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.DiffeoSymEqns

Contents

Lorentz Invariance
- Area Metric
Perturbative Diffeomorphism Equivariance
- Area Metric
- Metric

Description

This module collects several tensorial equations. These equations arise in the research topic of perturbative constructive gravity and are used there to encode perturbative diffeomorphism invariance.

Providing further usage examples of the sparse-tensor package the equations included here nicely illustrate the syntax that is used when entering tensors. They also show how the sparse-tensor package can be used to manipulate not only tenors but linear tensorial equations. The sparse-tensor package can for instance be used to extract the information that is contained in such a tensorial equation in matrix form. This then obviously allows one to computate the rank of the linear tensorial equation or even explicitly solve it.

All equations that are contained in this module are functions that take possibly several tensors of type ATens AnsVarR as input. These tensors then represent the individual unknown tensors in the equation. The output that is computed by the functions is also of this type.

When illustrating how the individual such equations that are included in this module are defined we will use again the same convention as in the Math.Tensor.Examples.Gravity module, i.e. we label indices of type Ind20 by $A,B,C,D,...$ , indices of type $I,J,K,L,...$ and spacetime indices of type ind3 are labeled by $a,b,c,d,...$ . Hence a general such tensor is displayed as $T^{A_1...A_m I_1...I_r a_1...a_p}_{B_1...B_n J_1...J_s b_1...b_s}$ . Such a tensor then has the type ATens m n r s p q.

Synopsis

ansatzA :: ATens 1 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 2 0 AnsVarR
ansatzAI :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 1 0 1 0 2 0 AnsVarR
ansatzAB :: ATens 2 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 2 0 AnsVarR
ansatzAaBb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 4 0 AnsVarR
ansatzABI :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 2 0 AnsVarR
ansatzAIBJ :: ATens 2 0 2 0 0 0 AnsVarR -> ATens 2 0 2 0 2 0 AnsVarR
ansatzABC :: ATens 3 0 0 0 0 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR
ansatzABCI :: ATens 3 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 2 0 AnsVarR
ansatzABbCc :: ATens 3 0 0 0 2 0 AnsVarR -> ATens 3 0 0 0 4 0 AnsVarR
ansatzAaBbCI :: ATens 3 0 1 0 2 0 AnsVarR -> ATens 3 0 1 0 4 0 AnsVarR
ansatzABICJ :: ATens 3 0 2 0 0 0 AnsVarR -> ATens 3 0 2 0 2 0 AnsVarR
ansatzAIBJCK :: ATens 3 0 3 0 0 0 AnsVarR -> ATens 3 0 3 0 2 0 AnsVarR
ansatzABCD :: ATens 4 0 0 0 0 0 AnsVarR -> ATens 4 0 0 0 2 0 AnsVarR
ansatzABCDJ :: ATens 4 0 1 0 0 0 AnsVarR -> ATens 4 0 1 0 2 0 AnsVarR
ansatzABCcDd :: ATens 4 0 0 0 2 0 AnsVarR -> ATens 4 0 0 0 4 0 AnsVarR
eqn1 :: ATens 0 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 0 0 AnsVarR -> ATens 0 0 0 0 1 1 AnsVarR
eqn3 :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 0 0 0 0 3 1 AnsVarR
eqn1A :: ATens 1 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 1 1 AnsVarR
eqn1AI :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 1 0 1 0 1 1 AnsVarR
eqn2Aa :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 0 0 2 0 AnsVarR -> ATens 1 0 0 0 3 1 AnsVarR
eqn3A :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 1 0 0 0 3 1 AnsVarR
eqn1AB :: ATens 2 0 0 0 0 0 AnsVarR -> ATens 3 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 1 1 AnsVarR
eqn1ABI :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 1 1 AnsVarR
eqn1AaBb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR
eqn2ABb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR
eqn3AB :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 0 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR
eqn1Met :: ATens 0 0 0 0 0 0 AnsVarR -> ATens 0 0 1 0 0 0 AnsVarR -> ATens 0 0 0 0 1 1 AnsVarR
eqn3Met :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 0 0 3 1 AnsVarR
eqn2AaMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 1 0 3 1 AnsVarR
eqn3AMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 1 0 3 1 AnsVarR
eqn1AMet :: ATens 0 0 1 0 0 0 AnsVarR -> ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 1 0 1 1 AnsVarR
eqn1AIMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 2 0 1 1 AnsVarR
eqn1ABMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 2 0 1 1 AnsVarR
eqn1ABIMet :: ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 4 0 0 0 AnsVarR -> ATens 0 0 3 0 1 1 AnsVarR
eqn1AaBbMet :: ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 3 0 2 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR
eqn2ABbMet :: ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 3 0 2 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR
eqn3ABMet :: ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 4 0 0 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR

Lorentz Invariance

Area Metric

The following equations encode the requirement that the input tensors are Lorentz invariant. They can be used to verify the Lorentz invariant tensors that are obtained by the Math.Tensor.LorentzGenerator module. If the input tensor is Lorentz invariant the functions return a tensor with all values being zero, and thus for instance applying tensorRank6' to these returns zero.

ansatzA :: ATens 1 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 2 0 AnsVarR Source #

ansatzAI :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 1 0 1 0 2 0 AnsVarR Source #

ansatzAB :: ATens 2 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 2 0 AnsVarR Source #

ansatzAaBb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 4 0 AnsVarR Source #

ansatzABI :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 2 0 AnsVarR Source #

ansatzAIBJ :: ATens 2 0 2 0 0 0 AnsVarR -> ATens 2 0 2 0 2 0 AnsVarR Source #

ansatzABC :: ATens 3 0 0 0 0 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR Source #

ansatzABCI :: ATens 3 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 2 0 AnsVarR Source #

ansatzABbCc :: ATens 3 0 0 0 2 0 AnsVarR -> ATens 3 0 0 0 4 0 AnsVarR Source #

ansatzAaBbCI :: ATens 3 0 1 0 2 0 AnsVarR -> ATens 3 0 1 0 4 0 AnsVarR Source #

ansatzABICJ :: ATens 3 0 2 0 0 0 AnsVarR -> ATens 3 0 2 0 2 0 AnsVarR Source #

ansatzAIBJCK :: ATens 3 0 3 0 0 0 AnsVarR -> ATens 3 0 3 0 2 0 AnsVarR Source #

ansatzABCD :: ATens 4 0 0 0 0 0 AnsVarR -> ATens 4 0 0 0 2 0 AnsVarR Source #

ansatzABCDJ :: ATens 4 0 1 0 0 0 AnsVarR -> ATens 4 0 1 0 2 0 AnsVarR Source #

ansatzABCcDd :: ATens 4 0 0 0 2 0 AnsVarR -> ATens 4 0 0 0 4 0 AnsVarR Source #

Perturbative Diffeomorphism Equivariance

Area Metric

The following equations can be used with the Lorentz invariant ansätze for the area metric $G_A = J_A^{abcd}G_{abcd}$ with the symmetries $G_{abcd} = G_{cdab} = -G_{bacd}$ as input tensors. In the following documentation these input tensors are labeled as $a_0,a^{A}, a^{AI}, a^{AB},$ etc. For the definition of the further included tensors see Math.Tensor.Examples.Gravity.

eqn1 :: ATens 0 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 0 0 AnsVarR -> ATens 0 0 0 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^A C_{An}^{Bm}N_B + a_0 \delta^m_n$ .

eqn3 :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 0 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = a^{AI}C_{An}^{B(m\vert }N_B J^{\vert pq)}_I$ .

eqn1A :: ATens 1 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 0 0 AnsVarR -> ATens 1 0 0 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^A C_{An}^{Bm} + 2 a^{AB}C_{An}^{Cm}N_C + a^B\delta^m_n$ .

eqn1AI :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 1 0 1 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^{AI}\left [C_{An}^{Bm}\delta^J _I- 2 \delta^B_A J_I^{pm}I^J_{pn} \right ] + a^{ABJ}C_{An}^{Cm}N_C + a^{BJ} \delta^m_n$ .

eqn2Aa :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 0 0 2 0 AnsVarR -> ATens 1 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2a^{A(p\vert Bq}C_{An}^{C\vert m)}N_C + a^{AI} \left [C_{An}^{B(m\vert} 2 J_{I}^{\vert p)q} - \delta_A^BJ_I^{pm}\delta^q_n \right ]$ .

eqn3A :: ATens 1 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 1 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = a^{BAI}C_{An}^{C(m\vert}N_CJ_I^{\vert pq)} + a^{AI}C_{An}^{B(m \vert} J_I^{\vert pq)}$ .

eqn1AB :: ATens 2 0 0 0 0 0 AnsVarR -> ATens 3 0 0 0 0 0 AnsVarR -> ATens 2 0 0 0 1 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{AC}C_{An}^{Bm} + 2a^{AB}C_{An}^{Cm} + 6 a^{ABC}C_{An}^{Dm} N_D + 2a^{BC} \delta^m_n$

eqn1ABI :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 0 0 AnsVarR -> ATens 2 0 1 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^{CAI} \left [C_{An}^{Bm}\delta^J _I- 2 \delta^B_A J_I^{pm}I^J_{pn} \right ] + 2 a^{ACBJ} C_{An}^{Dm} N_D + a^{CBJ} \delta ^m _n$ .

eqn1AaBb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{ApCr} \left [ C_{An}^{Bm} \delta ^q_p - \delta^B_A \delta^m_p \delta^q_n \right ] +2 a^{A Bq Cr} C_{An}^{Dm} N_D + 2 a^{BqCr} \delta^m_n$ .

eqn2ABb :: ATens 2 0 0 0 2 0 AnsVarR -> ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 0 0 2 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{C A(p \vert B q} C_{An}^{D \vert m )} N_D + a^{CAI} \left [C_{An}^{B(m\vert} 2 J_{I}^{\vert p)q} - \delta_A^BJ_I^{pm}\delta^q_n \right ]$ .

eqn3AB :: ATens 2 0 1 0 0 0 AnsVarR -> ATens 3 0 1 0 0 0 AnsVarR -> ATens 2 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{BCAI}C_{An}^{D(m \vert}N_DJ_I^{\vert pq)} + a^{CAI}C_{An}^{B(m \vert} J_I^{\vert pq)}$ .

Metric

The following equations can be used with the Lorentz invariant ansätze for the a traditional metric $g_I = J_I^{ab}g_{ab}$ as input. In the following documentation these input tensors are labeled as $a_0,a^{A}, a^{AJ}, a^{AB},$ etc. Care must be taken as for the case of the metric equations all indices that are labeled by $A,B,C,D,...$ are also of type Ind9, but are distinguished from the indices labeled by $I,J,K,L,...$ as they describe the metric components and the latter ones describe symmetric spacetime derivative pairs. For the definition of the further included tensors see Math.Tensor.Examples.Gravity.

eqn1Met :: ATens 0 0 0 0 0 0 AnsVarR -> ATens 0 0 1 0 0 0 AnsVarR -> ATens 0 0 0 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^A K_{An}^{Bm}\eta_B + a_0 \delta^m_n$ .

eqn3Met :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 0 0 3 1 AnsVarR Source #

The equation is given by: $0 = a^{AI}K_{An}^{B(m\vert }\eta_B J^{\vert pq)}_I$ .

eqn2AaMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 1 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2a^{A(p\vert Bq}K_{An}^{C\vert m)}\eta_C + a^{AI} \left [K_{An}^{B(m\vert} 2 J_{I}^{\vert p)q} - \delta_A^BJ_I^{pm}\delta^q_n \right ]$ .

eqn3AMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 1 0 3 1 AnsVarR Source #

The equation is given by: $0 = a^{BAI}K_{An}^{C(m\vert}\eta_CJ_I^{\vert pq)} + a^{AI}K_{An}^{B(m \vert} J_I^{\vert pq)}$ .

eqn1AMet :: ATens 0 0 1 0 0 0 AnsVarR -> ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 1 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^A K_{An}^{Bm} + 2 a^{AB}K_{An}^{Cm}\eta_C + a^B\delta^m_n$ .

eqn1AIMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 2 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^{AI}\left [K_{An}^{Bm}\delta^J _I- 2 \delta^B_A J_I^{pm}I^J_{pn} \right ] + a^{ABJ}K_{An}^{Cm}\eta_C + a^{BJ} \delta^m_n$ .

eqn1ABMet :: ATens 0 0 2 0 0 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 2 0 1 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{AC}K_{An}^{Bm} + 2a^{AB}K_{An}^{Cm} + 6 a^{ABC}K_{An}^{Dm} \eta_D + 2a^{BC} \delta^m_n$

eqn1ABIMet :: ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 4 0 0 0 AnsVarR -> ATens 0 0 3 0 1 1 AnsVarR Source #

The equation is given by: $0 = a^{CAI} \left [K_{An}^{Bm}\delta^J _I- 2 \delta^B_A J_I^{pm}I^J_{pn} \right ] + 2 a^{ACBJ} K_{An}^{Dm} \eta_D + a^{CBJ} \delta ^m _n$ .

eqn1AaBbMet :: ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 3 0 2 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{ApCr} \left [ K_{An}^{Bm} \delta ^q_p - \delta^B_A \delta^m_n \right ] +2 a^{A Bq Cr} K_{An}^{Dm} \eta_D + 2 a^{BqCr} \delta^m_n$ .

eqn2ABbMet :: ATens 0 0 2 0 2 0 AnsVarR -> ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 3 0 2 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{C A(p \vert B q} K_{An}^{D \vert m )} \eta_D + a^{CAI} \left [K_{An}^{B(m\vert} 2 J_{I}^{\vert p)q} - \delta_A^BJ_I^{pm}\delta^q_n \right ]$ .

eqn3ABMet :: ATens 0 0 3 0 0 0 AnsVarR -> ATens 0 0 4 0 0 0 AnsVarR -> ATens 0 0 2 0 3 1 AnsVarR Source #

The equation is given by: $0 = 2 a^{BCAI}K_{An}^{D(m \vert}\eta_DJ_I^{\vert pq)} + a^{CAI}K_{An}^{B(m \vert} J_I^{\vert pq)}$ .

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