Updated 1998 by PEG.
Thanks to Bill Woods for correcting the fuel
equation.
Original by Philip Gibbs 1996.
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 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) ]
c
sinh x
{ tanh 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
c
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 1,839 0.9999998 1,895
12 113,243 113,242 0.99999999996 116,641
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 around 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 times you will age when journeying to a few well known space marks, arriving at low speed:
4.3 ly nearest star 3.6 years 27 ly Vega 6.6 years 30,000 ly Center of our galaxy 20 years 2,000,000 ly Andromeda galaxy 28 years n ly anywhere, but see next paragraph 1.94 arccosh [n/1.94 + 1] years
For distances bigger than about a thousand million 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 you experience 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 photons or other massless particles which shoot out the back. Perhaps this may even be technically feasible if we ever produce an antimatter-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, we 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 before blast-off is
E = (M+m)c2
After the fuel has been used up it is
E = EL + mc2 gamma
where EL is the energy of the light.
EL is related to the light's momentum -p (where
p > 0) by
EL = pc
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 = mc2 gamma + mvc gamma
so that
M/m = gamma(1 + v/c) - 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(1 + v/c)
= cosh(aT/c)[ 1 + tanh(aT/c) ] = 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, using 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 possibilities 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. If there is a black hole at the centre of the Milky Way, as some astronomers think, then perhaps if you can get that far, 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.
For the derivation of the rocket equations see "Gravitation" by Misner, Thorne and Wheeler, section 6.2.