Lorentz Transformations as Rotations
In Thursday’s lecture, Tali started by describing how a rotation in Euclidean space left the distance between two points unchanged. In spacetime, the Lorentz transformation leaves the interval s between two events
s^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-c^2 (t_1-t_2)^2
unchanged. The Lorentz transformation is a hyperbolic rotation in space time.
To see how this works, let’s define an angle alpha such that tanh(alpha)=beta=v/c, where v is the relative velocity between the two frames. Recalling the relations between the hyperbolic functions *
cosh^2-sinh^2=1 and 1-tanh^2=1
you can show that gamma=1/sqrt(1-beta^2)=cosh(alpha) and beta*gamma=sinh(alpha). The transformation matrix lambda:
| gamma beta*gamma 0 0 |
| beta*gamma gamma 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
then looks like
| cosh(alpha) sinh(alpha) 0 0 |
| sinh(alpha) cosh(alpha) 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
which looks similar to the matrix for a rotation in Euclidean space (with some important sign differences).
So, Lorentz transformations just represent geometric operations in spacetime: rotation and translation in space, translation in space and time and hyperbolic rotation in spacetime. The represention above has no more practical value than using beta and gamma, but it does serve to illustrate the unified nature of the trnasformations. What Einstein really recognized was that his predecessors were just talking about the wrong space and, once you figure out the right space to use, it is all very simple. From this perspective, special relativity is just geometry.
What is also important to see is that time is now on a nearly equal footing with the the three spatial coordinates. Nearly equal because there is a minus sign in the metric tensor in the time component.
* Hyperbolic trigonometric functions: sinh x=(exp(x)-exp(-x))/2, cosh x=(exp(x)-exp(-x))/2, tanh x=sinh x/cosh x.
