fa هندسه مترهای نا متقارن و کاربرد آن در فیزیک نظری هندسه دیفرانسیل Differential Geometry مقاله مستقل Original Manuscript <p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">در این مقاله&rlm;، هندسه مترهای شبه-ریمانی و نامتقارن روی منیفلدها را در نظر می‌گیریم. رده خاصی از مترهای نامتقارن روی یک منیفلد&rlm;&rlm;، همزمان شامل یک متر شبه-ریمانی متقارن و یک ساختار همتافته هستند. یک التصاق با خواص التصاق لوی-چویتا برای این گونه مترهای نامتقارن را در نظر می‌گیریم. سپس&rlm;، تانسورهای انحنای این التصاق و انحنای اسکالر وابسته به آن را محاسبه کرده و به کمک یک لاگرانژین طبیعی (که تعمیم عمل هیلبرت-اینشتین است) و استفاده از حساب تغییرات&rlm;، معادلاتی بدست خواهیم آورد. معادلات بدست آمده نشان می‌دهند که قسمت متقارن&lrm;&lrm; متر شبه-ریمانی در ارتباط با مفهوم گرانش و قسمت همتافته آن در ارتباط با کمیات مربوط به ماده خواهد بود</p> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Introduction</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">General </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">relativity </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">is </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">model </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">nature, </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">especially, </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">gravity.</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;&lrm;</span></span> <span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">Its </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">cent</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">ral </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">assumption </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">is</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt"> that space, time, and gravity are all aspects of a single entity, </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">cal</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">led </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">space-</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">time, which is modeled by a 4-dimensional Lorentzian manifold</span><span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">.</span></span><span style="font-size:10.0pt"> It analyzes </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">space-</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">time, electromagnetism, matter, and their mutual </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">infl</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">uences. </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">But </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">the </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">effects </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">matter </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">and </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">electromagnetism </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">are </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">added </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">to </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">the </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">model </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">in a</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span> <span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">way</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span> <span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">which </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">is </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">not </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">directly </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">related </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">to </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">geometry </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">space-time </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">manifold. </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">In </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">fact, </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">in</span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">fluences </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">matter </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">and </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">electromagnetism </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">fields </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">are </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">added </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">to </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">theory </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">under </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">notion </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">of </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">stress-energy </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">tensor. </span><span lang="AR-SA" style="font-size:10.0pt"><span arial="" style="font-family:">&lrm;</span></span><span style="font-size:10.0pt">Hence, considering non-symmetric metrics extends this geometry and make a good apparatus to describe other physical quantities.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Material and Methods </span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">In this paper, we consider the geometry of a non-symmetric semi-Riemannian metric on a manifold M. A special class of such metrics contains a semi-Riemannian metric and a symplectic structure on M, simultaneously. Similar to the Levi-Civita connections in semi-Riemannain manifold, we define a new connection which is torsion free and compatible with our symplectic structure.<b> </b>With the help of semi-Riemannian metric, we define and compute the Rcci and scalar curvature of this new connection<b>.</b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"></span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and Discussion </span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Using a natural Lagrangian (which is a generalization of Hilbert-Einstein action) and calculus of variations we derive some new field equations. The equations show that the symmetric part of semi-Riemannain metrics is directly related to gravity and the symplectic part is capable of describing quantities related to matter.&nbsp; </span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"></span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:10.0pt"><span style="line-height:107%"><span calibri="" style="font-family:">In this work, we present a completely geometric theory of gravity. </span></span></span><span class="fontstyle01" style="font-family:CMR10"><span style="color:black"><span style="font-weight:normal"><span style="font-style:normal"><span style="font-size:10.0pt"><span style="line-height:107%">The Riemannian geometry, which is usually used to formulate gravitational theories adds the notion of matter to space time manifold as the way which is not directly related to geometry of the theory. In this framework, we retrieve Einstein&rsquo;s field equation</span></span></span></span></span></span> <span class="fontstyle01" style="font-family:CMR10"><span style="color:black"><span style="font-weight:normal"><span style="font-style:normal"><span style="font-size:10.0pt"><span style="line-height:107%">and we will show that the distribution of matter in space-time is directly related to symplectic part of our geometry</span></span></span></span></span></span> مترهای نامتقارن, تابعک هیلبرت-اینشتین, فضا-زمان Non-symmetric mterics, Hilbert-Enstein functional, space-time 146 156 http://mmr.khu.ac.ir/browse.php?a_code=A-13-454-3&slc_lang=fa&sid=1 Ghodratallah Fasihi-Ramandi قدرت اله فصیحی رامندی gh_fasihi@aut.ac.ir 10031947532846006432 10031947532846006432 Yes دانشگاه بین المللی امام خمینی