物理代写|广义相对论代写General relativity代考|PHYC90012

物理代写|广义相对论代写General relativity代考|Isotropic Coordinates

Isotropic coordinate system is a new coordinate system whose spatial distance is proportional to the Euclidean square of the distances. Usually, isotropic means that all three spatial dimensions are treated as identical. Thus, in isotropic coordinate, the line element takes the form
$$d s^2=A(r) d t^2-B(r) d \sigma^2 .$$

In Cartesian coordinates, the line element of Euclidean three space is
$$d \sigma^2=d x^2+d y^2+d z^2,$$
whereas in spherical polar coordinates
$$x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi, z=r \cos \theta,$$
the line element of Euclidean three space is
$$d \sigma^2=d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2 .$$
Here, the metric with $t=$ constant is conformally related to the metric of Euclidean space. Usually, isotropic coordinates are used when time $t=$ constant hypersurface, i.e., the three-dimensional subspace of spacetime requires to look like Euclidean. Generally, this type of coordinate system is used for modeling the gravitational field of a symmetrical object that does not discriminate between the $x, y$, or $z$ directions.

物理代写|广义相对论代写General relativity代考|Interaction between Gravitational

Electric and magnetic fields are generated when a charge is in motion, and it depends on space and time. This phenomenon is known as electromagnetism. The study of time-dependent electromagnetic fields and their behavior is described by a set of equations, known as Maxwell’s equations.

Before Maxwell, there were four fundamental equations of electromagnetism prescribed by several researchers. Maxwell improved those equations and composed them in the succeeding compact form known as Maxwell’s equations of electromagnetism.
(a) $\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho, \quad[=0$, without charge in a region $]$
(b) $\vec{\nabla} \cdot \vec{B}=0$,
(c) $\vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$
(d) $\vec{\nabla} \times \vec{B}-\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}=\mu_0 \vec{J}$
(Ampere’s law with Maxwell’s corrections)
Ampere’s law:
$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{J},$$
where, $E=$ electric field strength, $B=$ magnetic field strength, $J=$ current density, $\rho=$ charge density, and $\mu_0=$ magnetic permeability $=4 \pi \times 10^{-7}$ weber/amp-meter.

物理代写|广义相对论代写广义相对论代考|各向同性坐标

$$d s^2=A(r) d t^2-B(r) d \sigma^2 .$$

$$d \sigma^2=d x^2+d y^2+d z^2,$$
，而在球极坐标中
$$x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi, z=r \cos \theta,$$

$$d \sigma^2=d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2 .$$

物理代写|广义相对论代写广义相对论代考|引力之间的相互作用

(a) $\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho, \quad[=0$，在一个区域内无电荷$]$
(b) $\vec{\nabla} \cdot \vec{B}=0$，
(c) $\vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$
(d) $\vec{\nabla} \times \vec{B}-\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}=\mu_0 \vec{J}$
(带有麦克斯韦修正的安培定律)

$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{J},$$

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