Now that you know whether interstellar is travel possible and how fast can we travel in space, it's time for some formulas. In this section, you can find the "classical" and relativistic rocket equations that are included in the relativistic space travel calculator. There could be four combinations since we want to estimate how long it takes to arrive at the destination point at full speed as well as arrive at the destination point and stop. Every set contains distance, time passing on Earth and in the spaceship (only relativity approach), expected maximum velocity and corresponding kinetic energy (if you turn on the advanced mode), and the required fuel mass (see Intergalactic travel - fuel problem section for more information). The notation is:

Relativistic space travel calculator is dedicated to very long journeys, interstellar or even intergalactic, in which we can neglect the influence of the gravitational field, e.g., from Earth. We didn't include in the destination list our closest celestial bodies like Moon or Mars because it would be pointless. For them, we need different equations that also take into consideration gravitational force.

Newton's universe arrive at destination at full speed

It's the simplest case because here T equals t for any speed. To calculate distance covered, at constant acceleration during a certain time, you can use the following classical formula:

Since acceleration is constant, and we assume that the initial velocity equals zero, you can estimate the maximum velocity using this equation:

and the corresponding kinetic energy:

Newton's universe arrive at destination and stop

In this situation, we're accelerating to the half-way point, reaching maximum velocity and then decelerating to stop at the destination point. Distance covered during the same time is, as you may expect, smaller than before:

Acceleration remains positive until we're half-way there (then it is negative - deceleration), so the maximum velocity is:

and the kinetic energy equation is the same as the previous one.

Einstein's universe arrive at destination at full speed

The relativistic rocket equation has to consider the effects of light speed travel. These are not only speed limitations and time dilation, but also how every length becomes shorter for a moving observer, which is a phenomenon of special relativity called length contraction. If l is the proper length observed in rest frame and L is the length observed by a crew in a spaceship, then:

L = l / .

What does it mean? If a spaceship moves with the velocity of v = 0.995c, then = 10 and the length observed by a moving object is ten times smaller than the real length. For example, the distance to the Andromeda Galaxy equals about 2,520,000 light years with Earth as the frame of reference. For a spaceship moving with v = 0.995c, it will be "only" 252,200 light years away. That's a 90 percentage decrease or 164 percentage difference!

Now you probably understand why special relativity allows us for intergalactic travel. Below you can find the relativistic rocket equation for the case in which you want to arrive at the destination point at full speed (without stopping). You can find its derivation in the book by Messrs Misner, Thorne (Co-Winner of the 2017 Nobel Prize in Physics) and Wheller titled Gravitation, section 6.2. Hyperbolic motion. More accessible formulas are in the mathematical physicist's, John Baez, article The Relativistic Rocket:

t = c/a * sh[a*T/c] = [(d/c) + 2*d/a],

T = c/a * sh[a*t/c] = c/a * ch[a*d/c + 1],

d = c/a * [ch(a*T/c) - 1] = c/a * [(1 + (a*t/c)) - 1],

v = c * th[a*T/c] = a*t / [1 + (a*t/c)],

EK = mc * ( - 1)

The symbols sh, ch and th are respectively sine, cosine, and tangent hyperbolic functions, which are analogs of the ordinary trigonometric functions. In turn, sh and ch are the inverse hyperbolic functions that can be expressed with natural logarithms and square roots, according to the article Inverse hyperbolic functions on Wikipedia.

Einstein's universe arrive at destination point and stop

Most websites with relativistic rocket equations consider only arriving at desired place at full speed. If you want to stop there, you should start decelerating at the halfway point. Here, you can find set of equation estimating interstellar space travel parameters in situation when you want to stop at destination point:

t = 2*c/a * sh[a*T/(2*c)] = [(d/c) + 4*d/a],

T = 2*c/a * sh[a*t/(2*c)] = 2*c/a * ch[a*d/(2*c) + 1],

d = 2*c/a * [ch(a*T/(2*c)) - 1] = 2*c/a * [(1 + (a*t/(2*c))) - 1],

v = c * th[a*T/(2*c)] = a*t / (2 * [1 + (a*t/(2*c))]),

EK = mc * ( - 1)

Go here to see the original:

Space Travel Calculator | Relativistic Rocket Equation