古典的廣義相對論在解釋宇宙膨脹、水星進動、重力透鏡等現象上取得巨大的成功，最近觀測到的黑洞合併所放出的重力波更是支持廣義相對論的強力證據。從場論的觀點，廣義相對論可被描述為自旋為二的無質量規範粒子；長期以來，物理學家們試圖討論帶有質量的自旋二規範粒子，理論中卻會出現「鬼粒子」，意即理論會有無限負能量的模，導致不存在穩定的物理解。最近提出的dRGT理論證明可以去除帶質量重力子中的「鬼粒子」，並在稍後被推廣為雙度規的帶質量重力子模型。「鬼粒子」的問題在考慮重力與物質交互作用時又再度出現，其後在雙度規模型中證明了有一類等效度規與物質的交互作用並不會出現「鬼粒子」。在本論文中，我們利用了雙度規模型中一類已知的真空解，構建出一系列等效度規的精確解；在後半部分中，我們考慮一種規範作用，其中的物質粒子與一類等效度規交互作用；規範粒子則與另外一種等項度規作用，並且計算出這種模型底下等效的「庫倫位能」。

General Relativity(GR) has a great success to explain some astronomy problem such as universal expansion, Mercury precession, and gravitional lensing. The recent observation of grvitational wave from black hole converge also gives a strong evidence of GR. In the view of field theory, GR can be described as a massless spin-2 field. Physicists have been tried to construct a massive spin-2 gauge theory for a long time, however many of these attempts face the ghost problem, which involves a mode of infinite negative energy thus followed by inconsistence of the theory. A recent massive gravity theory called dRGT model and its extension model called bigravuty theory have been showed that they are ghost-free. The ghost may reappear when one concerns such theory coupling to matter fields, but recent study shows that a class of effective theory in bigravity model can be used to construct ghost-free interaction with matter. In this thesis we first take a brief review of dRGT model and bigravity, and then we solve a series of effective metrics by some known solution of bigravity theory. In the later part we consider a model in which a Dirac fermion couples to a effective metric and interacts with a U(1) gauge boson, which couples to another effective metric. Finally we give the modified Coulomb potential in this model.

Contents i
1 Introduction 1
2 Model 5
2.1 dRGT model and its bigravity extension . 5
2.2 degree of freedom counting by ADM formulism . 7
2.3 Some Known Solutions of Bigravity Model . 9
2.4 Matter coupling and effective metric . 13
3 Some Exact Solutions of Effective Metrics and Modified Coulomb
Potential in Curved Background 14
3.1 Effective Metrics Constructed by Konwn Solutions . 15
3.2 Modified Coulomb Potential in Curved Background . 16
4 Conclusion and Future Work 23
Reference 25

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