Metric

template<typename Type>
class Metric

The base class of all kinds of metrics.

Subclassed by SBody::Hayward< Type >, SBody::Kerr< Type >, SBody::KerrNewman< Type >, SBody::KerrTaubNUT< Type >, SBody::Newton< Type >, SBody::ReissnerNordstrom< Type >, SBody::Schwarzschild< Type >

Public Functions

virtual std::string Name() const = 0

Return the name of the metric.

Returns:

name of the metric.

virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) = 0

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

template<typename VecExp1, typename VecExp2>
inline Type DotProduct(const boost::numeric::ublas::bounded_vector<Type, 8> &position, const VecExp1 &x, const VecExp2 &y, bool include_time)

Dot product of vector x and y at position. \(g_{\mu\nu}x^\mu y^\nu\).

Parameters:

  • position – 4 dimensional vector, position to calcuate the dot product of x and y.

  • x – 4 dimensional vector

  • y – 4 dimensional vector

  • include_time – include time related terms if true.

Returns:

result

virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) = 0

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline int LocalInertialFrame(const boost::numeric::ublas::bounded_vector<Type, 8> &position, TimeSystem time, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &coordinate)

Calculate the local inertial frame coordinate of the object at position, stored in coordinate.

Parameters:

  • position – 8 dimensional vector

  • time – time systme of the object

  • coordinate – matrix with size 4×4

Returns:

status

virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) = 0

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) = 0

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) = 0

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) = 0

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) = 0

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) = 0

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type Redshift(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const boost::numeric::ublas::bounded_vector<Type, 8> &photon_position, TimeSystem object_time, TimeSystem photon_time)

Calculate the redshift of the object, \(1+z\).

Parameters:

  • y – 8 dimensional vector of the object

  • photon – photon traced to the object

  • object_time – time system of the object

  • photon_time – time system of the photon

Returns:

result

virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) = 0

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) = 0

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) = 0

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Newton & Post-Newtonian

template<typename Type>
class Newton : public SBody::Metric<Type>

Post-Newtonian.

Subclassed by SBody::PN1< Type >

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

inline virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

inline virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type Redshift(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const boost::numeric::ublas::bounded_vector<Type, 8> &photon_position, TimeSystem object_time, TimeSystem photon_time) override

Calculate the redshift of the object, \(1+z\).

Parameters:

  • y – 8 dimensional vector of the object

  • photon – photon traced to the object

  • object_time – time system of the object

  • photon_time – time system of the photon

Returns:

result

inline virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

template<typename Type>
class PN1 : public SBody::Newton<Type>

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Schwarzschild

template<typename Type>
class Schwarzschild : public SBody::Metric<Type>
\[\begin{split}g_{\mu\nu}=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}\end{split}\]

Reissner-Nördstrom

template<typename Type>
class ReissnerNordstrom : public SBody::Metric<Type>

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

inline virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

inline virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Kerr

template<typename Type>
class Kerr : public SBody::Metric<Type>

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

inline virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

inline virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Kerr-Newman

template<typename Type>
class KerrNewman : public SBody::Metric<Type>

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

inline virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

inline virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Kerr-Taub-NUT

template<typename Type>
class KerrTaubNUT : public SBody::Metric<Type>

Public Functions

inline virtual std::string Name() const override

Return the name of the metric.

Returns:

name of the metric.

inline virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

inline virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

inline virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

inline virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

inline virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

inline virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system

Hayward

template<typename Type>
class Hayward : public SBody::Metric<Type>

Public Functions

virtual int MetricTensor(const boost::numeric::ublas::bounded_vector<Type, 8> &position, boost::numeric::ublas::bounded_matrix<Type, 4, 4> &metric) override

Calculate the metric tensor at position, stored in metric.

Parameters:

  • position – 4 dimensional vector

  • metric – matrix with size 4×4

Returns:

status

virtual Type DistanceSquare(const boost::numeric::ublas::bounded_vector<Type, 8> &x, const boost::numeric::ublas::bounded_vector<Type, 8> &y, bool include_time) override

Calculate the square of the distance between x and y at x. \(g_{\mu\nu}(x^\mu-y^\mu)(x^\nu-y^\nu)\).

Parameters:

  • x – 8 dimensional vector

  • y – 8 dimensional vector

Returns:

result

virtual int LagrangianToHamiltonian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Lagrangian to Hamiltonian.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int HamiltonianToLagrangian(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Convert the coordinate system from Hamiltonian to Lagrangian.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int FastTrace(const Type r_observer, const Type theta_observer, const Type sin_theta_observer, const Type cos_theta_observer, const Type r_object, const Type theta_object, const Type phi_object, Type &alpha, Type &beta, boost::numeric::ublas::bounded_vector<Type, 8> &photon, Type &photon_time) override

Trace the photon from the observer to the target, using elliptic integrals.

Parameters:

  • r_observer – radius of the observer, \(r_\text{obs}\).

  • theta_observer – theta of the observer, \(\theta_\text{obs}\).

  • sin_theta_observer – std::sin(theta_observer), \(\std::sin\theta_\text{obs}\).

  • cos_theta_observer – std::cos(theta_observer), \(\std::cos\theta_\text{obs}\).

  • r_object – radius of the target, \(r_\text{tar}\).

  • theta_object – theta of the target, \(\theta_\text{tar}\).

  • phi_object – phi of the target, \(\phi_\text{tar}\).

  • alpha – x position of the target in the observer’s view.

  • beta – y position of the target in the observer’s view.

  • photon – 8 dimensional vector, position and the velocity of the photon traced to the target.

  • photon_time – look back time of the photon.

Returns:

status

virtual Type Energy(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the energy of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

virtual Type AngularMomentum(const boost::numeric::ublas::bounded_vector<Type, 8> &y, TimeSystem time, DynamicalSystem dynamics) override

Calculate the angular momentum of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

virtual Type CarterConstant(const boost::numeric::ublas::bounded_vector<Type, 8> &y, const Type mu2, TimeSystem time, DynamicalSystem dynamics) override

Calculate the Carter constant of the object.

Parameters:

  • y – 8 dimensional vector

  • time – time system of the object

  • dynamics – dynamical system of the object

Returns:

result

virtual int NormalizeTimelikeGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y) override

Normalize the timelike geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual int NormalizeNullGeodesic(boost::numeric::ublas::bounded_vector<Type, 8> &y, Type frequency = 1.) override

Normalize the null geodesic.

Parameters:

y – 8 dimensional vector

Returns:

status

virtual std::function<void(const boost::numeric::ublas::bounded_vector<Type, 8>&, boost::numeric::ublas::bounded_vector<Type, 8>&, const Type)> GetIntegrationSystem(TimeSystem time, DynamicalSystem dynamics, MotionMode motion = GEODESIC) override

Get the integration system to calculate the motion of the object.

Parameters:

  • time – time system of the object

  • dynamics – dynamical system of the object

  • motion – motion mode of the object

Returns:

integrator system