The Relativistic Rocket equation

Before diving into the rocket equations, first a reminder of relativistic mechanics. It us assumed that the concepts of four-vectors, covariant/contravariant vectors and intervals are known; if not they are given in any standard text book on Special Relativity. Also, please note that many authors give the four-velocity as

=

(in eg. Landau-Lifshitz) which differs from the definition here

=

where ds = c dτ = c dt =
The first component of the 4-velocity is

= = = = vx γ

where vx is the usual velocity in the x-direction. The full 4-velocity is thus (where indeces i = 1,2,3,4)

= (c γ, v γ). This is the "contravariant" components, and using a metric signature of +,-,-,- the covariant version is

= (c γ,-v γ)

The scalar product is then = (- ) = = const. The use of contravariant and covariant vectors is a way of expressing the fact the geometry of space is "pseudo-Euclidian" (or Minkowski) and that the interval d= - can be positive, zero or negative.

What is of interest here is the acceleration (arbitrariliy set in x-direction) in the instantaneous frame of the rocket ship:

= (0,a,0,0)

Given a rocket with negliable fuel mass - such as a Bussard Ramjet -  and constant acceleration (in the x-direction), the following equation holds:

F = d (γ v)/dt

where γ =        

Note that this relation only holds the frame at the instantaneous frame of the rocket. Since the acceleration a is constant this can be directly integrated:

F/ = a = ()               (1)

which gives (putting the integration constant to 0 since v=0, at t=0):

at  =  v γ

Solving for v gives:

v =   = c tanh ()        (2)

This is the speed of the rocket as experienced in the lab frame and also gives an expression for the relativistic γ-factor at constant acceleration:

=    =   cosh()                (2')

The relation between time an t in the frame at rest and proper time τ on board the rocket is given by the formula for relativistic time dilation:

dτ  = dt / γ =dt                           (3)

Inserting the expression for the speed (2) in (3) gives the proper time as a function of time at rest

τ = dt= arcsinh()        (4)

or inverting

t = sinh()                                                 (5)

Forumla (5) is the relation between the time t in the frame at rest (say the earth) and ship time τ.

The distance in the lab frame is found by integrating the velocity (2) and replacing lab frame time t with proper time τ given above (5).

s = vdt = (- 1)  = ( -1) = (cosh() - 1 )

Solving for τ finally gives the experienced time as a function of proper distance:

τ = arccosh(+1)                                    (6)

The more realistic form is accelerating half the distance, and breaking the other half, which means that the real distance is given by:

τ' = arccosh(+1)                                (6')


Let's do some numbers. Let s = 100000 light years (diameter of milky way), a = 1 g. One sees that the travel times in ship time are surprisingly realistic during a the life-span of an individual, but a single traveller would him/herself without friends or relatives in the frame at rest (Earth). Thus entire travelling communities in cold-sleep such as the "Queng-Ho" in Vernor Vinge's novels are probably more realistic.

A more realistic case is to assume some kind of fuel consumption function (see below) to indicate that the mass is a function of time. The two plots below show the dramatic consequences of the arccosh law (proper distance (ly)-> axis, ship time y-axis):Travelling to the nearest star Alpha Centauri will take almost the same amount of proper time as on Earth, whereas travelling across the galaxy will make the ship crew outlive people on Earth by thousands of generations.

The Following argument is an attempt to calculate the fuel consumption of the rocket. Here it is assumed (unrealistically of course) that the engine works at a 100 % efficiency such that the initial fuel mass M is entirely converted into radiation at the end of the trip.

Set the intial total energy of the ship in an orbit around earth at rest to , the fuel mass to M and the ship mass to m. Then
= (m+M)                                     (7)
At the end of the trip the final energy is

= γ m +                                   (8)

where is the radiation energy. This follows from the expression of the energy momentum 4-vector

P = (E/c,p) = (γ c,v) where is the rest mass and E is the total energy.

Conservation of energy gives

(m+M) = γ m +                           (9)

For the 3-momentum

= 0

and

= γm v - /c. The total forward momentum is given by the first term, and the last term is the final momentum as the "fuel" momentum (radiation). = gives

γm v - /c = 0.                                     (10)

Putting together (9) and (10) and eliminating gives

(m+M) = γ m + γ m v c

and finally solving for the fuel to mass ratio:
r = M/m = γ(1 + v/c) - 1

For the case of constant acceleration we have an expression for γ (2´) and v (2):
r = cosh() (1 + tanh()) - 1 = exp() - 1

We have two different cases:  accelerating/breaking (propT2) and just accelerating past the goal (propT)

Note the vastly different numbers here! This essentially means that the fuel mass ratio essentially prevents starting and stopping, at least at distances on the order of size of the Milky Way. Thus it would seem that the only realistic alternatives are rocket ships using sun sails or propulsion systems such as a Bussard Ramjet.

Created by Mathematica  (October 1, 2005)