smash::EnergyMomentumTensor Class Reference

#include <energymomentumtensor.h>

The EnergyMomentumTensor class represents a symmetric positive semi-definite energy-momentum tensor \( T^{\mu \nu}\).

Definition at line 29 of file energymomentumtensor.h.

Public Types
typedef std::array< double, 10 >	tmn_type
	The energy-momentum tensor is symmetric and has 10 independent components. More...
using	iterator = tmn_type::iterator
	Iterator over the components. More...
using	const_iterator = tmn_type::const_iterator
	Constant iterator over the components. More...

Public Member Functions
	EnergyMomentumTensor ()
	Default constructor (nulls all components) More...
	EnergyMomentumTensor (const tmn_type &Tmn)
	Copy constructor. More...
double	operator[] (std::size_t i) const
	Return ith component of the tensor. More...
EnergyMomentumTensor	operator+= (const EnergyMomentumTensor &Tmn0)
	Addition of \(T^{\mu \nu}_0\) to tensor. More...
EnergyMomentumTensor	operator-= (const EnergyMomentumTensor &Tmn0)
	Subtraction of \(T^{\mu \nu}_0\) of tensor. More...
EnergyMomentumTensor	operator*= (double a)
	Scaling of the tensor by scalar \(a\). More...
EnergyMomentumTensor	operator/= (double a)
	Division of the tensor by scalar \(a\). More...
FourVector	landau_frame_4velocity () const
	Find the Landau frame 4-velocity from energy-momentum tensor. More...
EnergyMomentumTensor	boosted (const FourVector &u) const
	Boost to a given 4-velocity. More...
void	add_particle (const FourVector &mom)
	Given momentum of the particle adds \(p^{\mu}p^{\mu}/p^0\) to the energy momentum tensor. More...
void	add_particle (const ParticleData &p, double factor)
	Same, but \( p^{\mu}p^{\mu}/p^0\) times factor is added. More...
void	add_particle_for_derivatives (const ParticleData &, double, ThreeVector)
	Dummy function need for update_general_lattice. More...
iterator	begin ()
	Returns an iterator starting at the (0,0) component. More...
iterator	end ()
	Returns an iterator pointing after the (3,3)th component. More...
const_iterator	begin () const
	Const overload of the above. More...
const_iterator	end () const
	Const overload of the above. More...
const_iterator	cbegin () const
const_iterator	cend () const

Static Public Member Functions
static std::int8_t	tmn_index (std::int8_t mu, std::int8_t nu)
	Access the index of component \( (\mu, \nu) \). More...

Private Attributes
tmn_type	Tmn_
	The internal storage of the components. More...

Member Typedef Documentation

tmn_type

typedef std::array<double, 10> smash::EnergyMomentumTensor::tmn_type

The energy-momentum tensor is symmetric and has 10 independent components.

Definition at line 32 of file energymomentumtensor.h.

iterator

using smash::EnergyMomentumTensor::iterator = tmn_type::iterator

Iterator over the components.

Definition at line 79 of file energymomentumtensor.h.

const_iterator

using smash::EnergyMomentumTensor::const_iterator = tmn_type::const_iterator

Constant iterator over the components.

Definition at line 81 of file energymomentumtensor.h.

Constructor & Destructor Documentation

EnergyMomentumTensor() [1/2]

smash::EnergyMomentumTensor::EnergyMomentumTensor ( )

inline

Default constructor (nulls all components)

Definition at line 34 of file energymomentumtensor.h.

{ Tmn_.fill(0.); }

EnergyMomentumTensor() [2/2]

smash::EnergyMomentumTensor::EnergyMomentumTensor ( const tmn_type &  Tmn )

inlineexplicit

Copy constructor.

Parameters

[in]

Tmn

Components of the energy-momentum tensor

Definition at line 40 of file energymomentumtensor.h.

                                                     {
    for (size_t i = 0; i < 10; i++) {
      Tmn_[i] = Tmn[i];
    }
  }

Member Function Documentation

operator[]()

double smash::EnergyMomentumTensor::operator[] ( std::size_t  i ) const

inline

Return ith component of the tensor.

Definition at line 47 of file energymomentumtensor.h.

{ return Tmn_[i]; }

tmn_index()

static std::int8_t smash::EnergyMomentumTensor::tmn_index ( std::int8_t  mu, std::int8_t  nu  )

inlinestatic

Access the index of component \( (\mu, \nu) \).

Parameters

[in]	mu	\(\mu\) is the row index (0 to 3)
[in]	nu	\(\nu\) is the line index (0 to 3)

Definition at line 54 of file energymomentumtensor.h.

                                                         {
    // clang-format off
    constexpr std::array<std::int8_t, 16> indices = {0, 1, 2, 3,
                                                     1, 4, 5, 6,
                                                     2, 5, 7, 8,
                                                     3, 6, 8, 9};
    // clang-format on
    if (mu < 4 && nu < 4 && mu >= 0 && nu >= 0) {
      return indices[mu + 4 * nu];
    } else {
      throw std::invalid_argument("Invalid indices: " + std::to_string(mu) +
                                  ", " + std::to_string(nu));
    }
  }

operator+=()

EnergyMomentumTensor smash::EnergyMomentumTensor::operator+= ( const EnergyMomentumTensor &  Tmn0 )

inline

Addition of \(T^{\mu \nu}_0\) to tensor.

Definition at line 154 of file energymomentumtensor.h.

                                      {
  for (size_t i = 0; i < 10; i++) {
    Tmn_[i] += Tmn0[i];
  }
  return *this;
}

operator-=()

EnergyMomentumTensor smash::EnergyMomentumTensor::operator-= ( const EnergyMomentumTensor &  Tmn0 )

inline

Subtraction of \(T^{\mu \nu}_0\) of tensor.

Definition at line 168 of file energymomentumtensor.h.

                                      {
  for (size_t i = 0; i < 10; i++) {
    Tmn_[i] -= Tmn0[i];
  }
  return *this;
}

operator*=()

EnergyMomentumTensor smash::EnergyMomentumTensor::operator*= ( double  a )

inline

Scaling of the tensor by scalar \(a\).

Definition at line 182 of file energymomentumtensor.h.

                                                                           {
  for (size_t i = 0; i < 10; i++) {
    Tmn_[i] *= a;
  }
  return *this;
}

operator/=()

EnergyMomentumTensor smash::EnergyMomentumTensor::operator/= ( double  a )

inline

Division of the tensor by scalar \(a\).

Definition at line 199 of file energymomentumtensor.h.

                                                                           {
  for (size_t i = 0; i < 10; i++) {
    Tmn_[i] /= a;
  }
  return *this;
}

landau_frame_4velocity()

FourVector smash::EnergyMomentumTensor::landau_frame_4velocity

(

)

const

Find the Landau frame 4-velocity from energy-momentum tensor.

IMPORTANT: resulting 4-velocity is fourvector with LOWER index

Definition at line 23 of file energymomentumtensor.cc.

                                                              {
  using Eigen::Matrix4d;
  using Eigen::Vector4d;
  /* We want to solve the generalized eigenvalue problem
     T^{\mu \nu} h_{nu} = \lambda g^{\mu \nu} h_{nu}, or in the other way
     T_{\mu}^{\nu} h_{nu} = \lambda h_{mu}. The eigenvector
     corresponding to the largest (and the only positive) eigenvalue is
     proportional to 4-velocity of the Landau frame.
     Denote T^{\mu}_{\nu} as A and g^{\mu \nu} as B.
     A is symmetric and positive semi-definite. B is symmetric.
     A x = \lambda B x. I have to solve generalized eigenvalue
     problem, because A can be not positively definite (e.g. if
     energy-momentum tensor is computed for particles with momenta lying
     in one plane). For positively definite A a more efficient solution
     is possible, but I (oliiny) do not consider it, until it becomes
     important for SMASH performance.
     */
  Matrix4d A;
  // A = T_{\mu}^{\nu} = g_{\mu \mu'} T^{\mu' \nu}
  // clang-format off
  A <<  Tmn_[0],  Tmn_[1],  Tmn_[2],  Tmn_[3],
       -Tmn_[1], -Tmn_[4], -Tmn_[5], -Tmn_[6],
       -Tmn_[2], -Tmn_[5], -Tmn_[7], -Tmn_[8],
       -Tmn_[3], -Tmn_[6], -Tmn_[8], -Tmn_[9];
  // clang-format on
 
  logg[LTmn].debug("Looking for Landau frame for T_{mu}^{nu} ", A);
  Eigen::EigenSolver<Matrix4d> es(A);
 
  Vector4d eig_im = es.eigenvalues().imag();
  Vector4d eig_re = es.eigenvalues().real();
  size_t i_maxeigenvalue = 0;
  for (size_t i = 0; i < 4; i++) {
    if (eig_re(i_maxeigenvalue) < eig_re(i)) {
      i_maxeigenvalue = i;
    }
  }
 
  // Sanity checks
  // Eigen values of A should be strictly real, the largest one corresponding
  // to energy density should be non-negative, the other ones
  // corresponding to pressure should be non-positive, because of the
  // metric tensor gmunu = (1, -1, -1, -1) convention.
  if (i_maxeigenvalue != 0) {
    logg[LTmn].warn(
        "The Tmn diagonalization code previously relied on assumption that"
        " 0th eigenvalue is the largest one. It seems to be always fulfilled "
        "in practice, but not guaranteed by Eigen. Here is Tmn * gmn, ",
        A, " for which it is not fulfilled. Please let Dima(oliiny) know.");
  }
  for (size_t i = 0; i < 4; i++) {
    if (std::abs(eig_im(i)) > really_small) {
      logg[LTmn].error("Tmn*gmn\n ", A, "\n has a complex eigenvalue ",
                       eig_re(i), " + i * ", eig_im(i));
    }
    if (i == i_maxeigenvalue && eig_re(i) < -really_small) {
      logg[LTmn].error("Tmn*gmn\n", A,
                       "\nenergy density eigenvalue is not positive ",
                       eig_re(i), " + i * ", eig_im(i));
      logg[LTmn].error("i_max = ", i_maxeigenvalue);
    }
    if (i != i_maxeigenvalue && eig_re(i) > really_small) {
      logg[LTmn].error("Tmn*gmn\n", A, "\npressure eigenvalue is not negative ",
                       eig_re(i), " + i * ", eig_im(i));
    }
  }
 
  Vector4d tmp = es.eigenvectors().col(i_maxeigenvalue).real();
  // Choose sign so that zeroth component is positive because we want
  // 4-velocity to have 0-component positive
  if (tmp(0) < 0.0) {
    tmp = -tmp;
  }
 
  FourVector u(tmp(0), tmp(1), tmp(2), tmp(3));
  const double u_sqr = u.sqr();
  if (u_sqr > really_small) {
    u /= std::sqrt(u_sqr);
  } else {
    logg[LTmn].error(
        "Landau frame is not defined.", " Eigen vector", u, " of ", A,
        " is not time-like and",
        " cannot be 4-velocity. This may happen if energy-momentum",
        " tensor was constructed for a massless particle.");
    u = FourVector(1., 0., 0., 0.);
  }
  return u;
}

boosted()

EnergyMomentumTensor smash::EnergyMomentumTensor::boosted

(

const FourVector &

u

)

const

Boost to a given 4-velocity.

IMPORTANT: boost 4-velocity is fourvector with LOWER index

Parameters

[in]

u

4-velocity vector

Definition at line 112 of file energymomentumtensor.cc.

                                                                            {
  using Eigen::Matrix4d;
  Matrix4d A, L, R;
  // Energy-momentum tensor
  // clang-format off
  A << Tmn_[0], Tmn_[1], Tmn_[2], Tmn_[3],
       Tmn_[1], Tmn_[4], Tmn_[5], Tmn_[6],
       Tmn_[2], Tmn_[5], Tmn_[7], Tmn_[8],
       Tmn_[3], Tmn_[6], Tmn_[8], Tmn_[9];
  // clang-format on
  // Compute Lorentz matrix of boost
  const ThreeVector tmp = u.threevec() / (1.0 + u[0]);
  // clang-format off
  L << u[0], u[1], u[2], u[3],
       u[1], u[1] * tmp.x1() + 1.0, u[2] * tmp.x1(), u[3] * tmp.x1(),
       u[2], u[1] * tmp.x2(), u[2] * tmp.x2() + 1.0, u[3] * tmp.x2(),
       u[3], u[1] * tmp.x3(), u[2] * tmp.x3(), u[3] * tmp.x3() + 1.0;
  // clang-format on
  // Boost
  R = L * A * L;
  // clang-format off
  return EnergyMomentumTensor({R(0, 0), R(0, 1), R(0, 2), R(0, 3),
                                        R(1, 1), R(1, 2), R(1, 3),
                                                 R(2, 2), R(2, 3),
                                                          R(3, 3)});
  // clang-format on
}

add_particle() [1/2]

void smash::EnergyMomentumTensor::add_particle

(

const FourVector &

mom

)

Given momentum of the particle adds \(p^{\mu}p^{\mu}/p^0\) to the energy momentum tensor.

Parameters

[in]

mom

Momentum 4-vector with upper index, as all 4-momenta in SMASH

Definition at line 140 of file energymomentumtensor.cc.

                                                             {
  const ThreeVector tmp = mom.threevec() / mom[0];
  Tmn_[0] += mom[0];
  Tmn_[1] += mom[1];
  Tmn_[2] += mom[2];
  Tmn_[3] += mom[3];
  Tmn_[4] += mom[1] * tmp.x1();
  Tmn_[5] += mom[1] * tmp.x2();
  Tmn_[6] += mom[1] * tmp.x3();
  Tmn_[7] += mom[2] * tmp.x2();
  Tmn_[8] += mom[2] * tmp.x3();
  Tmn_[9] += mom[3] * tmp.x3();
}

add_particle() [2/2]

void smash::EnergyMomentumTensor::add_particle	(	const ParticleData &	p,
		double	factor
	)

Same, but \( p^{\mu}p^{\mu}/p^0\) times factor is added.

Parameters

[in]

p

Reference to the data information of the particle

See also

ParticleData

Parameters

[in]

factor

Usually a smearing factor

Definition at line 154 of file energymomentumtensor.cc.

                                                                            {
  if (factor != 0) {
    add_particle(p.momentum() * factor);
  }
}

add_particle_for_derivatives()

void smash::EnergyMomentumTensor::add_particle_for_derivatives ( const ParticleData &  , double  , ThreeVector    )

inline

Dummy function need for update_general_lattice.

Definition at line 111 of file energymomentumtensor.h.

                                                                               {
  }

begin() [1/2]

iterator smash::EnergyMomentumTensor::begin ( )

inline

Returns an iterator starting at the (0,0) component.

The iterator implements the randomIterator concept. Thus, you can simply write begin() + 1 to get an iterator that points to the 1st component.

Definition at line 120 of file energymomentumtensor.h.

{ return Tmn_.begin(); }

end() [1/2]

iterator smash::EnergyMomentumTensor::end ( )

inline

Returns an iterator pointing after the (3,3)th component.

Definition at line 125 of file energymomentumtensor.h.

{ return Tmn_.end(); }

begin() [2/2]

const_iterator smash::EnergyMomentumTensor::begin ( ) const

inline

Const overload of the above.

Definition at line 128 of file energymomentumtensor.h.

{ return Tmn_.begin(); }

end() [2/2]

const_iterator smash::EnergyMomentumTensor::end ( ) const

inline

Const overload of the above.

Definition at line 130 of file energymomentumtensor.h.

{ return Tmn_.end(); }

cbegin()

const_iterator smash::EnergyMomentumTensor::cbegin ( ) const

inline

See also

begin

Definition at line 133 of file energymomentumtensor.h.

{ return Tmn_.cbegin(); }

cend()

const_iterator smash::EnergyMomentumTensor::cend ( ) const

inline

See also

end

Definition at line 135 of file energymomentumtensor.h.

{ return Tmn_.cend(); }

Member Data Documentation

Tmn_

tmn_type smash::EnergyMomentumTensor::Tmn_

private

The internal storage of the components.

Tensor has 16 components, but it is symmetric, so number of independent components reduces to 10.

Definition at line 143 of file energymomentumtensor.h.

---

The documentation for this class was generated from the following files:

• src/include/smash/energymomentumtensor.h
• src/energymomentumtensor.cc