Einstein's Field Equations

gentillabdulla

Can anyone give me a thorough explanation on Einstein's field equations, and how they relate to an electric field?(I.e. how to calculate the energy density of an electric field)

Thanks!

Enkidu

gentillabdulla wrote: »

Can anyone give me a thorough explanation on Einstein's field equations, and how they relate to an electric field?(I.e. how to calculate the energy density of an electric field)

Thanks!

They don't have anything to do with electric fields in particular. They related the geometry of spacetime to the stress-energy of matter and its forces.

Morbert

Enkidu wrote: »

Can anyone give me a thorough explanation on Einstein's field equations, and how they relate to an electric field?(I.e. how to calculate the energy density of an electric field)

Thanks!

It is more natural to speak of an electromagnetic field, rather than an electric field or a magnetic field. The electromagnetic field posesses energy, momentum, and stress, so it can curve spacetime I suppose, but could you be more specific about the questions?

gentillabdulla

Morbert wrote: »

It is more natural to speak of an electromagnetic field, rather than an electric field or a magnetic field. The electromagnetic field posesses energy, momentum, and stress, so it can curve spacetime I suppose, but could you be more specific about the questions?

I did mean electromagnetic field.Must of not noticed what I was saying.

I meant "Could any of you explain thoroughly the Einstein field equations?"

And "Could they be used to calculate the energy density of an electromagnetic field, and if so how?"

Enkidu

gentillabdulla wrote: »

I did mean electromagnetic field.Must of not noticed what I was saying.

I meant "Could any of you explain thoroughly the Einstein field equations?"

And "Could they be used to calculate the energy density of an electromagnetic field, and if so how?"

You can't use Einstein's field equations to calculate the energy density of an electromagnetic field. However if you already know the energy density of an electromagnetic you can use the field equations to figure out the geometry of space time.

The basic idea of Einsteins Field Equations is that the Stress-Energy of matter, a material quantitiy, is equal (up to a constant) to the Einstein Tensor, a geometrical quantity. This way the geometry of spacetime is determined by matter.

gentillabdulla

Enkidu wrote: »

You can't use Einstein's field equations to calculate the energy density of an electromagnetic field. However if you already know the energy density of an electromagnetic you can use the field equations to figure out the geometry of space time.

The basic idea of Einsteins Field Equations is that the Stress-Energy of matter, a material quantitiy, is equal (up to a constant) to the Einstein Tensor, a geometrical quantity. This way the geometry of spacetime is determined by matter.

So what you are saying is that if I have energy density already I can calculate geometry of spacetime?

(i.e. I can see how much spacetime is displaced by a certain electromagnetic field. Gravity from an electromagnetic field.)

Enkidu

gentillabdulla wrote: »

So what you are saying is that if I have energy density already I can calculate geometry of spacetime?

(i.e. I can see how much spacetime is displaced by a certain electromagnetic field. Gravity from an electromagnetic field.)

Yes, exactly.

gentillabdulla

Enkidu wrote: »

Yes, exactly.

How would I begin to do such a process, I mean I can do all the way up to calculus at my current state of understanding?

Do you have any suggestions?

For example say I have a magnetic field with strength of about 3 tesla, could I somehow relate 3 Tesla to a certain curvature int he spacetime continuum, ignoring the amount of energy density from the object causing the field of course.

Professor_Fink

gentillabdulla wrote: »

How would I begin to do such a process, I mean I can do all the way up to calculus at my current state of understanding?

Do you have any suggestions?

General relativity in its most usual form is expressed using differential geometry, which is a fairly advanced topic in maths (usually taken near the end of an undergrad degree if at all). It has somewhat different notation from what you will have seen before, so if is worth reading up on, since there is not really a substitute for working with the field equations. (Well, there is a substitute, but it's equally as advanced, and less common, so I'll ignore it)

Basically, to determine the space time, what you want to do is determine what is called the line element for the spacetime. To do this you need to fix the stress-energy-momentum tensor for the spacetime. This tensor is what encodes the setup.

Once you have the line element, you essentially have what you need. The paths of freely falling particles are given by the geodesics. Further, if you are trying to show that this setup allows time travel, then the spacetime must have a closed time-like curve. This is a path within the space time along which travel is always time-like, but which is closed.

Learning how to do this basically amounts to a full course in general relativity and differential geometry, so it's not really something that anyone here will be able to go through with you step by step (there is simply too much to cover). However, there is an excellent and surprisingly cheap book on general relativity by Paul Dirac which covers the topics needed for general relativity in very concise chunks. Make sure you understand and can reproduce the derivation of the line element for the Schwarzschild black hole.

I should warn you that very often the line element for some complicated spacetime will not have a closed form, and so it might not be possible to fully calculate by hand. In such a case, using a computer to approximate the answer is usually necessary, which can be a very non-trivial task.