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Abstrak :
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.

Abstrak :
ABSTRACT
Teori Gravitasi Termodifikasi MOG memodifikasi persamaan Poisson dalam limit Newtonian. Modifikasi gravitasi ini juga penting sebagai alat uji untuk menganalisa validasi Relativitas Umum GR apakah konsisten terhadap observasi, salah satunya dapat diuji melalui properti bintang. Bintang Katai Coklat atau Brown Dwarf merupakan objek substellar dan bintang VLM Very Low Mass yang sulit dideteksi karena luminositas yang rendah. MOG akan diuji dalam bintang katai coklat untuk mempelajari efek modifikasi gravitasi terhadap relasi massa-radius. Kami menurunkan secara analitik persamaan keadaan untuk bintang katai coklat untuk tekanan Gas Fermi Ideal non-relativistik pada temperatur berhingga. Kami juga menurunkan persamaan struktur bintang termodifikasi dan menyelesaikannya secara numerik untuk mempelajari relasi massa-radius dan profile densitas energi bintang terhadap masing-masing variasi parameter dari tiap teori MOG. Kami juga mengkonstrain parameter teori-teori ini dengan membandingkan Data Observasi dari Relasi Massa-Radius pada paper D.Bayliss dengan Relasi Massa-Radius teoritis yang telah dikerjakan, sehingga kesalahan dapat diketahui. Perbandingan ini dapat dilakukan dengan tes chi-squared dengan mendapatkan nilai minimum dari chi-squared sehingga kami dapat mengkonstrain parameter ruang dari model- model gravitasi.

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ABSTRACT
Modified Gravity MOG can modify Poisson equation in the Newtonian limit. This MOG is also important as test bed to analyze whether General Relativity GR is consistent with observation. Some test beds include stellar properties. Brown Dwarf is substellar object and Very Low Mass VLM object that is difficult to be detected because of their low luminosity. MOG will be tested in brown dwarf in order to study their effects on mass and radius relation. We derive analytically the Brown Dwarf equation of state for non relativistic ideal Fermi gas pressure at finite temperature. We also derive modified hydrostatic stellar equation and solve it numerically to study mass and radius relation and also energy density profile for each parameter on MOG. We constrain these parameters with comparing our theoretical result to observational data from D.Bayliss paper. We do chi squared test to get minimum value of chi squared and as a result we get to constraint the parameters from MOG.