Starship Travel Planner - This form returns the time that passes on Earth during the round-trip time you spend on your starship.
Choose your g-Force: Standard Gravity is used in this form = 9.80665 m/s2.

The speed of light, c = 299792458 m/s.
Enter starship's round-trip travel time in years: 365.242 days = 3.15569 x 107seconds is used in this form.

Earth time as a function of starship round-trip time:

t = 4c/a sinh(aτ/4c),

where t is the rest frame (Earth) time, τ is the accelerated frame's time, a is the acceleration, and c is the speed of light.

Earth years (t) that will pass during your trip: years
Earth days that will pass during your trip: days
Earth seconds that will pass during your trip: seconds

 MAXIMUM SPEED AND FARTHEST DISTANCE Given the Earth time that passes during the starship's round-trip, what was the ship's maximum speed and how far out did it travel? The outward bound half of your round-trip consists of two equal blocks of time (half of four equal blocks that make-up the round-trip). At the launch of the starship, the Earth time t and starship time τ equal zero (t0 = 0), the ship's initial speed equals zero (v0 = 0), its acceleration a is constant, and it will reach its maximum speed vmax just before the engines are turned around midway to its destination (at the end of the first of the four equal blocks of time the make up the total round-trip time; again, the engines are turned around so as to decelerate the ship at a constant rate and bring it to rest at its destination). From the definition of relativistic uniform acceleration, the speed v of the starship in the frame of the Earth (see below) is found to be v = dx/dt = at/√(1 + a2t2/c2), where dx/dt is the instantaneous speed of the ship and t is the time that passes in the frame of the Earth; dx and dt are the infinitesimal change in position and time in the frame of the Earth, respectively. However, the contant acceleration a of the ship will not be the acceleration of ship as "seen" within the rest frame of the Earth; this acceleration, dv/dt = a(1 - v2/c2)3/2, approaches zero as the speed of the ship v approaches the speed of light c. From the definition of relativistic uniform acceleration aR, the ship's maximum speed and farthest distance reached during its round-trip can be expressed as function of Earth time t: aR ≡ dv/dt = a(1 - v2/c2)3/2. Rearranging and integrating (at x = 0, t = 0), at = v/√(1 - v2/c2)  →  v = dx/dt = at/√(1 + a2t2/c2)  →  x = (c2/a)[√(1 + a2t2/c2) - 1] ⇒  farthest distance = 2(c2/a)[√(1 + a2t2/c2) - 1)] ⇒  maximum speed = at/√(1 + a2t2/c2) Choose your g-Force: Standard Gravity is used in this form = 9.80665 m/s2 Your Spaceship's acceleration will be: m/s2 Years on Earth you'd like to pass while you're gone: This form uses 1 year = 3.15569 x 107 seconds. Ship time τ as function of Earth time t: τ = 4c/a sinh-1(at/4c). Earth round-trip time in seconds (t): seconds Ship's round-trip time in years (τ): years Ship's round-trip time in seconds (τ): seconds Farthest distance in meters (m): meters Farthest distance = xmax = 2(c2/a)[√(1 + a2t2/c2) - 1)] Maximum speed = vmax = at/√(1 + a2t2/c2) 1 light-year = 9.4605284 x 1015 meters 1 Parsec = 3.08567758 x 1016 meters BACK TO TOP Farthest distance in light-years (ly): light-years Maximum speed reached in meters per second: m/s Maximum speed in fraction of light speed: c Distance reached in parsecs: pc

DERIVATIONS, REFERENCES, & LINKS OF INTEREST

The forms above use the following equations, respectively:

STARSHIP TIME (τ) TO EARTH TIME (t): T = 4c[eaT'/4c - e-aT'/4c]/2a = (4c/a)sinh(aT'/4c),

EARTH TIME (t) TO STARSHIP TIME (τ): T' = (4c/a) ln[aT/4c + sqrt( 1 + a2T2/16c2] = 4c/a sinh-1(aT/4c),

EARTH TIME (t) MINUS STARSHIP TIME (τ): T - T' = T -   [4c/a sinh-1(aT/4c)] = (4c/a)sinh(aT'/4c)   - T'.

T is the time that passes within the rest frame (Earth frame), a is the acceleration, c is the speed of light, and T is accelerated frame (the starship) time. The round-trip journey is broken into four parts (note the 4’s in the equations) allowing for constant acceleration. The ship leaves Earth with constant acceleration, briefly turns around its rockets at the halfway point on its outward journey, and decelerates until it comes to rest at its destination. The ship then turns its rockets around again and returns to Earth in the same fashion.

The following is a derivation of a relationship between the time that passes with a rest frame (Earth) and the time that passes within a moving frame (a starship) that starting from rest, moves with a predetermined constant acceleration to a predetermined destination and back.

Here's a more detailed derivation in a pdf format:
An Introduction to Acceleration in Special Relativity.

Let v be the velocity of a point in S, and v' the velocity in S’.

v = (v' + V)/(1 + Vv'/c2)

t = (t' + Vx'/c2)/sqrt(1 - V2/c2),

Taking the differentials of each,

dv = dv'/(1 + Vv'/c2) - (v' + V)(V/c2) dv'/(1 + Vv'/c2)2 = (1- V2/c2) dv'/(1 + Vv'/c2)2

dv = (1- V2/c2) dv'/(1 + Vv'/c2)2;

dt = (dt' + Vdx'/c2) /sqrt(1 - V2/c2)= dt'[1 + Vdx'/(dt'c2)] /sqrt(1 - V2/c2)

v'= dx'/ dt'

dt = (1 + Vv'/c2) dt' /sqrt(1 - V2/c2)

Dividing dv by dt, we find the acceleration in S; dv’/dt’ is the acceleration experienced in S’.

dv/dt = (1 – V2/c2)3/2 dv'/[(1 + Vv'/c2)3dt']

If S’ is the instantaneous co-moving system of the point P, then v’ = 0, V = v; and if dv’/dt’ = a', then

dv/dt = (1 - V2/c2) 3/2a'

a = (1 - V2/c2) 3/2a'

Rearranging,

(1 - V2/c2)-3/2dv = a'dt

Integrating,

v(1 - V2/c2)-1/2dv = a't

Rearranging, recalling SR's Clock Hypothesis, and integrating,

T' = (c/a) ln{aT/c + sqrt( 1 + a2T2/c2} = c/a sinh-1(aT/c)

This represents the time that goes by on Earth (rest frame) as function (transform) of the time the goes by within the starship (moving frame), moving with a constant acceleration (a'). At the moment of the engine's first turn-around (and its assumed that the duration of the moment is neglible), halfway to the ship's outward destination, the the ship's acceleration is revearsed so the it can slow and come to rest at the end of the second quarter of its four part round-trip.

T'/4 = (c/a) ln{aT/4c + sqrt( 1 + a2T2/16c2} = 4c/a sinh-1(aT/4c)

REFERENCES

Acosta, V., Cowan, C. L., & Graham, B. J. (1973) Essentials of Modern Physics. Harper & Row: New York.

Arzeleiès, Henri, (1972) Relativistic Point Dynmaics . trans. by P. W. Hawkes, Pergamon P: Oxford.

DeWitt, Bryce (2011) Bryce DeWitt’s Lectures on Gravitation, Edited by Steven M. Christensen, Lecture Notes in Physics, 826. Springer-Verlag: Berlin-Heidelberg.

Einstein, Albert (1952) The Principle of Relativity (On the Electrodynamics of Moving Bodies, trans. by W. Perrett and G. B. Jefferey from “Zur Elecktrodynamik bewegter Körper,” Annalen der Physik, 17, 1905), Dover: New York.

(Einstein, Albert, 1918, Dialogue about Objections to the Theory of Relativity/Dialog über Einwände gegen die Relativitätstheorie . Naturwissenschaften, 6, 697–702.)

Marder, L., (1971) Time and the Space-Traveller . U of Pennsylvania P: Philadelphia.

Møller, C., (1962) The Theory of Relativity . Oxford UP: Amen House, London.

Powers, Robert M., (1981) The Coatails of God: The Ultimate Spaceflight - The Trip to the Stars. Warner Bks: New York.

Schlegel, Richard, (1968) Time and the Physical World . Dover Publications, Inc. (1961, Michigan State University Press): New York.

Susskind Lectures on Relativistic Kinematics

George Smoot on Relativity: Physics 139 Relativity, UC Berkeley

The Original Usenet Physics FAQ: The Relativistic Rocket

scienceworld.wolfram: "Proper Time"

scienceworld.wolfram: "Velocity Four-Vector"

Leonard Susskind on Special Relativity (YouTube)

NASA Science News: Scientists Examine Using Antimatter and Fusion to Propel Future Spacecraft