[Relativity FAQ] - [Copyright]

Updated 15 May 1998 by PEG
thanks to Bill Woods for correcting the fuel equation
Original by Philip Gibbs 21 September 1996

The Relativistic Rocket

The theory of relativity sets a severe limit to our ability to explore the galaxy in space-ships. As an object approaches the speed of light, more and more energy is needed to accelerate it further. To reach the speed of light an infinite amount of energy would be required. It seems that the speed of light is an absolute barrier which cannot be reached or surpassed by massive objects (see relativity FAQ article on faster than light travel). Given that the galaxy is about 100,000 light years across there seems little hope for us to get very far in galactic terms unless we can overcome our own mortality.

Science fiction writers can make use of wormholes, or warp drives to overcome this restriction but it is not clear that such things can ever be made to work in reality. Another way to get round the problem may be to use the relativistic effects of time dilation and length contraction to cover large distances within a reasonable time span for those aboard a space-ship. If a rocket accelerates at 1g (9.81 m/s2) the crew will experience the equivalence of a gravitational field the same as that on Earth. If this could be maintained for long enough they would eventually receive the benefits of the relativistic effects which improve the effective rate of travel.

What then, are the appropriate equations for the relativistic rocket?

First of all we need to be clear what we mean by continuous acceleration at 1g. The acceleration of the rocket must be measured at any given instant in a non-accelerating frame of reference travelling at the same instantaneous speed as the rocket. This acceleration will be denoted by a. The proper time as measured by the crew of the rocket will be denoted by T and the time as measured in a the non-accelerating frame of reference in which they started will be denoted by t. We assume that the stars are essentially at rest in this frame. The distance covered as measured in this frame of reference will be denoted by d and the speed v. The time dilation or length contraction factor at any instant is gamma

The relativistic equations for a rocket with constant acceleration a are,

            c      a                   2     
      t =   - sinh - T  =   sqrt[ (d/c) + 2d/a ]
            a      c        

      { sinh(x) = (ex + e-x)/2 }           

             2                       2
            c                       c                   2
      d =   - ( cosh(aT/c) - 1 ) =  - ( sqrt[ 1 + (at/c) ] - 1 )
            a                       a

      { cosh(x) = (ex - e-x)/2 }           

                   a                             2
      v =   c tanh - T    = at / sqrt[ 1 + (at/c) ]

      { tanh(x) = sinh(x)/cosh(x) }           

            c         a        c              2
      T =   - arcsinh - t   =  - arccosh[ ad/c + 1 ] 
            a         c        a

                   a                     2        2
      gamma = cosh - T = sqrt[ 1 + (at/c) ] = ad/c + 1

These equations are valid in any consistent system of units such as seconds for time, metres for distance, metres per second for speeds and metres per second squared for accelerations. In these units c = 3 x 108 m/s (approx). To do some example calculations it is easier to use units of years for time and light years for distance. Then c = 1 lyr/yr and g = 1.03 lyr/yr2. Here are some typical answers for a = 1g.

    T          t         d          v                gamma
    1 year    1.19 yrs  0.56 lyrs  0.77c              1.58  
    2         3.75      2.90       0.97               3.99
    5        83.7      82.7        0.99993            86.2
    8     1,840     1,840          0.9999998         1,890
   12   113,000   113,000          0.99999999996   117,000

So in theory you can travel across the galaxy in just 12 years of your own time. If you want to arrive at your destination and stop then you will have to turn your rocket round half way and decelerate at 1g. In that case it will take nearly twice as long for the longer journeys. Here are some of the apparent times required to get to a few well-known spacemarks to arrive at low speed:

4.3 ly        nearest star:         3.6 years
27 ly         Vega                  6.6 years
30,000 ly     Center of our galaxy: 21 years
2,000,000 ly  Andromeda galaxy:     29 years
n ly          anywhere              1.94 arccosh[n/1.94 + 1] years

For distances bigger than about a billion light years the formulas given here are inadequate because the universe is expanding. General Relativity would have to be used to work out those cases.

If you wish to pass by a distant star and return to Earth, but you don't need to stop there, then a looping route is better than a straight-out-and-back route. A good course is to head out at constant acceleration in a direction at about 45 degrees to your destination. At the appropriate point you start a long arc such that the centrifugal acceleration is also equivalent to earth gravity. After 3/4 of a circle you decelerate in a straight line until you arrive home.

Sadly there are a few technical difficulties you will have to overcome before you can head off into space. One is to create your propulsion system and generate the fuel. The most efficient theoretical way to propel the rocket is to use a "photon drive". It would convert mass to light photons or other massless particles which shoot out the back. Perhaps this may even be technically feasible if they ever produce an anti-matter driven graser (gamma ray laser).

Remember that energy is equivalent to mass according to the formula E = mc2 so provided mass can be converted to 100% radiation by means of matter-antimatter annihilation you just want to know what is the mass M of the fuel required to accelerate the payload m. The answer is most easily worked out by conservation of energy and momentum. The total energy

            E = (M+m)c2. 

After the fuel has been used up it is

            E = EL + mc2 gamma   

where EL is the energy in the light.

EL is related to its momentum -p by

            EL = |p|c

Since everything started at rest in the Earth frame, the total momentum is zero and the momentum of the rocket is always the negative of that of the light so

            p = mv gamma

Now just eliminate p, EL and E from these equations to get

            (M+m)c2 = mvc gamma + mc2 gamma


            M/m = gamma(v/c + 1) - 1 

This equation is true irrespective of how the ship accelerates to velocity v but if it accelerates at constant rate a then

            M/m + 1 = gamma(v/c + 1) 
          = cosh(aT/c)( tanh(aT/c) + 1 ) = exp(aT/c)

If this is too much fuel for your requirements then there are a limited number of solutions which do not violate energy-momentum conservation or require hypothetical entities such as tachyons or wormholes.

It may be possible to scoop up hydrogen as the rocket goes through space and use fusion to drive the rocket. Another possibility would be to push the rocket away using an Earth-bound grazer directed onto the back of the rocket. There are a few extra technical difficulties but expect NASA to start looking at the possibility soon :-).

You might also consider using a large rotating black hole as a gravitational catapult but it would have to be very big to avoid the rocket being torn apart by tidal forces or spun at high angular velocity. Perhaps if you can get as far as the centre of the Milky way you can use this effect to shoot you off to the next galaxy.

The next problem you have to solve is shielding. As you approach the speed of light you will be heading into an increasingly energetic and intense bombardment of cosmic rays and other particles. After only a few years of 1g acceleration even the cosmic background radiation is Doppler shifted into a lethal heat bath hot enough to melt all known materials.

ref: for the derivation of the rocket equations see "Gravitation" by Misner, Thorn and Wheeler, section 6.2