Relativistic Time Dilation at Accelerated Velocities (With Side Benefit of Artificial Gravity)
Starship Calculator is a Java Script calculator that computes the
times to reach distant points with a space vehicle that can accelerate
continuously. Such a vehicle can approach but never exceed the speed
of light in vacuum, C. The calculator computes travel times for this
mission profile:
The starship accelerates continuously from the origin to the midpoint
of the mission.
At
the midpoint, the ship turns its thrusters to face the destination.
The
ship decelerates continuosly from the midpoint to the destination.
The
observed elapsed time of the mission is computed for two cases:
For
the astronauts in the starship who are in the moving frame.
For
the clocks at the origin and destination which are in the rest
frame.
The
maximum velocity, V, of the starship is reached at the midpoint
of the mission. The calculator shows this as the ratio of maximum
velocity to the speed of light.
Some
representative distances:
Mars 
1.5 
Alpha Centauri 
4.3 
Hyades Star Cluster 
150 
Large Magellanic Cloud 
163,000 
Jupiter 
5.2 
Barnard's Star 
6.0 
Betelgeuse 
309 
Small Magellanic Cloud 
196,000 
Saturn 
9.6 
Wolf 359 
7.7 
Pleiades Star Cluster 
408 
Andromeda galaxy 
2,000,000 
Uranus 
19.2 
Sirius 
8.6 
Rigel 
913 
Spiral galaxy M101 
25,000,000 
Neptune 
30.0 
Ross 154 
9.4 
Crab Nebula 
6,000 
Galaxy M87 
54,800,000 
Pluto 
39.2 
Altair 
16.6 
Star Cluster M13 
21,000 
Perseus cluster 
239,000,000 


Vega 
26.4 
Center 
30,000 
Ursa Major cluster 
670,000,000 


Arcturus 
35.8 


Quasar 3C 273 
1,900,000,000 






Quasar 3C 309.1 
7,400,000,000 






Hubble radius 
15,000,000,000 
Some
constants to consider when using the calculator:
Earth's gravity = 9.80665 [meters/sec/sec]
astronomical unit = 1.496 x 10^{11} [meters] {average distance from Sun to Earth}
light year = 9.460 x 10^{15} [meters] {distance light travels in one year}
To
operate the calculator, first select the desired distance, acceleration,
and time units using the radio buttons. Next, enter acceleration
and distance quantities that are greater than 0. Last, press the
Calculate button. All entries are cleared by pressing the
Clear button. On invalid entries, the the output windows
will display: NaN  Not a Number
Notes
In Einstein's universe, the relativistic effect known as time
dilation may allow long journeys in human lifetimes. Time moves
more slowly in a moving frame than a stationary frame. A clock
in the moving starship will run more slowly than a clock on
Earth according to the equation:
δT_{earth} = γ·δT_{ship}
where,
__________
γ = 1 / √ 1  v^{2}/c^{2}
v  velocity of the starship
c  speed of light in vacuum = 299,792,458 [meters/second]
Note that as v gets closer to c, the term γ
approaches infinity. The effect of time dilation is negligible
at small velocities but increases asymptotically as the velocity
of the starship approaches the speed of light. Note how the
kinetic energy becomes extremely large. Besides this, there
are many other practical problems in realizing a starship
(such as hitting small dust particles while moving at relativistic
velocities). If a practical starship could be constructed,
the term 'astronaut' (which means starvoyager) would be literally
correct.
Additionaly, it is worth noting that if a practical starship could
be constructed, while traveling significant distances at accelerated
velocities nearing the speed of light, the age of the surrounding
universe relative to the traveler's ages would also have been rapidly
accelerated. For example, persons traveling to the Andromeda Galaxy
2million light years from Earth at 1g acceleration would age
approximately 28years during the journey. During that same time, the
universe would have aged over 2million years! So, at the end of their
voyages, not only would great distances have been travesed, the
voyagers would leave their starships far into their respective futures,
essentially also making them 'tempronauts' (time travelers).
Copyright © 2004, Stephen R. Schmitt

