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Monge's transport problem in synthetic Lorentzian spacetimes

  • Afiny Akdemir
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Abstract

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property TMCP+(K,N). The cost for this problem is the time separation function ℓ(x,y) (the Lorentz distance), which represents the maximum amount a particle can age when traveling from x to y. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property TMCP+(K,N) is equivalent to the TMCP+rLip(K,N) condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.

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Cite

BibTeXRIS
BibTeX
@misc{akdemir2026monges,
  title = {{Monge's transport problem in synthetic Lorentzian spacetimes}},
  author = {Afiny Akdemir},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17592},
  eprint = {2609.17592},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17592},
}
RIS
TY  - UNPB
TI  - Monge's transport problem in synthetic Lorentzian spacetimes
AU  - Afiny Akdemir
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.17592
UR  - https://arxiv.org/abs/2609.17592
AB  - We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property TMCP^+(K,N). The cost for this problem is the time separation function ℓ(x,y) (the Lorentz distance), which represents the maximum amount a particle can age when traveling from x to y. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property TMCP^+(K,N) is equivalent to the TMCP^+_rLip(K,N) condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.
ER  -