Ricci tensor from Ricci 1-form

In summary, an equation for the Ricci 1-form is Ric=R^{a}\otimes e_{a} where R^{a} is the Ricci 1-form and e_{a} are the elements of an orthonormal basis. My confusion pertains to the multiplication between the elements. I think it should be (using the Einstein summation convention for repeated indices): Ric=R^{a}\otimes e_{a} := R^0 \otimes e_0 + R^1 \otimes e_1 + R^2 \otimes e_2 = (Ae_{0} + Be_{1}) \otimes e_0 + (Be_{0} -
  • #1
graupner1000
16
1
Hi all,

once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation

[itex]Ric=R^{a}\otimes e_{a}[/itex]

where [itex]Ric[/itex] is the Ricci tensor, [itex]R^{a}[/itex] is the Ricci 1-form and [itex]e_{a}[/itex] are the elements of an orthonormal basis.

Now, let's say for arguments sake that [itex]a=0,1,2[/itex] and I have a Ricci 1-form that looks something like this (What I'm actually trying to work out is a lot larger but follows a similar pattern)

[itex]R^{a}=\left[ \begin{array}{c} Ae_{0} + Be_{1} \\ Be_{0} - Ae_{1} \\ e_{2} \end{array} \right][/itex]

where [itex]A[/itex] and [itex]B[/itex] are constants. The next step would be to take the tensor product of [itex]R^{a}[/itex] and [itex]e_{a}[/itex] and this is where the problem lies. My instinct would be to treat this as an outer product so you end up with something like

[itex]R^{a}\otimes e_{a}=\left[ \begin{array}{ccc} (Ae_{0} + Be_{1})e_{0} & (Ae_{0} + Be_{1})e_{1} & (Ae_{0} + Be_{1})e_{2} \\ (Ae_{0} - Be_{1})e_{0} & (Ae_{0} - Be_{1})e_{1} & (Ae_{0} - Be_{1})e_{2} \\ e_{2}e_{0} & e_{2}e_{1} & e_{2}e_{2} \end{array} \right][/itex]

But that seems to be ignoring the sum over [itex]a[/itex] (or is this the operation it implies?) and more importantly, I really doubt there should be multiplication between the elements, i.e does

[itex](Ae_{0} + Be_{1})e_{0}[/itex]
imply
[itex](Ae_{0} + Be_{1})\otimes e_{0}[/itex]
or
[itex](Ae_{0} + Be_{1})\wedge e_{0}[/itex]

As said, this is a really silly thing to be stuck with and probably means that I've missed(read not paid attention to) something really basic so any help would be very much appreciated.
 
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  • #2
graupner1000 said:
Hi all,

once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation

[itex]Ric=R^{a}\otimes e_{a}[/itex]

where [itex]Ric[/itex] is the Ricci tensor, [itex]R^{a}[/itex] is the Ricci 1-form and [itex]e_{a}[/itex] are the elements of an orthonormal basis.

Now, let's say for arguments sake that [itex]a=0,1,2[/itex] and I have a Ricci 1-form that looks something like this (What I'm actually trying to work out is a lot larger but follows a similar pattern)

[itex]R^{a}=\left[ \begin{array}{c} Ae_{0} + Be_{1} \\ Be_{0} - Ae_{1} \\ e_{2} \end{array} \right][/itex]

where [itex]A[/itex] and [itex]B[/itex] are constants. The next step would be to take the tensor product of [itex]R^{a}[/itex] and [itex]e_{a}[/itex] and this is where the problem lies. My instinct would be to treat this as an outer product so you end up with something like

[itex]R^{a}\otimes e_{a}=\left[ \begin{array}{ccc} (Ae_{0} + Be_{1})e_{0} & (Ae_{0} + Be_{1})e_{1} & (Ae_{0} + Be_{1})e_{2} \\ (Ae_{0} - Be_{1})e_{0} & (Ae_{0} - Be_{1})e_{1} & (Ae_{0} - Be_{1})e_{2} \\ e_{2}e_{0} & e_{2}e_{1} & e_{2}e_{2} \end{array} \right][/itex]

But that seems to be ignoring the sum over [itex]a[/itex] (or is this the operation it implies?) and more importantly, I really doubt there should be multiplication between the elements, i.e does

[itex](Ae_{0} + Be_{1})e_{0}[/itex]
imply
[itex](Ae_{0} + Be_{1})\otimes e_{0}[/itex]
or
[itex](Ae_{0} + Be_{1})\wedge e_{0}[/itex]

As said, this is a really silly thing to be stuck with and probably means that I've missed(read not paid attention to) something really basic so any help would be very much appreciated.

I think it should be (using the Einstein summation convention for repeated indices):

[tex]Ric=R^{a}\otimes e_{a} := R^0 \otimes e_0 + R^1 \otimes e_1 + R^2 \otimes e_2 = (Ae_{0} + Be_{1}) \otimes e_0 + (Be_{0} - Ae_{1}) \otimes e_1 + e_2 \otimes e_2 = \dots [/tex]

This gives you a 2-tensor, as you are supposed to get.
 
  • #3
Back again. Thanks for your answer, that was one thing I was thinking about. But is there any way to write that in a "traditional" matrix form?
 
  • #4
Just use the correspondence between the coefficients and basis-expansion of a 2-tensor.

[tex]
A = A_{\mu\nu} \omega^\mu \otimes \omega^\nu
[/tex]

to indentify the matrix components as the coefficients in this expansion.

Btw, I'm not sure what you meant by "[itex]R^a[/itex] is the Ricci 1-form"? Do you have three "Ricci 1-forms", one for each value of a, or are these the components of one "Ricci 1-form". If the latter is the case, your 3-component expression for [itex]R^a[/itex] doesn't make sense, since you have put basis elements in the components.

Sorry, this terminology is a bit unusual for me, I'm used to the curvature forms like what is done here:

http://www.uio.no/studier/emner/mat...dervisningsmateriale/Kursmateriell/fys307.pdf

I haven't heard of a Ricci 1-form before.
 
  • #5
This is the terminology I have been taught, but it might have other names elsewhere. The Ricci 1-form is the contraction of the curvature form (or Ricci 2-form):

[itex]R_{a}=i_{b}R^{b}_{ a}[/itex]

(Using R twice might not be the best convention) where [itex]R^{b}_{ a}[/itex] is given by Cartan's second structure equation.

My example has three components just because I needed an example. What I am actually trying to work out is considerably larger and I couldn't be asked to write out the entire thing.
 

What is the Ricci tensor?

The Ricci tensor is a mathematical object used in the study of the curvature of a space. It is defined as a set of numbers that describe how the curvature of a space changes in different directions.

What is a Ricci 1-form?

A Ricci 1-form is a mathematical object that is used to define the Ricci tensor. It is a one-dimensional array of numbers that represents the curvature of a space in a specific direction.

How is the Ricci tensor calculated from the Ricci 1-form?

The Ricci tensor is calculated by taking the derivative of the Ricci 1-form. This involves finding the change in the curvature of the space in different directions and then using these values to construct the tensor.

What is the significance of the Ricci tensor from a physical perspective?

The Ricci tensor is an important tool in general relativity, a theory that describes the gravitational force. It is used to calculate the curvature of space and time, which is related to the distribution of matter and energy in the universe.

Why is the Ricci tensor important in mathematics?

The Ricci tensor is important in mathematics because it is used to study the geometry and topology of curved spaces. It provides a way to measure and understand the curvature of a space, which has implications in various branches of mathematics, including differential geometry and topology.

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