Space time Curvature around the Earth

[GRstudent]

Hi all,

I am wondering if it is possible to calculate, using Einstein Tensor, the space time curvature around the Earth. As far as I understand, Einstein Field Equations tell us that the presence of a matter curves the space time. So space time curvature is gravity and gravity is space time curvature.

So the questions are:

1) Can we get Einstein Tensor for the Earth (Ricci+1/2metric*Ricci Scalar)

2) Can we calculate the Stress-Energy-Momentum Tensor of the Earth?

Basically, zero-zero component of Stress-Energy Tensor is energy density. So according to famous E=mc^2, energy density is equal to mass density times c^2. But I have no idea how to calculate the Momentum or Stress of the Earth.

Thank you!

Joe W.

 

[Mentz114]

1) Can we get Einstein Tensor for the Earth (Ricci+1/2metric*Ricci Scalar)

We can get one for the inside of the earh, and one for the empty space around the earth. They are called the Schwarzschild interior and Schwarzschild vacuum solutions.

2) Can we calculate the Stress-Energy-Momentum Tensor of the Earth?

Not exactly, but stellar interiors have been modeled using the Schwarzschild interior solution.

See http://en.wikipedia.org/wiki/Schwarzschild_metric and linked pages.

 

[GRstudent]

Why we cannot calculate Stress-Energy Tensor for the Earth? Thank you.

 

[Mentz114]

Why we cannot calculate Stress-Energy Tensor for the Earth? Thank you.

In principle it could be done, and maybe it has been done but the exterior solution is not dependent on the SET, only on spherical symmetry. Calculating the SET of the Earth wouldn't be very useful in GR.

 

[GRstudent]

In principle it could be done, and maybe it has been done but the exterior solution is not dependent on the SET, only on spherical symmetry. Calculating the SET of the Earth wouldn't be very useful in GR.

Still is it possible to solve the Einstein Field Equations for Earth? If yes, how?

 

[Mentz114]

Still is it possible to solve the Einstein Field Equations for Earth? If yes, how?

Not exactly, but we can solve the EFE for a spherically symmetric gravitational source and a rotating spherically symmetric source. These solutions are good approximations for planets and stars.

Any good textbook will show how to derive the metric from the field equations in the spherically symmetric case.

 

[GRstudent]

So what inputs should I put into EFE to get a solution for a rotating spherically symmetric source. I would like to see how it is obtained from G(mu,nu)=8piT(mu,nu) to a real solution for a rotating spherically symmetric source. Thank you.

 

[WannabeNewton]

What textbook are you using? It should have a section on the Kerr metric. That is what you want to look at.

 

[Mentz114]

So what inputs should I put into EFE to get a solution for a rotating spherically symmetric source. I would like to see how it is obtained from G(mu,nu)=8piT(mu,nu) to a real solution for a rotating spherically symmetric source. Thank you.

I suggest you start doing some research and look for the derivation of the Kerr metric ( rotating spherical source).

 

[GRstudent]

What textbook are you using? It should have a section on the Kerr metric. That is what you want to look at.

I am not using the textbook. I just started watching Leonard Susskind's lectures on GR.

 

[GRstudent]

I looked up to Kerr metric, and I have 2 questions.

1) It tells about tau, the proper time? Is it time which is "true" or what?

2) Is Kerr metric a solution of EFE? How is it derived from EFE?

Appreciate your help.

 

[Mentz114]

I looked up to Kerr metric, and I have 2 questions.

1) It tells about tau, the proper time? Is it time which is "true" or what?

2) Is Kerr metric a solution of EFE? How is it derived from EFE?

Appreciate your help.

I don't think you're learning much from those TV lectures.

You should read Carrolls lecture notes which are freely available, and understand special relativity before tackling GR.

 

[GRstudent]

Your advice is well noted! Thanks! I'll certainly review them.

 

[elfmotat]

I am not using the textbook. I just started watching Leonard Susskind's lectures on GR.

I find Susskind's youtube lectures to be awful. He tends to go off on tangents, ramble, and doesn't really explain things clearly. My advice is to get a good intro textbook. The first one I used was Schutz.

 

[GRstudent]

I find Susskind's youtube lectures to be awful. He tends to go off on tangents, ramble, and doesn't really explain things clearly. My advice is to get a good intro textbook. The first one I used was Schutz.

Yes, I agree. The main problem I found with Susskind's lectures is lack of examples. I mean, in real classes a teacher gives out a formula then gives an example of that formula in use. But Susskind just talks about theory. And his GR lectures are not the best way to understand the differential geometry but one thing which he does well is explaining Black Holes. I mean he spent most time of his carer as a physicist tackling the Black Hole Information Paradox so...it not unusual.

 

[WannabeNewton]

Yes, I agree. The main problem I found with Susskind's lectures is lack of examples. I mean, in real classes a teacher gives out a formula then gives an example of that formula in use. But Susskind just talks about theory. And his GR lectures are not the best way to understand the differential geometry but one thing which he does well is explaining Black Holes. I mean he spent most time of his carer as a physicist tackling the Black Hole Information Paradox so...it not unusual.

Try Hartle's Gravity. It's an introductory GR text with tons of examples and tons of concrete problems that range from very easy to annoying so you get a varied palette. Schutz is good too but he doesn't have as many examples so you can figure out what's going on before you jump in yourself. He has a great chapter on gravitational waves though if that is of great interest to you.

 

[GRstudent]

Yes, I know Introduction to GR by Hartle is a very good choice (probably the BEST!).

Also, which book would you suggest to review all of undergrad physics?