# 物理代写| Gravitational Redshift Pound-Rebka Experiment 相对论代考

## 物理代写

7.4 Gravitational Redshift: Pound-Rebka Experiment
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7.4 Gravitational Redshift: Pound-Rebka Experiment
Gravitational redshift is a fundamental prediction of GR, which not only provides an important validation of the theory, but is crucial in day-to-day life, specifically in Global Positioning System (GPS) primarily used for navigation. This topic has been covered in the previous chapters and the expressions were derived for Schrawzschild geometry in Chapter 6 . Here we describe the first gravitational redshift measurement due to earth’s gravitational field.

In 1959 Robert Pound and Glenn Rebka devised an experiment at Harvard University to test the gravitational redshift prediction of General Relativity. They placed atomic gamma ray emitters on top of a tower and the same material at the bottom of the tower which would absorb those photons. However, due to gravitational blue shift, the frequency of the emitted photons increases when they reach the bottom of the tower, which does not match the difference in energy levels required for the atoms are given the right amount of upward velocity (that is, away from the receiver atoms) that could compensate for the gravitational blueshift and then the photons would get absorbed. This is the basic principle for the experiment.

If $R$ is the radius of the earth, the $h(h \ll R)$ height of the tower, $\lambda_{e}$ the wavelength of the emitted photons and $\lambda_{r}$ the wavelength when the photons reach the receiver atoms, then from the gravitational redshift formula, Eq. (6.5.3), we obtain:
$\frac{\lambda_{r}}{\lambda_{e}}=\left[\frac{\left(g_{00}\right){r}}{\left(g{00}\right){e}}\right]^{1 / 2}=\frac{\left(1-\frac{2 m}{R}\right)^{1 / 2}}{\left(1-\frac{2 m}{R+h}\right)^{1 / 2}} \approx 1+\frac{m}{R+h}-\frac{m}{R} \approx 1-\frac{m h}{R^{2}}$ where, as usual $m=G M{\oplus} / c^{2}$, and $M_{\oplus}$ is the mass of the earth. If the emitter atoms are given a velocity $v$ upwards and if $\lambda_{0}$ is the wavelength of the photons in the rest frame of the atoms, due to special relativistic Doppler shift the effective emitted wavelength $\lambda_{e}$ becomes,
$$\lambda_{e}=\sqrt{\frac{1+v / c}{1-v / c}} \lambda_{0} \approx(1+v / c) \lambda_{0}$$
Thus the wavelength of the photons received at the bottom of the tower is,
$$\lambda_{r}=\left(1-\frac{m h}{R^{2}}\right)\left(1+\frac{v}{c}\right) \lambda_{0} .$$
In order for the received photons to have the same wavelength, $\lambda_{r}=\lambda_{0}$, so that they could be absorbed by the receiving atoms in their rest frame, requires, to the first order, $v / c=m h / R^{2}=G M_{\oplus} h /(R c)^{2}=g h / c^{2}$, where $g$ is the acceleration due to gravity at the surface of the earth.
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7 Classical Tests of General Relativity
In the set up used by Pound and Rebka, the emitters were kept at a height of $h=22.5 \mathrm{~m}$ with respect to the receiving atoms. Hence, the velocity required for the emitters to satisfy the absorption criteria described above, $v \approx 2.5 \times 10^{-15} \mathrm{c}=$ $7.5 \times 10^{-7} \mathrm{~m} / \mathrm{s}$. However, this velocity is much smaller (by five orders of magnitude) than the recoil velocity of each atom when it emits or absorbs a photon. To alleviate this problem, Pound and Rebka made use of Mössbauer spectroscopy, where the that, as a whole, reacted to the recoil momentum, significantly reducing the recoil velocity. In the experiment, the emitter was mounted on a loudspeaker cone which moved the emitting atoms with sinusoidally varying velocity. When the velocity matched the required value to compensate for the gravitational redshift, the photons would get absorbed by the receiving atoms. This showed a variation in the reading of a scintillation counter kept below the receiver. To rule out spurious correlations, the experiment was performed for different loudspeakers for a range of oscillation frequencies. The results matched the prediction and GR successfully passed yet another test.

## 物理代考

7.4 引力红移：Pound-Rebka 实验
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7.4 引力红移：Pound-Rebka 实验

1959 年，罗伯特·庞德和格伦·雷布卡在哈佛大学设计了一个实验来测试广义相对论的引力红移预测。他们在塔顶放置了原子伽马射线发射器，在塔底放置了相同的材料来吸收这些光子。然而，由于引力蓝移，发射光子的频率在到达塔底时会增加，这与给定适当的向上速度（即远离来自接收器原子）可以补偿引力蓝移，然后光子会被吸收。这是实验的基本原理。

$\frac{\lambda_{r}}{\lambda_{e}}=\left[\frac{\left(g_{00}\right){r}}{\left(g{00}\right) {e}}\right]^{1 / 2}=\frac{\left(1-\frac{2 m}{R}\right)^{1 / 2}}{\left(1-\frac {2 m}{R+h}\right)^{1 / 2}} \约 1+\frac{m}{R+h}-\frac{m}{R} \约 1-\frac{mh }{R^{2}}$ 其中，与往常一样，$m=G M{\oplus} / c^{2}$，$M_{\oplus}$ 是地球的质量。如果给定发射器原子向上的速度 $v$ 并且如果 $\lambda_{0}$ 是原子静止框架中光子的波长，由于特殊的相对论多普勒频移，有效发射波长 $\lambda_{e }$ 变成，
$$\lambda_{e}=\sqrt{\frac{1+v / c}{1-v / c}} \lambda_{0} \approx(1+v / c) \lambda_{0}$$

$$\lambda_{r}=\left(1-\frac{m h}{R^{2}}\right)\left(1+\frac{v}{c}\right) \lambda_{0} 。$$

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Matlab代写