PPWaveMetric · section of the QIQT-H book

QIQTH.PPWaveMetric

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PPWaveMetric · entries 534–545 of 1000

Definition 534 (ppMetric).  source ↗

The pp-wave metric ds² = 2 du dv + H du² + dx² + dy², coords (u,v,x,y) = (0,1,2,3): g₀₀ = H, g₀₁ = g₁₀ = 1, g₂₂ = g₃₃ = 1, rest 0.

gpp H x a b  :=  if a=0∧b=0 then H x else if a=0∧b=1∨a=1∧b=0 then 1 else if a=2∧b=2∨a=3∧b=3 then 1 else 0g^{\mathrm{pp}}\,H\,x\,a\,b \;:=\; \text{if }a = 0 \wedge b = 0\text{ then }H\,x\text{ else }\text{if }a = 0 \wedge b = 1 \vee a = 1 \wedge b = 0\text{ then }1\text{ else }\text{if }a = 2 \wedge b = 2 \vee a = 3 \wedge b = 3\text{ then }1\text{ else }0

Used by ppMetric_symm, ppMetric_inv, ppMetric_contDiff, ppFrame_cong, qiqt_gr_ppwave, qiqt_gr_ppwave_showcase.

Definition 535 (ppMetricInv).  source ↗

The inverse pp-wave metric: gi₀₁ = gi₁₀ = 1, gi₁₁ = −H, gi₂₂ = gi₃₃ = 1, rest 0.

(gpp)−1 H x a b  :=  if a=0∧b=1∨a=1∧b=0 then 1 else if a=1∧b=1 then −H x else if a=2∧b=2∨a=3∧b=3 then 1 else 0(g^{\mathrm{pp}})^{-1}\,H\,x\,a\,b \;:=\; \text{if }a = 0 \wedge b = 1 \vee a = 1 \wedge b = 0\text{ then }1\text{ else }\text{if }a = 1 \wedge b = 1\text{ then }-H\,x\text{ else }\text{if }a = 2 \wedge b = 2 \vee a = 3 \wedge b = 3\text{ then }1\text{ else }0

Used by ppMetric_inv, ppMetricInv_symm, ppMetricInv_contDiff, qiqt_gr_ppwave, qiqt_gr_ppwave_showcase.

Lemma 536 (ppMetric_symm).  source ↗

gpp H x a b=gpp H x b a\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x\,a\,b = \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x\,b\,a

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 537 (ppMetric_inv).  source ↗

∑σgpp H x a σ⋅(gpp)−1 H x σ b=δab\sum_{\sigma} \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x\,a\,\sigma \cdot \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetricinv}{(g^{\mathrm{pp}})^{-1}}\,H\,x\,\sigma\,b = \delta_{ab}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 538 (ppMetricInv_symm).  source ↗

(gpp)−1 H x a b=(gpp)−1 H x b a\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetricinv}{(g^{\mathrm{pp}})^{-1}}\,H\,x\,a\,b = \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetricinv}{(g^{\mathrm{pp}})^{-1}}\,H\,x\,b\,a

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 539 (ppMetric_contDiff).  source ↗

The pp-wave metric components are C^∞ when H is (hCg): each is H or a constant.

(H)∈C∞→∀(ab:Fin 4),(λy↦gpp H y a b)∈C∞({H})\in C^{\infty} \to \forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,y\,a\,b})\in C^{\infty}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 540 (ppMetricInv_contDiff).  source ↗

The inverse pp-wave metric components are C^∞ when H is (hCgi): each is −H or a constant.

(H)∈C∞→∀(ab:Fin 4),(λy↦(gpp)−1 H y a b)∈C∞({H})\in C^{\infty} \to \forall (a b : \mathrm{Fin}\,4), ({\lambda y \mapsto \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetricinv}{(g^{\mathrm{pp}})^{-1}}\,H\,y\,a\,b})\in C^{\infty}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Definition 541 (ppFrame).  source ↗

The pp-wave coframe P_{ab} = e^a_b.

ppFrame H x a b  :=  if a=0∧b=0 then (H x−1)/2 else if a=1∧b=0 then (H x+1)/2 else if a=0∧b=1∨a=1∧b=1 then 1 else if a=2∧b=2∨a=3∧b=3 then 1 else 0\mathrm{ppFrame}\,H\,x\,a\,b \;:=\; \text{if }a = 0 \wedge b = 0\text{ then }(H\,x - 1) / 2\text{ else }\text{if }a = 1 \wedge b = 0\text{ then }(H\,x + 1) / 2\text{ else }\text{if }a = 0 \wedge b = 1 \vee a = 1 \wedge b = 1\text{ then }1\text{ else }\text{if }a = 2 \wedge b = 2 \vee a = 3 \wedge b = 3\text{ then }1\text{ else }0

Used by ppFrame_cong, ppFrame_pp, ppFrame_pp', qiqt_gr_ppwave.

Definition 542 (ppFrameInv).  source ↗

The inverse frame Pinv = P⁻¹.

Used by ppFrame_pp, ppFrame_pp', qiqt_gr_ppwave.

Lemma 543 (ppFrame_cong).  source ↗

hcong for the pp-wave: g_{ij} = ∑_{kl} P_{ki} gm_{kl} P_{lj} (the metric is the Minkowski reference pulled back by the tetrad).

gpp H x i j=∑k∑lppFrame H x k i⋅ηkl⋅ppFrame H x l j\href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppmetric}{g^{\mathrm{pp}}}\,H\,x\,i\,j = \sum_{k} \sum_{l} \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframe}{\mathrm{ppFrame}}\,H\,x\,k\,i \cdot \href{/browser/qiqth-einsteinequationofstate#d-qiqth-einsteineos-gm}{\eta_{{k}{l}}} \cdot \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframe}{\mathrm{ppFrame}}\,H\,x\,l\,j

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 544 (ppFrame_pp).  source ↗

hPP for the pp-wave: ∑_k P_{ik} Pinv_{kj} = δ_{ij}.

∑kppFrame H x i k⋅ppFrameInv H x k j=δij\sum_{k} \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframe}{\mathrm{ppFrame}}\,H\,x\,i\,k \cdot \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframeinv}{\mathrm{ppFrameInv}}\,H\,x\,k\,j = \delta_{ij}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

Lemma 545 (ppFrame_pp').  source ↗

hPP' for the pp-wave: ∑_k Pinv_{ik} P_{kj} = δ_{ij}.

∑kppFrameInv H x i k⋅ppFrame H x k j=δij\sum_{k} \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframeinv}{\mathrm{ppFrameInv}}\,H\,x\,i\,k \cdot \href{/browser/qiqth-ppwavemetric#d-qiqth-curvature-ppframe}{\mathrm{ppFrame}}\,H\,x\,k\,j = \delta_{ij}

Proof. Immediate from the definitions. □\square

Used by qiqt_gr_ppwave.

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