# Annoying Differential Geometry/tensor question in GR

• latentcorpse
In summary, the conversation is about working through appendix A of the paper "Vacuum Spacetimes with Two-Parameter Spacelike Isometry Groups and Compact Invariant Hypersurfaces: Topologies and Boundary Conditions" by Robert H. Gowdy. The speaker is using an orthonormal basis method to obtain curvature tensors and solve the Einstein equations. They are discussing the structure equations and their rewritten form in terms of one-form matrices. The conversation also touches on antisymmetric derivatives and solving for connection forms. There are questions about the notation and specific calculations involved in the process.
latentcorpse
i'm working through appendix A of the paper "Vacuum Spacetimes with Two-Parameter Spacelike Isometry Groups and Compact Invariant Hypersurfaces:Topologies and Boundary Conditions" by Robert H. Gowdy

anyway i. using a orthonormal basis method to get the curvature tensors and hence the einstein eqns

i have $w^0=Le^adt, w^1=Le^a d \theta, w= \left( \begin{array}{c} w^2 \\ w^3 \end{array} \right)=L \left( \begin{array}{cc} M_{22} & M_{23} \\ M_{32} & M_{33} \end{array} \right)$

the structure eqn is $dw^{\mu}=w^\mu_\nu \wedge w^\nu$ and is rewritten in terms of the one-form matrices $\Omega_0=\left( \begin{array}{c} w^2_0 \\ w^3_0 \end{array} \right), \Omega_1 \left( \begin{array}{c} w^2_1 \\ w^3_1 \end{array} \right), \Omega=w^2_3 \epsilon$ where $\epsilon= \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$

the structure eqns become

$dw^0=w^0_1 \wedge w^1 + \hat{\Omega_0} \wedge w$ (1)
$dw^1=w^0_1 \wedge w^0 - \hat{\Omega_1} \wdge w$ (2)
$dw=\Omega_0 \wedge w^0 + \Omega_1 \wedge w^1 + \Omega \wedge w$
where the antisymmetry $w_{\mu \nu}=-w_{\nu \mu}$ has been used.

the antisymmetric derivatives are copmuted to be

$dw^0=L^{-1} a' e^{-a} w^1 \wedge w^0$
$dw^1=L^{-1} \dot{a} e^{-a} w^0 \wedge w^1$
$dw=L^{-1} e^{-a} (Pw^0+uw^1) \wedge w$
where $P=\dot{M}M^{-1},U=M'M^{-1}$

i should mention that L is a constant and a is a function of $\theta,t$
so I'm having a few problems with what should be straightforward differential geometry:

(i) in (1) and (2) why is there a hat on the Omega matrices? what does this signify? some sort of antisymmetric version of Omega?

(ii) in (2), if you plug the numbers into the structure eqns, the first term should just be $w^1_0 \wedge w^1$ but they somehow rearrange to $w^0_1 \wedge w^1$. How does this work?

(iii) finally, and probably most importantly, how do they get $dw^0,dw^1$ and $dw$?

ok. I've actually managed (iii) myself. i still don't get (i) or (ii) though.

i have a bit of difficulty with another bit though:
we then solve the structure eqns for the connection forms giving
$w^0{}_1=-L^{-1}e^{-a}(a'w^0+ \dot{a} w^1)$ i can get this one

i can't get these next three though:
$\Omega_0=-L^{-1}e^{-a}P_{sym}w, \Omega_1=-L^{-1}e^{-a}U_{sym}w, \Omega=L^{-1}e^{-a}(Pw^0+Uw^1)_{asym}$
in particular why we take the symmetric and asymmetric components of those matrices.
can anybody help here?

thanks.

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## 1. What is Differential Geometry?

Differential Geometry is a branch of mathematics that deals with the study of curves and surfaces in a higher-dimensional space. It involves the use of calculus and linear algebra to understand the geometric properties of these objects.

## 2. What is a tensor in General Relativity?

In General Relativity, a tensor is a mathematical object that represents the curvature of spacetime. It is a multi-dimensional array of numbers that describes the relationship between different points in space and time.

## 3. How is Differential Geometry used in General Relativity?

Differential Geometry is the mathematical framework used in General Relativity to describe the curvature of spacetime and the effects of gravity. It allows us to understand the behavior of objects in a gravitational field and make predictions about their motion.

## 4. What is the most annoying question about Differential Geometry and tensors in GR?

The most annoying question about Differential Geometry and tensors in GR is often related to the complicated mathematical notation and terminology used. Many people find it difficult to understand and apply these concepts without a strong background in mathematics.

## 5. How can I better understand Differential Geometry and tensors in GR?

To better understand Differential Geometry and tensors in GR, it is important to have a solid foundation in mathematics, particularly in calculus and linear algebra. It also helps to read textbooks and articles that explain these concepts in simpler terms and to practice solving problems and applying these concepts to real-world scenarios.

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