### 物理代写|宇宙学代写cosmology代考|Scalar fields

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## 物理代写|宇宙学代写cosmology代考|Scalar fields

For a scalar field $\phi$ with arbitrary self-interaction, described by a potential $V(\phi)$, if it is minimally coupled to gravity, the Lagrangian in curved space is written in precisely the same way as in Minkowski space
$$L_{s}=\frac{1}{2} g^{\mu \nu} \phi_{, \mu} \phi_{, \nu}-V(\phi),$$
and its variation with respect to $\phi$ leads to an equation that generalizes the Klein-Gordon relativistic equation
$$\square \phi+d V / d \phi=0,$$
with the general-relativistic d’Alembert operator applied to a scalar
$$\square=\nabla^{\alpha} \nabla_{\alpha}=\frac{1}{\sqrt{-g}} \partial_{\alpha}\left(\sqrt{-g} g^{\alpha \beta} \partial_{\beta}\right)$$
(the same operator $\nabla^{\alpha} \nabla_{\alpha}$ applied to a vector or to a higher-rank tensor will have a more complicated expression due to more Christoffel symbols involved, see (2.34) and (2.37)).

The scalar field SET is obtained from the Lagrangian (2.72) by its variation according to (2.56):
$$T_{\mu s}^{\nu}=\phi_{\mu} \phi^{\nu}-\delta_{\mu}^{\nu} L_{s} .$$

More complex forms of scalar fields are also considered in the problems of gravitation and cosmology. For instance, the so-called $k$-essence with Lagrangians of the general form $L_{s}=L(\phi, X)$ (with $X=(\partial \phi)^{2}$ ) does not violate the minimal coupling principle.

Some of these Lagrangians as well as those with nonminimal coupling will be considered later.

## 物理代写|宇宙学代写cosmology代考|The electromagnetic field

The electromagnetic (massless vector) field is characterized by the vector potential $A_{\mu}$ and by the strength tensor, also called the Maxwell tensor
$$F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} .$$
The electromagnetic field Lagrangian directly generalizes the corresponding flat-space expression. For a field with sources, the Lagrangian reads
$$L_{\mathrm{e}-\mathrm{m}}=-\left(\frac{1}{4} F_{\mu \nu} F^{\mu \nu}-j^{\mu} A_{\mu}\right),$$
where $j^{\mu}$ is the electric charge current density. Its variation with respect to $A_{\mu}$ gives a dynamic equation that corresponds to the second pair of Maxwell equations in usual electrodynamics,
$$\nabla_{\nu} F^{\mu \nu} \equiv \frac{1}{\sqrt{-g}} \partial_{\nu}\left(\sqrt{-g} F^{\mu \nu}\right)=j^{\mu},$$
and due to $(2.78)$, the electric charge conservation law automatically holds:
$$\nabla_{\mu} j^{\mu} \equiv \frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g} j^{\mu}\right)=0$$
The first pair of Maxwell equations finds its analogue in the identity that follows from (2.76),
$$\nabla_{\mu} F_{\nu \sigma}+\nabla_{\sigma} F_{\mu \nu}+\nabla_{\nu} F_{\sigma \mu}=0,$$
or, equivalently,
$$\nabla_{\mu}^{*} F^{\mu \nu}=0$$

## 物理代写|宇宙学代写cosmology代考|Energy conditions

The energy conditions are certain invariant inequalities, mostly characterizing the known and conventional forms of matter, which are used in many studies and lead to many well-known results of outstanding significance, see, e.g., $[230,285,549]$. Let us enumerate and briefly discuss them.

Weak energy condition (WEC): Let $\xi^{\mu}$ be any timelike or null vector field $\left(\xi^{\mu} \xi_{\mu} \geq 0\right)$, and $T_{\mu \nu}$ the SET of matter or a particular kind of matter, then
$$T_{\mu \nu} \xi^{\mu} \xi^{\nu} \geq 0 .$$
If $\xi^{\mu}$ is timelike, we can choose the reference frame where this vector has the direction of 4 -velocity of matter, and then (2.84) reduces to $T_{0}^{0} \geq 0$, that is, the energy density of this kind of matter is nonnegative.

If $\xi^{\mu}$ is null, then the inequality (2.84) has its own separate name.
Null energy condition (NEC): It is the inequality whose violation is a necessary condition at wormhole throats in GR and other “exotic” objects and phantom dark energy in cosmology. If $T_{\mu \nu}$ is the SET of an anisotropic fluid $(2.68)$ in its comoving frame, then (2.84) takes the form
$$\rho+p_{i} \geq 0, \quad i=1,2,3$$
Example: The cosmological constant $\Lambda>0$, if treated as a kind of matter, satisfies the WEC, but respects the NEC only marginally since for it $\rho=\Lambda$ and $p=-\Lambda$.

## 物理代写|宇宙学代写cosmology代考|Scalar fields

φ+d在/dφ=0,

=∇一个∇一个=1−G∂一个(−GG一个b∂b)
（同一个运算符∇一个∇一个由于涉及更多 Christoffel 符号，应用于向量或更高阶张量将具有更复杂的表达式，请参见 (2.34) 和 (2.37))。

## 物理代写|宇宙学代写cosmology代考|The electromagnetic field

Fμν=∂μ一个ν−∂ν一个μ.

∇νFμν≡1−G∂ν(−GFμν)=jμ,

∇μjμ≡1−G∂μ(−Gjμ)=0

∇μFνσ+∇σFμν+∇νFσμ=0,

∇μ∗Fμν=0

ρ+p一世≥0,一世=1,2,3

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