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

2023年3月29日

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物理代写|广义相对论代写General relativity代考|Cylindrically Symmetric Spacetime

If a system is such that a rotation through any angle about an axis is invariant, then it is called cylindrically symmetric spacetime. The conditions for cylindrically symmetric spacetime are as follows:
(i) $\exists$ two KV in which one is time-like and other space-like
(ii) Orbits of space-like $\mathrm{KV}$ are closed.
Note 5.4
The study of conformal symmetries is important as it helps us know more about the internal structure of the spacetime geometry, whenever we need to resolve the geodesic equations of motion for the concerning spacetimes. To examine the usual connection between geometry and matter, this symmetry also helps a lot. The deportment of the metric is significant while progressed together with curves on a manifold in relativity. Conformal KV $\xi$ is defined as a vector field on a manifold when a metric is pulled along the curves produced by $\xi$. Lie derivative of the metric is straight proportionate to itself, i.e.,
$$\mathcal{L}{\xi} g{i k}=\psi g_{i k}$$

where the scalar field $\psi$ is termed as a conformal factor and $\mathcal{L}$ is the Lie derivative operator. The physical significance of this requirement is that when the metric is pulled along precise congruence of curves it perseveres itself modulo certain scale factor, $\psi$, that might fluctuate from location to location within the manifold. It is to be noticed that $\psi$ is not haphazardly chosen rather it is dependent on the conformal KV $\xi$ as $\psi\left(x^k\right)=\frac{1}{4} \xi_{; i}^i$ for four-dimensional Riemannian space. The vector $\xi$ illustrates the conformal symmetry, however, the metric tensor $g_{i k}$ is conformally haggard against itself along $\xi$.

Here, the conformal KV $\xi$ are called homothetic motions or homothetic vector (HV) fields if $\psi$ is constant, and for $\psi=0$ one will get KV fields.

物理代写|广义相对论代写General relativity代考|Spherically Symmetric Line Element

Spherically symmetric means an invariance under any arbitrary rotation of axes at a particular point, called the center of symmetry. Using $\theta$ and $\phi$ (polar coordinates) and choosing the center of symmetry at origin, we have the general form of the line element with spherical symmetry.
$$d s^2=A(r, t) d t^2+2 H(r, t) d r d t-B(r, t) d r^2-F(r, t)\left(d \theta^2+\sin ^2 \theta d \phi^2\right) .$$
For the surfaces $r=$ constant and $t=$ constant, the line elements reduces to form two spheres on which a typical point is labeled by coordinate $\theta$ and $\phi$ and line element takes the form
$$d s^2=d \theta^2+\sin ^2 \theta d \phi^2 .$$
This spherical symmetric line element is invariant when $\theta$ and $\phi$ are varied. The center of symmetry is the point $\mathrm{O}$, which is given by $r=0$.
Now we introduce new coordinates by the transformations:
$$r=r^{\prime}, \quad t=K\left(r^{\prime}, t^{\prime}\right)$$
where the function $K$ will be chosen later.
From the above transformation equations, we have
$$d r=d r^{\prime} ; d t=\frac{\partial K}{\partial r^{\prime}} d r^{\prime}+\frac{\partial K}{\partial t^{\prime}} d t^{\prime}$$
Then the line element becomes
$$d s^2=A\left(\frac{\partial K}{\partial r^{\prime}} d r^{\prime}+\frac{\partial K}{\partial t^{\prime}} d t^{\prime}\right)^2+2 H d r^{\prime}\left(\frac{\partial K}{\partial r^{\prime}} d r^{\prime}+\frac{\partial K}{\partial t^{\prime}} d t^{\prime}\right)-B d\left(r^{\prime}\right)^2-F\left(d \theta^2+\sin ^2 \theta d \phi^2\right) .$$
Now we choose $K$ such that coefficient of $d r^{\prime} d t^{\prime}$ is zero.
Thus, we have
$$A \frac{\partial K}{\partial r^{\prime}}+H=0 .$$

广义相对论代考

物理代写|广义相对论代写General relativity代考|Cylindrically Symmetric Spacetime

(i) ∃ 两个 KV，其中一个是时间类的，另一个是类空间的 (
ii) 类空间的 KV 的轨道是闭合的。

$$\mathcal{L}{\xi} g{ik}=\psi g_{ik}$$

物理代写|广义相对论代写General relativity代考|Spherically Symmetric Line Element

$$A \frac{\partial K}{\partial r^{\prime}}+H=0$$

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