[A_Messenger.]

The 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.

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

Lf = (1 - v^2/c^2) -1/2There 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 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

s^2 = x^2 + y^2 + z^2 - ct^2

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 (too 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, long enough for the ‘size zero’ human to evolve as the average size.