Cut Your Travel Time in Half

There are some aspects of the universe that are exceedingly difficult to imagine properly. For instance, if you were in a spaceship traveling at half the speed of light, and you were to measure the speed of the light you shot out ahead of you, you'd find that it was traveling exactly the speed of light both from your standpoint, and from the standpoint of the stationary observer. This is true for the spacecraft traveling at 99% of the speed of light as well. Special relativity- Einstein's description of the universe in which there are NO special, or privileged reference frames (one's where light doesn't go the speed of light), leaves us with the understanding that in order for this to be the case- in order for light to always travel light speed- other things we consider to be constant get warped and squished and stretched. The rate of the passage of time. The physical dimensions of objects. Mass. Even the very concept of simultaneity.

While it's difficult to imagine, it's not difficult to calculate.

The factor, called gamma, by which we divide the passage of time for the moving reference frame, or multiply for the rest frame, is a simple calculation.

gamma = 1 / ( square root of ( 1 - ( v^2 / c^2 ))

v is velocity expressed as a decimal of the speed of light, c. C, is 1. 1 squared is still 1. So you can disregard that part of the equation. So...

gamma = 1 / square root of (1 - v^2)

And now for some examples:

v/c___________GAMMA__________percent relative to rest frame
0.0___________1________________100%
0.00001_______1.00000000005_____99.99999999995%
0.0001________1.000000005_______99.999999995%
0.001_________1.0000005_________99.9999995%
0.01__________1.00005___________99.99995%
0.1___________1.005_____________99.49874%
0.2___________1.021_____________97.979736%
0.25__________1.033_____________96.82458%
0.3___________1.048_____________95.39392%
0.33333_______1.061_____________94.28090%
0.4___________1.091_____________91.65151%
0.41660_______1.1_______________90.90909%
0.5___________1.155_____________86.60254%
0.55277_______1.2_______________83.33333%
0.6___________1.25______________80%
0.63897_______1.3_______________76.92308%
0.66666_______1.342_____________74.53556%
0.69985_______1.4_______________71.42857%
0.7___________1.40028___________71.41428%
0.74535_______1.5_______________66.66666%
0.75__________1.512_____________66.14378%
0.78062_______1.6_______________62.5%
0.8___________1.666_____________60%
0.80869_______1.7_______________58.82353%
0.83148_______1.8_______________55.55556%
0.85029_______1.9_______________52.63158%
0.8660254038__2________________50%
0.9___________2.294_____________43.58899%
0.91__________2.412_____________41.46082%
0.91652_______2.5_______________40%
0.92__________2.552_____________39.19184%
0.93__________2.721_____________36.75595%
0.94__________2.931_____________34.11744%
0.94281_______3________________33.33333%
0.95__________3.203_____________31.22499%
0.96__________3.571_____________28%
0.96825_______4________________25%
0.97__________4.113_____________24.31049%
0.97980_______5________________20%
0.98__________5.025_____________19.89975%
0.98601_______6________________16.66667%
0.98974_______7________________14.28571%
0.99__________7.089_____________14.10674% (two nines of c)
0.9949874371__10_______________10%
0.999_________22.366____________4.47101% (three nines of c)
0.9999________70.712____________1.41418% (four nines of c)
0.9999499987__100______________1%
0.99999_______223.607___________0.44721% (five nines of c)
0.999999______707.107___________0.14142% (six nines of c)
0.9999994999__1000_____________0.1%
0.9999999_____2236.068__________0.04472% (seven nines of c)
0.99999999____7071.068__________0.01414% (eight nines of c)
0.999999999___22360.680_________0.00447% (nine nines of c)
0.9999999999__70710.678_________0.00141% (ten nines of c)
1.0000000000__(infinite)__________0%

Notice anything interesting happening at high c? Makes it easy to remember, doesn't it?

Let's put gamma to work in some simple operations.

Q. A ship with a 1 gram rest weight is traveling at ten nines of the speed of light. How much does it weigh?

A. 1g x 70710.678 = 70.7kg.

Q. A neutrino with negligible mass is moving exactly the speed of light. How much does it weigh?

A. (almost nothing) x (infinity) = infinite weight. In other words, neutrinos *don't* quite make light speed. Light, which can only travel at light speed, is completely massless.

Q. A one-kilometer long ship is traveling at 0.8c. How long is it when viewed from the rest frame?

A. 1000m x 0.6 = 600m.

Q. A starship is traveling at four nines of the speed of light for ten years according to the rest frame. How many years ass on board the ship?

A. 10 x 0.014142 = 0.14142 years or 51.65 days.

Q. A ship is traveling at 96% of the speed of light for ten years of shipboard time. How many years pass on earth?

A. 1 / 0.28 x 10 = 35.71 years.

Q. At what speed does relativistic time dilation cause shipboard travel time to be exactly half that of the rest frame observer?

A. To find the answer, we set gamma equal to 2 and solve for v.

2 = 1 / square root of ( 1 - v^2)

Divide both sides by (square root of ( 1 - v^2)
Reset.
Divide both sides by two.
Reset. Get 0.5 = square root of ( 1 - v^2)
Square both sides.
Reset. Get 0.25 = 1 - v^2
Subtract 1 from both sides.
Reset. Get -0.75 = -v^2
Take square root of both sides.
Assume the answer we desire is positive.
Get 0.8660254038 c.

v = square root of [ - {(1/gamma)^2 - 1} ]

That's how fast you'd have to travel to cut your travel time in half.