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Homework Statement:

[B]Question[/B]
[Question Context: Consider the motion of a test particle of (constant) mass ##m## inside the gravitational field produced by the sun in the context of special relativity.]
Consider the equations of motion for the test particle, which can be written as $$\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F},$$
OR
$$\frac{d(m\gamma \vec{v})}{dt} = \vec{F},$$
where ##\vec{v}## is the speed of the test particle, ##c## is the (constant) speed of light, and by definition, $$\gamma \equiv \frac{1}{\sqrt{1 \frac{\vec{v}^2}{c^2}}} .$$
In addition, the gravitational force is given by $$\vec{F} \equiv \frac{GMm}{r^2} \hat{e}_r$$
where ##\hat{e}_r## is the unit vector in the direction between the Sun (of mass M) and the test particle (of mass ##m##).
Now, integrate the first equation above  that is, ##\frac{d(m\gamma c)}{dt} = \frac{\vec{v}}{c} \cdot \vec{F}##  to find ##\gamma## as a function of ##r##, by using the property that $$\frac{\dot{r}}{r^2} = \frac{d}{dt} \Big(\frac{1}{r}\Big).$$
You may need to introduce a constant of integration. This will be a free parameter of the solution.
Homework Equations:
 Please refer below ##\Longrightarrow##
Below is an attempted solution based off of another user's work on StackExchange:
Source: [https://physics.stackexchange.com/questions/525169/specialrelativitytestparticleinsidethesunsgravitationalfield/525212#525212]
To begin with, I will be using the following equation mentioned in the question  that is, $$\vec{F} \equiv \frac{GMm}{r^2} \hat{e}_r.$$
Using this, I get the following by plugging the equation for gravitational force to the other one shown above in the question section:
$$\frac{d(m\gamma c)}{dt}=\frac{\dot{r}}{c}\cdot \frac{GMm}{r^2} \hat{e}_r$$
where both ##m## and ##c## are constants. With this in mind, I could pull those two constants out of the deferential before rewriting the given formula to:
$$c^2\frac{d\gamma}{dt}=GMm\frac{d}{dt}\left(\frac{1}{r}\right) \hat{e}_r$$
Now, I can integrate both sides to get the following answer:
$$c^2\gamma=\frac{GMm}{r} \hat{e}_r+k,$$
where ##k## the constant of integration.
Comment
As mentioned before, this is a solution that is largely based upon the work by another used named "fielder". However, after looking through the problem, did this user succeeded in finding ##\gamma## as a function of ##r##? If the user is correct in his approach, however, why is that?
Therefore, to put it simply, if the attempted solution is incorrect, any amount of guidance to help me reach the correct answer is much appreciated. However, if the attempted solution is correct, I would greatly appreciate it if anybody here on the forum could briefly explain why the attempted solution above succeeded in finding ##\gamma## as a function of ##r##.
Thank you for reading through this and I would sincerely appreciate any amount of assistance!
Source: [https://physics.stackexchange.com/questions/525169/specialrelativitytestparticleinsidethesunsgravitationalfield/525212#525212]
To begin with, I will be using the following equation mentioned in the question  that is, $$\vec{F} \equiv \frac{GMm}{r^2} \hat{e}_r.$$
Using this, I get the following by plugging the equation for gravitational force to the other one shown above in the question section:
$$\frac{d(m\gamma c)}{dt}=\frac{\dot{r}}{c}\cdot \frac{GMm}{r^2} \hat{e}_r$$
where both ##m## and ##c## are constants. With this in mind, I could pull those two constants out of the deferential before rewriting the given formula to:
$$c^2\frac{d\gamma}{dt}=GMm\frac{d}{dt}\left(\frac{1}{r}\right) \hat{e}_r$$
Now, I can integrate both sides to get the following answer:
$$c^2\gamma=\frac{GMm}{r} \hat{e}_r+k,$$
where ##k## the constant of integration.
Comment
As mentioned before, this is a solution that is largely based upon the work by another used named "fielder". However, after looking through the problem, did this user succeeded in finding ##\gamma## as a function of ##r##? If the user is correct in his approach, however, why is that?
Therefore, to put it simply, if the attempted solution is incorrect, any amount of guidance to help me reach the correct answer is much appreciated. However, if the attempted solution is correct, I would greatly appreciate it if anybody here on the forum could briefly explain why the attempted solution above succeeded in finding ##\gamma## as a function of ##r##.
Thank you for reading through this and I would sincerely appreciate any amount of assistance!
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