Swartzchild Ricci tensor?

[Endercreeper01]

Does anyone know what R_ab is in the swartzchild solution?

[imatfaal]

it is a vacuum solution so T_ab = 0 and thus R_ab = 0 (if no cosmological constant) or R_ab=Lamba . g_ab (with a cosmological constant)

[Endercreeper01]

it is a vacuum solution so T_ab = 0 and thus R_ab = 0 (if no cosmological constant) or R_ab=Lamba . g_ab (with a cosmological constant)

what is a vacuum solution?

[xyzt]

what is a vacuum solution?

absence of any gravitational bodies bar one.

[timo]

A "vacuum solution" is the solution to a differential equation in the case that there are no objects (more precisely: no objects relevant for the particular differential equation) in the region (note that there may be objects outside the region, as in this case), i.e. the solution in a vacuum (sub-) volume.

 

absence of any gravitational bodies bar one.

huh?

Edited August 15, 2013 by timo

[xyzt]

A "vacuum solution" is the solution to a differential equation in the case that there are no objects (more precisely: no objects relevant for the particular differential equation) in the region (note that there may be objects outside the region, as in this case), i.e. the solution in a vacuum (sub-) volume.

 

huh?

Let me illustrate with some examples:

 

• Schwarzschild vacuum (which describes the spacetime geometry around a spherical mass),
• Kerr vacuum (which describes the geometry around a rotating spherical mass)
• Reissner-Nordstrom vacuum (which describes the spacetime geometry around a charged spherical mass),
• Kerr-Newman vacuum (which describes the spacetime geometry around a rotating charged spherical mass)

So, there are objects, more precisely there is one very specific object in the region, which is exactly what I said.

Edited August 15, 2013 by xyzt

[timo]

Let me illustrate with a single example: http://en.wikipedia.org/wiki/Vacuum_solution

[ajb]

A vacumm solution is when the energy-momentum tensor is zero in the field equations. This means that there are no masses and no fields other than the gravitational field on the space-time.

 

Now things like the Schwarzschild metric and so on describe the vacumm around some massive object. You need to also consider "interior solutions" carefully matched up with the vacumm solution to describe the whole space-time.

[xyzt]

Let me illustrate with a single example: http://en.wikipedia.org/wiki/Vacuum_solution

Not very interesting. Actually not interesting at all.

[ajb]

 

Not very interesting. Actually not interesting at all.

It is however, straight to the point. We have no gravitational sources.

Edited August 15, 2013 by ajb

[xyzt]

It is however, straight to the point. We have no gravitational sources.

I understand the confusion: I cited the interesting cases, you two are citing the trivial case.

The term "vacuum solution" has multiple connotations, I prefer the interesting one.

Edited August 15, 2013 by xyzt

[ajb]

I understand the confusion: I cited the interesting cases, you two are citing the trivial case.

The term "vacuum solution" has multiple connotations, I prefer the interesting one.

The cases you list are some standard exact vaccum solutions in the sense we usually mean. The space-time contains no gravitational sources. The solutions you list describe the space-time around some objetcs not the space-time inside that object.

 

I am not aware of any other use of the term vacuum solution. Can you ellaborate?

[xyzt]

The cases you list are some standard exact vaccum solutions in the sense we usually mean. The space-time contains no gravitational sources. The solutions you list describe the space-time around some objetcs not the space-time inside that object.

 

I am not aware of any other use of the term vacuum solution. Can you ellaborate?

 

 

I already explained :

Schwarzschild vacuum solution

Kerr vacuum solution

Kerr-Newman vacuum solution

Riesner-Nordstrom vacuum solution

 

They are all vacuum solutions, they describe the gravitational field in the vacuum neighboring an interesting object, like an uncharged, non-rotating sphere (Schwarzschild), charged non-rotating sphere (Riessner), etc. This is basic GR, I do not understand why you have so much difficulty understanding.

 

 

 

The solutions you list describe the space-time around some objetcs not the space-time inside that object.

 

Huh? None of the so-called "interior solutions" claims to be a "vacuum solution", so your objection totally misses the point.

Edited August 15, 2013 by xyzt

[ajb]

They are all vacuum solutions, they describe the gravitational field in the vacuum neighboring an interesting object....

Right, which is what I have been saying all the time.

 

Basically, you have to "cut out" the object in question from your space-time and then describe what is left with such a vacuum solution.

 

This is the point; the space-time described by a vacuum solution is source free.

 

This is basic GR, I do not understand why you have so much difficulty understanding.

It is, but the problem is your explanations. I see what you are saying, but see what I wrote above.

 

 

Huh? None of the so-called "interior solutions" claims to be a "vacuum solution", so your objection totally misses the point.

I have no objection here, again read above.

 

So as I see it there is no other use of the term vacuum solution in GR? Or is there?

[xyzt]

Right, which is what I have been saying all the time.

 

Basically, you have to "cut out" the object in question from your space-time and then describe what is left with such a vacuum solution.

 

This is the point; the space-time described by a vacuum solution is source free.

 

 

It is, but the problem is your explanations. I see what you are saying, but see what I wrote above.

 

 

 

I have no objection here, again read above.

 

So as I see it there is no other use of the term vacuum solution in GR? Or is there?

If you "cut out" the object, then you get no gravitational field. I take it that you are a mathematician, not a physicist.

That also explains your confusion about the interior solutions: of course that the interior solutions cannot be vacuum solutions, the interior is obviously not a vacuum. Anyways, you are splitting hairs and you are splitting them badly.

 

 

 

So as I see it there is no other use of the term vacuum solution in GR? Or is there?

 

You must have missed my answer to timo, check out :

 

Schwarzschild vacuum

Kerr vacuum

etc, etc.

Edited August 15, 2013 by xyzt

[imatfaal]

Right, which is what I have been saying all the time.

 

Basically, you have to "cut out" the object in question from your space-time and then describe what is left with such a vacuum solution.

 

This is the point; the space-time described by a vacuum solution is source free....

 

nice phrasing - sums it up perfectly.

 

If you "cut out" the object, then you get no gravitational field. I take it that you are a mathematician, not a physicist.

That also explains your confusion about the interior solutions: of course that the interior solutions cannot be vacuum solutions, the interior is obviously not a vacuum. Anyways, you are splitting hairs and you are splitting them badly.

 

 

 

 

You must have missed my answer to timo, check out :

 

Schwarzschild vacuum

Kerr vacuum

etc, etc.

 

ajb and timo know a lot more about this than me - but you seem to be ignoring their explanations and getting a bit personal so I will try to explain.

 

vacuum solutions have to "cut out" the object they are interested in from the region for which the solution holds; the influence of that object is what is important - but the solution cannot cover the region of that object. As I said in my first response the Einstein tensor and the Ricci tensor must both vanish to zero - thus the gravitational source cannot be within the region of the solution. All the solutions you have listed explicitly remove the area of the massive/energetic object and focus on the empty space around it.

Its this bit that is the trouble...

 

.... snipped

 

So, there are objects, more precisely there is one very specific object in the region, which is exactly what I said.

[ajb]

If you "cut out" the object, then you get no gravitational field.

 

The point is you cannot describe the entire space-time with a single isolated massive object with a vaccum solution. You basically remove the space-time where that object sits and describe the space-time around the object which will be a vacuum solution. All the examples you have given are like this, or really lets say that there physical interpretations are like this.

 

I take it that you are a mathematician, not a physicist.

A bit of both, but leaning more towards mathematics, not that this has anything to do with this thread.

 

That also explains your confusion about the interior solutions: of course that the interior solutions cannot be vacuum solutions, the interior is obviously not a vacuum.

I am not confused.

 

Anyways, you are splitting hairs and you are splitting them badly.

Well you did claim that vacumm solutions contain sources.

 

So, there are objects, more precisely there is one very specific object in the region, which is exactly what I said.

[xyzt]

The point is you cannot describe the entire space-time with a single isolated massive object with a vaccum solution. You basically remove the space-time where that object sits and describe the space-time around the object which will be a vacuum solution. All the examples you have given are like this, or really lets say that there physical interpretations are like this.

 

 

A bit of both, but leaning more towards mathematics, not that this has anything to do with this thread.

 

 

I am not confused.

 

 

Well you did claim that vacumm solutions contain sources.

 

Sure they do, otherwise you are stuck with your totally trivial solution, no gravitational sources, no gravitational field :

 

• Schwarzschild vacuum (which describes the spacetime geometry around a non-charged, non-rotating spherical mass),
• Kerr vacuum (which describes the geometry around a rotating spherical mass)
• Reissner-Nordstrom vacuum (which describes the spacetime geometry around a charged spherical mass),
• Kerr-Newman vacuum (which describes the spacetime geometry around a rotating charged spherical mass)

You seem very unfamiliar with GR, have you taken any classes on this subject?

Edited August 15, 2013 by xyzt

[ajb]

Sure they do, otherwise you are stuck with your totally trivial solution, no gravitational sources, no gravitational field :

 

It seems that you have not taken in anything I have written here. Read it all again carefully and sleep on it.

 

 

You seem very unfamiliar with GR, have you taken any classes on this subject?

 

I don't think I will dignify that with an answer.

 

Are you now going to attack my credentials rather than what I have stated here?

[xyzt]

It seems that you have not taken in anything I have written here. Read it all again carefully and sleep on it.

I have taken in what you wrote, you don't know what you are talking about. You may want to sleep on it. Alternatively, you may want to click on the links I provided you with. BTW, as credentials go, I have many more papers published than you do. I wasn't attacking your credentials, I was just pointing out that you have serious misconceptions about GR, that's all <shrug>.

Edited August 15, 2013 by xyzt

[ajb]

I have taken in what you wrote, you don't know what you are talking about. You may want to sleep on it. Alternatively, you may want to click on the links I provided you with. BTW, as credentials go, I have many more papers published than you do.

You are now going to try some argument on why you are right and I am wrong based on the number of papers you have written?

 

I think we have to leave it there.

... I was just pointing out that you have serious misconceptions about GR, that's all <shrug>.

Okay then.

[xyzt]

 

 

vacuum solutions have to "cut out" the object they are interested in from the region for which the solution holds; the influence of that object is what is important - but the solution cannot cover the region of that object. As I said in my first response the Einstein tensor and the Ricci tensor must both vanish to zero - thus the gravitational source cannot be within the region of the solution. All the solutions you have listed explicitly remove the area of the massive/energetic object and focus on the empty space around it.

 

 

I think that you and I are saying the same exact thing.

 

I never said that the solutions cover the object itself, I said that exactly the opposite is true: the interior solutions are NOT vacuum solutions, the EXTERIOR ones , by contrast, ARE vacuum solutions. They all describe the gravitational field in the vacuum AROUND the respective OBJECT, OUTSIDE the object proper. One doesn't "cut out" the object lest the effects of the object charge, mass, momentum are nullified, one SEPARATES the external solution from the internal one. Maybe it is just an issue of terminology but one needs to be precise.

The problem is one of semantics, I am trying to explain to ajb that one MUST have different sources PRESENT (charged, uncharged, rotating and non-rotating) in order to have interesting solutions to the EFEs. In ADDITION, contrary to his claims, they are ALL called.... VACUUM solutions. This is because they all describe thee vacuum surrounding the object. The vacuum solutions do NOT reduce to the ONLY ONE, the MOST TRIVIAL ONE, the one that has total absence of sources. This reduces to the flat spacetime solution, the Minkowski metric, totally uninteresting (and not the only vacuum solution known to physicists).

Edited August 15, 2013 by xyzt

[Endercreeper01]

Then what is the riemann tensor for the interior?

[ajb]

Then what is the riemann tensor for the interior?

The interior are stars can be described using fluid solutions. For the Schwarzchild metric one uses the static spherically symmetric perfect fluid solution as this can always be matched with the Schwarzchild solution. Interestingly, the fluid solution is an exact solution.

 

I can't recall the details here, so maybe someone else can, or you can google "fluid solutions".

Edited August 15, 2013 by ajb

[xyzt]

The interior are stars can be described using fluid solutions. For the Schwarzchild metric one uses the static spherically symmetric perfect fluid solution as this can always be matched with the Schwarzchild solution. Interestingly, the fluid solution is an exact solution.

 

I can't recall the details here, so maybe someone else can, or you can google "fluid solutions".

Detailed solution is given in R.C. Tolman "Relativity, Thermodynamics and Cosmology", p. 245.

Edited August 15, 2013 by xyzt