Basic questions about raising and lowering indices

[Ibix]

I simplified the equation with

and got the last equation in post26. I am not sure if I did it correctly.

I think the thing that's confusing us is that this isn't a simplification. You've replaced one sum with two.

You seem to be fighting very hard against the notation. The whole point of it is to allow you to express gigantic error-prone horrors like your #26 in a clean and manageable way, with algebraic complexity "boxed away" inside simple symbols.

Have you ever programmed a computer? What you are doing is a bit like insisting on trying to write a full-featured word processor directly in machine code.

 

[olgerm]

I like that form. maybe after simplifiyng it more it becomes sipmler. the form that includes Ricci tensor and other things that I do not know has jus no intuitive meaning to me.
would like to know whether I used

$T^\mu{}_\mu=T_{\mu\rho}g^{\rho\mu}$

correctly.

 

[weirdoguy]

I like that form.

Ok, but keep in mind that you are alone with that. Don't be surprised with lack of help. Especially since you've been told multiple times in multiple threads of yours that what you write is unreadable. You've been told at least twice that star is not a symbol of multiplication, use \cdot command if necessarry (and quite often it is not necessary at all). And yet you ignore everything, because you like it that way. That's ok in your private files, but you are not writing it for yourself, you do this for other members to help you. Keep that in mind.

 

[olgerm]

try to understand my notation. Are the equations in post26 correct? Have any recomendations to simplify einstein fild equation füther?

 

[PeterDonis]

the original indices $\mu$ and $\nu$ are still upper indices

What original indices?

the last equation in post26. I am not sure if I did it correctly

That equation is such a mess that I am unable to tell whether it is correct or not.

 

[Ibix]

try to understand my notation.

I can't even count the number of brackets you've used without computer support, let alone see if you've nested them correctly. Neither can you, which is why you are asking for help. Shouldn't that tell you something about your method?

 

[robphy]

I like that form. maybe after simplifiyng it more it becomes sipmler. the form that includes Ricci tensor and other things that I do not know has jus no intuitive meaning to me.
would like to know whether I used correctly.

Although you may be reducing the field equation written as a single tensor equation
in terms of more primitive terms involving the metric and coordinate-dependent Christoffel symbols,
I don't think simpler-as-more-primitive
is more intuitive than simpler-as-tensors[-as-stuctures]
because you are missing sight of all of the symmetries and structures that went into the tensor formulation.

Those primitives are needed to build up the structures.
But it's the structures that provide the intuition behind the field equations.

Yes, you'll need to know how the Einstein tensor is built and what it means
in terms of the more primitive ideas.
But I don't think it's helpful, especially as a first step,
to write it out fully in terms of metrics and Christoffel symbols.
Trust us,... you won't gain any intuition by doing this.

That's why we don't write down Maxwell's Equations in its original form (as 20 coupled PDEs
https://en.wikipedia.org/wiki/Histo...Dynamical_Theory_of_the_Electromagnetic_Fieldhttps://en.wikisource.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field/Part_III) but we instead use the vector calculus notation of Heaviside... or, fancier, a tensor equation
... or even more fancy: with differential forms, or with geometric-algebra
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_fieldTensor notation was partially designed to avoid seeing explicit summation signs.
Sub-subscripts x_{i_1}y_{i_2} (or indices-with-indices) are unnecessarily taxing on the reader,
unless there are important relations among the indices being suggested.Possibly useful advice:
https://aapt.scitation.org/doi/abs/10.1119/1.5111838?journalCode=ajp"Low-entropy expressions"
American Journal of Physics 87, 613 (2019); https://doi.org/10.1119/1.5111838
Sanjoy Mahajan

Possibly more useful use of time and effort:
http://pages.pomona.edu/~tmoore/LesHouches/DiagMetric.pdf

 

[olgerm]

I still want to see this equation in form that includes more easily comprihensable quantities.

I remove sum symbols if it is easier for you:

$g^{\mu m_1}*g^{\nu m_2} g^{j_1 m_3}(\frac{\partial \Gamma_{m_3 m_2 m_1}}{\partial x^{j_1}}-\frac{{\partial \Gamma_{m_3 j_1 m_1}}}{\partial x^{m_2}}+g^{j_2 m_4}({\Gamma_{m_3 j_1 j_2}}{\Gamma_{m_4 m_2 m_1}}-{\Gamma_{m_3 m_2 j_2}}{\Gamma_{m_4 j_1 m_1}}))+ (\Lambda-\frac{1}{2}*g_{i_1i_2}*g^{i_1m_1}*g^{i_2m_2}*g^{m_3 j_1}(\frac{\partial{\Gamma}_{m_3 m_2 m_1}}{\partial x^{j_1}}-\frac{\partial \Gamma_{m_3 j_1 m_1}}{\partial x^{m_2}}+g^{j_2m_4}(\Gamma_{m_3 j_1 j_2}\Gamma_{m_4 m_2 m_1 }-\Gamma_{m_3 m_2 j_2} \Gamma_{m_4 j_1 m_1})))*g^{\mu \nu}=\frac{8*\pi*G}{c^4}*T^{\mu \nu}$

maybe it is easier understand my notation without sum symbols. is the equation correct? it is same equation as last equation in post26. Have any recomendations to simplify einstein fild equation füther?

 

[PeterDonis]

I still want to see this equation in form that includes more easily comprihensable quantities.

The standard form of the equation is already more comprehensible to everyone except you.

maybe it is easier understand my notation without sum symbols. is the equation correct?

It's still too much of a mess for me to tell whether it's correct.

Have any recomendations to simplify einstein fild equation füther?

What you are doing is not "simplifying" to anyone except you. To everyone else, the standard form of the equation is the "simple" one.

 

[PeterDonis]

This thread is going nowhere and is now closed.