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Weyl transformation
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<h2 id="conformal-weight">Conformal weight</h2> <p>A quantity φ {\displaystyle \varphi } has <a href="/facts/Conformal_weight/McVlk4hR">conformal weight</a> k {\displaystyle k} if, under the Weyl transformation, it transforms via </p> φ → φ e k ω . {\displaystyle \varphi \to \varphi e^{k\omega }.} <p>Thus conformally weighted quantities belong to certain <a href="/facts/Density_bundle/WMAT62QK">density bundles</a>; see also <a href="/facts/Conformal_dimension/m3S57nHU">conformal dimension</a>. Let A μ {\displaystyle A_{\mu }} be the <a href="/facts/Connection_one-form/Hnn6FJFN">connection one-form</a> associated to the Levi-Civita connection of g {\displaystyle g} . Introduce a connection that depends also on an initial one-form ∂ μ ω {\displaystyle \partial _{\mu }\omega } via </p> B μ = A μ + ∂ μ ω . {\displaystyle B_{\mu }=A_{\mu }+\partial _{\mu }\omega .} <p>Then D μ φ ≡ ∂ μ φ + k B μ φ {\displaystyle D_{\mu }\varphi \equiv \partial _{\mu }\varphi +kB_{\mu }\varphi } is covariant and has conformal weight k − 1 {\displaystyle k-1} . </p> <h2 id="formulas">Formulas</h2> <p>For the transformation </p> g a b = f ( ϕ ( x ) ) g ¯ a b {\displaystyle g_{ab}=f(\phi (x)){\bar {g}}_{ab}} <p>We can derive the following formulas </p> g a b = 1 f ( ϕ ( x ) ) g ¯ a b − g = − g ¯ f D / 2 Γ a b c = Γ ¯ a b c + f ′ 2 f ( δ b c ∂ a ϕ + δ a c ∂ b ϕ − g ¯ a b ∂ c ϕ ) ≡ Γ ¯ a b c + γ a b c R a b = R ¯ a b + f ″ f − f ′ 2 2 f 2 ( ( 2 − D ) ∂ a ϕ ∂ b ϕ − g ¯ a b ∂ c ϕ ∂ c ϕ ) + f ′ 2 f ( ( 2 − D ) ∇ ¯ a ∂ b ϕ − g ¯ a b ◻ ¯ ϕ ) + 1 4 f ′ 2 f 2 ( D − 2 ) ( ∂ a ϕ ∂ b ϕ − g ¯ a b ∂ c ϕ ∂ c ϕ ) R = 1 f R ¯ + 1 − D f ( f ″ f − f ′ 2 f 2 ∂ c ϕ ∂ c ϕ + f ′ f ◻ ¯ ϕ ) + 1 4 f f ′ 2 f 2 ( D − 2 ) ( 1 − D ) ∂ c ϕ ∂ c ϕ {\displaystyle {\begin{aligned}g^{ab}&={\frac {1}{f(\phi (x))}}{\bar {g}}^{ab}\\{\sqrt {-g}}&={\sqrt {-{\bar {g}}}}f^{D/2}\\\Gamma _{ab}^{c}&={\bar {\Gamma }}_{ab}^{c}+{\frac {f'}{2f}}\left(\delta _{b}^{c}\partial _{a}\phi +\delta _{a}^{c}\partial _{b}\phi -{\bar {g}}_{ab}\partial ^{c}\phi \right)\equiv {\bar {\Gamma }}_{ab}^{c}+\gamma _{ab}^{c}\\R_{ab}&={\bar {R}}_{ab}+{\frac {f''f-f^{\prime 2}}{2f^{2}}}\left((2-D)\partial _{a}\phi \partial _{b}\phi -{\bar {g}}_{ab}\partial ^{c}\phi \partial _{c}\phi \right)+{\frac {f'}{2f}}\left((2-D){\bar {\nabla }}_{a}\partial _{b}\phi -{\bar {g}}_{ab}{\bar {\Box }}\phi \right)+{\frac {1}{4}}{\frac {f^{\prime 2}}{f^{2}}}(D-2)\left(\partial _{a}\phi \partial _{b}\phi -{\bar {g}}_{ab}\partial _{c}\phi \partial ^{c}\phi \right)\\R&={\frac {1}{f}}{\bar {R}}+{\frac {1-D}{f}}\left({\frac {f''f-f^{\prime 2}}{f^{2}}}\partial ^{c}\phi \partial _{c}\phi +{\frac {f'}{f}}{\bar {\Box }}\phi \right)+{\frac {1}{4f}}{\frac {f^{\prime 2}}{f^{2}}}(D-2)(1-D)\partial _{c}\phi \partial ^{c}\phi \end{aligned}}} <p>Note that the Weyl tensor is invariant under a Weyl rescaling. </p> <ul><li>Weyl, Hermann (1993) [1921]. <i>Raum, Zeit, Materie</i> [<i>Space, Time, Matter</i>]. Lectures on General Relativity (in German). Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-56978-2.</li></ul>