[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 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 1*g* (9.81 m/s^{2}) 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 1*g*. 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) = (e^{x}+ e^{-x})/2 } 2 2 c c 2 d = - ( cosh(aT/c) - 1 ) = - ( sqrt[ 1 + (at/c) ] - 1 ) a a { cosh(x) = (e^{x}- e^{-x})/2 } a 2 v = c tanh - T = at / sqrt[ 1 + (at/c) ] 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 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 10^{8} 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/yr^{2}. Here are some
typical answers for *a = *1*g*.

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
1*g*. 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 yearsnly 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 = mc ^{2}* so provided mass can be
converted to 100% radiation by means of matter-antimatter
annihilation you just want to know what is the mass

E = (M+m)c^{2}.

After the fuel has been used up it is

E = E_{L}+ mc^{2}gamma

where *E _{L}* is the energy in the light.

E_{L}= |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*, *E _{L}* and

(M+m)c^{2}= mvc gamma + mc^{2}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 1*g* 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*