Two persons headed towards the black hole horizon

[Aristarchus_]

Homework Statement

Person A and Person B are twins and are both 2m tall. They each fall towards their own black holes, legs first. Person A falls towards the black hole with a mass of $8*10^30kg$ and person B towards the one with the mass of $8*10^36kg$. I am supposed to find the difference in gravitational force(gravitational acceleration) exerted on the lower part of their body and on the upper part. That is, in those 2meters difference...

Relevant Equations

$g=\frac{GM}{r^2}$

I have tried inserting in the above formula 2m for r, but I get a huge answer. The correct answer is $1.28*10^9 N/kg$ for person A and $1.28*10^-3 N/kg$ for person B. I also suspect that the formula for the Lorentz factor ($1/\sqrt{1-2GM/r*c^2}$) has some relevancy here, but I cannot figure it out...

 

[malawi_glenn]

Relevant Equations:: $g=\frac{GM}{r^2}$

I have tried inserting in the above formula 2m for r, but I get a huge answer.

That is wrong. r in that formula means something else. Can you figure out what? Hint: Go back and study Newtons gravitational law.

At what distance are those differences in g to be calculated btw? At the scwharchild radius?

Also a LaTeX tip: you must enclose Sub- and superscripts with { } brackets. Also use \cdot instead of asterix for multiplication. And, have units outside the LaTeX environment or use \text{kg}. Units should not be written in italics.

Where does that formula for the Lorentz factor come from? Have you been given it in class like that?

Btw: you are given the mass of the Black hole with one significant digit? and the length of the persons as well? Answer should not be given with three significant digits. If this is the case, point that out to your teacher.

 

[Aristarchus_]

That is wrong. r in that formula means something else. Can you figure out what? Hint: Go back and study Newtons gravitational law.

At what distance are those differences in g to be calculated btw? At the scwharchild radius?

Also a LaTeX tip: you must enclose Sub- and superscripts with { } brackets. Also use \cdot instead of asterix for multiplication. And, have units outside the LaTeX environment or use \text{kg}. Units should not be written in italics.

Where does that formula for the Lorentz factor come from? Have you been given it in class like that?

Btw: you are given the mass of the Black hole with one significant digit? and the length of the persons as well? Answer should not be given with three significant digits. If this is the case, point that out to your teacher.

Yes, r means the radial distance to the mass center, but I do not have that information given in the problem...Yes at the Scwharchild radius they are to be calculated

 

[malawi_glenn]

Yes, r means the radial distance to the mass center, but I do not have that information given in the problem...Yes at the Scwharchild radius they are to be calculated

Then first calculate $r_s$ for both BH's and use that to calculate $\Delta g$ for each.

I just did the calculations myself and got those answers you quoted.

 

[Aristarchus_]

Then first calculate $r_s$ for both BH's and use that to calculate $\Delta g$ for each.

I just did the calculations myself and got those answers you quoted.

I managed it. Once you said calculate $r_s$ I came back to the previous problem I have done. There, the formula for the distance from to event horizon was given, namely r = 2GM/c^2. Which was not a formula I was familiar with prior to this. How does one derive it?
Never mind... found it. It involves setting the escape velocity equal to the speed of light :) Thank you in any case

 

[malawi_glenn]

I managed it

You solved the entire problem or just the $r_s$?

The amateur way to derive $r_s$ (which is not how you do it in a General Relativity class) is to find at what distance from the center the escape velocity is equal to the speed of light $c$.

 

[Aristarchus_]

You solved the entire problem or just the $r_s$?

The amateur way to derive $r_s$ (which is not how you do it in a General Relativity class) is to find at what distance from the center the escape velocity is equal to the speed of light $c$.

I solved for the r_s and then inserted that into the formula for the gravitational acceleration, and then compared the two (one with r_s and the other r_2 +2m). Although I am excited about the General relativity course, I will not be taking it any time soon, as I am going to my senior year of hs next year. You could, however, provide me with a link to the more advanced derivation of the distance from the center...

 

[malawi_glenn]

You could, however, provide me with a link to the more advanced derivation of the distance from the center...

It is covered in all intro GR books. Consider Schutz book or Kip Thornes new book "relativity and cosmology" or Susskinds "General relativity the theoretical minumums" (new book will be released next year.

Note that you can write $\Delta g \approx \dfrac{4GM}{r_s{}^3}$ here because $2$ m $\ll r_s$

 

[malawi_glenn]

What you could take away from this problem is that the "Spaghettification" (tidal gravity effect) is much more severe for a stellar black hole ($M \sim 10^{30}$ kg) than for a super massive BH (found in the center of many galaxies, such as in our own) ($M \sim 10^{36}$ kg)

 

[Aristarchus_]

It is covered in all intro GR books.

I see in your bio that you specialize in theoretical particle physics, but that you also have some knowledge of differential geometry/ topology in physics. As I have a similar interest and I am wondering what I should major in after next year, could I ask you if you double majored in math and physics, or was the physics degree enough to cover all the abstract mathematical courses, such as diff geometry and topology? I am asking, again, since I have a keen interest in theoretical physics but also a love for math. I would like to study physics but also topology and algebraic, diff geometry, so could I still major in physics and cover all of these mathematical courses, or do I need to transfer to math instead? I read on some other forums that it is usually more difficult to specialize in physics than it is in math, therefore it was suggested that one should major in physics if one wants to go theoretical, regardless of the interest in pure maths (such as topology in this case). Is this true? And thank you for the feedback in advance...

 

[malawi_glenn]

I see in your bio that you specialize in theoretical particle physics, but that you also have some knowledge of differential geometry/ topology in physics. As I have a similar interest and I am wondering what I should major in after next year, could I ask you if you double majored in math and physics, or was the physics degree enough to cover all the abstract mathematical courses, such as diff geometry and topology? I am asking, again, since I have a keen interest in theoretical physics but also a love for math. I would like to study physics but also topology and algebraic, diff geometry, so could I still major in physics and cover all of these mathematical courses, or do I need to transfer to math instead? I read on some other forums that it is usually more difficult to specialize in physics than it is in math, therefore it was suggested that one should major in physics if one wants to go theoretical, regardless of the interest in pure maths (such as topology in this case). Is this true? And thank you for the feedback in advance...

I think it is better that you ask this in subforum: https://www.physicsforums.com/forums/stem-academic-advising.139/ instead.

Short answer: you will hit a wall in theoretical physics not knowing enough math.

 

[PeroK]

Homework Statement:: Person A and Person B are twins and are both 2m tall. They each fall towards their own black holes, legs first. Person A falls towards the black hole with a mass of $8*10^30kg$ and person B towards the one with the mass of $8*10^36kg$. I am supposed to find the difference in gravitational force(gravitational acceleration) exerted on the lower part of their body and on the upper part. That is, in those 2meters difference...
Relevant Equations:: $g=\frac{GM}{r^2}$

I have tried inserting in the above formula 2m for r, but I get a huge answer. The correct answer is $1.28*10^9 N/kg$ for person A and $1.28*10^-3 N/kg$ for person B. I also suspect that the formula for the Lorentz factor ($1/\sqrt{1-2GM/r*c^2}$) has some relevancy here, but I cannot figure it out...

Where are you getting this homework? In a previous thread I thought we had established that you were only beginning your studies of SR and had little knowledge of calculus? Now you are studying GR? But using the equations of Newtonian gravity?

 

[PeroK]

Short answer: you will hit a wall in theoretical physics not knowing enough math.

Even if one has a Texas TI-82 calculator at one's disposal?

 

[malawi_glenn]

Even if one has a Texas TI-82 calculator at one's disposal?

Ti-84 Python edition can take you further ;)

 

[Aristarchus_]

Where are you getting this homework? In a previous thread I thought we had established that you were only beginning your studies of SR and had little knowledge of calculus? Now you are studying GR? But using the equations of Newtonian gravity?

It is true that I have finished with hs calculus and have little knowledge beyond that, and I am not studying GR yet

 

[malawi_glenn]

So in what class did you encounter this problem?

 

[Aristarchus_]

So in what class did you encounter this problem?

AP Physics

 

[PeroK]

Homework Statement:: Person A and Person B are twins and are both 2m tall. They each fall towards their own black holes, legs first. Person A falls towards the black hole with a mass of $8*10^30kg$ and person B towards the one with the mass of $8*10^36kg$. I am supposed to find the difference in gravitational force(gravitational acceleration) exerted on the lower part of their body and on the upper part. That is, in those 2meters difference...
Relevant Equations:: $g=\frac{GM}{r^2}$

I have tried inserting in the above formula 2m for r, but I get a huge answer. The correct answer is $1.28*10^9 N/kg$ for person A and $1.28*10^-3 N/kg$ for person B. I also suspect that the formula for the Lorentz factor ($1/\sqrt{1-2GM/r*c^2}$) has some relevancy here, but I cannot figure it out...

This question seems to omit the distance to the massive object where the calculation is to be done.

Note that a black hole is a relativistic object. If you replace that by simply a star of the same mass, then you can use Newtonian mechanics. That said, if you are far enough from the event horizon of the black hole it makes no difference, as the Newtonian approximation is generally good enough.

 

[Orodruin]

Yes, r means the radial distance to the mass center

It does not. It is unfortunately more complicated in general relativity due to the spacetime curving a lot near and within the black hole. Technically the radial coordinate r is related to the area of a sphere $4\pi r^2$ at that coordinate (and rotated using the spherical symmetry).

 

[malawi_glenn]

This question seems to omit the distance to the massive object where the calculation is to be done.

We figured out that they meant the Scwharchild radius.
OP figured out the "correct" solution (the answer given in his/her answer key).

It does not. It is unfortunately more complicated in general relativity due to the spacetime curving a lot near and within the black hole. Technically the radial coordinate r is related to the area of a sphere $4\pi r^2$ at that coordinate (and rotated using the spherical symmetry).

Yeah but this is not a problem from a GR class/book, it is an AP physics problem so it is very heurestic.
What is interesting here is the result that the tidal effect is much more severe for a stellar black hole vs. supermassive BH. This you would find in a more sound calculation in GR as well.

 

[Orodruin]

very heurestic

Physics curriculum to English dictionary:
”Very heuristic” - ”Wrong”

What is interesting here is the result that the tidal effect is much more severe for a stellar black hole vs. supermassive BH.

… at the event horizon, which is much larger in the case of a SMBH. The tidal forces will approach infinity in both cases when you go closer to r=0. It does however make intuitive sense. To confine light to a smaller spherical surface you intuitively need more local curvature than to a large spherical surface.

 

[malawi_glenn]

Physics curriculum to English dictionary:
”Very heuristic” - ”Wrong”

The OP is just the messenger here. Who is to blaim is his/her teacher who assign the problem :(

And OP is not even a student at university. The point of high school physics is partly to make pupils interested in pursuing physics at higher eduaction. As I tell my students "high school physics are just a bunch of lies " ;)

High school natural science is not an academic subject. It is a mix of academic, popular science and other aspects such as ethics, gender and other social aspects.

If you want to survive as a high school physics/math teacher, you have keep your academic snobbery at home.

 

[Orodruin]

The OP is just the messenger here. Who is to blaim is his/her teacher who assign the problem :(

I don’t think I blamed the student for the problem here. What I objected to was r being a distance.

And OP is not even a student at university. The point of high school physics is partly to make pupils interested in pursuing physics at higher eduaction. As I tell my students "high school physics are just a bunch of lies " ;)

I’d argue teaching GR at high school level is not necessarily a good thing. Most high school teachers will not have a sufficient grasp of GR even to convey even the popularised versions. (Many physics teachers at university level don’t either…)

I do remember coming to high school myself and becoming very disappointed that GR was not part of the curriculum. Having learned it later on, that was probably a good thing.

 

[malawi_glenn]

What I objected to was r being a distance.

Its not going to help OP solve the problem (which is not rigorously constructed, similar to OPs recent problem with time dilation on the ISS)

I’d argue teaching GR at high school level is not necessarily a good thing.

There is some very rudimentary SR in high school, I do not think that is a good thing except that some students think its cool and want to learn more about it in their free-time. Also we are supposed to cover "the standard model of particle physics" in the second semester...

I do remember coming to high school myself and becoming very disappointed that GR was not part of the curriculum

Every year, there are a few such students starting in my school. Last year, I had a student who was very upset that we are not using differential equations and integrals (which he apparently taught himself on the summer break before starting high school since he did have grades in those courses already) in the intro physics class . He said he was going to report me to the Swedish School board :-) I told him he could use some of my old e-mails where I basically raise the same complaint :-)

 

[Orodruin]

Also we are supposed to cover "the standard model of particle physics" in the second semester...

Do you start by constructing classical Yang-Mills theory or do you jump straight into the quantised theory?

Last year, I had a student who was very upset that we are not using differential equations and integrals (which he apparently taught himself on the summer break before starting high school since he did have grades in those courses already) in the intro physics class . He said he was going to report me to the Swedish School board :-) I told him he could use some of my old e-mails where I basically raise the same complaint :-)

Arguably those are not the students in need of teaching. What they do need is encouragement and stimulation to not get bored. Taking a university class or two in parallel doesn’t hurt. In my case, my physics and math teacher quickly realized and let me read ahead and not pay attention in class (he would ask me where he went wrong if he lost the thread though). In problem solving time he would let me help other students as an extra teacher. Great for really making the foundation solid and others got more help. Everyone wins. One of my teacher role models. Sadly he likely passed away by now. He was very close if not above retirement age when I was in high school…

 

[malawi_glenn]

Arguably those are not the students in need of teaching. What they do need is encouragement and stimulation to not get bored. Taking a university class or two in parallel doesn’t hurt.

Hard to get enrolled formally at a university (at least in sweden) without your high school dimploma. We do have a deal with SU that they can start the "Matematik 1, 30hp" course package given that they have completed all high school math classes with highest grade and have a letter of recommendation from their math teacher (usually me) and principals signature/approval.

Regarding physics and these kind of kids, I usually point them to Lennys books (cheap paperback books) and/or let them borrow "university physics" books from the physics department library. Sometimes even Sören Holst book (whom you know). Furthermore I encourage them to participate in math and physics competitions (which I supervise 90min/week outside of lessons hours) or the "mathematical essey" competition.

Some are also quite happy with typing my solutions to problems in various courses in LaTeX (running out of material there soon) or programming/numerical solutions/visualizations (this summer I have two students working with a project for 3Blue1Browns summer math challenge). Different things ticks for different students. I am glad we have the Linear Algebra course and "Fysik 3" at my school at least.