# Source code for geomstats.geometry.minkowski

```"""Minkowski space.

"""

import math

import geomstats.backend as gs
from geomstats.algebra_utils import from_vector_to_diagonal_matrix
from geomstats.geometry.euclidean import Euclidean
from geomstats.geometry.riemannian_metric import RiemannianMetric

[docs]class Minkowski(Euclidean):
"""Class for Minkowski space.

This is the Euclidean space endowed with the inner-product of signature (
dim-1, 1).

Parameters
----------
dim : int
Dimension of Minkowski space.
"""

def __new__(cls, dim, **kwargs):
"""Instantiate a Minkowski space.

This is an instance of the `Euclidean` class endowed with the
`MinkowskiMetric`.
"""
space = Euclidean(dim)
space._metric = MinkowskiMetric(dim)
return space

[docs]class MinkowskiMetric(RiemannianMetric):
"""Class for the pseudo-Riemannian Minkowski metric.

The metric is flat: the inner product is independent of the base point.

Parameters
----------
dim : int
Dimension of the Minkowski space.
"""

def __init__(self, dim):
super(MinkowskiMetric, self).__init__(dim=dim, signature=(dim - 1, 1))

[docs]    def metric_matrix(self, base_point=None):
"""Compute the inner product matrix, independent of the base point.

Parameters
----------
base_point : array-like, shape=[..., dim]
Base point.

Returns
-------
inner_prod_mat : array-like, shape=[..., dim, dim]
Inner-product matrix.
"""
q, p = self.signature
diagonal = gs.array([-1.0] * p + [1.0] * q)
return from_vector_to_diagonal_matrix(diagonal)

[docs]    def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Inner product between two tangent vectors at a base point.

Parameters
----------
tangent_vec_a: array-like, shape=[..., dim]
Tangent vector at base point.
tangent_vec_b: array-like, shape=[..., dim]
Tangent vector at base point.
base_point: array-like, shape=[..., dim]
Base point.
Optional, default: None.

Returns
-------
inner_product : array-like, shape=[...,]
Inner-product.
"""
q, p = self.signature
diagonal = gs.array([-1.0] * p + [1.0] * q, dtype=tangent_vec_a.dtype)
return gs.einsum("...i,...i->...", diagonal * tangent_vec_a, tangent_vec_b)

[docs]    def exp(self, tangent_vec, base_point, **kwargs):
"""Compute the Riemannian exponential of `tangent_vec` at `base_point`.

The Riemannian exponential is the addition in the Minkowski space.

Parameters
----------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at base point.
base_point : array-like, shape=[..., dim]
Base point.

Returns
-------
exp : array-like, shape=[..., dim]
Riemannian exponential.
"""
exp = base_point + tangent_vec
return exp

[docs]    def log(self, point, base_point, **kwargs):
"""Compute the Riemannian logarithm of `point` at `base_point`.

The Riemannian logarithm is the subtraction in the Minkowski space.

Parameters
----------
point : array-like, shape=[..., dim]
Point.
base_point : array-like, shape=[..., dim]
Base point.

Returns
-------
log : array-like, shape=[..., dim]
Riemannian logarithm.
"""
log = point - base_point
return log

"""Compute the radius of the injectivity domain.

This is is the supremum of radii r for which the exponential map is a
diffeomorphism from the open ball of radius r centered at the base point onto
its image.
In the case of the Minkowski space, it does not depend on the base point and
is infinite everywhere, because of the flat curvature.

Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold.

Returns
-------