Draft:Alexandrov's theorem on Lorentz transformations

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Alexandrov's theorem on Lorentz transformations is a fundamental result in the foundations of special relativity and chronogeometry, proved by Alexander Danilovich Alexandrov. The theorem states that Lorentz transformations can be derived from the invariance of light cones or, more generally, from the preservation of the causal structure of spacetime.

Statement of the theorem

Main formulation

Let $f:\mathbb {R} ^{4}\to \mathbb {R} ^{4}$ be a bijective mapping of Minkowski space onto itself that preserves light cones:

$f(C_{x})=C_{f(x)}\quad {\text{for all }}x\in \mathbb {R} ^{4}$

where $C_{x}$ is the light cone with vertex at point $x$.

Then $f$ is a Lorentz transformation up to homothety:

$f(x)=\lambda Lx+a$

where $\lambda \neq 0$, $L$ is a Lorentz transformation, and $a$ is a translation vector.

Local formulation

In later works, Alexandrov generalized the theorem to the case of local mappings of spacetime domains. If a mapping $f:G\to \mathbb {R} ^{4}$ of a domain $G\subset \mathbb {R} ^{4}$ preserves the causal structure, then it is a combination of:

• Lorentz transformations with homotheties ($HL$)
• Inversions ($HLI$)
• Special double inversions ($HLJ$)

Historical context and development

Alexandrov's works

• 1949-1950 — First results on deriving Lorentz transformations from the constancy of the speed of light
• 1967 — "A contribution to chronogeometry" — general theory of cone systems in affine space
• 1976 — "On the foundations of relativity theory" — complete classification of local causal automorphisms

Relation to other works

• 1964 — E. C. Zeeman independently proved that causal automorphisms of Minkowski space coincide with the Lorentz group
• Alexandrov's theorem is more general as it considers both global and local mappings and does not assume the Minkowski space structure a priori

Significance and consequences

Philosophical significance

The theorem shows that Lorentz transformations and, consequently, special relativity can be derived from the principle of causality alone, without additional assumptions about the metric or linearity of transformations.

Physical interpretation

• Preservation of light cones means invariance of the speed of light
• Preservation of causal structure means that if event A can influence event B, then after transformation f(A) can also influence f(B)

Mathematical significance

The theorem establishes a deep connection between:

• Chronogeometry (geometry determined by the precedence relation)
• Conformal geometry
• Lorentz group

Proof

Main stages of the proof: 1. Proof of the preservation of isotropic lines 2. Proof of local linearity using inversions 3. Proof of global linearity 4. Establishing that a linear mapping preserving light cones is a Lorentz transformation

See also

• Lorentz transformation
• Causality
• Chronogeometry
• Zeeman's causality theorem

References

E. C. Zeeman. Causality Implies the Lorentz Group. J. Math. Phys. 5, 490 (1964) https://doi.org/10.1063/1.1704140;

A. D. Alexandrov. A Contribution to Chronogeometry. Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 1119 - 1128 DOI: https://doi.org/10.4153/CJM-1967-102-6

External links

• Alexandrov's original works in MathNet.Ru
• Lorentz transformation in Wikipedia

Category:Theory of relativity Category:Mathematical physics Category:Geometry Category:Theorems