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"Studi tentang lensa gravitasi sudah dimulai sejak diperkenalkannya teori relativitas umum. Tetapi sudut de eksi cahaya oleh benda masif seperti matahari hanya terjadi pada beberapa arc sekon. Seiiring dengan berkembangnya observasi pada benda supermasif seperti neutron star atau lubang hitam, hal ini memberikan peluang untuk menguji teori relativitas umum lebih lanjut. Lubang hitam reguler merupakan lubang hitam yang menarik, karena lubang hitam ini tidak memiliki singularitas di seluruh koordinat. Pada riset kali ini, kami menghitung sudut de eksi cahaya dari lubang hitam reguler statik speris simetrik. Kami mengaproksimasi sudut de eksi cahaya pada medan lemah dengan mengekspansi persamaan integral sudut de eksi cahaya hingga orde keempat. Semen- tara pada medan kuat kami membagi persamaan integral menjadi dua bagian, bagian konvergen dan bagian divergen. Kami mengisolasi bagian divergen dari persamaan integral sudut de eksi cahaya. Persamaan yang tidak lagi mengandung bagian divergen kemudian dikalkulasi secara numerik sementara bagian divergen diaproksimasi dengan mengekspansi persamaan pada daerah sekitar radius of photon sphere.

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The study of gravitational lensing has been started since the beginning of the introduction of General Relativity. But Light De ection Angle by massive object such as our Sun is only in a few arc second. As the observation of Super Massive Objects such as Neutron Star or Black Hole becomes more common, this provides an opportunity to test General Relativity in a new ground. A regular Black Hole is an interesting Black Hole as it is lack of essential singularity. On this research we try to calculate the de ection angle of a spher- ically symmetric Regular Charged Black Hole. We rst approximate the de ection angle in Weak Field Limit by expanding the integral from the light equation of motion. As we approach the Strong Field Limit, we isolate the divergence from the integral. The integral containing no divergence part then calculated numerically while the integral containing the divergence part is approximated by expanding the integrand around the photon sphere."

"Studi tentang lensa gravitasi sudah dimulai sejak diperkenalkannya teori relativitas umum. Tetapi sudut de eksi cahaya oleh benda masif seperti matahari hanya terjadi pada beberapa arc sekon. Seiiring dengan berkembangnya observasi pada benda supermasif seperti neutron star atau lubang hitam, hal ini memberikan peluang untuk menguji teori relativitas umum lebih lanjut. Lubang hitam reguler merupakan lubang hitam yang menarik, karena lubang hitam ini tidak memiliki singularitas di seluruh koordinat. Pada riset kali ini, kami menghitung sudut de eksi cahaya dari lubang hitam reguler statik speris simetrik. Kami mengaproksimasi sudut de eksi cahaya pada medan lemah dengan mengekspansi persamaan integral sudut de eksi cahaya hingga orde keempat. Semen- tara pada medan kuat kami membagi persamaan integral menjadi dua bagian, bagian konvergen dan bagian divergen. Kami mengisolasi bagian divergen dari persamaan integral sudut de eksi cahaya. Persamaan yang tidak lagi mengandung bagian divergen kemudian dikalkulasi secara numerik sementara bagian divergen diaproksimasi dengan mengekspansi persamaan pada daerah sekitar radius of photon sphere.

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The study of gravitational lensing has been started since the beginning of the introduction of General Relativity. But Light De ection Angle by massive object such as our Sun is only in a few arc second. As the observation of Super Massive Objects such as Neutron Star or Black Hole becomes more common, this provides an opportunity to test General Relativity in a new ground. A regular Black Hole is an interesting Black Hole as it is lack of essential singularity. On this research we try to calculate the de ection angle of a spher- ically symmetric Regular Charged Black Hole. We rst approximate the de ection angle in Weak Field Limit by expanding the integral from the light equation of motion. As we approach the Strong Field Limit, we isolate the divergence from the integral. The integral containing no divergence part then calculated numerically while the integral containing the divergence part is approximated by expanding the integrand around the photon sphere."

"Persamaan-persamaan medan gravitasi dalam teori relativitas umum dapat dikonstruksi dengan dna pendekatan berbeda. Pendekatan pertama dengan metode tensor klasik di mana simbol Christoffel (koneksi affin) diturunkan untuk mendapatkan tensor Ricci. Pendekatan kedua dengan menggunakan prinsip variasi. Dari kedua pendekatan tersebut didapatkan persamaan medan gravitasi yang dikenal dengan persamaan medan Einstein, Persamaan tersebut berupa persamaan diferensial pasial non-liniear. Dengan metode solusi Schwarzschild persamaan medan Einstein dapat dipecahkan untuk kasus ruang vakum yang statis dan simetris sferis, Metrik tensor yang dihasilkan berupa (element jarak) yang menggambarkan struktur ruang di sekitar bintang. Solusi ini dapat digunakan untuk menjelaskan fenomena fisika berupa pergeseran (presisi) orbit planet-planet di dalam tata surya dan pergeseran merah gravitasi.

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Gravitational field equations on general theory of relatifity can be constructed with two different approaches. First it uses classical method of tensor where the Christiffel symbol (affine connection) is derived to obtain the Ricci tensor. The second approach is by using, vamanional principle. The gralntatiorial lield equation which has been obtained front ming, both of approach is called Eirtsteirfs Held equation. This equation is in the fcrm of ncn-liniear partial of differential equaticn (non-liniear PDE). The Schwarzschild solution method will be used to save this field equation in the special case. The case is static and spherically symmetric on the vacuum field. The metric tensor om' that soluuon describes the field outside a rotating star. Thar solution could be used to explain some physical phenomenon as the examples are precision of orlbital planets and gravitational red shift."

"The book starts by establishing the mathematical background (differential geometry, hypersurfaces embedded in space-time, foliation of space-time by a family of space-like hypersurfaces), and then turns to the 3+1 decomposition of the Einstein equations, giving rise to the Cauchy problem with constraints, which constitutes the core of 3+1 formalism. The ADM Hamiltonian formulation of general relativity is also introduced at this stage. Finally, the decomposition of the matter and electromagnetic field equations is presented, focusing on the astrophysically relevant cases of a perfect fluid and a perfect conductor (ideal magnetohydrodynamics). The second part of the book introduces more advanced topics: the conformal transformation of the 3-metric on each hypersurface and the corresponding rewriting of the 3+1 Einstein equations, the Isenberg-Wilson-Mathews approximation to general relativity, global quantities associated with asymptotic flatness (ADM mass, linear and angular momentum) and with symmetries (Komar mass and angular momentum). In the last part, the initial data problem is studied, the choice of spacetime coordinates within the 3+1 framework is discussed and various schemes for the time integration of the 3+1 Einstein equations are reviewed. "

"The general theory of relativity : a mathematical exposition will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) are furnished. Topics in this book include, but are not limited to tensor analysis, the special theory of relativity, the general theory of relativity and Einstein’s field equations, spherically symmetric solutions and experimental confirmations, static and stationary space-time domains, black holes, cosmological models, algebraic classifications and the Newman Penrose equations, and the coupled Einstein-Maxwell-Klein-Gordon equations appendices covering mathematical supplements and special topics."