Galactic rotation curves in conformal scalar-tensor gravity

共形不变的标量-张量引力中的星系旋转曲线

李强

11849167

硕士

物理学

Leonardo Modesto

2020-05-27

2020-05-27

哈尔滨工业大学

深圳

Conformal gravity is a compelling alternative to Einstein’s general relativity in some cases. Although there are many precise experimental tests like gravitational waves and precession of apsides that have proved the validity of Einstein’s gravity, this theory also suffers several challenges such as the singularities at small scale and large discrepancies between its predictions of galactic rotation curves and observations if only baryonic matter is considered. In this thesis, we study these issues within the frame of conformal gravity. We briefly review the idea that using conformal gauge transformations to transfer the singularity in the spacetime metric to the unobserved dilaton fields. An example of the Schwarzschild metric is discussed. And we also show how the formula for calculating the black holes’ entropy should be modified in the conformal scalar-tensor gravity. We also propose to explain the characteristics of the rotation curves of observed galactic bulge and halo within conformal scalar-tensor gravity. An analytic rotation velocity formula for general spacetime metric is derived. For spherically symmetric baryonic matter distribution models of galactic bulge and halo. we show that after choosing a specific kind of scale factor, the observed shape of rotation curve can arise naturally. Besides, we found a spherically symmetric solution of conformal scalar-tensor gravity in the presence of a cosmological constant compatible with two minimal assumptions: (i) the metric is written in Schwarzschild coordinates in which the two dimensional transverse area equals 4Pi times r square, (ii) the time-like and space-like components of the metric satisfy the relation 𝑔_{00} = -1/g_{11}. The resulting geometry captures both the local and global structure of the universe. We find a formula which has three universal parameters for the rotation velocity with the metric we found. The first of the three parameters (gamma star) is related to the only ambiguity present in the metric describing the local geometry, while the second parameter (gamma_0) appears in a similar metric, but now describing the global geometry due to all the other galaxies in the Universe. The third parameter is the cosmological constant. From the geometry of the spacetime we have extracted the local and global contributions to the velocity of a probe star, the two universal parameters are fixed as gamma star = 2.56 times 10 to (-39) m to (-1) and gamma_0 = 2.94 times 10 to (-28) m to (-1). We used the MCMC method to fit the value M/L for each galaxy by fitting the actual rotation curves of a sample of 104 galaxies. The results are shown in the Fig.s. 4-2-4-8 and Table (4-1) and (4-2). The plots show an extraordinary agreement between the theoretical model and observational data. We also compared conformal gravity with metric skew tensor gravity (MSTG) and modified Newtonian dynamics(MOND) for the case of a spherically distribution of matter. It turns out that at small distances, the three theories are consistent with Newtonian gravity, but at large distance the rotation velocities behave very different: in MSTG and conformal gravity the velocity decrease but in a very different way, while the velocity is constant in MOND. Although we can fit the observed rotation curves very well with the only baryonic matter, the existence of dark matter is not excluded. When compared to models in general relativities, the needed amount of dark matter is much less. We consider the outcome of our work a good reason to further reflect the dark matter problem.

在一些情况下，共形引力被认为是爱因斯坦的广义相对论的有力的替代理论。 虽然目前已经有很多相当精确的实验测试比如引力波和近日点的进动证明了广义 相对论的有效性，这个理论仍然面临一些挑战，比如在小尺度的奇点以及它关于 星系旋转曲线的预言和观测数据之间的巨大差距（如果仅考虑重子物质）。本毕业 论文在共形引力的框架下研究了这些问题。 首先，我们简单介绍了利用共形规范变换来将时空奇点转移到辅助场的方案。 然后以史瓦西度规为例做了具体介绍。此外还证明了在共形引力中，计算黑洞熵的 公式应该如何被修正。本文用共形不变的标量-张量引力解释观测到的星系球核以 及星系晕中旋转曲线的特征。对于一般时空度规解析地推导了旋转速度公式。本 文还证明了在具有球对称重子物质分布的星系球核以及晕中，某些特殊选取的共 形因子可以给出具有观测特征的旋转曲线。 另外，在考虑宇宙学常数的情况下，本文发现共形不变的标量-张量引力的一 个考虑宇宙学常数的球对称解满足如下的两个假设：(1). 在具有二维横向面积为 4Pi乘r平方的史瓦西坐标系中，(2). 度规的类时和类空分量满足这样的关系:g_00 = -1/g_{11}。 这个度规解可以反映出宇宙的局域和全局几何的性质。我们用这个度规得到了含 有三个普适参数的星系旋转速度公式。第一个参数(gamma star) 存在于描述局域几何的度 规中，第二个参数gamma_0 来源于一个和局域度规类似但是被用来描述受宇宙中所有其 他物质影响的全局度规，第三个参数是宇宙学常数。我们将三个普适参数中的两个固定为gamma star= 2.56 乘 10的(-39)次方米的(-1)次方和gamma_0 = 2.94 乘10的(-28)次方米的(-1)次方。本文对104 个星系的旋转 曲线数据使用蒙特卡洛马尔可夫链方法来拟合光质比参数。结果被列在表格(6-1) 和(6-2) 中，拟合的旋转曲线被画在图(6-1) 到(6-7) 中。我们可以看到，理论模型 拟合出的曲线和观测到的数据符合的很好。 本文还在球对称物质分布的情况下对比了共形引力，倾斜度规引力以及修正 牛顿动力学这三种理论。我们发现，在小尺度的情况下，这三种理论都可以给出 和牛顿引力同样的结果。但是在大距离的情况下，他们给出的旋转曲线表现的很 不一样。具体来说，倾斜度规理论和共形引力中旋转速度都会在大距离时随着距 离增加而下降，不过是以不同的规律；然而修正牛顿动力学会给出常数旋转速度 的结果。 虽然在仅考虑重子物质的情况下共形引力可以给出和观测数据接近的星系旋转曲线，但是暗物质存在的可能性并没有被排除。和广义相对论中的模型相比，我 们的模型需要少得多的暗物质。我们认为这些结果可以作为更详细地探讨暗物质 疑难的一个理由。

Conformal gravity galactic rotation curves singularity black hole’s entropy

共形引力 星系旋转曲线 时空奇点 黑洞熵

英语

联合培养

南方科技大学

Li Q. Galactic rotation curves in conformal scalar-tensor gravity[D]. 深圳. 哈尔滨工业大学,2020.