Solve Another GR Problem: Interval of Signals Received from Tower

[quasar987]

Homework Statement

An observer atop a tower of 10m on Earth sends a signal towards the base of the tower every second according to his clock. The gravitational field corresponds to the metric.

$$d\tau^2=(1+zg)dt^2+\frac{1}{1+zg}(dx^2+dy^2+dz^2)$$

(where g~10m/s²)

An observer on the ground receives the signals on what interval?

Homework Equations

The Attempt at a Solution

Since the signals are sent at the same place (x=0,y=0,z=10), the metric atop the tower for these events is

$$d\tau^2=(1+10g)dt^2$$

But what is dt, what is dtau exactly? How do they relate to the 1 second interval? Is dtau supposed to be the same for both observer or something?

 

[Magister]

Have you tried to compute the gravitational red shift? I guess this will solve the problem...

 

[nrqed]

Homework Statement

An observer atop a tower of 10m on Earth sends a signal towards the base of the tower every second according to his clock. The gravitational field corresponds to the metric.

$$d\tau^2=(1+zg)dt^2+\frac{1}{1+zg}(dx^2+dy^2+dz^2)$$

(where g~10m/s²)

An observer on the ground receives the signals on what interval?

Homework Equations

The Attempt at a Solution

Since the signals are sent at the same place (x=0,y=0,z=10), the metric atop the tower for these events is

$$d\tau^2=(1+10g)dt^2$$

But what is dt, what is dtau exactly? How do they relate to the 1 second interval? Is dtau supposed to be the same for both observer or something?

Good questions to ask. $d \tau$ is the physical time measured by the observer. So you do it in two steps.

At the top, you fix dtau = physical time interval of emission and you use th emetric to find the coordinate interval dt.

Now you use the key fact that the dt will be the same at the bottom.

use then the metric at the bottom to then find dtau there.

 

[quasar987]

So exactly the same principle can be applied to my other thread about the orbit can it not? except here, the interval is 10 hours on earth, and we want to know what it is on the ship (atop the tower). This allows one to know how long a day is on the ship.

I don't know if this helps though.

 

[nrqed]

So exactly the same principle can be applied to my other thread about the orbit can it not? except here, the interval is 10 hours on earth, and we want to know what it is on the ship (atop the tower). This allows one to know how long a day is on the ship.

I don't know if this helps though.

yes, but $d\phi$ is not zero.

 

[quasar987]

But can't the "orientation" of the coordinate system be chosen such that $\theta=0$, so that sinO=0 and the fact that $d\phi \neq 0$ is not "apparent"?

(Because the term in $d\phi^2$ in the schwartzschild metric has coefficient $\cos\theta$)

 

[quasar987]

Homework Statement

An observer atop a tower of 10m on Earth sends a signal towards the base of the tower every second according to his clock. The gravitational field corresponds to the metric.

$$d\tau^2=(1+zg)dt^2+\frac{1}{1+zg}(dx^2+dy^2+dz^2)$$

(where g~10m/s²)

An observer on the ground receives the signals on what interval?

There are also two other questions:

If the signals are reflected back at the top of the tower.

b) At what rate does the observer atop the tower receive the reflexions (on his clock)?

c) What is the delay in his receiving the reflexions?


I said

b) same rate as emission: 1 pulse/second

c) zero...

Am I missing something?

(Btw, these are exams from years past I'm doing... and thanks again for helping me with this nrqed. Where exactly are you located? I live in montreal but my hometown is on the south shore in St-Lambert.)

 

[quasar987]

c) zero...

Let me take that back. Actually, what does he mean by "the delay at which the observer receives the signals"?

 

[nrqed]

There are also two other questions:

If the signals are reflected back at the top of the tower.

b) At what rate does the observer atop the tower receive the reflexions (on his clock)?

c) What is the delay in his receiving the reflexions?


I said

b) same rate as emission: 1 pulse/second

c) zero...

not zero! it's the time for the signal to go and come back.

Am I missing something?

(Btw, these are exams from years past I'm doing... and thanks again for helping me with this nrqed. Where exactly are you located? I live in montreal but my hometown is on the south shore in St-Lambert.)

really? I am in Sherbrooke! where do you study?

 

[quasar987]

Université de Montréal.. where do you teach? :)

 

[quasar987]

not zero! it's the time for the signal to go and come back.

ok then the tower is 10 meters high and speed of light is c so I'm going to go with a naive 20/c.

 

[quasar987]

Are there places/books that have solved example in them? That's my problem, I know the math fairly well but have no idea how it applies to the physics. :grumpy:

 

[nrqed]

Are there places/books that have solved example in them? That's my problem, I know the math fairly well but have no idea how it applies to the physics. :grumpy:

This is a problem with GR, there are good books about the mathematical concepts but most books spend very little time (if at all) on applying the equations to actual calculations of physical importance. There is the book "Relativity Demystified" by McMahon which some people hate because there is almost no background given and many examples are done in excruciating details which means that in the end, there is not a lot of content. But I personally learn a lot by seeing explicit examples before jumping into general theory (i understood Cartan's sctructure equations only when I saw the examples of this book worked out). But you would still need other books to complement. Although it is elementary, I like "exploring black holes" by wheeler and Taylor, which really gets into the *meaning* of the metric and how to relate it to physical measurements.

Who is teaching you GR? I am teaching in a small CEGEP in Sherbrooke. I did a postdoc at McGill and knew some people at UdM.

Regards

Patrick

 

[quasar987]

It's Manu Paranjape.

 

[nrqed]

It's Manu Paranjape.

Ok. he is a very nice person.

Do you use a textbook for the class?

 

[quasar987]

Nope, but I self-taught myself the math from the first two chapters of 'De Felice & Clark's 'Relativity on Curved Manifold'. However, Manu tells us that his notes are inspired mainly from Hawking & Ellis and Weinberg.

I had a physics teacher while I was in CEGEP (Edouard-Montpetit) who migrated to Sherbrooke for 2-3 years. He name is Yves Charbonneau. I see he is no longer at Sherbrooke now though. Did you know him? He was the best teacher I had in CEGEP.

 

[nrqed]

Nope, but I self-taught myself the math from the first two chapters of 'De Felice & Clark's 'Relativity on Curved Manifold'. However, Manu tells us that his notes are inspired mainly from Hawking & Ellis and Weinberg.

Ah ok.

One problem with many books in GR is that there is little time devoted to the interpretation of the metric.

I forgot to mention an excellent book to build physical intuition and understanding of GR: the book by Hartle. I highly recommend that book!

I had a physics teacher while I was in CEGEP (Edouard-Montpetit) who migrated to Sherbrooke for 2-3 years. He name is Yves Charbonneau. I see he is no longer at Sherbrooke now though. Did you know him? He was the best teacher I had in CEGEP.

Ah...No. I never met him.

Best luck!

 

[quasar987]

I'm taking note of your recommendations. Thx.

How do you like MWT? It seems like another good book for physical interpretation from what I've seen by flipping the pages.