This number is creeping me out

[Corkeyno2]

I've been messing around with that time dilation formula that you should probably know if you are going to answer this question. Lorentz factor=1/sqrt(1-beta^2). I've been getting this pattern:

Speed in c:                                  0.9                             0.99              0.999                    0.9999                                                                  

Time dilation ratio to 1:               3.16227766017         10                 31.6227766          100

I'm getting that odd number and then 1 with a number of zeros after it. I did some messing around and I found that the square root of 0.1 was 0.316227766017.

I'm currently recording my findings down in a table and I think they might explain why time dilation increases more rapidly the closer you go to the speed of light. 

I might be being completely stupid but if anyone is experiencing the same thing or has any advice, replies would be greatly appreciated.

Corkeyno2  

[Steve Ward]

0.316227766017 x  0.316227766017 = 0.1 .....             Square root of 1 is 1.

[Tiny Small]

If you use the full formula and put in actual speeds (in m/s) rather than ratios or percentages, you will get accurate results:

1/root(1-(v^2/c^2))

So, going with 0.99 x c:

1/root(1-(296794533.42^2/299792458^2)) = 7.088

Wit v = 0.9999c:

1/root(1-(299762479^2/299792458^2)) = 70.71

Changing c by 1 m/s:

1/root(1-(299792457^2/299792458^2)) = 12243.2116

As the numerator 1 is T0  (1 unit of time... seconds, minutes, hours or years... doesn't matter which as long as you use the same throughout) this means that a year travelling at 0.99 x c would be experienced as 7 years to an observer outside your frame of reference, whilst travelling 1 m/s slower than c would result in the same year being experienced as 12243 years by the same observer.

This was using the calculator on my phone rather than my actual calculator which is at home, so how my phone handles parentheses may have an effect on the results... but the principle should be obvious.

[Corkeyno2]

oh guys im so sorry i meant to put the square root of 0.1. I am kicking myself.

[Corkeyno2]

Thanks so much for the advice guys.

[Tiny Small]

7 minutes ago, Corkeyno2 said:

Thanks so much for the advice guys.

That's why I use the actual numbers. It saves making that type of error. Learned the hard way though.