Einstein's Field Equations and Poisson's Equation

[lugita15]

Firstly in(3) is incorrect it should be GM/Sqrt r rather than GM/r

GM/Sqrt r wouldn't even be the right units!

 

[Mueiz]

Now the difference is clear between density in EFE which is local(so equal zero outside the source) and density in PE which is related to the region which include the point and the source(equal M/v outside the source)
I want to remind you my question; Is it not true that using poisson equation as a model to derive field equations is confusing ?

 

[lugita15]

But no problem that is easy suppose we want to apply it in the pont (1,0,0) which is outside the source;
Then we have out(5)=-3GM-3GM-3GM+3GM=-6GM not equal 0

No, you would get -3GM-0-0+3GM=0.

 

[Mueiz]

GM/Sqrt r wouldn't even be the right units!

We want to calculate the lablacian of what?
of the gravitational field
How gravitational field is measured ?
GM/Sqrt r

 

[lugita15]

We want to calculate the lablacian of what?
of the gravitational field
How gravitational field is measured ?
GM/Sqrt r

Gravitational field is inversely proportional to r^2. Gravitational potential is inversely proportional to r.

 

[Mueiz]

Gravitational field is inversely proportional to r^2. Gravitational potential is inversely proportional to r.

Poission Equation is conserned with the Lablacian of gravitational field and not the potential see any textbook about poission equation

 

[lugita15]

Poission Equation is conserned with the Lablacian of gravitational field and not the potential see any textbook about poission equation

No, that's wrong. See this link.

 

[Mueiz]

No, you would get -3GM-0-0+3GM=0.

your calculation here is correct according to the incorrect form of Mr Dale
but not according to the correct form

 

[lugita15]

your calculation here is correct according to the incorrect form of Mr Dale
but not according to the correct form

What do you think the correct form is?

 

[Mueiz]

No, that's wrong. See this link.

your link of wikipeadia says;
"This provides an alternate means of calculating the gravitational potential and gravitational field. "
which one to chose ?

 

[lugita15]

your link of wikipeadia says;
"This provides an alternate means of calculating the gravitational potential and gravitational field. "
which one to chose ?

It's quite simple. You solve Poisson's Equation to get the gravitational potential, then you take the gradient to get the gravitational field.

 

[Dale]

Firstly in(3) is incorrect it should be GM/Sqrt r rather than GM/r

-GM/r is correct:
http://www.diracdelta.co.uk/science/source/g/r/gravitational potential/source.html

As lugita points out the using the square root would not have the correct units.

Secondly what is the result of out(5)? you did not write it !

Obviously not. It is output. It is the Laplacian of the potential in Cartesian coordinates. It is undefined for x=y=z=0 and it simplifies to 0 for all other points.

But no problem that is easy suppose we want to apply it in the pont (1,0,0) which is outside the source;
Then we have out(5)=-3GM-3GM-3GM+3GM=-6GM not equal 0
-6GM equal 4 *pi *G *Density as I said
4* pi *G *density=-4 pi G M/(4/3 pi *1)=-6GM

It does equal zero. Check your arithmetic.

I will not say you need to learn some basic DE but I think you just did a mistake.

Not only do you need to learn some basic differential equations, but you need to learn how to do simple algebraic substitution.

I have no idea why you bother posting this continuing line of nonsense. You are not able to contribute anything useful to others and you are not willing to learn anything useful to yourself.

 

[Dale]

Poission Equation is conserned with the Lablacian of gravitational field and not the potential see any textbook about poission equation

This doesn't even make any sense. The density is a scalar field, it cannot possibly be proportional to the Laplacian of a vector field. See a little further down in the link I originally posted.

http://en.wikipedia.org/wiki/Gauss'...vitational_potential_and_Poisson.27s_equation

 

[Mueiz]

What do you think the correct form is?

second d =dg/dr of (gravitational field)=
second d =(GM/sqr r)= -d/dr((GM/r^2)+(4*pi*dr*Density/r^2))
where r=X^2+Y^2+Z^2 ,Density=M/V, V is volume
this give poission (4*pi*G*D) equation in far distance of r
the first part is the result of the change of gravitational field due to differential change in r
the second part is the result of the change of gravitational field due to additional spherical slide

 

[Mueiz]

This doesn't even make any sense. The density is a scalar field, it cannot possibly be proportional to the Laplacian of a vector field. See a little further down in the link I originally posted.

http://en.wikipedia.org/wiki/Gauss'...vitational_potential_and_Poisson.27s_equation

I mean the second derivative of poisson equation not Lablacian (this was a mistake)

 

[Mueiz]

what is equal to 4*pi*G*M is the second dervative of gravitational field and not Lablacian of the potential

 

[Dale]

I mean the second derivative of poisson equation not Lablacian (this was a mistake)

Huh?

Poisson's equation is:
\nabla^2 \phi = 4 \pi G \rho

And you want to talk about some sort of second derivative of that, which would be some sort 4th derivative of the potential? Can you be explicit about what you are talking about? Are you perhaps talking about Gauss' law for gravity:
\nabla \cdot \mathbf g = -4 \pi G \rho

 

[lugita15]

what is equal to 4*pi*G*M is the second dervative of gravitational field and not Lablacian of the potential

The only thing that is equal to 4piGM is the flux of a gravitational field through a closed surface, in which case M is the mass enclosed by the surface. But the flux of the gravitational field involves a double integral (a surface integral to be precise), not a second derivative.

 

[Born2bwire]

We want to calculate the lablacian of what?
of the gravitational field
How gravitational field is measured ?
GM/Sqrt r

This is ridiculous. Please pick up a textbook. This is a problem that is addressed in most undergraduate physics and electrodynamic texts. But at this point the information would not be much different from what you have been shown via Wikipedia and other links.

 

[Mueiz]

The second d of gravitational field does not equal zero outside
The lablacian of the potential equals zero outside and undefined at the origin
If poission equation relates Lablacian of Potential which is always zero why not just write it ; L(potential of GF)=zero ?
also if poisson equation relates lablacian of the potential to the quantity 4*pi*G*dinsity (as you all claim) it can not be used as a model to find the constant "4PiG'' in EFEbecause Poission equation -as you understand it -can not relate the density of matter in a point to the properties of gravitational field in the absence of matter . the only thing it say is that; outside the source
the lablacian of the potential equals zero and it cannot be applied in the presence of matter ,(in both cases there is no method to characterize the differences in gravitational properties and relate it quantitatively to matterial properties While in EFE both in the presence and absense of matter in a point there exist a method to characterize gravitational field
(reply this question and leave my mistake in previous posts of using the word "lablacian of gravitational field'' instead of "second d of gravitational field dg/dr, g=dr/dt '')

 

[Dale]

The second d of gravitational field does not equal zero outside

Well, the vector Laplacian of the gravitational field (one possible meaning of "the second d") is in fact 0 outside a point mass. However, since there are three dimensions there are an infinite number of possible second derivatives of the gravitational field. I don't know which one you are specifically referring to as "the second d", but you are correct that many of the possible second derivatives (besides the Laplacian) may be non-zero in vacuum.

The lablacian of the potential equals zero outside and undefined at the origin

For a point mass, yes. What is the density of a point mass?

If poission equation relates Lablacian of Potential which is always zero why not just write it ; L(potential of GF)=zero ?

Why don't you answer this question yourself by doing the following exercise. Start with the known expression for the gravitational potential inside a solid uniform spherical mass (i.e. a solid ball, not a hollow shell) of unit density (http://en.wikipedia.org/wiki/Gravitational_potential#Spherical_symmetry), take the Laplacian of that expression and see what you get. And yes, I can do this exercise but then how will you learn.

the only thing it say is that; outside the source
the lablacian of the potential equals zero and it cannot be applied in the presence of matter ,(in both cases there is no method to characterize the differences in gravitational properties and relate it quantitatively to matterial properties

I don't know how you come up with any of this. This whole paragraph is simply incorrect. Please do the exercise above to see for yourself.

 

[lugita15]

The second d of gravitational field does not equal zero outside
The lablacian of the potential equals zero outside and undefined at the origin
If poission equation relates Lablacian of Potential which is always zero why not just write it ; L(potential of GF)=zero ?
also if poisson equation relates lablacian of the potential to the quantity 4*pi*G*dinsity (as you all claim) it can not be used as a model to find the constant "4PiG'' in EFEbecause Poission equation -as you understand it -can not relate the density of matter in a point to the properties of gravitational field in the absence of matter . the only thing it say is that; outside the source
the lablacian of the potential equals zero and it cannot be applied in the presence of matter ,(in both cases there is no method to characterize the differences in gravitational properties and relate it quantitatively to matterial properties While in EFE both in the presence and absense of matter in a point there exist a method to characterize gravitational field
(reply this question and leave my mistake in previous posts of using the word "lablacian of gravitational field'' instead of "second d of gravitational field dg/dr, g=dr/dt '')

I can't make sense of a lot of what you're saying, but I think you have one major misunderstanding about Poisson's equation which is leading to a lot of smaller confusions. It is true that for each point where there is no matter, Poisson's equation simply states that the Laplacian of the potential is zero. But the mere fact that the Laplacian is zero is not sufficient information to determine what the value of the potential is at that point, because there are an infinite number of possible functions whose Laplacian is zero. So you require some additional information, known as boundary conditions, to solve Laplace's equation (which is just Poisson's equation with the right hand side equal to zero). The boundary conditions are in the form of the distribution of mass throughout space; in other words you need to specify density as a function of position. Once you know the mass distribution, then you have sufficient information to solve the potential, because there is a theorem that says that there exists a unique solution to Laplace's equation for each set of boundary conditions.

The bottom line is that even though the Laplacian of the potential at a point of empty space is always zero, still the potential itself can and does depend on the magnitude and distribution of the surrounding matter.

 

[Dale]

The bottom line is that even though the Laplacian of the potential at a point of empty space is always zero, still the potential itself can and does depend on the magnitude and distribution of the surrounding matter.

I made this point back in post 15 (https://www.physicsforums.com/showpost.php?p=3086463&postcount=15) but hopefully he will pay attention now that more than one person is saying the same thing. Although, I am not placing any bets on it

 

[lugita15]

I made this point back in post 15 (https://www.physicsforums.com/showpost.php?p=3086463&postcount=15) but hopefully he will pay attention now that more than one person is saying the same thing. Although, I am not placing any bets on it

Well, I guess it bears repeating.

 

[Mueiz]

1\ I agree that the Lablacian outside the source =0
2\ I disagree that the left hand side of poisson equation is the Lablacian of the Potential
3\ I claim that the left hand side of poisson equation is dg/dr g is acceleration of gravitational field .r^2 = x^2+Y^2+Z^2 (also = C^2t^2)
and I can prove that dg/dr =4*pi*G*density
4\I want anyone who disagree just to prove that Ladlacian of Potential =4*pi*G*densitynoone so far did this and is not found in all links you posted to me
5\Noone reply my basic argument in my last post that ;Possion equation as you understand it can not be useful as a model to EFE
(These are the points of Discussion but not the zero value of L of potential and whether it means that the gravitational field is zero or my mistake of using the wrong phrase.)

 

[Dale]

2\ I disagree that the left hand side of poisson equation is the Lablacian of the Potential
3\ I claim that the left hand side of poisson equation is dg/dr

Are you blind?

http://en.wikipedia.org/wiki/Gauss'...vitational_potential_and_Poisson.27s_equation
gives Poisson's equation as \nabla^2 \phi = 4 \pi G \rho

http://en.wikipedia.org/wiki/Poisson's_equation
gives it as \nabla^2 \varphi = f

http://en.wikipedia.org/wiki/Electrostatics#Poisson.27s_equation
gives it as \nabla^2 \phi = -\frac{\rho}{\epsilon_0}

http://farside.ph.utexas.edu/teaching/em/lectures/node31.html
gives it as \nabla^2 \phi = -\frac{\rho}{\epsilon_0} and also gives its most general form as \nabla^2 u = v

All of these are direct links already posted in this thread and all of them confirm that the left hand side of Poisson's equation is the Laplacian of the potential whether you are dealing with a gravitational potential, an electrostatic potential, or some arbitrary potential. You have absolutely no excuse to be posting nonsense like this after so many references have already been provided.

Please retract your statements and then do the exercise I suggested and post your work.

 

[Mueiz]

Are you blind?

I am not blind but I see what is in front of me rather than what I wish to see as you do which is worse than being blind!
None of these Pages contain a proof that Lablacian(GM/r)=4*pi*G*density
If wikipeadia say A becomes B this is not a proof
Not all that Wikipedia says is correct ..see what the ( Holy) wikipeadia says about Faster-than-light spots of light.

 

[Dale]

I see what is in front of me

Apparently not given the number of links to Poisson's equations that you didn't see even after we put them in front of you.

None of these Pages contain a proof that Lablacian(GM/r)=4*pi*G*density

I was responding to your erroneous claims 2 and 3 regarding Poisson's equation, which is clearly defined in those links and in any textbook on the subject. We need to clear those up and you need to post the results of the exercise before going on to claim 4. The left hand side of Poisson's equation is clearly the Laplacian of the potential. If you cannot even understand that with so much evidence provided then any further discussion is pointless.

Are you ready to retract points 2 and 3 and post your work on the exercise?

 

[Mueiz]

This is the exersise:



Code:
In[1]:= << VectorAnalysis`

In[2]:= SetCoordinates[Spherical[r, \[Theta], \[Phi]]];

In[3]:= Laplacian[-((G M)/r)]

Out[3]= 0
(this is your work in Post #28)
Everyone -even the blind - can see that Laplacian[-((G M)/r)] =0 and not 4*pi*G*Density
Now can anyone prove that Laplacian[-((G M)/r)]=4*pi*G*Density?
If you do this (and you and everyone can not) I will stop the discusson and believe in your opinion.

 

[Dale]

Everyone -even the blind - can see that Laplacian[-((G M)/r)] =0

Excellent copying and pasting, but that wasn't the exercise. The exercise was to find the Laplacian of the potential inside a spherical mass, not outside. The potential inside a spherical mass is not -((G M)/r). See the link which I gave above in post 51:

http://en.wikipedia.org/wiki/Gravitational_potential#Spherical_symmetry

Mueiz, the gravitational potential inside a spherical mass is first-semester freshman physics material. If you don't know that, then you have no business trying to learn the EFE. I applaud your ambition, but you need to go back and learn basic Newtonian physics first. I mean, if you don't even know freshman physics nor differential equations then you are simply not equipped to understand the proof that you are asking for. We can post it (it is easy enough to find on the internet) but you won't be able to understand it.

Anyway, please complete the exercise and post your results and then we will proceed from there. Hopefully the exercise itself will be a valuable learning experience and will help prepare you for the proof.