Math 2.0

Relativity and gravitational theory Spacetime geometry and causal structure Conjecture

Chronology protection conjecture

Builds on Closed timelike curveCauchy surfaceEnergy conditionsStress–energy tensor

Statement

The laws of physics do not allow the appearance of closed timelike curves. In a more precise form, consider a spacetime that starts out without closed timelike curves. For example, it may have a partial Cauchy surface in its past. Under physically reasonable laws, including quantum effects, it cannot evolve into a region that contains closed timelike curves.

Remarks

The conjecture is open, and it is not a precise mathematical statement: it depends on what counts as physically reasonable. Hawking studied, in classical general relativity, closed timelike curves that appear in a bounded region, with a compactly generated Cauchy horizon. He showed that if they develop from a noncompact initial surface, then the weak energy condition must be violated. He also argued from semiclassical gravity that the expected stress–energy tensor of quantum fields tends to diverge near the boundary of the region with closed timelike curves. This divergence would prevent the region from forming. Kay, Radzikowski and Wald proved a related result for quantum fields on such a compactly generated Cauchy horizon. There, the two-point function cannot be a well-defined Hadamard distribution at every point of the horizon. On the other hand, there are examples of spacetimes and fields for which the expected divergence does not occur, so the semiclassical evidence is not conclusive.

Reading path

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

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Cauchy surface, concept, no evidence levelCauchy surfaceconceptClosed timelike curve, concept, no evidence levelClosed timelike curveconceptEnergy conditions, concept, no evidence levelEnergy conditionsconceptStress–energy tensor, concept, no evidence levelStress–energy tensorconceptChronology protection conjecture, conjecture, no evidence levelChronology protection…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cauchy surface concept
  2. Closed timelike curve concept
  3. Energy conditions concept
  4. Stress–energy tensor concept
  5. Chronology protection conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Chronology_protection_conjecture
  2. Web https://doi.org/10.1103/PhysRevD.46.603
About this page
TagWVSG6FFD5S
URIm2://WVSG6FFD5S
Kindconjecture
WritingW0 stub
MSC83C99, 83C47
StatusThis copy of the site has no build data. Run m2 status WVSG6FFD5S.
AboutClosed timelike curve
UsesClosed timelike curve, Cauchy surface, Energy conditions, Stress–energy tensor
Changed3 November 2026

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