scri.asymptotic_bondi_data.transformations

Functions

boosted_grid(frame_rotation, boost_velocity, ...)

conformal_factors(boost_velocity, ...)

Compute various combinations of the conformal factor

transform(self, **kwargs)

Apply BMS transformation to AsymptoticBondiData object
scri.asymptotic_bondi_data.transformations.boosted_grid(frame_rotation, boost_velocity, n_theta, n_phi)[source]
scri.asymptotic_bondi_data.transformations.conformal_factors(boost_velocity, distorted_grid_rotors)[source]

Compute various combinations of the conformal factor

This is primarily a utility function for use in the transform function, pulled out so that it can be tested separately.

Parameters:
boost_velocity: array of 3 floats

Three-velocity of the new frame relative to the old frame

distorted_grid_rotors: 2-d array of quaternions

Unit quaternions giving the rotation of the (x, y, z) basis onto the basis vectors with respect to which the output spin-weighted fields are evaluated

Returns:
k: spherical_functions.Grid
ðk_over_k: spherical_functions.Grid
one_over_k: spherical_functions.Grid
one_over_k_cubed: spherical_functions.Grid

These all have the same shape as distorted_grid_rotors except for an additional dimension of size 1 at the beginning, so that they can broadcast against the time dimension.

scri.asymptotic_bondi_data.transformations.transform(self, **kwargs)[source]

Apply BMS transformation to AsymptoticBondiData object

It is important to note that the input transformation parameters are applied in this order:

  1. (Super)Translations

  2. Rotation (about the origin)

  3. Boost (about the origin)

All input parameters refer to the transformation required to take the input data’s inertial frame onto the inertial frame of the output data’s inertial observers. In what follows, the coordinates of and functions in the input inertial frame will be unprimed, while corresponding values of the output inertial frame will be primed.

The translations (space, time, spacetime, or super) can be given in various ways, which may override each other. Ultimately, however, they are essentially combined into a single function α, representing the supertranslation, which transforms the asymptotic time variable u as

u’(u, θ, ϕ) = u(u, θ, ϕ) - α(θ, ϕ)

A simple time translation by δt would correspond to

α(θ, ϕ) = δt # Independent of (θ, ϕ)

A pure spatial translation δx would correspond to

α(θ, ϕ) = -δx · n̂(θ, ϕ)

where · is the usual dot product, and n̂ is the unit vector in the given direction.

Parameters:
abd: AsymptoticBondiData

The object storing the modes of the original data, which will be transformed in this function. This is the only required argument to this function.

time_translation: float, optional

Defaults to zero. Nonzero overrides corresponding components of spacetime_translation and supertranslation parameters. Note that this is the actual change in the coordinate value, rather than the corresponding mode weight (which is what supertranslation represents).

space_translationfloat array of length 3, optional

Defaults to empty (no translation). Non-empty overrides corresponding components of spacetime_translation and supertranslation parameters. Note that this is the actual change in the coordinate value, rather than the corresponding mode weight (which is what supertranslation represents).

spacetime_translationfloat array of length 4, optional

Defaults to empty (no translation). Non-empty overrides corresponding components of supertranslation. Note that this is the actual change in the coordinate value, rather than the corresponding mode weight (which is what supertranslation represents).

supertranslationcomplex array [defaults to 0]

This gives the complex components of the spherical-harmonic expansion of the supertranslation in standard form, starting from ell=0 up to some ell_max, which may be different from the ell_max of the input abd object. Supertranslations must be real, so these values should obey the condition

α^{ℓ,m} = (-1)^m ᾱ^{ℓ,-m}

This condition is actually imposed on the input data, so imaginary parts of α(θ, ϕ) will essentially be discarded. Defaults to empty, which causes no supertranslation. Note that some components may be overridden by the parameters above.

frame_rotationquaternion [defaults to 1]

Transformation applied to (x,y,z) basis of the input mode’s inertial frame. For example, the basis z vector of the new frame may be written as

z’ = frame_rotation * z * frame_rotation.inverse()

Defaults to 1, corresponding to the identity transformation (no rotation).

boost_velocityfloat array of length 3 [defaults to (0, 0, 0)]

This is the three-velocity vector of the new frame relative to the input frame. The norm of this vector is required to be smaller than 1.

output_ell_max: int [defaults to abd.ell_max]

Maximum ell value in the output data.

working_ell_max: int [defaults to 2 * abd.ell_max]

Maximum ell value to use during the intermediate calculations. Rotations and time translations do not require this to be any larger than abd.ell_max, but other transformations will require more values of ell for accurate results. In particular, boosts are multiplied by time, meaning that a large boost of data with large values of time will lead to very large power in higher modes. Similarly, large (super)translations will couple power through a lot of modes. To avoid aliasing, this value should be large, to accomodate power in higher modes.

Returns:
abdprime: AsymptoticBondiData

Object representing the transformed data.