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Copyright © 2003 jsd

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1  Higher Dimensions

In the discussion of rapidities, boosts, and rotations we considered only rotations in two dimensions (the XY plane) and boosts in two dimensions (the TX plane). Let us now extend this to four dimensions.

The basic two-dimensional equations were:

⎡
⎢
⎣
PX(A)
PY(A)
⎤
⎥
⎦		  =  ⎡
⎢
⎣
cos(φ) −sin(φ)
sin(φ) cos(φ)
⎤
⎥
⎦		  ⎡
⎢
⎣
PX(B)
PY(B)
⎤
⎥
⎦		              (1)

  

⎡
⎢
⎣
PT(A)
PX(A)
⎤
⎥
⎦		  =  ⎡
⎢
⎣
cosh(ρ) sinh(ρ)
sinh(ρ) cosh(ρ)
⎤
⎥
⎦		  ⎡
⎢
⎣
PT(B)
PX(B)
⎤
⎥
⎦		              (2)

In four dimensions we can still have rotation in the XY plane (rotation around the +Z axis),

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 cos(φ) −sin(φ) 0
0 sin(φ) cos(φ) 0
0 0 0 1
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (3)

which has the same meaning as equation 1; the T and Z components are just spectators, unchanged by the transformation.

  We can still have a boost in the TX plane (X direction),

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
cosh(ρ) sinh(ρ)00
sinh(ρ) cosh(ρ)00
0 0 10
0 0 01
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (4)

which has the same meaning as equation 2; the Y and Z components are just spectators, unchanged by the transformation.

Some other rotation matrices of interest include rotations in the ZX plane (rotation around the +Y axis):

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 cos(θ) 0 sin(θ)
0 0 1 0  
0 −sin(θ) 0 cos(θ)
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (5)

   Some other boost matrices of interest include boosts in the TY plane (Y direction),

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
cosh(ρ) 0  sinh(ρ)0
0 1 00
sinh(ρ) 0 cosh(ρ)0
0 0 01
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (6)

and rotation in the YZ plane (rotation around the +X axis),

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 cos(ψ) −sin(ψ)
0 0 sin(ψ) cos(ψ)
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (7)

   and boosts in the TZ plane (Z direction):

⎡
⎢
⎢
⎢
⎣
PT(A)
PX(A)
PY(A)
PZ(A)
⎤
⎥
⎥
⎥
⎦		  =  ⎡
⎢
⎢
⎢
⎣
cosh(ρ) 0  0 sinh(ρ) 
0 1 00
0 0 10
sinh(ρ) 0 0 cosh(ρ)
⎤
⎥
⎥
⎥
⎦		  ⎡
⎢
⎢
⎢
⎣
PT(B)
PX(B)
PY(B)
PZ(B)
⎤
⎥
⎥
⎥
⎦		              (8)

The matrices appearing above correspond to the rotation operators R(ψ; xy), R(θ; zx), and R(φ; yz), respectively. The first argument specifies the amount of rotation, while the second argument specifies the plane of rotation. They are called yaw, pitch, and roll respectively. The angles can be called Euler angles, but beware that there is a plethora of mutually-inconsistent conventions as to which angle goes with which axis, and which operators get applied in which order.   The matrices appearing above correspond to the boost operators B(ρ; tx), B(ρ; ty), B(ρ; tz) respectively.

Note that the matrix for R(; zx) looks different from the others because it has the minus sign in the lower-left corner, and similarly note that its orientation is spelled ZX not XZ. This is necessary to preserve the right-hand rule; we want a rotation around the +Y axis.    

An arbitrary rotation can be described by combining the aforementioned matrices in the following order: Alice is in the lab frame. Bob is rotated relative to Alice by a yaw angle (ψ). Carol is rotated relative to Bob by a pitch angle (θ). Doug is rotated relative to Carol by a roll angle (φ). Then Alice’s measurements are related to Doug’s measurements by

P(A) = R(ψ; xy) R(θ; zx) R(φ; yz) P(D)              (9)

    

The ordering is important here. Although rotations in any given plane commute with each other, they don’t commute with rotations in any other plane, or with boosts.   Although boosts in any given direction commute with each other, they don’t commute with boosts in any other direction, or with rotations.

This representation, while conventional and widely used, is not trouble-free. For one thing, it treats the XY, YZ, and ZX planes on unequal footing; one might have hoped for more symmetry. More specifically, there is a singularity when the pitch angle reaches 90 degrees. This problem can be overcome by representing things in terms of Clifford algebra (or, equivalently, quaternions) or in terms of matrices, but that’s beyond the scope of this document, at least for now.    

We should mention the rotation group. The the group elements are rotation-operators. They can be represented by 3x3 matrices, or equivalently by 4x4 matrices in which the T component is a spectator. The group-operator is composition, i.e. carrying out one rotation then another, which can be represented by matrix multiplication. The set of all rotations is closed under this operation.   There is no group consisting just of boosts, because the boosts are not closed under composition. If you apply a sequence of boosts such as B(−є; ty) B(−1; tx) B(є; ty) B(1; tx) (corresponding to going around a thin rectangle in velocity-space) the net result will be a pure rotation. The smallest group containing the boosts is the Lorentz group, which consists of the boosts and the rotations.

[Contents]
Copyright © 2003 jsd