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arXiv 1507.04076gr-qc
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标量-张量引力中的引力自作用力

Gravitational self-force in scalar-tensor gravity

Peter Zimmerman

摘要:

受浮动轨道的发现和对其他理论提供额外约束的可能性的启发,本文推导了背景时空中在加速世界线上运动的小型紧凑物体的自作用力方程,它是标量张量理论中引力场和标量场耦合方程的解。在爱因斯坦框架中,由粒子引起的扰动的耦合场方程与标量真空时空的扰动场方程具有相同的形式,两者都属于齐默尔曼和泊松研究的一般双曲场方程。在这里,我们解决了正式的延迟格林函数,其中有明确的陈述阿达玛形式的世界线附近的场方程。利用Hadamard形式的准局域展开,根据Detweiler-Whiting公式,我们得到了费米正规坐标下的正则解。为了计算运动方程,我们用质量和“电荷”来参数化世界线,我们用原始约旦框架质量、其导数和将约旦框架中的固有时间转换为爱因斯坦框架的参数来定义世界线。这些参数取决于背景标量场的大小及其自场修正。由规则场导出的运动方程与广义相对论标量真空几何中作用于带电大质量粒子的自作用力方程非常相似。与标量真空情形不同,标量张量自作用力方程中的“电荷”参数是随时间变化的,并导致额外的局部和尾部项。我们还提供了自场影响下的世界线参数的演化方程。

英文摘要:

Motivated by the discovery of floating orbits and the potential to provide extra constraints on alternative theories, in this paper we derive the self-force equation for a small compact object moving on an accelerated world line in a background spacetime which is a solution of the coupled gravitational and scalar field equations of scalar-tensor theory. In the Einstein frame, the coupled field equations governing the perturbations sourced by the particle share the same form as the field equations for perturbations of a scalarvac spacetime, with both falling under the general class of hyperbolic field equations studied by Zimmerman and Poisson. Here, we solve the field equations formally in terms of retarded Green functions, which have explicit representations as Hadamard forms in the neighbourhood of the world line. Using a quasi-local expansion of the Hadamard form, we derive the regular solutions in Fermi normal coordinates according to the Detweiler-Whiting prescription. To compute the equation of motion, we parameterize the world line in terms of a mass and "charge", which we define in terms of the original Jordan frame mass, its derivative, and the parameter which translates the proper time in the Jordan frame to the Einstein frame. These parameters depend on the value of the background scalar field and its self-field corrections. The equation of motion which follows from the regular fields strongly resembles the equation for the self-force acting on a charged, massive particle in a scalarvac geometry of general relativity. Unlike the scalar vacuum scenario, the "charge" parameter in the scalar-tensor self-force equation is time variable and leading to additional local and tail terms. We also provide evolution equations for the world line parameters under the influence of the self-fields.

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