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...

Detailed Description

Contains functions used in special relativity calculations.

Function Documentation

lorentz_boost_matrix() [1/2]

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 \(\Lambda^{a}{}_{\bar{a}}\) for a Lorentz boost with that velocity [e.g. Eq. (2.38) of [180]]:

\begin{align} \Lambda^t{}_{\bar{t}} &= \gamma, \\ \Lambda^t{}_{\bar{i}} = \Lambda^i{}_{\bar{t}} &= \gamma v^i, \\ \Lambda^i{}_{\bar{j}} = \Lambda^j{}_{\bar{i}} &= [(\gamma - 1)/v^2] v^i v^j + \delta^{ij}. \end{align}

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 \(v \rightarrow -v\)) 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() [2/2]

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 \(\Lambda^{a}{}_{\bar{a}}\) for a Lorentz boost with that velocity [e.g. Eq. (2.38) of [180]]:

\begin{align} \Lambda^t{}_{\bar{t}} &= \gamma, \\ \Lambda^t{}_{\bar{i}} = \Lambda^i{}_{\bar{t}} &= \gamma v^i, \\ \Lambda^i{}_{\bar{j}} = \Lambda^j{}_{\bar{i}} &= [(\gamma - 1)/v^2] v^i v^j + \delta^{ij}. \end{align}

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 \(v \rightarrow -v\)) 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.