Time and space have fascinated man since the dawn of civilisation. People have spent aeons thinking about these concepts and the ideas behind them. The Greeks, the Romans, the English, all have stared at the heavens and wondered. And not without reason! As the boundaries of physics are pushed back and back it is becoming clear that whoever understands the laws of physics best will be able to travel through time and space, easily gaining a dominant position in the known Universe. Indeed that race which can move from universe to universe will get to control all universes! We must therefore double our efforts to ensure that it is we and not some other race who gains understanding first! Or we will end up playing second fiddle for rest of time!

So where do we start? Well let us start with one of the greatest triumphs of the human mind, the great theorem of Pythagoras, a true pillar of all mathematics and physics. The theorem, which is applicable to right angled triangles in flat Cartesian (Newtonian) space takes the form of:

c^2 = a^2 + b^2

where a, b and c are the lengths of the sides of the triangle.

Next we will jump straight to Einstein's theory of Relativity which states that neither time, length, or indeed mass remain constant additive quantities when approaching the speed of light c. Our simple ideas of time and space come from the fact the we are so used to living in a three dimensional universe. Einstein showed that this was simply not true and in fact all the "foundational" three laws of Newton have to be fudged by the Lorentz factor

L_f = (1 - v^2/c^2)^1/2

There are, however, certain quantities that do remain constant. These constants are related to four-dimensional quantities known as metric tensors. From this Einstein proved that space and time are two aspects of the same thing and that matter and energy are also two aspects of the same thing. From the second of these concepts we get the most famous equation in physics

E = mc^2

Now since time and space are aspects of space-time and we wish to travel through time and not build atom bombs we will leave E=mc^2 for the moment. To illustrate this, look at the extension of Pythagorean theorem for the distance, d, between two points in space:

d^2 = x^2 + y^2 + z^2

where x, y and z are the lengths,  or more correctly the difference in the co ordinates, in each of the  three spatial directions. This distance remains constant for fixed  displacements of the origin.

In Einstein's relativity the same equation is modified to remain constant with  respect to displacement (and rotation), but  not with respect to motion. For a moving object, at least one of the  lengths from which the distance, d, is calculated is contracted relative to a stationary observer. The equation now becomes:

d^2 = x^2 + y^2 + z^2 (1-v^2/c^2)^1/2

and this new distance does indeed remain constant  for all who are in relative motion. This distance is said to be a Lorentz  transformation invariant and has the same value for all inertial observers. Since the equation mixes time and space up we have to think in terms of a new concept: space-time!

What does this have to do with time travel? Imagine  an imaginary journey to Andromeda, some 2.2 million light years away. For  the time being ignore the problems of propulsion (like they do in all Sci-Fi films!). Firstly, lets assume a Newtonian Universe and we'll ignore the effects of gravity and friction (not much of this in space anyway). The first problem: how fast do we accelerate? Well, if we could accelerate at an infinite rate we could reach an infinite speed instantly and reach our  destination in no time at all! Unfortunately, if we were to do so we would experience infinite acceleration forces and be crushed to an infinitesimally  thin film instantly. Not much use. The gravitational field of the Earth of 1g (9.81 m/s per second) is however a comfortable acceleration to subject us to, so lets assume the acceleration of our spaceship will be 1g.

So how long to Andromeda at 1g using Newton's theory?  We will add the condition that we wish to stop when we get there, if only to turn around and come back. The best time we can make is achieved by accelerating for the first half of the journey and decelerating for the second. The total time for the trip can be calculated to be some 2,065 years. Rather a long time really.  Consider the same journey in an Einsteinian Universe. We now have a limited maximum speed (the speed of light), which at 1g is reached in 30,000,000 seconds, or a little under 354 days. After we reach this speed, how much longer will it take to reach Andromeda? The answer is no time at all! For the distance to Andromeda will have shrunk to zero for the spacecraft. However to the people back on Earth a considerable length of time would pass: some 2.2 million years.

For practical reasons, such as having no way of navigating  in an infinitely thin universe, we would stop just short of the  speed of light at the halfway point and reverse engines to come to a halt  at Andromeda. The entire trip would have taken a little less than 2 years  at a comfortable 1g. The same is true of a trip to anywhere within the  Universe. We can get literally anywhere in a little under two years: four for the round trip. The main problem is the ageing of the rest of the universe while we are travelling. The other problem is that the mass of the spacecraft rises greatly as it approaches the speed of light, so an enormous power source would be required.

What about power? For the Newtonian case the power requirement is enormous, even assuming perfect efficiency. The energy required would be 5.1*10^26 Joules for a 10 tonne spaceship. This is rather a lot. NASA's space shuttles are not really up to it I'm afraid. In the Einsteinian Universe the power is still huge since the mass of the ship rises and becomes so large! And unless you get very close to c the distance to Andromeda will not shrink very much and it will take ages to get there.

Unfortunately there are still a few little problems: the friction associated with such a high speed and of course the dreaded time dilation effects.

Friction: though space is essentially empty, with increasing speed the little matter that there is would become increasingly concentrated. In addition, its relative mass increases, consequently, we would encounter it with ever greater density. Indeed as we approach the speed of light the whole Universe becomes concentrated into an infinitely thin, and consequently infinitely dense, barrier in the direction in which we are travelling. The force of friction would increase accordingly.