Swartzchild metric?

[Endercreeper01]

The swartzchild metric gab=-r^4sin^2θ. Why doesn't this have to do with mass?

[timo]

Not sure what "gab" is supposed to be, but what you posted seems like a single entry of the Schwarzschild metric, at best. The Schwarzschild metric is a tensor (written in particular coordinates), and some of its entries do depend on the mass.

[Endercreeper01]

Not sure what "gab" is supposed to be, but what you posted seems like a single entry of the Schwarzschild metric, at best. The Schwarzschild metric is a tensor (written in particular coordinates), and some of its entries do depend on the mass.

g_ab is the meric tensor

Edited August 13, 2013 by Endercreeper01

[imatfaal]

[latex]g_{ab} = \left( \begin{matrix}-(1-2M/r) & 0 & 0 & 0 \\0 & (1-2M/r)^{-1} & 0 & 0 \\0 & 0 & r^2 & 0\\0 & 0 & 0 & r^2sin^2\theta \end{matrix} \right)[/latex]

 

It is slightly more complicated than you first posted

[Endercreeper01]

the Swartzchild metric (using spherical coordinates are t, r, Θ, Φ) ds²=(1-2GM/rc²)dt²- (1-2GM/rc²)^-1dr²-r²(dΘ²+r²sin²ΘdΦ). The metric tensor, g_ab=ds²/Σdx^adx^b. If you work everything out, then it becomes -r⁴sin²θ

[latex]g_{ab} = \left( \begin{matrix}-(1-2M/r) & 0 & 0 & 0 \\0 & (1-2M/r)^{-1} & 0 & 0 \\0 & 0 & r^2 & 0\\0 & 0 & 0 & r^2sin^2\theta \end{matrix} \right)[/latex]

 

It is slightly more complicated than you first posted

(1-m/r) and (1-m/r)^-1 cancel out and make -1, and the other part becomes r⁴sin²θ and so it all becomes -r⁴sin²θ

[imatfaal]

Do you know what a matrix and/or a tensor are?

 

http://en.wikipedia.org/wiki/Tensor

 

you wrote this ds²=(1-2GM/rc²)dt²- (1-2GM/rc²)^-1dr²-r²(dΘ²+r²sin²ΘdΦ)

 

that is the flat space limit which shows we get agreement with newtonian gravitation - although I think you are missing a c^2 from the first part. you can think of it - in a hazy way - of multiplying through that tensor and each section is multiplied by a different factor c^2dt^2, dr^2 dphi^2 and dtheta^2. what it actually is, i hope, is a line element; which is the summation of products of the tensor and four coordinate vectors (phi, theta, radius, and time*c^2) giving the square of the arc length

[Endercreeper01]

Do you know what a matrix and/or a tensor are?

 

http://en.wikipedia.org/wiki/Tensor

 

you wrote this ds²=(1-2GM/rc²)dt²- (1-2GM/rc²)^-1dr²-r²(dΘ²+r²sin²ΘdΦ)

 

that is the flat space limit which shows we get agreement with newtonian gravitation - although I think you are missing a c^2 from the first part. you can think of it - in a hazy way - of multiplying through that tensor and each section is multiplied by a different factor c^2dt^2, dr^2 dphi^2 and dtheta^2. what it actually is, i hope, is a line element; which is the summation of products of the tensor and four coordinate vectors (phi, theta, radius, and time*c^2) giving the square of the arc length

then how would you calculate the determinant of a 4x4 matrix? and yes, I do

and it is actually the swartzchild metric:

Schwarzschild metric[edit source | edit]

Besides the flat space metric the most important metric in general relativity is the Schwarzschild metric which can be given in one set of local coordinates by

[xyzt]

the Swartzchild metric (using spherical coordinates are t, r, Θ, Φ) ds²=(1-2GM/rc²)dt²- (1-2GM/rc²)^-1dr²-r²(dΘ²+r²sin²ΘdΦ). The metric tensor, g_ab=ds²/Σdx^adx^b. If you work everything out, then it becomes -r⁴sin²θ

(1-m/r) and (1-m/r)^-1 cancel out and make -1, and the other part becomes r⁴sin²θ and so it all becomes -r⁴sin²θ

Nonsense, you are confusing a matrix with its determinant.

[imatfaal]

agree with xyzt - you seem to be thinking a matrix is merely its determinant.

 

What you have is the line element - which is the distance along an arc squared.

 

Your second equation is exactly the same as the first just represented in terms of solid angle rather than azimuth and inclination. If you really do understand matrices and tensors - then here is a page that describes line elements which is what you are getting with your ds^2 equations

 

http://en.wikipedia.org/wiki/Line_element

[Endercreeper01]

Schwarzschild metric[edit source | edit]

Besides the flat space metric the most important metric in general relativity is the Schwarzschild metric which can be given in one set of local coordinates by

where

from Wikipedia

agree with xyzt - you seem to be thinking a matrix is merely its determinant.

 

What you have is the line element - which is the distance along an arc squared.

 

Your second equation is exactly the same as the first just represented in terms of solid angle rather than azimuth and inclination. If you really do understand matrices and tensors - then here is a page that describes line elements which is what you are getting with your ds^2 equations

 

http://en.wikipedia.org/wiki/Line_element

oh

[Widdekind]

the exterior Schwarzschild metric is "four-volume preserving", i.e. space stretches, but time compresses, such that (c dt) dx dy dz = [math](c dt) r^2 dr sin(\theta) d\theta d\phi[/math] = constant

 

trying to take the Trace or determinant, and observing cancelations of crucial coefficients, is essentially the same, as noting that the warping of space-time about the body is "four-volume preserving", space stretches, but time "thins out", preserving the over-all (four) volume, a little like a piece of putty stretching in one dimension, but thinning out in orthogonal directions, preserving the overall volume of putty