Conformally related metrics have the same null geodesics

In summary, a conformal transformation does not change the null cones of a metric. To show that two metrics, g and 𝛽𝑔, have the same null geodesics, it is necessary to show that n^a 𝛽∇_a n^b = 0 implies n^a ∇_a n^b = 𝛼 n^b, where ∇_a and 𝛽∇_a are the covariant derivative operators adapted to g and 𝛽𝑔, respectively. After some calculations, it can be shown that the last term in the parenthesis of the expression for n^a 𝛽∇_a n^b becomes zero when contracted
  • #1
etotheipi
Homework Statement:: i) If ##\bar{g} = \Omega^2 g## for some positive function ##\Omega##, show that ##\bar{g}## and ##g## have the same null geodesics.

ii) Let ##\psi## solve ##g^{ab} \nabla_a \nabla_b \psi + \xi R \psi = 0##. Determine ##\xi## such that ##\bar{\psi} = \Omega^p \psi## for some ##p## solves the equation in a spacetime with metric ##\bar{g} = \Omega^2 g## if ##\psi## solves the equation in a spacetime with metric ##g##.
Relevant Equations:: N/A

A conformal transformation doesn't change the null cones of the metric, so if ##n^a## is a null vector of ##g## then it is also a null vector of ##\bar{g}##. So it's necessary to show that ##n^a \bar{\nabla}_a n^b = 0 \implies n^a \nabla_a n^b = \alpha n^b## where ##\nabla_a## and ##\bar{\nabla}_a## are the covariant derivative operators adapted to ##g## and ##\bar{g}## respectively. We have\begin{align*}

\bar{\Gamma}^i_{kl} &= \frac{1}{2} \bar{g}^{im} (\partial_l \bar{g}_{mk} + \partial_k \bar{g}_{ml} - \partial_m \bar{g}_{kl}) \\

&= \frac{1}{2} \Omega^{-2} g^{im}(\Omega^2 \left[ \partial_l g_{mk} + \partial_k g_{ml} - \partial_m g_{kl}\right] + 2\Omega [g_{mk} \partial_l \Omega + g_{ml} \partial_k \Omega - g_{kl} \partial_m \Omega]) \\

&= \Gamma^{i}_{kl} + \Omega^{-1} (\delta^i_k \partial_l \Omega + \delta^i_l \partial_k \Omega - g^{im} g_{kl} \partial_m \Omega)

\end{align*}It follows that\begin{align*}

n^a \bar{\nabla}_a n^b &= n^a (\partial_a n^b + \bar{\Gamma}^b_{ac} n^c) \\

&= n^a\left( \partial_a n^b + \Gamma^b_{ac} n^c \right) + n^a n^c \Omega^{-1} \left( \delta^b_a \partial_c \Omega + \delta^b_c \partial_a \Omega - g^{bm}g_{ac} \partial_m \Omega \right) \\

&= n^a \nabla_a n^b + n^a n^c \Omega^{-1} \left( 2 \delta^b_a \partial_c \Omega - g^{bm}g_{ac} \partial_m \Omega \right)

\end{align*}I can't see how to tidy up the right hand side; a hint would be appreciated. Thanks!
 
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  • #2
The last term in the parenthesis of the last expression becomes zero when contracted with ##n^a n^c## due to ##n## being a null vector. The other term in the parenthesis is proportional to ##n^b## (also when contracted with ##n^a n^c##).

Edit: You'll obtain
$$
n^a \bar\nabla_a n^b = n^a \nabla_a n^b + n^b n^a \partial_a \ln(\Omega^2).
$$
 
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Likes strangerep and etotheipi
  • #3
Thanks! I see, ##n^b n^a \partial_a \mathrm{ln}(\Omega^2) = 2 \Omega^{-1} n^b n^a \partial_a \Omega =2\Omega^{-1} n^c n^a \delta^b_c \partial_a \Omega## which under the replacement ##a \leftrightarrow c## gives the first term in my OP. And as you said the second contains ##n^a n^c g_{ac} = n^a n_a = 0## since ##n## is null, so vanishes. And that's it, because we can identify the scalar ##\alpha = n^a \partial_a \mathrm{ln}(\Omega^2)##.

I'll have a go at part ii) probably tomorrow. :smile:
 

Related to Conformally related metrics have the same null geodesics

1. What are conformally related metrics?

Conformally related metrics refer to two metrics that can be transformed into each other through a conformal transformation. This means that the two metrics have the same geometric structure, but differ in their overall scale.

2. How are conformally related metrics related to null geodesics?

Conformally related metrics have the same null geodesics, meaning that the paths of light rays in these metrics are identical. This is because null geodesics are determined by the metric's conformal structure, which is preserved under conformal transformations.

3. Why is the concept of conformally related metrics important?

Conformally related metrics are important because they allow us to study different metrics that share the same geometric properties. This can be useful in simplifying calculations and understanding the behavior of certain physical systems.

4. Can all metrics be conformally related to each other?

No, not all metrics can be conformally related to each other. For two metrics to be conformally related, they must have the same underlying conformal structure. This is not always the case for all metrics.

5. What are some examples of conformally related metrics?

Some examples of conformally related metrics include the Minkowski metric and the conformally flat metric in general relativity, as well as the Euclidean metric and the conformally flat metric in two-dimensional space. Additionally, any metric that is a multiple of another metric is also conformally related to it.

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