### 物理代写|宇宙学代写cosmology代考|Riemannian space–time — curvature

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## 物理代写|宇宙学代写cosmology代考|Riemannian space–time — curvature

Let us present some definitions and relations, important for the further content of the book.

To begin with, the contravariant components of the metric $g^{\mu \nu}$ form a matrix reciprocal to the matrix $g_{\mu \nu}$ :
$$g_{\mu \alpha} g^{\alpha \nu}=\delta_{\mu}^{\nu},$$

where $\delta_{\mu}^{\nu}$ is the Kronecker symbol, a tensor whose components form the unity matrix: they are equal to unity for coinciding values of the indices and to zero for noncoinciding ones. The tensors $g_{\mu \nu}$ and $g^{\mu \nu}$ are used for raising and lowering arbitrary tensor indices.

Partial derivatives of any scalar function (that is, a function whose values are the same in all coordinate systems) $f\left(x^{\mu}\right)$ with respect to the coordinates, $\partial_{\mu} f$, form a covariant vector called the gradient of $f$. Partial derivatives of the components of a vector $A_{\mu}$ or $A^{\mu}$, in general, do not form a tensor because coordinate transformations are, in general, nonlinear. To obtain a covariant form of physical equations and for many other purposes it is thus necessary to generalize the notion of a derivative, to make it a tensor. This goal is achieved by introducing the covariant derivatives
\begin{aligned} &\nabla_{\mu} A_{\nu}=\partial_{\mu} A_{\nu}-\Gamma_{\mu \nu}^{\alpha} A_{\alpha} \ &\nabla_{\mu} A^{\nu}=\partial_{\mu} A^{\nu}+\Gamma_{\mu \alpha}^{\nu} A^{\alpha} \end{aligned}
where the quantities $\Gamma_{\mu \nu}^{\alpha}$ (they do not form a tensor!) are called the Christoffel symbols, or affine connection coefficients agreeing with the metric (the metric connection, or the metric affinity). They are expressed in terms of the metric tensor and its first-order partial derivatives:
$$\Gamma_{\mu \nu}^{\sigma}=\frac{1}{2} g^{\sigma \alpha}\left(\partial_{\nu} g_{\alpha \mu}+\partial_{\mu} g_{\alpha \nu}-\partial_{\alpha} g_{\mu \nu}\right)$$

## 物理代写|宇宙学代写cosmology代考|The Einstein equations

In GR, the dynamic variables characterizing the gravitational field are the metric tensor components $g_{\mu \nu}$. The dynamic equations of GR are derived from Hilbert’s variation principle
$$\delta S=0, \quad S=\int \frac{R-2 \Lambda}{2 x} \sqrt{-g} d^{4} x+S_{m},$$
where $S_{m}=\int L_{m} \sqrt{-g} d^{4} x$ is the action of matter, i.e., substance and all fields except the gravitational field, and $\Lambda$ is the cosmological constant, which is usually negligible when considering “local” configurations (up to the scale of a cluster of galaxies) but is manifestly important on the cosmological scale. The condition $\delta S=0$ leads to the Hilbert-Einstein equations (more frequently they are simply called the Einstein equations)
$$R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R+g_{\mu \nu} \Lambda=-\varkappa T_{\mu \nu} .$$
Here, $\varkappa=8 \pi G / c^{4}$ is the Einstein gravitational constant ( $G$ is the Newtonian gravitational constant), and $T_{\mu \nu}$ is the (metric) stressenergy tensor (SET) of matter:
$$T_{\mu \nu}=\frac{2}{\sqrt{-g}} \frac{\delta S_{m}}{\delta g^{\mu \nu}} .$$
In (2.55), there are in general ten nonlinear partial differential equations. However, first, the freedom of choosing a coordinate system makes it possible to impose four arbitrary coordinate conditions which can be formulated as equalities involving the coefficients $g_{\mu \nu}$, and therefore remain only six independent equations. Second, among the remaining equations there are four differential dependences related to the identities (2.46), which results in only two real dynamic equations. The other four are constraint equations which do not contain second-order time derivatives. These circumstances are of importance for all dynamic processes in GR, and above all for gravitational waves, which can have only two independent polarizations.

## 物理代写|宇宙学代写cosmology代考|Geodesic equations

The equations of motion for free particles in Riemannian space-times can be obtained by varying the action (2.21) (now written for a Riemannian linear element). The variation equation, as in Minkowski space, has the meaning of a trajectory (geodesic) equation, describing the extremum of the world line length between two given points. It can be written as
$$\frac{d u^{\alpha}}{d s}+\Gamma_{\mu \nu}^{\alpha} u^{\mu} u^{\nu}=0 .$$
Here, $s$ is the interval which coincides with the proper time of an observer moving along the geodesic if it is timelike and the proper length along the geodesic if it is spacelike. In all cases $s$ is a canonical parameter. ${ }^{5}$

Let us derive Eq. (2.58) from the conservation law (2.57). It will be a good illustration of the possibility to derive the equations of motion for matter from the Einstein equations.

Let us begin with the expression for the SET of a perfect fluid, which can be obtained as a natural extension of the corresponding expression from SR [366] to Riemannian spaces:
$$T_{\mu \nu}=(\varepsilon+p) u_{\mu} u_{\nu}-p g_{\mu \nu},$$

where $u_{\mu}$ is the 4 -velocity of particles of the fluid, $\varepsilon=\rho c^{2}$ is its energy density, and $p$ its pressure. In particular, for dustlike matter, consisting of noninteracting particles,
$$p=0, \quad T_{\mu \nu}=\varepsilon u_{\mu} u_{\nu}$$

## 物理代写|宇宙学代写cosmology代考|Riemannian space–time — curvature

gμαgαν=δμν,

∇μAν=∂μAν−ΓμναAα ∇μAν=∂μAν+ΓμανAα

Γμνσ=12gσα(∂νgαμ+∂μgαν−∂αgμν)

## 物理代写|宇宙学代写cosmology代考|The Einstein equations

δS=0,S=∫R−2Λ2x−gd4x+Sm,

Rμν−12gμνR+gμνΛ=−ϰTμν.

Tμν=2−gδSmδgμν.

## 物理代写|宇宙学代写cosmology代考|Geodesic equations

duαds+Γμναuμuν=0.

Tμν=(ε+p)uμuν−pgμν,

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