Relativistic Interstellar Missions: the physical principles

One light year equals 9.46 10¹⁵ meters, and the closest star is over 4 light years away from us. These are such enormous distances that visiting other stellar systems is currently science fiction. (By comparison, Pluto is -on average- at ‘only’ 5.9 10¹² meters, i.e. at around 263.2 light-minutes away from us, and no astronaut visits are planned yet.) However, it is worth considering what we could achieve if we were able to build a spaceship capable of keeping an acceleration a=10 m/s² (similar to that of gravity that we have on Earth and quite comfortable for the crew) for years. In fact, currently, we can develop much higher accelerations, but only for short times.

In the next section, as a preparation, I review the simpler problem of a spaceship starting with constant acceleration from an inertial space reference system; this exercise can be solved using special relativity.

1-Accelerated motion

Consider a space station which is an inertial reference system with space-time coordinates (x,y,z,t); a spaceship with space coordinates x’,y’.z’ and time coordinate

The space ship starts from rest from the station and moves along the x=x’ axis in such a way that the crew experiences a constant acceleration a, as an artificial gravity; therefore we can drop the y and z coordinates. What is the motion of the spaceship in the system of the station? The station master measures the velocity V of the spaceship as a function of the station time t. In the spaceship, the crew agrees with the station about the relative velocity V but experiences a constant gravity field

(I am assuming no other source of gravity exists). According to the station master, the spaceship constantly increases its speed, but as it approaches the speed of light, its acceleration goes to zero. Now we examine this fact in detail. In the fixed reference, in the infinitesimal time interval dt, the rocket travels the distance dx. Since the interval s is invariant,

that is,

with the consequence that we get the well known time dilation:

(1)

This allows to write formally the proper time of the crew as a function of t,

(2)

although V(t) is still unknown. To find V(t), we substitute Equation (1) into

and we find the law for the speed in the fixed system,

(3)

Integrating and imposing V=0 for t=0, one finds:

(4)

Hence, in the station, the velocity grows with time as:

(5)

For long times, this goes to c, as expected. Integrating again and putting x=0 for t=0 one finds the position of the rocket as a function of time, as judged from the station:

(6a)

This relation is algebraic and can be inverted: the station time t as a function of x reads

(6b)

again we can see that for large enough x, the second term in the root becomes negligible and the rocket goes at a velocity approaching c. Equation (2 ) now allows calculating the proper time as a function of t,

(7)

Solving equation (7) for t, one finds the time t in the station as a function of the astronaut’s proper time. It reads:

(8)

Putting together (5) and (8) one finds:

This gives the velocity V as a function of the proper time:

(9)

2-Missions to Proxima Centauri and other nearby stars

Actually, the space ship must stop at the destination; using the above equations, one can consider a variety of alternative flight plans that allow this. Here, I suppose that the space ship starts from an inertial near Earth station with constant acceleration a, directed towards the target star (I am neglecting the star’s proper motion for simplicity); at half way, the Captain orders the vessel to capsize, so that acceleration becomes -a. The second part of the journey takes place symmetrically to the first, the speed decreases symmetrically and the space ship arrives at its destination with zero speed. The second part of the trip is symmetric to the first part and requires no more calculations.

Let us see what happens when we put some numbers in. We need the following data: one year= 3.154 10⁷ seconds, 1 LightYear =9.46 10¹⁵ m, the velocity of light c=3 10⁸ m/s, and the acceleration a=10 m/s². Then, Equation 6b) becomes, with the time (in years) taken to reach the star measured in the station and writing the distance of the star from the earth in light years,

To find the duration of the station-star trip as measured by the crew one first uses equation (6b) to obtain the capsize time as measured in the station; the capsize time for the crew is obtained by equation (7) and the result must be multiplied by 2. The result (in years) is:

Some results are summarized in the following Table (with times in years).

Such missions are not feasible with the current technology. For one thing, we lack motors. Moreover, the maximum speed at capsize can be obtained using equation (5) and the above data, and one finds 0.95c in the case of Proxima Centauri and up to 0.9988c in the case of Arcturus. At such speeds, even a small stone is a big problem. However, such missions are not physically impossible.

There are 103 known main sequence stars within 20 light years, and sooner or later we could have a strong motivation, or even the need, to develop interstellar flights. Using the above theory, one can easily model many alternative flight plans that could be more feasible. Moreover, after all, modern science is a relatively recent achievement of our race. The results show that if an advanced civilization exists over there, the idea that aliens are visiting the Earth is not so unreasonable.