Kinematics and SL(2,R)
Kinematics is the study of motion without regard to force or momentum. SL(2,R) is the special linear group of 2 x 2 real matrices with determinant 1. It is a continuous group of three dimensions as the four matrix entries are constrained by determinant fixed at 1.
The coordinate plane may be taken as a frame of reference (t, x). When an element of SL(2, R) acts on the coordinate plane, the area of figures is preserved due to the determinant condition. Some matrices yield a rotation of the plane, which preserves distance as well as area. Kinematic transformations with zero acceleration are expressed by matrices that preserve area but not distance.
The first instance is linear motion of velocity v where a new position of x, after time t, is x + vt. The linear transformation is expressed:
It is called a shear transformation in linear algebra. This shear transformation is part of classical mechanics and has been replaced in the electronic age as follows:
The supreme velocity in kinematics is the velocity of light c, so a kinematic velocity v satisfies v < c or v /c < 1. The concept of rapidity is introduced using hyperbolic angles and the hyperbolic tangent function. This function is also bounded above by 1, so rapidity w satisfies tanh w = v /c. Change of frame of reference is accomplished by hyperbolic rotation:
where cosh w is hyperbolic cosine of w and sinh w is hyperbolic sine of w. The trigonometric identity
- confirms that the transformation is in SL(2, R).
Velocity measured with hyperbolic angle uses wings for units, where one wing is the speed of light in water as w:Ludwik Silberstein notes on page 181 of The Theory of Relativity (1914). More modest speeds are commonly described in miles per hour (mph) in English measure. Naturally one mph is 1 mile/3600 seconds compared to c = 186,000 miles per second. The ratio 1 mph/c is about 1.5 x 10−9. Thus the rapidity w for one mph satisfies tanh w = 1.5 x 10−9.
Since the derivative of tanh w is sech2 w, the maximum rate of change of tanh w is one when w=0, and away from zero the slope of tanh nears zero. So for small values of w, tanh behaves as the identity function. Thus the rapidity corresponding to a mile per hour is 1.5 nanowings where a nanowing is 10−9 wings. A hundred mile per hour fastball has 150 nanowings rapidity. A kilometer per hour is 0.9375 nanowings.
Recall that rotations are in SL(2,R) and note the angular measure of these transformations. The classical and modern transformations for linear motion also have angular measures: differences of slope and hyperbolic angle. See Geometry/Unified Angles for details.