## Wednesday, June 2, 2010

### Back to the Past

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 length 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 infers 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 distances are 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 think always in terms of this new concept: space-time! This means that time is not 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