توصيفگر ها :
شار ريچي , منيفلد در بينهايت مسطح , جرم ADM , قضيهي جرم مثبت , نظريهي جراحي
چكيده فارسي :
ﺷﺎﺭ ﺭﯾچي ﯾك ﻣﻌﺎﺩﻟﻪ ﺩﯾﻔﺮﺍﻧﺴﯿﻞ پﺎﺭﻩﺍي ﻫﻨﺪﺳي پﺮكﺎﺭﺑﺮﺩ ﺍﺳﺖ كﻪ ﺗﺤﻮﻝ ﻣﺘﺮﯾك ﺭﯾﻤﺎﻧي ﺩﺭ ﺭﺍﺳﺘﺎي ﻗﺮﯾﻨﻪي ﺍﻧﺤﻨﺎي ﺭﯾچي
ﺭﺍ ﺑﻪﺩﺳﺖ ﻣيﺩﻫﺪ. ﺍﺯ ﻃﺮﻓي ﺩﺭ ﻧﺴﺒﯿﺖ ﻋﺎﻡ، ﺟﺮﻡ ﯾك ﻓﻀﺎ‑ﺯﻣﺎﻥ ﺩﺭ ﺑيﻧﻬﺎﯾﺖ ﻣﺴﻄﺢ )ﻣﺪﻝ ﺷﺪﻩ ﺑﻪ ﻭﺳﯿﻠﻪي ﯾك ﻣﻨﯿﻔﻠﺪ
ﺭﯾﻤﺎﻧي( ﻣﻮﺳﻮﻡ ﺑﻪ ﺟﺮﻡ ﺁﺭﻧﻮﻭﯾﺖ‑ﺩﺯﺭ‑ﻣﯿﺴﻨﺮ، ﺑﻪﺻﻮﺭﺕ ﯾك ﺍﻧﺘگﺮﺍﻝ ﺍﺯ ﻣﻘﺎﺩﯾﺮي كﻪ ﺑﻪ ﻣﺘﺮﯾك ﺭﯾﻤﺎﻧي ﻭ ﻣﺸﺘﻘﺎﺕ ﺁﻥ
ﻭﺍﺑﺴﺘﻪ ﺍﺳﺖ، ﺗﻌﺮﯾﻒ ﻣيﺷﻮﺩ. ”ﻗﻀﯿﻪي ﺟﺮﻡ ﻣﺜﺒﺖ” ﺑﺮﺍي چﻨﯿﻦ ﻓﻀﺎ‑ﺯﻣﺎﻥﻫﺎﯾي، ﺗﻌﻤﯿﻤي ﺍﺯ ﺍﯾﻦ ﺣﻘﯿﻘﺖ ﻧﺴﺒﯿﺘي ﺍﺳﺖ
كﻪ ﺟﺮﻡ ﯾك ﺳﯿﺴﺘﻢ، ﻧﺎﻣﻨﻔي ﻣيﺑﺎﺷﺪ ﻭ ﺑﺮﺍﺑﺮ ﺻﻔﺮ ﺍﺳﺖ ﺍگﺮ چگﺎﻟي ﺩﺭ ﻫﻤﻪﺟﺎي ﻓﻀﺎ ﺻﻔﺮ ﺑﺎﺷﺪ. ﺍﺯ ﺁﻧﺠﺎ كﻪ ﺷﺎﺭ ﺭﯾچي ﺩﺭ
ﺣﺎﻟﺖ كﻠي ﯾك ﻣﻌﺎﺩﻟﻪي گﺮﻣﺎ ﺍﺳﺖ ﻭ ﺧﺎﺻﯿﺖ ﻫﻤگﻦﺳﺎﺯي ﻣﺘﺮﯾك ﺭﯾﻤﺎﻧي ﺭﺍ ﺩﺍﺭﺩ، ﺍﺑﺰﺍﺭي كﺎﺭﺑﺮﺩي ﺑﺮﺍي ﺍﺛﺒﺎﺕ ﺍﯾﻦ ﻗﻀﯿﻪ
ﺑﻪ ﻧﻈﺮ ﻣيﺭﺳﺪ.
ﺩﺭ ﺍﯾﻦ پﺎﯾﺎﻥﻧﺎﻣﻪ، ﻗﺼﺪ ﺩﺍﺭﯾﻢ ﺍﺛﺒﺎﺕ ”ﻗﻀﯿﻪ ﺟﺮﻡ ﻣﺜﺒﺖ” ﺭﺍ ﺑﻪﻭﺳﯿﻠﻪي ﺷﺎﺭ ﺭﯾچي ﻣﺮﻭﺭ كﻨﯿﻢ. ﺩﻭ ﺍﺛﺒﺎﺕ ﺑﺮﺍي ﺍﯾﻦ ﻗﻀﯿﻪ
‑ﺑﻌﺪ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ. ﻣﻔﺎﻫﯿﻢ3 ‑ﺑﻌﺪ ﻭ ﺩﯾگﺮي ﺑﺮﺍي ﻓﻀﺎ‑ﺯﻣﺎﻥ ﺩﺭn ﺑﻪﻭﺳﯿﻠﻪي ﺷﺎﺭ ﺭﯾچي، ﯾكي ﺑﺮﺍي ﻓﻀﺎ‑ﺯﻣﺎﻥ ﺩﺭ
ﺍﺳﺎﺳي ﻫﻨﺪﺳﻪي ﺭﯾﻤﺎﻧي ﻭ ﻓﻀﺎﻫﺎي ﺗﺎﺑﻌي، ﺑﻪﺧﺼﻮﺹ ﻓﻀﺎي ﺳﻮﺑﻮﻟﻒ ﺭﺍ ﺑﻪﺗﺮﺗﯿﺐ ﺑﺮﺍي ﺷﻨﺎﺧﺖ ﻫﻨﺪﺳﻪي ﻓﻀﺎ‑ﺯﻣﺎﻥ ﻭ
‑ﺑﻌﺪ ﺑﺎ ﻓﺮﺽ ﻭﺟﻮﺩ ﺩﺭﺍﺯﻣﺪﺕ ﺷﺎﺭn ﻭﯾژگيﻫﺎي ﻣﻘﺪﻣﺎﺗي ﺟﻮﺍﺏﻫﺎي ﺷﺎﺭ ﺭﯾچي ﻣﻄﺮﺡ ﻣيكﻨﯿﻢ. ﺩﺭ ﻣﺴﯿﺮ ﺍﺛﺒﺎﺕ ﻗﻀﯿﻪ ﺩﺭ
ﺭﯾچي، ﺑﺮﺍي ﺗﺤﻠﯿﻞ ﺗكﯿﻨگيﻫﺎ، ﺗﺎﺑﻌك ﺍﻧﺘﺮﻭپي پﺮﻟﻤﺎﻥ ﺭﺍ ﻣﻌﺮﻓي ﻭ ﺑﻪﻃﻮﺭ گﺴﺘﺮﺩﻩ ﺑﺮﺭﺳي ﻣيكﻨﯿﻢ. ﺳپﺲ ﺑﺎ ﺑﻪكﺎﺭ ﺑﺮﺩﻥ
‑ﺑﻌﺪ ﻣﺮﻭﺭ ﻣيكﻨﯿﻢ.3 ﻧﻈﺮﯾﻪي ﺟﺮﺍﺣي ﺩﺭ ﺗكﯿﻨگيﻫﺎي ﺷﺎﺭ ﺭﯾچي، ﺍﺛﺒﺎﺕ ﻗﻀﯿﻪ ﺭﺍ ﺩﺭ
چكيده انگليسي :
The Ricci flow is a widely geometric partial differential equation that describes the evolution of a Riemannian metric
in the direction of the Ricci curvature tensor. In general relativity, the mass of an asymptotically flat spacetime
(modeled by a Riemannian manifold), known as the Arnowitt-Deser-Misner (ADM) mass, is defined as an integral that
dependes on the Riemannian metric and its derivatives. The Positive Mass Theorem for such spacetimes generalizes
the relativistic fact that the mass of a system is non-negative and equals zero if the density is zero everywhere in space.
Since the Ricci flow, in general, behaves like a heat equation and has a homogenizing effect on the Riemannian metric,
it appears to be a useful tool for proving this theorem.
In this thesis, we aim to review the proof of the Positive Mass Theorem using the Ricci flow. Two proofs are presented:
one for spacetime in n-dimensions and another for spacetime in three dimensions. We introduce fundamental concepts
of Riemannian geometry and function spaces, particularly Sobolev spaces, to understand the geometry of spacetime
and the preliminary properties of Ricci flow solutions. In proving the theorem in n-dimensions, assuming the long-
time existence of the Ricci flow, we introduce and extensively examine Perelman’s entropy functional to analyze
singularities. Then, using the surgery theory for Ricci flow singularities, we review the proof of the theorem in three
dimensions.
