Required math: calculus
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 3; Problem 3.6.
As a rather fanciful example of using the conservation of four-momentum, suppose we have a spaceship of total mass (including fuel) , initially at rest on Earth. The fuel consists of matter/anti-matter which when mixed, produces photons that are ejected out of the back of the ship. If the ship burns enough fuel to accelerate to , then travels to some star system such as Alpha Centauri, then decelerates to zero for a landing, then, after some time at its destination it reverses the trip by again accelerating to , returning to Earth, and decelerating to rest, what is its final mass as a fraction of its initial mass?
We can assume that all motion takes place along the axis, and treat the problem in the Earth’s frame. Then the initial momentum is
After accelerating, the combined momentum of the ship + ejected photons is
where , is the energy of the ejected photons and is the mass of the ship after burning the fuel needed to accelerate.
By the conservation of momentum, we have so
Adding these 2 equations, we get
Now to decelerate the ship, we eject the photons ahead of the ship, and we start with a momentum of . After deceleration, the mass is now and the ship is at rest, so we must have
Again, conservation of momentum requires , so
Subtracting these equations we get
On the return trip, we go through exactly the same procedure, except we now start with a mass rather than . Thus on the return to Earth, the ship’s mass will be
Thus the ship’s initial mass is
Virtually all the initial mass is fuel.
Incidentally, if we do this calculation for an arbitrary velocity , we get
Thus each acceleration or deceleration multiplies the previous mass by a factor of .