Accelerating Spacetimes With Cosmological Constant And Their Properties

Ever since general relativity has been promoted, people tried to find out variousexact solutions of Einstein equation in order to understand General Relativity moreprofoundly. Exact analytic solutions and the properties of whom are very important forus to understand the properties of spacetime, the characteristic of strong coupling ofgravity, the relationship between thermodynamics and gravity, even quantum gravity.Among various analytic solutions of Einstein equation, we are very interested in accel-erating spacetime in particular, a distinguishing feature of whom is, the static observersin accelerating spacetime will observe accelerating horizons, which correspond respec-tively to the accelerating observers in Minkowski spacetime and the Rindler horizonsthey test, thus accelerating spacetime has non-trial global structure.In addition, the Lovelock family of gravity theories-of which Einstein-Gauss-Bonnet gravity is a good model among various ones of generalized gravity. It is theghost-free extenstion of general relativity with the left hand side of the field equationsbeing symmetric conserved tensor and containing no more than second order deriva-tives of the metric. Thus searching for various solutions as well as studying the proper-ties of the solutions will help for understanding the relationships between Einstein andGauss-Bonnet gravity. The main content of the thesis are as follows:A three-dimensional accelerating spacetime of Einstein gravity is given, and it isfound to be a accelerating BTZ black hole at proper ranges of the parameters. Thetemperature of the black hole are calculated. It is found that the horizon of theblack hole does not intersect with conformal infinity in de Sitter background, butthe two causal boundaries does intersect with each other in anti-de Sitter back-ground. Thus, while evaluating the area of such horizons, one should excludethe part of horizon that is hidden beyond the conformal infinity. In the samesense, since conformal infinity as a causal boundary will influence the causalstructure of the spacetime, we analyze the global structure of the black hole indifferent background and different ranges of angles. Since our solution representa accelerating black hole that are asymptotically non-flat, the way ever knows todetermine the mass of asymptotically flat black holes does not work, then we cannot check the first law of black hole thermodynamics. Furthermore, if consider-ing the cosmological constant as being the pressure of some liquid component,the first law of black hole thermodynamics should be modified, even if we accept the correctness of the first law by ad hoc assumption, it is still insufficient todetermine the mass of the black hole.A kind of ring accelerating vacua of Einstein gravity are given, the horizons ofwhom are found to possess non-trivial topology of SD3bundle over S1, in factit is conformally distorted SD3bundle over S1. After working out the curvatureinvariants, it’s found that they are all constants with respective to the cosmolog-ical constant. Accelerating parameter can still be interpreted as the accelerationof the origin. During studying the causal structure of the spacetime, we find thatconformal infinity separate the spacetime into two patches, and that whether ornot the conformal infinity can be reached depends on the range of the correspond-ing angle. Thus we draw the global of the spacetime corresponding to differentranges of angle. Taking various parameters limits will get several spacetime thatare well known. After Euclideanization, we get a smooth, compact and inho-mogeneous Einstein manifold with the topology of conformally distorted SD1bundle over S1, which can be regarded as analogue of Don Page’s gravitationalinstanton or its generalization. Taking the five-dimensional Euclideanized ring asan example, we compute the volume of whom, and find it is zero. However, thecompact Einstein metrics of zero volume are not rare, there are other examples.The accelerating vacuum solutions of Einstein-Gauss-Bonnet gravity in five andsix dimensions are given, and this kind of vacua are found also to be solutionsof pure Einstein equation. Thus we work out the effective cosmological con-stant, and find that it is dependent of the accelerating parameter but independentof the Gauss-Bonnet parameter. Take proper limit one will find that, in orderto get a small positive value of the effective cosmological constant the Gauss-Bonnet parameter must be required to be negative and have a very large absolute.Those are mysterious figure for us at the beginning that the coefficients repre-senting the relationship between effective cosmological constant and the Gauss-Bonnet parameter, which promote us to search the solutions of the field equationsof Einstein-Gauss-Bonnet gravity in any dimensions. After working out the ef-fective cosmological constants, we get the exact expressions of the coefficientsrepresenting the relationship between effective cosmological constants and theGauss-Bonnet parameters expressed with the dimensions of spacetime. We takethe five-dimensional vacuum solution as an example to analyze the causal struc-ture of the accelerating spacetime in detail in different ranges of the parameters and angle. We provide a clear physical meaning for the accelerating parameter,i.e., the amplitude of the proper acceleration of the origin, through working outthe acceleration of the static observers in the spacetime. Finally, we analyze akind of geodesics of the particles in the spacetime, from the diagram of the ef-fective potential it can be seen that, the particles tend to go to the horizon of thespacetime.Finally, we take the Ricci-squared gravity as an example to illustrate the warpeddecomposition method can be widely used without worrying about what themodel of gravity is and what the dimensions of the spacetime are. A kind of so-lutions of Ricci-squared gravity in any-dimensions are given, although they sharethe same form as the ones of Einstein gravity, they describe different spacetimein different models of gravity, as the physical quantities are different in differentmodels of gravity.