Vanishing twist of time - like killing field

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In summary, Wald's proof shows that the two orthogonal killing fields are integrable in the sense of Frobenius' theorem. One of the end of chapter problems is to show from direct calculation that \mathcal{L}_{\psi }\omega _{a} = 0 where \omega _{a} = \epsilon _{abcd}\xi ^{b}\triangledown^{c}\xi ^{d} is the twist of \xi ^{a} ,\mathcal{L}_{\psi } is the lie derivative along the flow of \psi ^{a}, and \epsilon _{abcd} is the levi
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WannabeNewton
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Hi there. This is regarding a proof from Wald that involves showing that for an axisymmetric, static space - time, the 2 - planes orthogonal to the (commuting) time - like and closed space - like killing vector fields [itex]\xi ^{a},\psi ^{a}[/itex], respectively, are integrable in the sense of Frobenius' theorem. One of the end of chapter problems is to show from direct calculation that [itex]\mathcal{L}_{\psi }\omega _{a} = 0[/itex] where [itex]\omega _{a} = \epsilon _{abcd}\xi ^{b}\triangledown^{c}\xi ^{d}[/itex] is the twist of [itex]\xi ^{a}[/itex] ,[itex]\mathcal{L}_{\psi }[/itex] is the lie derivative along the flow of [itex]\psi ^{a}[/itex], and [itex]\epsilon _{abcd}[/itex] is the levi - civita symbol. I was hoping someone could check my work to see if it is all legal.

We have that [itex]\mathcal{L}_{\psi}\omega _{a} = \mathcal{L}_{\psi}(\epsilon _{abcd}\xi ^{b}\triangledown^{c}\xi ^{d}) = \epsilon _{abcd}\mathcal{L}_{\psi}(\xi ^{b}\triangledown^{c}\xi ^{d}) + \xi ^{b}\triangledown^{c}\xi ^{d}\mathcal{L}_{\psi}\epsilon _{abcd}[/itex]. Starting with the first expression, we have [itex]\mathcal{L}_{\psi}(\xi ^{b}\triangledown^{c}\xi ^{d}) = \triangledown ^{c}\xi ^{d}\mathcal{L}_{\psi}\xi ^{b} + \xi ^{b}\mathcal{L}_{\psi}(\triangledown ^{c}\xi ^{d}) = \xi ^{b}\mathcal{L}_{\psi}(\triangledown ^{c}\xi ^{d}) [/itex] where I have used the fact that [itex][\xi ,\psi ] = 0[/itex].

[itex]\mathcal{L}_{\psi}(\triangledown ^{c}\xi ^{d}) = (\psi ^{e}\triangledown _{e}\triangledown ^{c}\xi ^{d} - \triangledown ^{c}\xi ^{d}\triangledown _{e}\psi ^{c} - \triangledown ^{c}\xi ^{e}\triangledown _{e}\psi ^{d}) [/itex]. We have that [itex]\psi ^{e}\triangledown _{e}\triangledown ^{c}\xi ^{d} = R^{dc}{}_{ef}\psi ^{e}\psi ^{f} = R^{dc}{}_{[ef]}\psi ^{(e}\psi ^{f)} = 0[/itex] therefore [itex]\mathcal{L}_{\psi}(\triangledown ^{c}\xi ^{d}) = - \triangledown ^{c}\xi ^{d}\triangledown _{e}\psi ^{c} - \triangledown ^{c}\xi ^{e}\triangledown _{e}\psi ^{d} = \triangledown ^{d}\xi ^{e}\triangledown _{e}\psi ^{c} - \triangledown ^{d}\psi ^{e}\triangledown _{e}\xi ^{c}[/itex]. Now, [itex]\mathcal{L}_{\psi}\xi ^{c} = 0 \Rightarrow \xi ^{e}\triangledown _{e}\psi ^{c} = \psi^{e}\triangledown _{e}\xi ^{c}\Rightarrow \triangledown ^{d}( \xi ^{e}\triangledown _{e}\psi ^{c}) = \triangledown ^{d}(\psi^{e}\triangledown _{e}\xi ^{c})[/itex] so we can conclude that [itex]\triangledown ^{d}\xi ^{e}\triangledown _{e}\psi ^{c} - \triangledown ^{d}\psi ^{e}\triangledown _{e}\xi ^{c} = \psi ^{e}\triangledown ^{d}\triangledown _{e}\xi ^{c} - \xi ^{e}\triangledown ^{d}\triangledown _{e}\psi ^{c} = \psi ^{e}\xi ^{f}(R^{d}{}_{ef}{}^{c} + R^{d}{}_{f}{}^{c}{}_{e}) = -\psi ^{e}\xi ^{f}R^{dc}{}_{ef} = 0[/itex], where I have used the Bianchi identity, thus [itex]\mathcal{L}_{\psi}(\triangledown ^{c}\xi ^{d}) = 0[/itex].

Now for the second expression, first I'll prove the following claim: let [itex]\mathbf{T}[/itex] be a rank 2 tensor then [itex]\sum_{k}\epsilon _{a_1...a_{k-1}ja_{k+1}...a_{n}}T^{j}_{a_k} = \epsilon _{a_1...a_n}T^{j}_j[/itex]. To do this, first let [itex]k\in \left \{ 1,...,n \right \}[/itex] be fixed but arbitrary. Note that [itex]\forall j\neq a_k[/itex], [itex]\epsilon _{a_1...a_{k-1}ja_{k+1}...a_{n}}T^{j}_{a_k} = 0[/itex] because we would have a repeated index and by definition of the levi - civita symbol, a component with two or more repeated indices vanishes. Therefore, [itex]\epsilon _{a_1...a_{k-1}ja_{k+1}...a_{n}}T^{j}_{a_k} = \epsilon _{a_1...a_{k-1}a_ka_{k+1}...a_{n}}T^{a_k}_{a_k} =\epsilon _{a_1...a_{n}}T^{a_k}_{a_k} [/itex]. Because [itex]k[/itex] was arbitrary, [itex]\sum_{k}\epsilon _{a_1...a_{k-1}ja_{k+1}...a_{n}}T^{j}_{a_k} = \epsilon _{a_1...a_n}\sum_{k}T^{a_k}_{a_k} =\epsilon _{a_1...a_n}T^{j}_j [/itex]. Using this, [itex]\mathcal{L}_{\psi}\epsilon _{abcd} = \psi ^{e}\triangledown _{e}\epsilon _{abcd} + \sum_{k}\epsilon _{a..e..d}\triangledown _{a_k}\psi ^{e} = \epsilon _{abcd}\triangledown _{e}\psi ^{e} = 0[/itex] where the facts that the covariant derivative of the levi - civita symbol vanishes and that killing fields are divergence free have been used. This finally gives us [itex]\mathcal{L}_{\psi}\omega _{a} = 0[/itex].

I really, really appreciate any and all help on finding mistakes in the solution and/or flaws in any arguments. Sorry for the long winded post. Cheers!
 
Physics news on Phys.org

1. What is the "vanishing twist of time" phenomenon?

The "vanishing twist of time" refers to a theoretical concept in which the flow of time suddenly stops or slows down, similar to a black hole's event horizon. This could potentially occur in certain extreme conditions, such as near the singularity of a black hole or during the collapse of a star.

2. How does the "vanishing twist of time" relate to the "killing field" theory?

The "killing field" theory, also known as the "event horizon censorship" theory, suggests that the laws of physics prevent observers from seeing beyond a black hole's event horizon. The "vanishing twist of time" is a similar concept, as both involve a sudden and dramatic change in the flow of time near a black hole's event horizon.

3. What are the potential consequences of the "vanishing twist of time"?

If the "vanishing twist of time" were to occur, it could have significant consequences for our understanding of the universe and the laws of physics. It could also lead to the breakdown of causality and the possibility of time travel, as well as the destruction of matter and energy.

4. Is there any evidence for the existence of the "vanishing twist of time"?

Currently, there is no direct evidence for the "vanishing twist of time" phenomenon. However, some theories, such as the "firewall" theory, suggest that it could be a possible outcome of the black hole information paradox, which is still a topic of debate among scientists.

5. How are scientists studying the "vanishing twist of time"?

Scientists are using a combination of theoretical models and observations of black holes to study the "vanishing twist of time". They are also conducting experiments and simulations to better understand the nature of black holes and their event horizons, which could provide more insights into this phenomenon.

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