lorentz_group – SPHINCS

This module contains the definitions of the TYPES representing objects in the Lorentz group, that is, the isometry group of the Lorentz metric

FT 23.02.2022

Uses

- tensor
- utility
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Used by

- Descendants:
- actions
- constructors
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Graph Key

Nodes of different colours represent the following:

Solid arrows point from a submodule to the (sub)module which it is descended from. Dashed arrows point from a module or program unit to modules which it uses.

Variables

eta

Variables

Type	Visibility	Attributes		Name		Initial
double precision,	public,	parameter, DIMENSION(n_sym4x4)	::	eta	=	[-one, zero, zero, zero, one, zero, zero, one, zero, one]	Minkowski metric

Interfaces

interface

public module function apply_as_congruence_to_symrank2_tensor(this, t) result(transformed_t)

Action of the lorentz_transformation on a $4$ -vector

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_transformation),	intent(in)			::	this	lorentz_transformation object to apply
double precision,	intent(in),		DIMENSION(n_sym4x4)	::	t	$10$ -vector storing the components of the symmetric $4\times 4$ tensor to be boosted

Return Value double precision, DIMENSION(n_sym4x4)

$10$ -vector storing the components of the boosted symmetric $4\times 4$ tensor

interface

public module function apply_as_congruence_to_tensor(this, t) result(transformed_t)

Action of the lorentz_transformation on a $4$ -vector

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_transformation),	intent(in)			::	this	lorentz_transformation object to apply
double precision,	intent(in),		DIMENSION(4,4)	::	t(0:3,0:3)	$4\times 4$ tensor to be boosted

Return Value double precision, DIMENSION(4,4), (0:3,0:3)

Boosted $4\times 4$ tensor

interface

public module function apply_as_similarity_to_symrank2_tensor(this, t) result(transformed_t)

Action of the lorentz_transformation on a $4$ -vector

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_transformation),	intent(in)			::	this	lorentz_transformation object to apply
double precision,	intent(in),		DIMENSION(n_sym4x4)	::	t	$10$ -vector storing the components of the symmetric $4\times 4$ tensor to be boosted

Return Value double precision, DIMENSION(n_sym4x4)

$10$ -vector storing the components of the boosted symmetric $4\times 4$ tensor

interface

public module function apply_as_similarity_to_tensor(this, t) result(transformed_t)

Action of the lorentz_transformation on a $4$ -vector

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_transformation),	intent(in)			::	this	lorentz_transformation object to apply
double precision,	intent(in),		DIMENSION(4,4)	::	t(0:3,0:3)	$4\times 4$ tensor to be boosted

Return Value double precision, DIMENSION(4,4), (0:3,0:3)

Boosted $4\times 4$ tensor

interface

public module function apply_to_vector(this, u) result(transformed_u)

Action of the lorentz_transformation on a $4$ -vector

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_transformation),	intent(in)			::	this	lorentz_transformation object to apply
double precision,	intent(in),		DIMENSION(4)	::	u(0:3)	$4$ -vector to be boosted

Return Value double precision, DIMENSION(4), (0:3)

Boosted $4$ -vector

interface

public pure module subroutine compute_boost_matrices(this, p, lambda_s, matrix)

Compute the matrices for the lorentz_boost

Arguments

Type	Intent	Attributes		Name
class(lorentz_boost),	intent(inout)		::	this
double precision,	intent(in),	DIMENSION(3)	::	p	lambda $\times$ v
double precision,	intent(out),	DIMENSION(n_sym3x3)	::	lambda_s	Spatial part of the Lorentz boost
double precision,	intent(out),	DIMENSION(4,4)	::	matrix(0:3,0:3)	$4\times 4$ matrix representing the Lorentz boost

interface

public module subroutine compute_rotation_matrices(this, euler_angles, r_x, r_y, r_z, r, matrix, inv_r_x, inv_r_y, inv_r_z, inv_r, inv_matrix)

Compute the matrices for the spatial_rotation

Arguments

Type	Intent	Attributes		Name
class(spatial_rotation),	intent(inout)		::	this	spatial_rotation object to compuet the matrices for
double precision,	intent(in),	DIMENSION(3)	::	euler_angles	lambda $\times$ v
double precision,	intent(out),	DIMENSION(3,3)	::	r_x	Rotation operator around the $x$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	r_y	Rotation operator around the $y$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	r_z	Rotation operator around the $z$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	r	Full $3\times 3$ rotation operator
double precision,	intent(out),	DIMENSION(4,4)	::	matrix(0:3,0:3)	$4\times 4$ matrix representing the Lorentz boost
double precision,	intent(out),	DIMENSION(3,3)	::	inv_r_x	Inverse rotation operator around the $x$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	inv_r_y	Inverse rotation operator around the $y$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	inv_r_z	Inverse rotation operator around the $z$ axis
double precision,	intent(out),	DIMENSION(3,3)	::	inv_r	Inverse of the full $3\times 3$ rotation operator
double precision,	intent(out),	DIMENSION(4,4)	::	inv_matrix(0:3,0:3)	Inverse of the $4\times 4$ matrix representing the Lorentz boost

interface

public pure module function get_lambda(this) result(lambda)

Returns the Lorentz factor lambda

Arguments

Type	Intent	Optional	Attributes		Name
class(lorentz_boost),	intent(in)			::	this	lorentz_boost object owning this FUNCTION

Return Value double precision

Lorentz factor lambda

public interface lorentz_boost

public module function construct_boost(v) result(boost)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(3)	::	v	Spatial velocity that determines the boost

Return Value type(lorentz_boost)

lorentz_boost object to be constructed

public module function construct_boost_components(vx, vy, vz) result(boost)

Arguments

Type	Intent		Name
double precision,	intent(in)	::	vx	$x$ component of the spatial velocity that determines the boost
double precision,	intent(in)	::	vy	$y$ component of the spatial velocity that determines the boost
double precision,	intent(in)	::	vz	$z$ component of the spatial velocity that determines the boost

Return Value type(lorentz_boost)

lorentz_boost object to be constructed

public interface spatial_rotation

public module function construct_rotation(euler_angles) result(rotation)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(3)	::	euler_angles	Euler angles that define the rotation around the $x,y,z$ axes, in this order

Return Value type(spatial_rotation)

spatial_rotation object to be constructed

public module function construct_rotation_angles(alpha, beta, gamma) result(rotation)

Arguments

Type	Intent		Name
double precision,	intent(in)	::	alpha	Euler angle that defines the rotation around the $x$ axis
double precision,	intent(in)	::	beta	Euler angle that defines the rotation around the $y$ axis
double precision,	intent(in)	::	gamma	Euler angle that defines the rotation around the $z$ axis

Return Value type(spatial_rotation)

spatial_rotation object to be constructed

Derived Types

type, public, extends(lorentz_transformation) :: lorentz_boost

TYPE representing a Lorentz boost

Components

Type	Visibility	Attributes		Name
double precision,	private,	DIMENSION(n_sym3x3)	::	inv_lambda_s	Spatial part of the inverse Lorentz boost
double precision,	private		::	lambda	Lorentz factor
double precision,	private,	DIMENSION(n_sym3x3)	::	lambda_s	Spatial part of the Lorentz boost
double precision,	private,	DIMENSION(3)	::	p	Spatial vector equal to $\lambda \,v$
double precision,	private,	DIMENSION(3)	::	v	Spatial velocity that determines the boost
double precision,	private		::	v_speed	Euclidean norm of v (its speed)

Constructor

public module function construct_boost (v)
public module function construct_boost_components (vx, vy, vz)

Type-Bound Procedures

generic, public :: apply_as_congruence => apply_as_congruence_to_tensor, apply_as_congruence_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a congruence
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_symrank2_tensor Interface	Action of the lorentz_transformation as a congruence on a $10$ -vector storing the components of a symmetric, purely covariant, $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_tensor Interface	Action of the lorentz_transformation as a congruence on a generic purely covariant tensor
generic, public :: apply_as_similarity => apply_as_similarity_to_tensor, apply_as_similarity_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a similarity
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_symrank2_tensor Interface	Action of the lorentz_transformation as a similarity on a $10$ -vector storing the components of a symmetric $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_tensor Interface	Action of the lorentz_transformation as a similarity on a generic tensor
procedure , public , NON_OVERRIDABLE :: apply_to_vector Interface	Action of the lorentz_transformation on a $4$ -vector
procedure , public :: compute_boost_matrices Interface	Computes the spatial part of the matrix of the Lorentz boost, and its whole matrix, starting from the vector $p$
procedure , public :: get_lambda Interface	Returns the Lorentz factor lambda

type, public :: lorentz_transformation

TYPE representing a 4D, proper, orthochronous Lorentz transformation

Components

Type	Visibility	Attributes		Name
double precision,	private,	DIMENSION(4,4)	::	inv_matrix(0:3,0:3)	$4\times 4$ matrix representing the inverse Lorentz transformation
double precision,	private,	DIMENSION(4,4)	::	matrix(0:3,0:3)	$4\times 4$ matrix representing the Lorentz transformation
double precision,	private,	DIMENSION(4,4)	::	tr_matrix(0:3,0:3)	Transpose of the $4\times 4$ matrix representing the Lorentz transformation

Type-Bound Procedures

generic, public :: apply_as_congruence => apply_as_congruence_to_tensor, apply_as_congruence_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a congruence
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_symrank2_tensor Interface	Action of the lorentz_transformation as a congruence on a $10$ -vector storing the components of a symmetric, purely covariant, $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_tensor Interface	Action of the lorentz_transformation as a congruence on a generic purely covariant tensor
generic, public :: apply_as_similarity => apply_as_similarity_to_tensor, apply_as_similarity_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a similarity
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_symrank2_tensor Interface	Action of the lorentz_transformation as a similarity on a $10$ -vector storing the components of a symmetric $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_tensor Interface	Action of the lorentz_transformation as a similarity on a generic tensor
procedure , public , NON_OVERRIDABLE :: apply_to_vector Interface	Action of the lorentz_transformation on a $4$ -vector

type, public, extends(lorentz_transformation) :: spatial_rotation

TYPE representing a spatial rotation

Components

Type	Visibility	Attributes		Name
double precision,	private,	DIMENSION(3)	::	euler_angles	Euler angles that define the rotation around the $x,y,z$ axes, in this order
double precision,	private,	DIMENSION(3,3)	::	inv_r	Full inverse rotation operator
double precision,	private,	DIMENSION(3,3)	::	inv_r_x	Inverse rotation operator around the $x$ axis
double precision,	private,	DIMENSION(3,3)	::	inv_r_y	Inverse rotation operator around the $y$ axis
double precision,	private,	DIMENSION(3,3)	::	inv_r_z	Inverse rotation operator around the $z$ axis
double precision,	private,	DIMENSION(3,3)	::	r	Full rotation operator
double precision,	private,	DIMENSION(3,3)	::	r_x	Rotation operator around the $x$ axis
double precision,	private,	DIMENSION(3,3)	::	r_y	Rotation operator around the $y$ axis
double precision,	private,	DIMENSION(3,3)	::	r_z	Rotation operator around the $z$ axis
double precision,	private,	DIMENSION(3,3)	::	tr_r	Transpose of the full inverse rotation operator

Constructor

public module function construct_rotation (euler_angles)
public module function construct_rotation_angles (alpha, beta, gamma)

Type-Bound Procedures

generic, public :: apply_as_congruence => apply_as_congruence_to_tensor, apply_as_congruence_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a congruence
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_symrank2_tensor Interface	Action of the lorentz_transformation as a congruence on a $10$ -vector storing the components of a symmetric, purely covariant, $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_congruence_to_tensor Interface	Action of the lorentz_transformation as a congruence on a generic purely covariant tensor
generic, public :: apply_as_similarity => apply_as_similarity_to_tensor, apply_as_similarity_to_symrank2_tensor	Generic procedure to apply the lorentz_transformation as a similarity
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_symrank2_tensor Interface	Action of the lorentz_transformation as a similarity on a $10$ -vector storing the components of a symmetric $4\times 4$ tensor
procedure , public , NON_OVERRIDABLE :: apply_as_similarity_to_tensor Interface	Action of the lorentz_transformation as a similarity on a generic tensor
procedure , public , NON_OVERRIDABLE :: apply_to_vector Interface	Action of the lorentz_transformation on a $4$ -vector
procedure , public :: compute_rotation_matrices Interface	Computes the spatial part of the matrix of the Lorentz

Functions

public pure function euclidean_inner_product(u, v) result(inner_product)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(3)	::	u
double precision,	intent(in),		DIMENSION(3)	::	v

Return Value double precision

public pure function minkowski_inner_product(u, v) result(inner_product)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(4)	::	u(0:3)
double precision,	intent(in),		DIMENSION(4)	::	v(0:3)

Return Value double precision

public pure function minkowski_sqnorm(u) result(sqnorm)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(4)	::	u(0:3)

Return Value double precision

public pure function row_by_column(u, v) result(res)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(:)	::	u	Row
double precision,	intent(in),		DIMENSION(:)	::	v	Column

Return Value double precision

public function square_matrix_multiplication(a, b) result(c)

Arguments

Type	Intent	Optional	Attributes		Name
double precision,	intent(in),		DIMENSION(:,:)	::	a	First matrix
double precision,	intent(in),		DIMENSION(:,:)	::	b	Second matrix

lorentz_group Module

Contents

Uses

Used by

Descendants:

Contents

Variables

Interfaces

interface

public module function apply_as_congruence_to_symrank2_tensor(this, t) result(transformed_t) Implementation →

Arguments

Return Value double precision, DIMENSION(n_sym4x4)

interface

public module function apply_as_congruence_to_tensor(this, t) result(transformed_t) Implementation →

Arguments

Return Value double precision, DIMENSION(4,4), (0:3,0:3)

interface

public module function apply_as_similarity_to_symrank2_tensor(this, t) result(transformed_t) Implementation →

Arguments

Return Value double precision, DIMENSION(n_sym4x4)

interface

public module function apply_as_similarity_to_tensor(this, t) result(transformed_t) Implementation →

Arguments

Return Value double precision, DIMENSION(4,4), (0:3,0:3)

interface

public module function apply_to_vector(this, u) result(transformed_u) Implementation →

Arguments

Return Value double precision, DIMENSION(4), (0:3)

interface

public pure module subroutine compute_boost_matrices(this, p, lambda_s, matrix) Implementation →

Arguments

interface

public module subroutine compute_rotation_matrices(this, euler_angles, r_x, r_y, r_z, r, matrix, inv_r_x, inv_r_y, inv_r_z, inv_r, inv_matrix) Implementation →

Arguments

interface

public pure module function get_lambda(this) result(lambda) Implementation →

Arguments

Return Value double precision

public interface lorentz_boost

public module function construct_boost(v) result(boost)

Arguments

Return Value type(lorentz_boost)

public module function construct_boost_components(vx, vy, vz) result(boost)

Arguments

Return Value type(lorentz_boost)

public interface spatial_rotation

public module function construct_rotation(euler_angles) result(rotation)

Arguments

Return Value type(spatial_rotation)

public module function construct_rotation_angles(alpha, beta, gamma) result(rotation)

Arguments

Return Value type(spatial_rotation)

Derived Types

type, public, extends(lorentz_transformation) :: lorentz_boost

Components

Constructor

Type-Bound Procedures

type, public :: lorentz_transformation

Components

Type-Bound Procedures

type, public, extends(lorentz_transformation) :: spatial_rotation

Components

Constructor

Type-Bound Procedures

Functions

public pure function euclidean_inner_product(u, v) result(inner_product)

Arguments

Return Value double precision

public pure function minkowski_inner_product(u, v) result(inner_product)

Arguments

Return Value double precision

public pure function minkowski_sqnorm(u) result(sqnorm)

Arguments

Return Value double precision

public pure function row_by_column(u, v) result(res)

Arguments

Return Value double precision

public function square_matrix_multiplication(a, b) result(c)

Arguments

Return Value double precision, DIMENSION(SIZE(a(1,:)),SIZE(a(1,:)))

public module function apply_as_congruence_to_symrank2_tensor(this, t) result(transformed_t)

public module function apply_as_congruence_to_tensor(this, t) result(transformed_t)

public module function apply_as_similarity_to_symrank2_tensor(this, t) result(transformed_t)

public module function apply_as_similarity_to_tensor(this, t) result(transformed_t)

public module function apply_to_vector(this, u) result(transformed_u)

public pure module subroutine compute_boost_matrices(this, p, lambda_s, matrix)

public module subroutine compute_rotation_matrices(this, euler_angles, r_x, r_y, r_z, r, matrix, inv_r_x, inv_r_y, inv_r_z, inv_r, inv_matrix)

public pure module function get_lambda(this) result(lambda)