Poisson Equation & Cosmological Term

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Peter Michalicka - 24 May 2005 12:40 GMT

laplace(phi) = 4*pi*G*rho
phi = -G*M/r

laplace(phi)=0 !!!

Proof with sample code from maple:
with(linalg);
v1 := [r, theta, phi];
laplacian(M*G/r, v1, coords=spherical);

So we have to introduce the Cosmological Term like A.Einstein:

laplace(phi) = 4*pi*G*rho + Lambda*phi = 0

Conclusion: Lambda = 3/r^2

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Tom Roberts - 24 May 2005 23:10 GMT

> laplace(phi) = 4*pi*G*rho
> phi = -G*M/r
> laplace(phi)=0 !!!

You plugged in the solution for phi corresponding to a point mass in
vacuum, so of course laplace(phi)=0 everywhere that rho=0, which is of
course everywhere except r=0. Note that laplace(phi) does _not_ equal
zero at r=0. It is undefined there, corresponding to an "infinite
density"....

> So we have to introduce the Cosmological Term like A.Einstein:

No, you just have to consider a realistic physical situation.

Tom Roberts tjroberts@lucent.com

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Peter Michalicka - 25 May 2005 10:44 GMT

>> laplace(phi) = 4*pi*G*rho
>> phi = -G*M/r
[quoted text clipped - 3 lines]
> vacuum, so of course laplace(phi)=0 everywhere that rho=0, which is of
> course everywhere except r=0.

For phi=-G*M/r you will get laplace(phi)=0
Proof by MAPLE:
with(linalg);
v:=[r,theta,phi];
laplacian(G*M/r,v,coords=spherical);

=> 0

Note that laplace(phi) does _not_ equal

> zero at r=0. It is undefined there, corresponding to an "infinite
> density"....
[quoted text clipped - 4 lines]
>
> Tom Roberts tjroberts@lucent.com

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Dirk Van de moortel - 25 May 2005 11:40 GMT

> >> laplace(phi) = 4*pi*G*rho
> >> phi = -G*M/r
[quoted text clipped - 11 lines]
>
> => 0

Can you define functions in Maple like for instance
f := x ==> x^2
?
If so, when you define
f := r ==> laplacian(G*M/r,v,coords=spherical),
what does it say when you let it calculate f(0) ?

Dirk Vdm

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pmichalicka - 25 May 2005 11:58 GMT

You are right:

f := r ==> laplacian(G*M/r,v,coords=spherical)
f(0) -> infinite

but

phi = -G*M/r is also infinite at 0

Peter Michalicka

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Dirk Van de moortel - 25 May 2005 12:10 GMT

> You are right:

I wasn't making a statement, but just asking a question
about Maple.

Dirk Vdm

> f := r ==> laplacian(G*M/r,v,coords=spherical)
> f(0) -> infinite
[quoted text clipped - 4 lines]
>
> Peter Michalicka

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Tom Roberts - 29 May 2005 18:44 GMT

> For phi=-G*M/r you will get laplace(phi)=0
> Proof by MAPLE:
> with(linalg);
> v:=[r,theta,phi];
> laplacian(G*M/r,v,coords=spherical);
> => 0

You have to _UNDERSTAND_ the math in order to use Maple in a reasonable
manner. Understanding of the problem domain is true for _any_ computer
program, and is known as GIGO: Garbage in => garbage out.

Tom Roberts tjroberts@lucent.com

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Natural Science Forum / Physics / Relativity / May 2005

Poisson Equation & Cosmological Term