----------------------------------------------------------------------------- -- | -- Module : Math.Tensor.Examples.Gravity.Schwarzschild -- Copyright : (c) 2019 Tobias Reinhart and Nils Alex -- License : MIT -- Maintainer : tobi.reinhart@fau.de, nils.alex@fau.de -- -- -- This module provides the metric, inverse metric, Christoffel symbol, Ricci tensor and Einstein tensor for the Schwarzschild spacetime as an -- example for tensor sections and partial derivatives thereof. -- ----------------------------------------------------------------------------- {-# LANGUAGE DataKinds #-} {-# LANGUAGE ScopedTypeVariables #-} module Math.Tensor.Examples.Gravity.Schwarzschild ( schwarzschild, schwarzschild', christoffel, ricci, einstein ) where import Math.Tensor import Numeric.AD.Internal.Forward (Forward(..)) -- | Schwarzschild metric $$g = (1-\frac{r_\text{s}}{r})\,\mathrm dt\otimes\mathrm dt - \frac{1}{1-\frac{r_\text{s}}{r}}\,\mathrm dr\otimes \mathrm dr - r^2\,\mathrm d\theta\otimes \mathrm d\theta - r^2\sin^2\theta\,\mathrm d\phi\otimes \mathrm d\phi$$. schwarzschild :: Floating a => a -> STTens 0 2 (CFun [a] a) schwarzschild rs = fromListT2 [ ((Empty, Ind3 0 Append singletonInd (Ind3 0)), CFun $$$_:r:_) -> r' r ), ((Empty, Ind3 1 Append singletonInd (Ind3 1)), CFun \(_:r:_) -> -1/r' r), ((Empty, Ind3 2 Append singletonInd (Ind3 2)), CFun \(_:r:_) -> -(r**2)), ((Empty, Ind3 3 Append singletonInd (Ind3 3)), CFun \(_:r:theta:_) -> -(r*sin theta)**2) ] where r' r = 1 - rs / r -- | Inverse Schwarzschild metric \( g = \frac{1}{1-\frac{r_\text{s}}{r}}\,\partial_t \otimes \partial_t - (1-\frac{r_\text{s}}{r})\,\partial_r \otimes \partial_r - \frac{1}{r^2}\,\partial_\theta \otimes \partial_\theta - \frac{1}{r^2\sin^2\theta}\,\partial_\phi \otimes \partial_\phi$$. schwarzschild' :: Floating a => a -> STTens 2 0 (CFun [a] a) schwarzschild' rs = fromListT2 [ ((Ind3 0 Append singletonInd (Ind3 0), Empty), CFun$ \(_:r:_) -> 1/r' r), ((Ind3 1 Append singletonInd (Ind3 1), Empty), CFun $\(_:r:_) -> - r' r), ((Ind3 2 Append singletonInd (Ind3 2), Empty), CFun$ \(_:r:_) -> -1/(r**2)), ((Ind3 3 Append singletonInd (Ind3 3), Empty), CFun $\(_:r:theta:_) -> -1/(r*sin theta)**2) ] where r' r = 1 - rs / r half :: Fractional a => SField a half = SField$ 1/2 -- | Christoffel symbol of the Schwarzschild metric. christoffel :: forall a.Floating a => a -> STTens 1 2 (CFun [a] a) christoffel rs = gamma where g = schwarzschild (Lift rs) g' = schwarzschild' rs :: STTens 2 0 (CFun [a] a) del_g = partial g :: STTens 0 3 (CFun [a] a) g'_del_g = g' &* del_g t1 = contrATens1 (0, 0) g'_del_g t2 = contrATens1 (0, 1) g'_del_g t3 = tensorTrans2 (0, 1) t2 s = t2 &+ (t3 &- t1) h = half :: SField a gamma = h &. s -- | Ricci tensor of the Schwarzschild metric. ricci :: forall a.Floating a => a -> STTens 0 2 (CFun [a] a) ricci rs = (term1 &- term2) &+ (term3 &- term4) where gamma1 = christoffel (Lift rs) gamma2 = christoffel rs del_gamma = partial gamma1 :: STTens 1 3 (CFun [a] a) gamma_gamma = contrATens1 (1,1) $gamma2 &* gamma2 :: STTens 1 3 (CFun [a] a) term1 = contrATens1 (0,0) del_gamma term2 = contrATens1 (0,1) del_gamma term3 = contrATens1 (0,0) gamma_gamma term4 = contrATens1 (0,1) gamma_gamma -- | Einstein tensor of the Schwarzschild metric. -- The component functions evaluate to zero: -- -- >>> let g = einstein 2 -- >>> g evalSec [1.1, 2.4, 1.7, 2.2] -- ZeroTensor einstein :: forall a.Floating a => a -> STTens 0 2 (CFun [a] a) einstein rs = r_ab &- (h &. r &* g) where r_ab = ricci rs :: STTens 0 2 (CFun [a] a) g = schwarzschild rs :: STTens 0 2 (CFun [a] a) g' = schwarzschild' rs :: STTens 2 0 (CFun [a] a) r = contrATens1 (0,0)$ contrATens1 (1,1) \$ g' &* r_ab h = half :: SField a