Copyright | (c) 2019 Tobias Reinhart and Nils Alex |
---|---|
License | MIT |
Maintainer | tobi.reinhart@fau.de, nils.alex@fau.de |
Safe Haskell | None |
Language | Haskell2010 |
Math.Tensor.Examples.Gravity.SchwarzschildSymbolic
Description
This module provides the Schwarzschild metric as an example for a tensor with symbolic values as well as functions to calculate Christoffel symbols, Ricci tensors and Einstein tensors from metric tensors with symbolic values.
Synopsis
- schwarzschildS :: STTens 0 2 SSymbolic
- schwarzschildS' :: STTens 2 0 SSymbolic
- christoffelS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 1 2 SSymbolic
- ricciS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 0 2 SSymbolic
- einsteinS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 0 2 SSymbolic
Documentation
schwarzschildS :: STTens 0 2 SSymbolic Source #
Schwarzschild metric g=(1−rsr)dt⊗dt−11−rsrdr⊗dr−r2dθ⊗dθ−r2sin2θdϕ⊗dϕ.
schwarzschildS' :: STTens 2 0 SSymbolic Source #
Inverse Schwarzschild metric g=11−rsr∂t⊗∂t−(1−rsr)∂r⊗∂r−1r2∂θ⊗∂θ−1r2sin2θ∂ϕ⊗∂ϕ.
christoffelS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 1 2 SSymbolic Source #
Christoffel symbols of any symbolic metric.
ricciS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 0 2 SSymbolic Source #
Ricci tensor of any symbolic metric.
einsteinS :: STTens 0 2 SSymbolic -> STTens 2 0 SSymbolic -> STTens 0 2 SSymbolic Source #
Einstein tensor of any symbolic metric. The components evaluate to zero:
>>>
let g = schwarzschildS
>>>
let g' = schwarzschildS'
>>>
let e = einsteinS g g'
>>>
print e
ZeroTensor -- modulo symbolic simplification, which is not implemented yet.