorentz's equation - Δt' = Δt / √(1 - v²/c²)

So I was curious about figuring out the amount of time it would take, assuming constant acceleration, I wrote a piece of code (C#), and I was curious if this was correct and if not, what I've done wrong. Just wondering, I was bored ig.

using System;

public class calc
{

// This is a calculator that will calculate how long it will take to get up
// to a certain speed, taking time dilation into account.

// Hopefully.

private const double c = 299792458; // Speed of light in meters

public static double accuracy = 0.05; // Measured in Meters
public static double speed; // Percentage of c
public static double acceleration = 999; // 1000 meters per second

//This is approximately equal to 1 meter as a percentage of the speed of light.
public static double time;

public static double LorentzFactor;

public static double speedPercentage = speed / c;

//    1 / SQRT( 1 - (u^2)  )  --- Lorentz Factor
// u being the percentage

static void Main(){

    time = 0;
    speed = 0;
    calculate();

}


static void calculate(){
    while( speed < (c - accuracy)){
      
    speedPercentage = speed / c;
      
    LorentzFactor = 1 / (Math.Sqrt( 1 - (speedPercentage*speedPercentage))  );
  
  
    // How many Seconds
    speed += accuracy; // m
    time += (accuracy / acceleration ) * LorentzFactor;
  
    // How many meters per second
    //speed += acceleration / (1 * LorentzFactor);
    //time += 1;
  
    }
  
    time = time / 60;
    time = time / 60;
    time = time / 24;
  
    Console.WriteLine("Accuracy: " + accuracy);
    Console.WriteLine("(m/s) Speed: " + speed);
    Console.WriteLine("");
    Console.WriteLine("Days: " + time);
    Console.WriteLine(" Speed: " + (speed / c));
    Console.WriteLine("Lorentz factor: " + LorentzFactor);
    Console.WriteLine("---");

}

}

` Basically the code take every specific distance traveled (Accuracy) so say for every meter accelerated, it calculates the time taken to to accelerate that meter, and puts the Lorentz Factor into affect. These are some of the results.

Accuracy: 0.05

(m/s) Acceleration: 1000 Days: 5.45031839906392 Speed: 0.999999999930029 Lorentz factor at end: 45955.4795189111

(m/s) Acceleration: 500

Days: 10.9006367981278 Speed: 0.999999999930029 Lorentz factor at end: 45955.4795189111

Accuracy: 0.1 (m/s) Acceleration: 1000

Days: 5.4503051461314 Speed: 0.999999999939864 Lorentz factor at end: 35637.0967410703

Accuracy: 1 (m/s) Acceleration: 1000

Days: 5.45003205655855 Speed: 0.999999996664359 Lorentz factor at end: 8657.2579371794

This is just for anyone whose interested, I'm not sure this feels correct, changing the accuracy changes the Lorentz affect by a lot, but hardly seemed to affect the time taken to accelerate to that speed. I'd been thinking it would start taking a very, very long time to accelerate to that speed, but that doesn't seem to be what its outputting. I really have no idea if this is correct or anything at all, just find the stuff fascinating.

 

[PeterDonis]

figuring out the amount of time it would take, assuming constant acceleration

Time according to whom? There is no absolute time in relativity. Do you want to calculate time according to a clock on board the ship? Or according to a clock that remains at rest at the launch point?

 

[PeterDonis]

using Lorentz's equation

That doesn't calculate what you actually are trying to calculate.

The equations you need can be found in this article on the relativistic rocket equation:

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

 

[Ibix]

speed += accuracy;

As Peter says, I don't think it's entirely clear what time you're calculating. Also, the quoted line from your code looks very wrong. The speed increase should be the coordinate acceleration (which probably isn't constant - proper and coordinate accelerations are different in relativity) times your time interval.

 

[TheStick]

Time according to whom? There is no absolute time in relativity. Do you want to calculate time according to a clock on board the ship? Or according to a clock that remains at rest at the launch point?

Sorry, I'd thought I had specified that but I guess not, I'm meaning from Earth's perspective, so like how much time would have passed on earth by the time they reached that speed.

 

[TheStick]

As Peter says, I don't think it's entirely clear what time you're calculating. Also, the quoted line from your code looks very wrong. The speed increase should be the coordinate acceleration (which probably isn't constant - proper and coordinate accelerations are different in relativity) times your time interval.

I don't understand coordinate acceleration very much. But what I was doing here was calculating how much time it would take to 'accelerate every say meter. Or you could change the accuracy, so how much time it would take for every centimeter. As I thought that would more accurately put Time Dilations affect.

 

[TheStick]

That doesn't calculate what you actually are trying to calculate.

The equations you need can be found in this article on the relativistic rocket equation:

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

So even though I'm only calculating the time it would take to accelerate to certain speeds, I need more than just Acceleration and Time Dilation? I guess one question I'm not sure of, is once you are travelling very fast, would Time Dilation affect say every single meter accelerated. So where it would get to a point where every single Meter you accelerate, would make it take even longer to accelerate to the next meter?
In the link, it has a table for T (years), t (years), d (ly), v/c, γ. But at points in the table T is say 5 years, then The Lorentz Factor is like 86.2. I'd thought it meant that for every five years on earth, the t would be 1. But instead t is 83.7 years. I'd thought it would be something like t = T*y. Which is I suppose is why I'd thought bringing it down to individual meters traveled, and how long that would take with Time Dilation, would be more accurate.

 

[PeterDonis]

I'm only calculating the time it would take to accelerate to certain speeds, I need more than just Acceleration and Time Dilation?

The problem is that the time dliation factor is not constant, because of the acceleration. In the general case, you need to do an integral. In the special case of constant acceleration, you can use the equations on the relativistic rocket page I linked to. But as you will see if you look, those equations aren't as simple as just plugging numbers into the time dilation formula you posted.

 

[TheStick]

The problem is that the time dliation factor is not constant, because of the acceleration. In the general case, you need to do an integral. In the special case of constant acceleration, you can use the equations on the relativistic rocket page I linked to. But as you will see if you look, those equations aren't as simple as just plugging numbers into the time dilation formula you posted.

By time dilation factor do you mean the Lorentz factor? In which case I never had it be constant. I had it change with every meter accelerated. I'll also likely research more into coordinate acceleration as that seems to be something I don't know much of either. Thank you for your help however.

 

[PeterDonis]

By time dilation factor do you mean the Lorentz factor?

Yes.

In which case I never had it be constant. I had it change with every meter accelerated.

It's not just the time dilation factor that changes. So does the coordinate acceleration, as @Ibix pointed out. Also, those things are not functions of the distance traveled (at least not in the formula you wrote down), so "with every meter" is not a correct way to try to do the calculation.

In any case, since there are closed form formulas for the scenario you are trying to calculate (on the relativistic rocket page I linked to), there is no need to write a complicated computer program to do the calculation by what amounts to a clunky approximation to an integral. Just use the formulas.

 

[TheStick]

Yes.


It's not just the time dilation factor that changes. So does the coordinate acceleration, as @Ibix pointed out. Also, those things are not functions of the distance traveled (at least not in the formula you wrote down), so "with every meter" is not a correct way to try to do the calculation.

In any case, since there are closed form formulas for the scenario you are trying to calculate (on the relativistic rocket page I linked to), there is no need to write a complicated computer program to do the calculation by what amounts to a clunky approximation to an integral. Just use the formulas.

Alright, Thanks for your explanations! I'll learn more about coordinate geometry and (attempt) to understand those formulas better. Appreciate your quick help.