[Special Relativity] Test Particle Inside the Sun's Gravitational Field

[Athenian]

I must confess that I didn't notice that you only need the energy equation. That is a shortcut. You don't even need the momentum equation!

Thank you! I'm glad it just worked out in the end. :)

 

[Orodruin]

I think it is very unfortunate if you need to do something that is wrong in order to pass a course. I would replace the gravitational force with an electromagnetic force with and state this in the beginning, but that is just me. It also unfortunately depends on how you think your professor would react.

Now, to the actual approach to this in GR, the problem is not well formulated because it is not clear what would be meant by $\gamma$. There are a few possibilities that include (but I probably one can think of others too):

1. The $t$-component of the 4-velocity of the particle, i.e., $\dot t = dt/d\tau$ where $\tau$ is the proper time and $t$ the time coordinate.
2. The $\gamma$ factor of the particle as measured by a static observer measuring its speed as it passes by (call this $\gamma_o$).

These two will give different final results. I would tend to favour number 2 here as it is more physical.

In the Schwarzschild solution (i.e., the spherically symmetric vacuum solution outside a spherically symmetric mass distribution), there is a constant of motion that is $(1-2GM/(c^2r)) \dot t = \gamma_\infty$, where $\gamma_\infty$ would be the $\gamma$ factor at infinity. This would also mean that, for $GM/(c^2 r) \ll 1$, you would have
$$
\dot t \simeq \gamma_\infty \left( 1 + \frac{2GM}{c^2 r}\right),
$$
which differs from the result of your problem by a factor $2\gamma_\infty$. However, as I said, my preferred interpretation would be 2, which would also require taking into account the gravitational time dilation of the static observer, leading to
$$
\gamma_o = \dot t \left( 1 - \frac{2GM}{c^2 r}\right)^{-1/2} = \gamma_\infty \left( 1 - \frac{2GM}{c^2 r}\right)^{-3/2} \simeq \gamma_\infty \left( 1 + \frac{3GM}{c^2 r}\right),
$$
which differs to the result of your problem by a factor $3\gamma_\infty$. (Assuming I did the math correctly here, it was kind of hastily between other tasks ...)

 

[Athenian]

I think it is very unfortunate if you need to do something that is wrong in order to pass a course. I would replace the gravitational force with an electromagnetic force with and state this in the beginning, but that is just me. It also unfortunately depends on how you think your professor would react.

Now, to the actual approach to this in GR, the problem is not well formulated because it is not clear what would be meant by $\gamma$. There are a few possibilities that include (but I probably one can think of others too):

1. The $t$-component of the 4-velocity of the particle, i.e., $\dot t = dt/d\tau$ where $\tau$ is the proper time and $t$ the time coordinate.
2. The $\gamma$ factor of the particle as measured by a static observer measuring its speed as it passes by (call this $\gamma_o$).

These two will give different final results. I would tend to favour number 2 here as it is more physical.

In the Schwarzschild solution (i.e., the spherically symmetric vacuum solution outside a spherically symmetric mass distribution), there is a constant of motion that is $(1-2GM/(c^2r)) \dot t = \gamma_\infty$, where $\gamma_\infty$ would be the $\gamma$ factor at infinity. This would also mean that, for $GM/(c^2 r) \ll 1$, you would have
$$
\dot t \simeq \gamma_\infty \left( 1 + \frac{2GM}{c^2 r}\right),
$$
which differs from the result of your problem by a factor $2\gamma_\infty$. However, as I said, my preferred interpretation would be 2, which would also require taking into account the gravitational time dilation of the static observer, leading to
$$
\gamma_o = \dot t \left( 1 - \frac{2GM}{c^2 r}\right)^{-1/2} = \gamma_\infty \left( 1 - \frac{2GM}{c^2 r}\right)^{-3/2} \simeq \gamma_\infty \left( 1 + \frac{3GM}{c^2 r}\right),
$$
which differs to the result of your problem by a factor $3\gamma_\infty$. (Assuming I did the math correctly here, it was kind of hastily between other tasks ...)

Wow, thank you for going through the calculation process here. I definitely learned a lot through this. Beyond that, I do agree that it is quite unfortunate that I have to answer something incorrectly to pass the course. While I did entertain the thought of replacing the gravitational force with the electromagnetic force upon your suggestion, I know with certainty that would guarantee me a failing mark on that specific problem. Therefore, even if the structure of the question is wrong to begin with, it is best I follow in accordance to the instructor's guidelines.

Beyond all that, though, I do sincerely appreciate the work you've showed here to help me gain a better understanding of what a structured problem and solution should look like here instead. Thank you very much!

 

[Athenian]

$\gamma (r) = \frac{GM}{c^2} + constant$

Ah, my apologies. I accidentally missed the $r$. The answer's supposed to be:
$$\gamma (r) = \frac{GM}{rc^2} + constant$$

Of course, please let me know if I'm mistaken. Thank you!

 

[Orodruin]

I do agree that it is quite unfortunate that I have to answer something incorrectly to pass the course.

Actually, it is beyond unfortunate as it puts you in a bit of moral dilemma also in regards to future students. If you just go along and hand in the expected solution then there will be no change for the next course offering and a new batch of students will face the same errors. On the other hand, it is not necessarily obvious that alerting your professor to this will have an impact either. You might want to consider your options based on the approachability of your teachers. I can understand that you as a student do not feel like you have the authority to challenge your professor, but you can always send your teachers here to discuss the issue with us. There is quite a number of people here that are quite well versed with regards to relativity.

 

[PeroK]

I'd say the OP has to live with the course he's on. Any moral dilemma rests with the university. The mathematics is almost identical to Newtonian gravitation, so it's not wasted effort in that sense.

I don't see it's any worse than teaching the Bohr atom. There are times when one, as a mere student, has to just get on with it!

 

[Orodruin]

I'd say the OP has to live with the course he's on. Any moral dilemma rests with the university. The mathematics is almost identical to Newtonian gravitation, so it's not wasted effort in that sense.

It is not a moral dilemma for the university or the professor if they are not aware of the issue. I do not know the professor so I do not know if this is a case of misplaced ”lies to children” or an actual misconception on the professor’s part.

I don't see it's any worse than teaching the Bohr atom. There are times when one has to just get on with it!

The Bohr model is not fundamentally incompatible with the theory it is being applied to. It is a model on its own that has been surpassed. You are not going to use the Bohr model as a part of a QFT problem. Here, Newtonian gravitation is fundamentally incompatible with SR and the problem is trying to force it in with hammer and nail.

 

[Ibix]

I don't see it's any worse than teaching the Bohr atom. There are times when one, as a mere student, has to just get on with it!

The Bohr atom is an inaccurate model and puts quantisation in by hand, but it's not self-contradictory. Using Newtonian gravity with its infinite propagation speed in a special relativistic framework is self-contradictory. It isn't even, so far as I know, a historical model in the sense that we occasionally discuss light as a wave in a medium with Fitzgerald contraction (arguably contradictory, given what we know now). That's my philosophical problem with this question.

Agree with @Orodruin that if there's some way @Athenian can (safely!) direct the prof here that would be good. Are the problem sheets online?

 

[Athenian]

Thank you all for your comments and the discussion over here.
However, considering that I am an exchange student in a foreign country and university, I find it quite difficult to approach the professor about this problem. I'm sure if I discuss the topic with the professor in an incredibly cautious and tactful manner, my argument may be accepted more or less well. However, I highly doubt that it will solve anything.

If anything, I think it would be best I approach the professor regarding the issue once the semester ends in give or take a week. Then, the grade will be finalized and I will not be taking any unnecessary risk.

In short, this is my opinion as I have very little idea of how the professor will react.
Regardless, thank you all for your thoughts and contributions on the matter!

 

[Athenian]

Are the problem sheets online?

Apologies for forgetting to answer this question. But, no, I highly doubt (and ultimately, do not believe) the problem sheets are online. However, thus far, I have posted 5 out of 8 of the questions here on the forum - all titled "Test Particle Inside the Sun's Gravitational Field".

 

[PeroK]

Apologies for forgetting to answer this question. But, no, I highly doubt (and ultimately, do not believe) the problem sheets are online. However, thus far, I have posted 5 out of 8 of the questions here on the forum - all titled "Test Particle Inside the Sun's Gravitational Field".

This problem intrigues me. First, it's a static gravitational field, so infinite propagation speed isn't an issue. Second, there's nothing wrong with investigating this and seeing what comes out. There is orbital precession, for example. I've looked up all the data for Mercury and estimated its precession based on this "Newtonian/SR" hybrid theory. I don't know if this is where your Prof is heading. If so, I see nothing invalid in showing that these calculations do not solve the gravity/SR conundrum.

The interesting thing I learned is that the precession of Mercury is dominated by a factor based on the gravitational pull of other planets. The GR correction is only about 10% of this. (How did the physicists of the past do all this without computers?)

Anyway, as long as you know you are only investigating a debunked hybrid theory, perhaps you can learn quite a lot about physics from this exercise.

 

[Athenian]

This problem intrigues me. First, it's a static gravitational field, so infinite propagation speed isn't an issue. Second, there's nothing wrong with investigating this and seeing what comes out. There is orbital precession, for example. I've looked up all the data for Mercury and estimated its precession based on this "Newtonian/SR" hybrid theory. I don't know if this is where your Prof is heading. If so, I see nothing invalid in showing that these calculations do not solve the gravity/SR conundrum.

The interesting thing I learned is that the precession of Mercury is dominated by a factor based on the gravitational pull of other planets. The GR correction is only about 10% of this. (How did the physicists of the past do all this without computers?)

Anyway, as long as you know you are only investigating a debunked hybrid theory, perhaps you can learn quite a lot about physics from this exercise.

Interesting! Thank you for the explanation. While I have no idea where my professor is heading with these questions, I suppose I'll find out by the end of all this.
But, perhaps it's just me, I do find it quite odd that students are tasked to "investigate a debunked hybrid theory" in a purely SR class (of course, with Newtonian background to reinforce understanding). Regardless, by the end of the day, I hope I learn something valuable.