Assuming you understand the basics, then let's look a little deeper.

Pythagoras into Einstein in Two Easy Steps

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:

c² = a² + b²

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 - v²/c²) ^-1/2

Elementary Guide to Relativity

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 = mc²

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:

d² = x² + y² + z²

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:

d² = x² + y² + z² (1-v²/c²)^1/2

and this implies that the distances all shrink as one moves faster, so does this mean there are no constant distances left in the universe? The answer is that there are because of Einstein's revolutionary concept of space-time where time is distance and distance is time! So now

s² = x² + y² + z² - ct²

and this new distance s (remember s stands for Space-time) 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 always think in terms of this new concept: space-time! This means that time isn't constant and that by simply increasing the velocity (to close to the speed of light for it to have an affect) significant time dilation effects can be seen. It will be a very long time indeed before we have the capability to build a time machine and travel the universe.

Where are the Time Travellers?

We now know that the theoretical evidence for time travel is considerable. There is, however, one question we have avoided until now. If time travel is possible, where are the time travellers? There are a number of possibilities. The most obvious and pessimistic of these is that life on Earth may simply not survive long enough for the technology to evolve. Nuclear wars, giant asteroids smashing into our planet, quantum instabilities developing within the Sun etc. could all stop mankind (or any other race) in its tracks.

Yet the absence of time travellers need not indicate anything nearly so sinister. It is possible that they have been here, and are here right now, but have been discreet about their presence in order to ensure that they return to something closely approximating the future universe of their departure. A further possibility is simply that none has arrived in this particular universe! Many conspiracy theories are based on the possibilities of time travelling aliens having the ability to control all world events on Earth while managing to remain completely hidden from our view. The CIA may be stranger than you think.

So to summarise, we have seen some simple ways of travelling vast distances and have noted some of their drawbacks. We have also looked at some of the more exciting possibilities, but some more real break-throughs are required. New mathematical methods need to be developed and new physical models of the universe must be conceived. Not a job for the faint hearted! Fortune it is said favour the brave. Nowhere will that be more true than in the race to understand the ultimate laws of physics. The race that achieves that first will go straight to top of the premiership of civilisations. Lets make sure it's going to be us!