Explicit form for the most general Lorentz transformation revisited
Howard E. Haber
备注：
26 pages; v2: typographical errors fixed and a minor improvement of notation is implemented; v3: further typographical errors fixed and a number of tweaks have been made; v4: final version, with additional edits, that (approximately) matches with the published version; one more reference (not in the published version) has been added
对应于纯洛伦兹增速和纯三维转动的 4×4 Lorentz 变换矩阵的显式公式是非常著名的。相比之下，对于具有任意非零增速参数和转动参数的一般 Lorentz 变换，其显式公式却远没有那么为人熟知。我们通过给出两种不同推导来重新考察这一更一般的公式。第一种推导（比文献中以往出现的推导略为简单）计算一个 4×4 实矩阵 A 的指数，其中 A 是对角矩阵 diag(+1,-1,-1,-1) 与任意 4×4 实反对称矩阵的乘积。exp A 的公式仅依赖于 A 的本征值，并使用 Lagrange 插值多项式。第二种推导利用如下观察：旋量积 η†\overlineσ^μχ 像 Lorentz 四矢量一样变换，其中 χ 和 η 是二分量旋量。后一种推导的优点在于，一般 Lorentz 变换 Λ 的相应公式可归结为计算 2×2 矩阵乘积的迹。文中表明两种计算给出关于 Λ 的等价表达式。
英文摘要：
Explicit formulae for the $4\times 4$ Lorentz transformation matrices corresponding to a pure boost and a pure three-dimensional rotation are very well-known. Significantly less well-known is the explicit formula for a general Lorentz transformation with arbitrary nonzero boost and rotation parameters. We revisit this more general formula by presenting two different derivations. The first derivation (which is somewhat simpler than previous ones appearing in the literature) evaluates the exponential of a $4\times 4$ real matrix $A$, where $A$ is a product of the diagonal matrix ${\rm diag}(+1, -1, -1, -1)$ and an arbitrary $4\times 4$ real antisymmetric matrix. The formula for $\exp A$ depends only on the eigenvalues of $A$ and makes use of the Lagrange interpolating polynomial. The second derivation exploits the observation that the spinor product $η^\dagger\overlineσ^{\lower3pt\hbox{$\scriptstyle μ$}}χ$ transforms as a Lorentz four-vector, where $χ$ and $η$ are two-component spinors. The advantage of the latter derivation is that the corresponding formula for a general Lorentz transformation $Λ$ reduces to the computation of the trace of a product of $2\times 2$ matrices. Both computations are shown to yield equivalent expressions for $Λ$.
On the Real and Apparent Positions of Moving Objects in Special Relativity: The Rockets-and-String and Pole-and-Barn Paradoxes Revisited and a New Paradox
