# 物理代写|广义相对论代写General relativity代考|Homogeneity and Isotropy

## 物理代写|广义相对论代写General relativity代考|Homogeneity and Isotropy

Optical and radio surveys of the sky confirm that the distribution of galaxies is the same in all directions. Also, the precise consistency of the CMBR confirms that, indeed, the universe is isotropic about our position on a large scale.
Based on the observations cosmologists introduced the following two postulates:
Weyl postulate: Let the galaxies be treated as particles (or points). The Weyl postulate states that the world lines of galaxies are recognized with a specific class of observers, known as fundamental observers, have a common point of intersection, form a bundle (or congruence) of geodesics. One can also describe a common time coordinate, which quantifies the proper time of each such observer.
Alternative: The Weyl postulate states that if one designates the world lines of galaxies as fundamental observers, then these world lines form a three bundle of nonintersecting geodesics orthogonal to a series of space-like hypersurfaces.

Now, we are interested in describing the Weyl postulate in terms of coordinate and metric of spacetime. Let us assume that galaxies, i.e., the particles are moving along nonintersecting world lines a, b, c, $\ldots$ which have no irregularities (see Fig. 99). We can also use $x^\mu(\mu=1,2,3)$ to label a typical world line in the three bundles of galaxy world lines. Here, the three coordinates $x^\mu$ are spacelike. Further, let the coordinate $x^0$ as time coordinate measures the proper time along each curve, $x^\mu=$ constant. Hence it is obvious that $x^0=$ constant is a usual spacelike hypersurface, which is orthogonal to the usual world line characterized by $x^\mu=$ constant. It is always possible to find a continuum from a discrete set of points (here galaxies) through smooth fluid approximation. Here we use four coordinate $x^i, i=0,1,2,3$ to describe space and time and in terms of these coordinates, we define the metric tensor as $g_{i k}$, where the line element is
$$d s^2=g_{i k} d x^j d x^k$$

According to Weyl postulate, the orthogonality condition yields
$$g_{0 \mu}=0 .$$
Also, the line $x^\mu=$ constant is a geodesic.
We know the geodesic equations as
$$\frac{d^2 x^i}{d s^2}+\Gamma_{k l}^i \frac{d x^k}{d s} \frac{d x^I}{d s}=0$$

## 物理代写|广义相对论代写General relativity代考|Robertson-Walker Metric

On the large scale (more than 100 million light years or so), the universe looks homogeneous and isotropic around us. At this scale, the density of galaxies is approximately the same and all directions from us appear to be equivalent. Using Weyl’s postulate one can write the general metric as
$$d s^2=c^2 d t^2-h_{i j} d x^i d x^j, \quad(i, j=1,2,3)$$
where the $h_{i j}$ are functions of $\left(t, x^1, x^2, x^3\right)$.
Now, we try to find the metric when the spacetime of the universe is homogeneous and isotropic. Let us consider two neighboring galaxies whose coordinates are $\left(x^1, x^2, x^3\right)$ and $\left(x^1+\Delta x^1, x^2+\right.$ $\left.\Delta x^2, x^3+\Delta x^3\right)$, respectively. The distance between two neighboring galaxies on the same hypersurface $t=$ constant is given by
$$d \sigma^2=h_{i j} \Delta x^i \Delta x^j$$
Consider three widely separated galaxies $A, B$, and $C$ shown in Fig. 100 at some time $t_i$. At a later time, $t$ another triangle $\left(A^{\prime} B^{\prime} C^{\prime}\right)$ is formed by the same galaxies.

The postulate of homogeneity and isotropy at all points and directions on a particular hypersurface implies that two triangles must be similar and, also, the increase in length must be independent of the position and direction of the triangle. Three galaxies at $A, B, C$ at time $t_i$ expand away from each other to $A^1, B^1, C^1$ at time $t$ and the sides of the triangle are expanded by the same factor. Thus, the increase in distance $A B \rightarrow A^1 B^1$ be the same as $A C \rightarrow A^1 C^1$ (look at the expansion from the point of view of the observer in A). It follows that the expansion is controlled by a single function of time, i.e., the functions $h_{i j}$ must involve the time coordinate $t$ through a common factor $R^2(t)$. Hence the metric (11.2) takes the form
$$d s^2=c^2 d t^2-R^2(t) \gamma_{i j} d x^i d x^j$$
where $\gamma_{i j}$ are the functions of $\left(x^1, x^2, x^3\right)$ only.
Now we give our attention to the homogeneous and isotropic three space given by
$$d \sigma^{1^2}=\gamma_{i j} d x^j d x^j$$
This space must be a space of constant curvature (standard theorem of differential geometry).

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|Homogeneity and Isotropy

Weyl 假设：将星系视为粒子 (或点) 。Weyl 假设指出，星系的世界线被一类特定的观 察者（称为基本观察者）所认可，它们有一个共同的交点，形成测地线束（或全等）。 人们还可以描述一个公共时间坐标，它量化每个这样的观察者的本征时间。

$$d s^2=g_{i k} d x^j d x^k$$

$$g_{0 \mu}=0$$

$$\frac{d^2 x^i}{d s^2}+\Gamma_{k l}^i \frac{d x^k}{d s} \frac{d x^I}{d s}=0$$

## 物理代写|广义相对论代写General relativity代考|Robertson-Walker Metric

$$d s^2=c^2 d t^2-h_{i j} d x^i d x^j, \quad(i, j=1,2,3)$$

$$d \sigma^2=h_{i j} \Delta x^i \Delta x^j$$

$$d s^2=c^2 d t^2-R^2(t) \gamma_{i j} d x^i d x^j$$

$$d \sigma^{1^2}=\gamma_{i j} d x^j d x^j$$

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