Special Relativity

Contains functions used in special relativity calculations. More...

Namespaces
namespace	sr
	Holds functions related to special relativity.
template<size_t SpatialDim>
tnsr::Ab< double, SpatialDim, Frame::NoFrame >	sr::lorentz_boost_matrix (const tnsr::I< double, SpatialDim, Frame::NoFrame > &velocity)
	Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field). More...
template<size_t SpatialDim>
void	sr::lorentz_boost_matrix (gsl::not_null< tnsr::Ab< double, SpatialDim, Frame::NoFrame > * > boost_matrix, const tnsr::I< double, SpatialDim, Frame::NoFrame > &velocity)
	Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field). More...
template<size_t SpatialDim>
tnsr::Ab< double, SpatialDim, Frame::NoFrame >	sr::lorentz_boost_matrix (const std::array< double, SpatialDim > &velocity)
	Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field). More...
template<typename DataType , size_t SpatialDim, typename Frame >
void	sr::lorentz_boost (gsl::not_null< tnsr::I< DataType, SpatialDim, Frame > * > result, const tnsr::I< DataType, SpatialDim, Frame > &vector, double vector_component_0, const std::array< double, SpatialDim > &velocity)
	Apply a Lorentz boost to the spatial part of a vector. More...
template<typename DataType , size_t SpatialDim, typename Frame >
void	sr::lorentz_boost (gsl::not_null< tnsr::a< DataType, SpatialDim, Frame > * > result, const tnsr::a< DataType, SpatialDim, Frame > &one_form, const std::array< double, SpatialDim > &velocity)
	Apply a Lorentz boost to a one form. More...

Detailed Description

Contains functions used in special relativity calculations.

Function Documentation

lorentz_boost() [1/2]

template<typename DataType , size_t SpatialDim, typename Frame >

void sr::lorentz_boost	(	gsl::not_null< tnsr::ab< DataType, SpatialDim, Frame > * >	result,
		const tnsr::ab< DataType, SpatialDim, Frame > &	tensor,
		const std::array< double, SpatialDim > &	velocity
	)

Apply a Lorentz boost to a one form.

Apply a Lorentz boost to each component of a rank-2 tensor with lower or covariant indices.

Note

In the future we might want to write a single function capable to boost a tensor of arbitrary rank.

lorentz_boost() [2/2]

template<typename DataType , size_t SpatialDim, typename Frame >

void sr::lorentz_boost	(	gsl::not_null< tnsr::I< DataType, SpatialDim, Frame > * >	result,
		const tnsr::I< DataType, SpatialDim, Frame > &	vector,
		double	vector_component_0,
		const std::array< double, SpatialDim > &	velocity
	)

Apply a Lorentz boost to the spatial part of a vector.

Details

This requires passing the 0th component of the vector as an additional argument.

lorentz_boost_matrix() [1/3]

template<size_t SpatialDim>

tnsr::Ab< double, SpatialDim, Frame::NoFrame > sr::lorentz_boost_matrix

(

const std::array< double, SpatialDim > &

velocity

)

Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field).

Details

Given a spatial velocity vector $v^i$ (with $c=1$), compute the matrix ${\mathrm{\Lambda }}^a{}_{\overline{a}}$ for a Lorentz boost with that velocity [e.g. Eq. (2.38) of [194]]:

$$\begin{array}{}\text{(1)}& {\mathrm{\Lambda }}^t{}_{\overline{t}}& =\gamma ,\text{(2)}& {\mathrm{\Lambda }}^t{}_{\overline{i}}={\mathrm{\Lambda }}^i{}_{\overline{t}}& =\gamma v^i,\text{(3)}& {\mathrm{\Lambda }}^i{}_{\overline{j}}={\mathrm{\Lambda }}^j{}_{\overline{i}}& =[(\gamma -1)/v^2]v^i{v}^j+{\delta }^{ij}.\end{array}$$

Here $v=\sqrt{{\delta }_{ij}{v}^i{v}^j}$, $\gamma =1/\sqrt{1-v^2}$, and ${\delta }^{ij}$ is the Kronecker delta. Note that this matrix boosts a one-form from the unbarred to the barred frame, and its inverse (obtained via $v\to -v$) boosts a vector from the barred to the unbarred frame.

Note that while the Lorentz boost matrix is symmetric, the returned boost matrix is of type tnsr::Ab, because Tensor does not support symmetric tensors unless both indices have the same valence.

lorentz_boost_matrix() [2/3]

template<size_t SpatialDim>

tnsr::Ab< double, SpatialDim, Frame::NoFrame > sr::lorentz_boost_matrix

(

const tnsr::I< double, SpatialDim, Frame::NoFrame > &

velocity

)

Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field).

Details

Given a spatial velocity vector $v^i$ (with $c=1$), compute the matrix ${\mathrm{\Lambda }}^a{}_{\overline{a}}$ for a Lorentz boost with that velocity [e.g. Eq. (2.38) of [194]]:

$$\begin{array}{}\text{(4)}& {\mathrm{\Lambda }}^t{}_{\overline{t}}& =\gamma ,\text{(5)}& {\mathrm{\Lambda }}^t{}_{\overline{i}}={\mathrm{\Lambda }}^i{}_{\overline{t}}& =\gamma v^i,\text{(6)}& {\mathrm{\Lambda }}^i{}_{\overline{j}}={\mathrm{\Lambda }}^j{}_{\overline{i}}& =[(\gamma -1)/v^2]v^i{v}^j+{\delta }^{ij}.\end{array}$$

Here $v=\sqrt{{\delta }_{ij}{v}^i{v}^j}$, $\gamma =1/\sqrt{1-v^2}$, and ${\delta }^{ij}$ is the Kronecker delta. Note that this matrix boosts a one-form from the unbarred to the barred frame, and its inverse (obtained via $v\to -v$) boosts a vector from the barred to the unbarred frame.

Note that while the Lorentz boost matrix is symmetric, the returned boost matrix is of type tnsr::Ab, because Tensor does not support symmetric tensors unless both indices have the same valence.

lorentz_boost_matrix() [3/3]

template<size_t SpatialDim>

void sr::lorentz_boost_matrix	(	gsl::not_null< tnsr::Ab< double, SpatialDim, Frame::NoFrame > * >	boost_matrix,
		const tnsr::I< double, SpatialDim, Frame::NoFrame > &	velocity
	)

Computes the matrix for a Lorentz boost from a single velocity vector (i.e., not a velocity field).

Details

Given a spatial velocity vector $v^i$ (with $c=1$), compute the matrix ${\mathrm{\Lambda }}^a{}_{\overline{a}}$ for a Lorentz boost with that velocity [e.g. Eq. (2.38) of [194]]:

$$\begin{array}{}\text{(7)}& {\mathrm{\Lambda }}^t{}_{\overline{t}}& =\gamma ,\text{(8)}& {\mathrm{\Lambda }}^t{}_{\overline{i}}={\mathrm{\Lambda }}^i{}_{\overline{t}}& =\gamma v^i,\text{(9)}& {\mathrm{\Lambda }}^i{}_{\overline{j}}={\mathrm{\Lambda }}^j{}_{\overline{i}}& =[(\gamma -1)/v^2]v^i{v}^j+{\delta }^{ij}.\end{array}$$

Here $v=\sqrt{{\delta }_{ij}{v}^i{v}^j}$, $\gamma =1/\sqrt{1-v^2}$, and ${\delta }^{ij}$ is the Kronecker delta. Note that this matrix boosts a one-form from the unbarred to the barred frame, and its inverse (obtained via $v\to -v$) boosts a vector from the barred to the unbarred frame.

Note that while the Lorentz boost matrix is symmetric, the returned boost matrix is of type tnsr::Ab, because Tensor does not support symmetric tensors unless both indices have the same valence.