Math 2.0

Relativity and gravitational theory Special relativity: kinematics Theorem

Herglotz–Noether theorem

Builds on Born rigidityMinkowski spaceFour-velocityKilling vector field

Statement

Let UU be a connected open subset of Minkowski space (ℝ4,η)(\R^4, \eta) with η=diag⁡(−1,1,1,1)\eta = \operatorname{diag}(-1, 1, 1, 1), in units where c=1c = 1. Let uu be a smooth vector field on UU with η(u,u)=−1\eta(u, u) = -1, the four-velocity field of a moving body. Assume that the motion is Born rigid, that is, the expansion and the shear of uu vanish on UU. Let ω\omega be the vorticity tensor of uu.

  1. If ω=0\omega = 0 on UU, then every hypersurface orthogonal to uu is a piece of a spacelike hyperplane.
  2. If ω≠0\omega \ne 0 at every point of UU, then there is a Killing vector field KK of Minkowski space such that u=K/−η(K,K)u = K / \sqrt{-\eta(K, K)} on UU.

Remarks

Born rigidity says that the distances between neighbouring particles of the body, measured in their momentary rest frames, do not change. In the rotational case, the integrability conditions of the rigidity equations use the flatness of Minkowski space. They force the flow of uu to be generated by an isometry, up to the normalisation of uu. So an irrotational rigid motion has the freedom of a family of moving hyperplanes. A rotational rigid motion is one of the isometric motions, such as a uniform rotation about a fixed axis. A body that rotates uniformly about the zz-axis with angular speed Ω\Omega has uu proportional to the Killing field ∂t+Ω(x∂y−y∂x)\partial_t + \Omega(x \partial_y - y \partial_x), on the region where Ω2(x2+y2)<1\Omega^2(x^2 + y^2) < 1. A consequence is that a body at rest cannot be brought into rotation while staying Born rigid, which is the core of the Ehrenfest paradox. The result is due to Gustav Herglotz and Fritz Noether.

Going deeper

For a student

Write h=η+u♭⊗u♭h = \eta + u^\flat \otimes u^\flat for the metric on the rest spaces; Born rigidity is the condition that the Lie derivative of hh along uu vanishes. This condition removes the expansion and shear parts of the covariant derivative of uu, so only the acceleration and the vorticity remain. The Herglotz–Noether theorem then splits Born rigid motions into the two classes of the statement. It covers the example of hyperbolic motion, which is irrotational, and of uniform rotation, which is a Killing motion.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Born rigidity, concept, no evidence levelBorn rigidityconceptFour-velocity, concept, no evidence levelFour-velocityconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptMinkowski space, concept, no evidence levelMinkowski spaceconceptHerglotz–Noether theorem, statement, no evidence levelHerglotz–Noether theo…statement
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  1. Born rigidity concept
  2. Four-velocity concept
  3. Killing vector field concept
  4. Minkowski space concept
  5. Herglotz–Noether theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Born_rigidity, Born rigid motions
About this page
TagDPKZYYBCCN
URIm2://DPKZYYBCCN
Kindstatement, theorem
WritingW0 stub
MSC83A05, 53B30
StatusThis copy of the site has no build data. Run m2 status DPKZYYBCCN.
AboutBorn rigidity
UsesBorn rigidity, Minkowski space, Four-velocity, Killing vector field
Changed3 November 2026

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